CBSE SAMPLE PAPERS (March 2018) MATHEMATICS

Transcription

1 CBSE SAMPLE PAPERS (March 2018) X MATHEMATICS

2 CONTENTS 1. Sample paper (By CBSE ) Sample paper Sample paper Sample paper Sample paper Sample paper Sample paper Sample paper Sample paper Sample paper Sample paper Sample paper

3 SAMPLE QUESTION PAPER (By CBSE) Class-X ( ) Mathematics General Instructions: 1. All questions are compulsory. 2. The question paper consists of 30 questions divided into four sections A, B, C and D. 3. Section A contains 6 questions of 1 mark each. Section B contains 6 questions of 2 marks each. Section C contains 10 questions of 3 marks each. Section D contains 8 questions of 4 marks each. 4. There is no overall choice. However, an internal choice has been provided in four questions of 3 marks each and three questions of 4 marks each. You have to attempt only one of the alternatives in all such questions. 5. Use of calculators is not permitted. Section A Question numbers 1 to 6 carry 1 mark each 1. Write whether the rational number will have a terminating decimal expansion or a nor-terminating repeating decimal expansion. Ans. Non terminating repeating decimal expansion. 2. Find the value(s) of k, if the quadratic equation has equal roots. Ans. 3. Find the eleventh term from the last term of the AP: 27, 23, 19,..., 65. Ans. 4. Find the coordinates of the point on y-axis which is nearest to the point ( 2, 5). Ans. 5. In given figure, and Find the ratio of the area of to the area of Material downloaded from mycbseguide.com. 1 / 16

4 Ans. 6. If find the value of Ans. 25 Section B Question numbers 7 to 12 carry 2 marks each. 7. If two positive integers p and q are written as are prime numbers, then verify: Ans. 8. The sum of first n terms of an AP is given by Find the sixteenth term of the AP. Ans. S n = 2n 2 + 3n S 1 = 5 = a 1 S 2 = a 1 + a 2 = 14 a 2 = 9 d = a 2 a 1 = 4 a 16 = a d = (4) = Find the value(s) of k for which the pair of linear equations have infinitely many solutions. Ans. For pair of equations kx + 1y = k 2 and 1x + ky = 1 We have: For infinitely many solutions, Material downloaded from mycbseguide.com. 2 / 16

5 From (i) and (ii), k = If is the mid-point of the line segment joining the points (2, 0) and then show that the line passes through the point Ans. Since is the mid-point of the line segment joining the points therefore, The line passes through the point 11. A box contains cards numbered 11 to 123. A card is drawn at random from the box. Find the probability that the number on the drawn card is (i) a square number (ii) a multiple of 7 Ans. (i) P(square number) (ii) P(multiple of 7) 12. A box contains 12 balls of which some are red in colour. If 6 more red balls are put in the box and a ball is drawn at random, the probability of drawing a red ball doubles than what it was before. Find the number of red balls in the bag. Ans. Let number of red balls be = x If 6 more red balls are added: The number of red balls = x + 6 Since, There are 3 red balls in the bag. Section C Question numbers 13 to 22 carry 3 marks each. 13. Show that exactly one of the numbers is divisible by 3. Ans. Let n = 3k, 3k + 1 or 3k + 2. (i) When n = 3k: n is divisible by 3. n + 2 = 3k + 2 n + 2 is not divisible by 3. n + 4 = 3k + 4 = 3(k + 1) + 1 n + 4 is not divisible by 3. Material downloaded from mycbseguide.com. 3 / 16

6 (ii) When n = 3k + 1: n is not divisible by 3. n + 2 = (3k + 1) + 2 = 3k + 3 = 3(k + 1) n + 2 is divisible by 3. n + 4 = (3k + 1) + 4 = 3k + 5 = 3(k + 1) + 2 n + 4 is not divisible by 3. (iii) When n = 3k + 2: n is not divisible by 3. n + 2 = (3k + 2) + 2 = 3k + 4 = 3(k + 1) + 1 n + 2 is not divisible by 3. n + 4 = (3k + 2) + 4 = 3k + 6 = 3(k + 2) n + 4 is divisible by 3. Hence exactly one of the numbers n, n + 2 or n + 4 is divisible by Find all the zeroes of the polynomial if two of its zeroes are Ans. Since are the two zeroes therefore, is a factor of given polynomial. We divide the given polynomial by For other zeroes, Zeroes of the given polynomial are 15. Seven times a two digit number is equal to four times the number obtained by reversing the order of its digits. If the difference of the digits is 3, determine the number. Ans. Let the ten s and the units digit be y and x respectively. So, the number is 10y + x. The number when digits are reversed is 10x + y. Material downloaded from mycbseguide.com. 4 / 16

7 Now, 7(10y + x) = 4(10x + y) 2y = x (i) Also x y = 3 (ii) Solving (1) and (2), we get y = 3 and x = 6. Hence the number is In what ratio does the x-axis divide the line segment joining the points Find the co-ordinates of the point of division. OR The points form a parallelogram. Find the length of the altitude of the parallelogram on the base AB. Ans. Let x-axis divides the line segment joining at the point P in the ratio 1 : k. Now, coordinates of point of division Since P lies on x-axis, therefore Hence the ratio is Now, the coordinates of P are OR Let the height of parallelogram taking AB as base be h. Now AB 17. In given figure then prove that OR Material downloaded from mycbseguide.com. 5 / 16

8 In an equilateral triangle ABC, D is a point on the side BC such that that Prove Ans. Since, Also And OR Construction: Draw Material downloaded from mycbseguide.com. 6 / 16

9 18. In given figure are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting at A and at B. Prove that Ans. Join OC In OP = OC (radii of same circle) PA = CA (length of two tangents) AO = AO (Common) Hence, Similarly Now, (By SSS congruency criterion) 19. Evaluate: OR Material downloaded from mycbseguide.com. 7 / 16

10 If then evaluate: Ans. OR 20. In given figure ABPC is a quadrant of a circle of radius 14 cm and a semicircle is drawn with BC as diameter. Find the area of the shaded region. Ans. We know, AC = r Required area = + ar(semicircle on BC as diameter) ar(quadrant ABPC Material downloaded from mycbseguide.com. 8 / 16

11 21. Water in a canal, 6 m wide and 1.5 m deep, is flowing with a speed of 10 km/h. How much area will it irrigate in 30 minutes, if 8 cm of standing water is needed? OR A cone of maximum size is carved out from a cube of edge 14 cm. Find the surface area of the remaining solid after the cone is carved out. Ans. Let the area that can be irrigated in 30 minute be A Water flowing in canal in 30 minutes Volume of water flowing out in 30 minutes Volume of water required to irrigate the field Equating (i) and (ii), we get Or Surface area of remaining solid where r and l are the radius and slant height of the cone. 22. Find the mode of the following distribution of marks obtained by the students in an examination: Marks obtained Number of students Material downloaded from mycbseguide.com. 9 / 16

12 Given the mean of the above distribution is 53, using empirical relationship estimate the value of its median. Ans. So, the mode marks is 68. Empirical relationship between the three measures of central tendencies is: Section D Question numbers 23 to 30 carry 4 marks each. 23. A train travelling at a uniform speed for 360 km would have taken 48 minutes less to travel the same distance if its speed were 5 km/hour more. Find the original speed of the train. OR Check whether the equation has real roots and if it has, find them by the method of completing the square. Also verify that roots obtained satisfy the given equation. Ans. Let original speed of the train be x km/h. Time taken at original speed Time taken at increased speed Now, OR Discriminant So, the given equation has two distinct real roots Multiplying both sides by 5. Material downloaded from mycbseguide.com. 10 / 16

13 Verification: Similarly, 24. An AP consists of 37 terms. The sum of the three middle most terms is 225 and the sum of the last three terms is 429. Find the AP. Ans. Let the three middle most terms of the AP be a d, a, a + d. We have, (a d) + a + (a + d) = 225 Now, the AP is a 18d,,a 2d, a d, a, a + d, a + 2d,, a + 18d Sum of last three terms: Now, first term The AP is 3, 7, 11,, Show that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. OR Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides. Ans. Given: A right triangle ABC right angled at B. To prove: Construction: Draw Proof: In Material downloaded from mycbseguide.com. 11 / 16

14 Now, (corresponding sides are proportional) Similarly Adding (1) and (2) OR Given: To prove: Construction: Draw In Therefore, But Hence, Material downloaded from mycbseguide.com. 12 / 16

15 26. Draw a triangle ABC with side Then, construct a triangle whose sides are times the corresponding sides of Ans. Draw in which and hence Construction of similar triangle as shown below: 27. Prove that Ans. 28. The angles of depression of the top and bottom of a building 50 metres high as observed from the top of a tower are respectively. Find the height of the tower and also the horizontal distance between the building and the tower. Material downloaded from mycbseguide.com. 13 / 16

16 Ans. Now, Now, Height of tower = TR = 75 m Distance between building and tower 29. Two dairy owners A and B sell flavoured milk filled to capacity in mugs of negligible thickness, which are cylindrical in shape with a raised hemispherical bottom. The mugs are 14 cm high and have diameter of 7 cm as shown in given figure. Both A and B sell flavoured milk at the rate of per litre. The dairy owner A uses the formula to find the volume of milk in the mug and charges for it. The dairy owner B is of the view that the price of actual quantity of milk should be charged. What according to him should be the price of one mug of milk? Which value is exhibited by the dairy owner B? Ans. Capacity of mug (actual quantity of milk) Amount dairy owner B should charge for one mug of milk Value exhibited by dairy owner B: honesty (or any similar value) Material downloaded from mycbseguide.com. 14 / 16

17 30. The following distribution shows the daily pocket allowance of children of a locality. The mean pocket allowance is Find the missing frequency k. Daily pocket allowance (in Rs.) Number of children k 5 4 OR The following frequency distribution shows the distance (in metres) thrown by 68 students in a Javelin throw competition. Distance (in m) Number of students Draw a less than type Ogive for the given data and find the median distance thrown using this curve. Ans. Daily pocket allowance Number of children Mid-point (in Rs.) K 20 1 k OR Less than Number of Students 10 4 Material downloaded from mycbseguide.com. 15 / 16

18 Median distance is value of x that corresponds to Cumulative frequency Therefore, Median distance = 36 m Material downloaded from mycbseguide.com. 16 / 16

19 SAMPLE QUESTION PAPER 01 Class-X ( ) Mathematics Time allowed: 3 Hours Max. Marks: 80 General Instructions: (i) All questions are compulsory. (ii) The question paper consists of 30 questions divided into four sections A, B, C and D. (iii) Section A contains 6 questions of 1 mark each. Section B contains 6 questions of 2 marks each. Section C contains 10 questions of 3 marks each. Section D contains 8 questions of 4 marks each. (iv) There is no overall choice. However, an internal choice has been provided in four questions of 3 marks each and three questions of 4 marks each. You have to attempt only one of the alternatives in all such questions. (v) Use of calculators is not permitted. Section-A 1. Can two numbers have 18 as their HCF and 380 as their LCM? Give reason. 2. Find the root of the equation. 3. Determine whether 50cm, 80cm, 100cm can be the sides of a right triangle or not. 4. The length of the shadow of a man is equal to the height of man. The angle of elevation is. 5. If the perimeter and area of a circle are numerically equal, then find the radius of the circle. 6. If three coins are tossed simultaneously, then find the probability of getting at least two heads. Section-B 7. Is a composite number? Justify your answer. Material downloaded from mycbseguide.com. 1 / 17

20 8. The 11 th term of an A.P. exceeds its 4 th term by 14. Find the common difference. 9. Find the relation between x and y, if the points, (1,2) and (7,0) are collinear. 10. Two tangents making an angle of 120 with each other are drawn to a circle of radius 6 cm, find the length of each tangent. 11. Prove that 12. A cone of height 20 cm and radius of base 5 cm is made up of modelling clay. A child reshapes it in the form of a sphere. Find the diameter of the sphere. Section-C 13. Use Euclid s division lemma to show that cube of any positive integer is either of the form 9q, 9q+1 or 9q + 8 for some integer q. 14. Obtain all other zeroes of x 4 + 5x 3-2x 2-40x -48 if two of its zeroes are 15. Solve for x :, x How many terms of the series 54, 51, 48,.be taken so that their sum is 513? Explain the double answer. Or In an AP p th, q th and r th terms are respectively a, b and c. Prove that p(b - c) + q(c - a) + r(a - b) = The point A(3, y) is equidistant from the points P(6,5) and Q(0, -3). Find the value of y. Or If A (4, 6), B (3, -2) and C (5, 2) are the vertices of Δ ABC, then verify the fact that a median of a triangle ABC divides it into two triangles of equal areas. 18. If prove that 19. An observer 1.5m tall is 28.5m away from a chimney. The angle of elevation of the top of Material downloaded from mycbseguide.com. 2 / 17

21 the chimney from her eyes is 45. What is the height of the chimney? Or From the top of a 7m high building, the angle of elevation of the top of a cable tower is 60 o and the angle of depression of the foot of the tower is 30 o. Find the height of the tower. 20. A boy is cycling such that the wheels of the cycle are making 140 revolutions per minute. If the diameter of the wheel is 60 cm, calculate the speed in Km per hour in which the boy is cycling. 21. The following table shows the gain in weight by 50 children in a year. Calculate modal gain in weight. Gain in weight (in kg) No. of children Or Compute the Median for the given data Class interval Frequency What is the probability that a leap year, selected at random will contain 53 Thursdays? Section-D 23. Solve graphically the following equations 2x + 3y = 9; x 2y =1. Shade the region bounded by the two lines and the x axis. Or Check graphically whether the pair of equations x + y = 8 and x 2y = 2 is consistent. If so, solve them graphically. Also find the coordinates of the points where the two lines meet the y-axis. Material downloaded from mycbseguide.com. 3 / 17

22 24. A thief away from a Police Station with a uniform speed 100m/minutes. After one minute a Policeman runs behind the thief to catch him. He goes at a speed of 100m/minute in first minute and increases the speed 10m/minute on each succeeding minute. After how many minutes the Policeman catches the thief. Now answer these questions: (i) Which mathematical concept is being used to solve the above problem? (ii) Which trait of personality of the policeman is showed? 25. In the adjoining figure, PQR, is a right triangle, right angled at Q. X and Y are the points on PQ and QR such that PX : XQ = 1 : 2 and QY : YR = 2 : 1. Prove that 9(PY 2 + XR 2 ) = 13 PR 2 Or A quadrilateral ABCD is drawn to circumscribe a circle (fig-2). Prove that, AB + CD = AD + BC. 26. Prove that the lengths of two tangents drawn from an external point to a circle are equal. 27. Construct an isosceles triangle whose base is 7cm and altitude 5 cm and then construct another triangle whose sides are times the corresponding sides of the isosceles triangle. 28. Suppose a person is standing on a tower of height m and observing a car coming towards the tower. He observed that angle of depression changes from 30 to 45, in 3 seconds. Find the speed of the car. 29. A container opens at the top and made up of metal sheet is in the form of a frustum of a Material downloaded from mycbseguide.com. 4 / 17

23 cone of height 16cm with diameters of its lower and upper ends as 16cm and 40cm respectively. Find the cost of metal sheet used to make the container, if it costs Rs.10 per 100cm 2. ( Use =3.14) 30. The mean of the following frequency distribution is 47. Find the value of p. Classes Frequency p 5 Or Compute the mode for the following frequency distribution. Size of items: Frequency: Material downloaded from mycbseguide.com. 5 / 17

24 CBSE SAMPLE PAPER 01 CLASS X MATHEMATICS Marking Scheme 1. No. Because HCF is always a factor of LCM but here 18 is not a factor of x 2-27x-10=0 (16x + 5) (x - 2)=0 x=, x = 2 3. Clearly, the sum of the squares of the lengths of two sides is not equal to the square of the length of the third side. Hence, given sides do not make a right triangle because it does not satisfy the property of Pythagoras theorem Perimetre of the Circle = Area of the Circle 6. Number of possible outcomes = 8 ( HHH, HHT, HTH,HTT, THH, THT, TTH, TTT) Number of favorable outcomes ( 2 head) = 4 So probability = 7. Yes, = 5045 It has more than two factors 8. Let the first term of AP is a and d is common difference then According to Question a 11 a 4 =14 ; => a + 10d (a + 3d) = 14 => a + 10d - a - 3d = 14 => 7d = 14 => d=2 9. [ ] Material downloaded from mycbseguide.com. 6 / 17

25 x = 7 3y 10. PT=PS ( length of tangents) 60 ΔOTP tan 60 = PT= cm 11. LHS = = RHS 12. Volume of cone = Volume of sphere r = 5 cm Material downloaded from mycbseguide.com. 7 / 17

26 13. a = bq + r ; b =3 ; r = 0, 1, 2 a 3 =(3m) 3 = 9 (3m 3 ) = 9 q a 3 =(3m+1) 3 = 27m m 2 + 9m +1 = 9 q + 1 a 3 =(3m+2) 3 = 27m m m +8 = 9 q Two zeroes are Therefore 15. Material downloaded from mycbseguide.com. 8 / 17

27 => 2x 2 +2ax +bx +ab = 0 => 2x( x+a ) + b ( x+a) = 0 => (x+a) (2x+b) = 0 => x = -a, 16. a = 54, d= -3, s n = 513 S n = 513 = => n = 18, 19 Since d is negative. we get double answer because sum of 18 terms and 19 terms is zero, as few terms are positive and few are negative. Or A + (p - 1)D = a..(i) A + (q-1) D=b. (ii) A + (r-1) D=c. (iii) (ii) - (iii) b - c = (q-1) D (r-1)d Similarly, Material downloaded from mycbseguide.com. 9 / 17

28 Adding (iv), (v) and (vi) 17. PA =QA 17. The point A(3, y) is equidistant from the points P(6,5) and Q(0, -3). Find the value of y Material downloaded from mycbseguide.com. 10 / 17

29 Given the height of the observer be DE = 1.5 m That is AB = 1.5 m Let BC = h is height of the chimney Hence AC = (h 1.5) m Given distance between the observer and the chimney is AD = BE = 28.5 m In right triangle DCA, θ = 45 tan 45 = h = = 30 m Thus the height of the chimney is 30 m. 20. Circumference of wheel = πd = 60 π cm Distance covered in 1 revolution = Distance covered in 140 revolution = km km (Distance covered in 1 min) Distance covered in 1 hr = Speed of cycle = km / hr 21. Modal class = 7-9 Mode- Material downloaded from mycbseguide.com. 11 / 17

30 = 22. There are 366 days in a leap year that contain 52 weeks and 2 more days. So, 52 Thursdays and 2 days. These 2 days can be: (Mon, Tue}, {Tue, Wed}, {Wed, Thu}, {Thu, Fri}, {Fri, Sat}, {Sat, Sun} and {Sun, Mon} (7 cases). In order to have 53 Thursdays we should have either {Thu, Fri} or {Wed, Thu} case. No. of sample spaces = 7. No. of event that gives 53 Thursdays in a leap Year = 2. Required Probability = 23. Table for Eqn.1 and its line Table for Eqn.2 and its line Solution x =3 and y =1 Shaded area is required solution. Or x Material downloaded from mycbseguide.com. 12 / 17

31 y = 8 x Three solutions for equation (i)are given in the table : Three solutions for equation (ii) are given in the table : x Drawing Line AC Drawing Line PR Plotting points A(0, 8), B(4, 4) and C(8, 0) on graph paper the straight line AC is obtained as graph of the equation (i) Plotting points P(0, 1), Q(2, 0) and R(8, 3) on graph paper the straight line PR is obtained as graph of the equation (ii). From the graph, it is clear that a point M(6, 2) common to both the lines AC and PR. So the pair of equations is consistent and the solutions of the equations are x = 6 and y = 2. From the graph it is seen that the coordinates of the points where the lines AC and PR meets the y-axis are (0, 8) and (0, 1) respectively. 24. Time taken by Thief before being caught = n+1 Material downloaded from mycbseguide.com. 13 / 17

32 Distance travelled by Thief = 100 ( n+ 1) 100 ( n+1) = n = 5 minutes i) Arithmetic Progression ii) Responsibility of their work ( duty) and honesty 25. Or Since, the lengths of tangents drawn from an external point to a circle are equal. AP = AS (i) BP = BQ (ii) Material downloaded from mycbseguide.com. 14 / 17

33 CQ = CR (iii) DR = DS (iv) Now, AB + CD = AP + PB + CR + RD = AS + BQ + CQ + DS = (AS + DS) + (BQ + CQ) = AD + BC Hence proved. 26. Construction: Draw a circle with centre O. From a point P outside the circle, draw two tangents P and R. To Prove: PQ = PR Proof: In Δ POQ and Δ POR OQ = OR (radii) PO = PO (common side) PQO = PRO (Right angle) Δ POQ Δ POR (By RHS Congruency rule) Hence proved Material downloaded from mycbseguide.com. 15 / 17

34 Δ PQB tan 45 =, x= 15( (1) Δ PQA tan 30 = x + y = 15 ( (2) From (1) and (2) y = 30 m Since the car moving from A to B in 3 seconds Speed = = 10 m / sec 29. R = 20 cm, r = 8 cm, h= 16 cm l = = 20 cm Material downloaded from mycbseguide.com. 16 / 17

35 Total surface area = CSA of frustum + area of base = πl(r+r) + = Rate of metal sheet used = Rs.10 per 100 cm 2 Cost of metal sheet used = = Rs (Approximately) 30. CI f i x i f i x i p 70 70p p p Material downloaded from mycbseguide.com. 17 / 17

36 SAMPLE QUESTION PAPER 02 Class-X ( ) Mathematics Time allowed: 3 Hours Max. Marks: 80 General Instructions: (i) All questions are compulsory. (ii) The question paper consists of 30 questions divided into four sections A, B, C and D. (iii) Section A contains 6 questions of 1 mark each. Section B contains 6 questions of 2 marks each. Section C contains 10 questions of 3 marks each. Section D contains 8 questions of 4 marks each. (iv) There is no overall choice. However, an internal choice has been provided in four questions of 3 marks each and three questions of 4 marks each. You have to attempt only one of the alternatives in all such questions. (v) Use of calculators is not permitted. SECTION - A 1. The sum and the product of zeroes of a quadratic polynomial p(x) are 7 and 10 respectively. Then find p(x). 2. Which term of the A.P 92, 88, 84, 80,.... is 0? 3. Find the ratio in which the Y axis divides the line segment joining the points (5, -6) and (-1, -4). 4. If PT is a tangent drawn from a point P to a circle touching it at T, and O is the centre of the circle, then OPT + POT is 5. If two towers of heights h 1 and h 2 subtend angles of 60 and 30 respectively, at the midpoint of the line joining their feet, then h 1 : h 2 is 6. In rolling a dice, the probability of getting a number less than 4 is SECTION - B Material downloaded from mycbseguide.com. 1 / 18

37 7. Solve for x by completing the square method: 8. Find the 7 th term from the end of the A.P. 7, 10, 13, Find a so that (3, a) lies on the line represented by 2x - 3y - 5 = 0. Also, find the coordinates of the point where the line cuts the x axis. 10. ABC is a right triangle right-angled at B. Let D and E be any points on AB and BC respectively. Prove that AE 2 + CD 2 = AC 2 + DE Two concentric circles have radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle. 12. A cylinder and a cone have base radii 5 cm and 3 cm, respectively, and their respective heights are 4 cm and 8 cm. Find the ratio of their volumes. SECTION - C 13. A part of monthly expenses of a family is constant and the remaining varies with the price of rice. When the cost of rice is Rs.250 per quintal, the monthly expenditure of the family is Rs.1000 and when the cost of rice is Rs.240 per quintal the monthly expenditure is Rs.980. Find the monthly expenditure of the family when the cost of rice is Rs.300 per quintal. 14. Prove that is irrational. 15. Use Euclid s division algorithm to find the HCF of 4052 and The Sum S n of first n even natural numbers is given by the relation S n = n(n+1). Find n, if the sum is 420. Or A man arranges to pay a debt of Rs.3600 in 40 monthly installments which are in a AP. When 30 installments are paid he dies leaving one third of the debt unpaid. Find the value of the first installment. 17. Find the coordinates of the points of trisection of the line segment joining the points A( 1, Material downloaded from mycbseguide.com. 2 / 18

38 -2) and B(-3, 4). Or If the points (p, q) (M, n) and (p m, q n) are collinear, show that pn = qm. 18. In an equilateral triangle ABC, if AD BC, then prove that 3AB 2 = 4AD In the adjoining figure, two tangents PQ and PR are drawn to a circle with Centre O from an external point P. Prove that Or A sector of circle of radius 12 cm has the angle 120⁰. It is rolled up so that the two bounding radii are formed together to form a cone. Find the volume of the cone. 20. Find the value of 21. Two different dice are thrown together. Write all the possible outcomes. Find the probability that the product of the numbers appearing on the top of the dice is less than 9. Or Anil bags contain 5 red marbles, 8 white marbles and 4 green marbles. One marble is taken out of the box at random. What is the probability that the marble taken out will be (i) Red (ii) White (iii) Not green 22. The king, queen and jack of clubs are removed from a deck of 52 playing cards. The remaining cards are well shuffled and one card is drawn at random from it. Find the probability of getting the selected card as (i)a heart (ii). a king (iii). a club Material downloaded from mycbseguide.com. 3 / 18

39 SECTION - D 23. A two-digit number is such that the product of its digits is 12. When 36 is added to this number, the digits interchange their places. Find the number. Or Solve for x, y: 24. A contract on construction job specifies a penalty for delay of completion beyond a certain date as follows: Rs. 200 for the first day, Rs. 250 for the second day, Rs.300 for the third day, etc., the penalty for each succeeding day being Rs.50 more that for preceding day. (i) How much money the contractor has to pay as penalty, if he has delayed the work by 30 days? (ii) Which moral value one can learn from above problem? 25. Draw a pair of tangents to a circle of radius 4 cm, which are inclined to each other at an angle of 60. Or Construct a tangent to a circle of radius 2 cm from a point on the concentric circle of radius 2.6 cm and measure its length. Also verify the measurements by actual caclulations.(length of tangent = 2.1 cm) 26. Prove that 27. In the adjoining figure four equal circles are described at the four corners of a square so that each touches two of the others. The shaded area enclosed between the circles is cm 2. Find the radius of each circle. Material downloaded from mycbseguide.com. 4 / 18

40 28. The angle of elevation of an aeroplane from a point A on the ground is After a flight of 30 seconds, the angle of elevation changes to If the plane is flying at a constant height of m, find the speed in km/hr of the plane. 29. A right triangle having sides 15cm and 20cm is made to revolve about its hypotenuse. Find the volume and Surface Area of the double cone so formed. (Use p =3.14) 30. During the medical check-up of 35 students of a class, their weights were recorded as follows Weight Less Less Less Less Less Less Less Less (in kg) than 38 than 40 than 42 than 44 than 46 than 48 than 50 than 52 No. of Students Draw a less than type ogive for the given data. Hence, obtain the median weight from the graph and verify the result by using the formula. Or Compute the Mean and Median for the given data. Class interval Frequency Material downloaded from mycbseguide.com. 5 / 18

41 CBSE SAMPLE PAPER 02 CLASS X - Mathematics Marking Scheme 1. x 2-7x th term is : : = 8. d = -3 ; a = 184 a 7 =a + 6d = (-3) = = Since (3, a) lies on 2x 3y -5 = 0, 2(3) 3a 5 =0 a = 1/3 Material downloaded from mycbseguide.com. 6 / 18

42 Let the co-ordinate of the point which cuts the x axis be (x, 0). 2x 3(0)-5 = 0. x = 5/2 (5/2, 0) is the point of intersection with x axis 10. Since ABE is right triangle, right-angled at B AE 2 = AB 2 + BE 2...(i) Again, DBC is right triangle right- angled at B CD 2 = BD 2 + BC 2...(ii) Adding (i) and (ii) we get AE 2 + CD 2 = (AB 2 + BE 2 )+( BD 2 + BC 2 ) AE 2 + CD 2 = (AB 2 + BC 2 )+( BE 2 + BD 2 ) Using Pythagoras theorem in ABC and DBE we have AE 2 + CD 2 = AC 2 +DE Material downloaded from mycbseguide.com. 7 / 18

43 Here PQ is tangent to smaller circle. OR PQ Now in PRO, PR 2 +OR 2 =OP 2 PR =5 2 PR= PR = QR (Perpendicular from the centre of the circle bisects the chord) Length of the chord of the larger circle = PQ = 2 PR = 2 4 = 8 cm. 12. Volume of cylinder : Volume of cone = 25 : Let the constant expenditure be Rs.x Let the consumption of rice by the family be y quintals. x y = (1) x y = (2) Material downloaded from mycbseguide.com. 8 / 18

44 Solving (1) and (2) x = 500; y = 2 Total expenditure when the cost of rice is Rs.300 = x +300y = Rs Suppose is rational But irrational while is rational and an irrational number can never be equal to a rational number. Thus our assumption is wrong and hence is irrational 15. Since > 4052, we apply the division lemma to and 4052, to get = = = = = = = The remainder has now become zero, so our procedure stops. HCF of and 4052 is 4. Material downloaded from mycbseguide.com. 9 / 18

45 16. n (n+1)=420 n 2 + n 420 = 0 n = -21, 20 n cannot be negative n= Let P(x 1,y 1 ), Q(x 2, y 2 ) divide AB into 3 equal parts. P divided AB in the ratio of 1 : 2 P(x 1, y 1 ) = = Since Q is the midpoint of PB, Q(x 2, y 2 ) = = 18. Given: ABC is equilateral and AD BC. To Prove: 3AB 2 = 4 AD 2 Material downloaded from mycbseguide.com. 10 / 18

46 Proof: In ABD, By Pythagoras theorem, AB 2 = AD 2 + BD 2 BD = AB 2 = AB Given: - (i) PQ and PR are the tangents of the circle (ii) O is the centre of the circle To prove: Construction : Join OR and OP Proof : - Now OR PR (radius from point of contact is to tangent) Similarly, OQ PQ In quadrilateral OPQR (angles in same segment).. (1) Also. (2) [from (1)] From (1) and (2) Material downloaded from mycbseguide.com. 11 / 18

47 Hence proved 20. = = = = S= { (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), ( 2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)} n(s) = 36 Favourable events:(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), ( 2,1), (2,2),(2,3), (2,4),(3,1), (3,2), (4,1), (4,2), (5,1), (6,1) n(e) = 16 P(E) = 22. Total number of cards = 52 Number of cards removed = 3 n(s) = 52-3 = 49 Material downloaded from mycbseguide.com. 12 / 18

48 i) P( heart) = ii) P(a king) = iii) P( a club ) = 23. Let Two digit number = 10x + y xy= (1) 10x + y + 36 = 10y + x x y = (2) from 1 and 2 we get x 2 + 4x 12 =0 x= -6, 2 x cannot be negative, so x= 2 y=12 / 2 = 6 Required number is The required sequence is 200, 250, 300, 350,... a = 200, d = 50, n = 30 The contractor has to pay Rs. 27,750. Value: One should be punctual and show dedication to his work, failing of which may result loss. 25. Material downloaded from mycbseguide.com. 13 / 18

49 Draw a circle with radius 4 cm. Draw 2 radius OQ and OR at 120. Construct s at Q, R to meet at P. Join OQ and OR and they are required tangents. 26. LHS = Material downloaded from mycbseguide.com. 14 / 18

50 27. Let r cm be the radius of each circle Area of square - Area of 4 sectors = Radius of each circle is 2 cm. (radius cannot be negative) 6r 2 =24 r 2 =4 r = ±2 28. Let P be the position of the flight initially. And BP be the height of the flight from the ground. Consider ABP, Material downloaded from mycbseguide.com. 15 / 18

51 Consider ACQ, BC = AC AB=10, = 7200 m Speed of the flight = = 864 km/hr 29. OA = 12 cm Material downloaded from mycbseguide.com. 16 / 18

52 30. By locating upper limits on X axis and number of students on Y axis, we get the following graph From the graph, we see that median is The frequency distribution table with the given cumulative frequencies becomes: Class Intervals Frequency Cumulative Frequency Below Material downloaded from mycbseguide.com. 17 / 18

53 Here n/2 = 35/2 = Now is the class whose cumulative frequency is 28 is greater than n/2 i.e17.5. So, is the median class. From the table f = 14, cf = 14, h =2 Median = = = Material downloaded from mycbseguide.com. 18 / 18

54 SAMPLE QUESTION PAPER 03 Class-X ( ) Mathematics Time allowed: 3 Hours Max. Marks: 80 General Instructions: (i) All questions are compulsory. (ii) The question paper consists of 30 questions divided into four sections A, B, C and D. (iii) Section A contains 6 questions of 1 mark each. Section B contains 6 questions of 2 marks each. Section C contains 10 questions of 3 marks each. Section D contains 8 questions of 4 marks each. (iv) There is no overall choice. However, an internal choice has been provided in four questions of 3 marks each and three questions of 4 marks each. You have to attempt only one of the alternatives in all such questions. (v) Use of calculators is not permitted. Section A Q.1 Find the value of so that the point, lies on the line represented by Q.2 Write the next term of the AP: Q.3 If the value of. Q.4 If areas of two similar triangles are in ratio 25: 64, write the ratio of their corresponding sides. Q.5 In figure, CP and CQ are tangents to a circle with centre O. ARB is another tangent touching the circle at R. If CP, BC = 7 cm, then find the length of BR. Q.6 A bag contains 4 red and 6 black balls. A ball is taken out of the bag at random. Find the probability of getting a black ball. Material downloaded from mycbseguide.com. 1 / 26

55 Section B Q.7 Prove that is an irrational number. Q.8 Solve: Q.9 Find the value of p so that the quadratic equation has two equal roots. Q.10 Evaluate : Q.11 The angle of elevation of the top of the tower from a point on the ground, which is 30 m away from the foot of the tower, is 30. Find the height of the tower. Q.12 A solid sphere of radius 10.5 cm is melted and recast into smaller solid cones, each of radius 3.5 cm and height 3 cm. Find the number of cones so formed. Section C Q.13 In a flight of 2800 km, an aircraft was slowed down due to bad weather. Its average speed is reduced by 100 km/h and time increased by 30 minutes. Find the original duration of the flight. Q.14 The sum of 4 th and 8 th term of an AP is 24 and sum of 6 th and 10 th terms is 44. Find AP. Q.15 If one diagonal of trapezium divides the other diagonal in the ratio 1: 3. Prove that one of the parallel sides is three times the other. Q.16 Find the area of quadrilateral ABCD whose vertices are. Or The points A(4, 2), B(7, 2), C(0, 9) and D( 3, 5) form a parallelogram. Find the length of the altitude of the parallelogram on the base AB. Material downloaded from mycbseguide.com. 2 / 26

56 Q. 17 An aeroplane, when 3000 m high passes vertically above plane at any instant, when the angle of elevation of the two aeroplanes from the same point on the ground are and respectively. Find the vertical distance between aeroplanes. Or The angles of depression of the top and bottom of a building 50 metres high as observed from the top of a tower are 30 and 60, respectively. Find the height of the tower and also the horizontal distance between the building and the tower. Q.18 If all sides of a parallelogram touch a circle, show that the parallelogram is rhombus. Q.19 Construct a triangle ABC in which BC = 8 cm, B = and C. Construct another triangle similar to ABC such that its sides are of the corresponding sides of ABC. Or Draw a triangle ABC with side BC = 7 cm,. Then, construct a triangle whose sides are times the corresponding sides of ABC. Q.20 Show that the point A (3, 5), B (6, 0), C (1, -3) and D (-2, 2) are the vertices of a square ABCD. Q.21 The median of distribution given below is Find the values of and, if the sum of frequency is 20. Class interval Frequency 4 x 5 y 1 Or Find the mean, mode and median for the following data: Classes Frequency Material downloaded from mycbseguide.com. 3 / 26

57 Q.22. Two dice are thrown simultaneously. What is the probability that a) 5 will not come up on either of them? b) 5 will come up on at least one? c) 5 will come up on at both Dice? Section D Q.23 Use Euclid s Division Algorithm to show that the square of any positive integer is either of the form or 3m+1 for some integer m. Or Show that exactly one of the numbers n, n + 2 or n + 4 is divisible by 3. Q.24 if the polynomial is divided by the reminder comes out to be Q.25 Draw the graphs of the following equations: x + y = 5, x - y = 5 (i) Find the solution of equation from the graph. (ii) Shade the triangular region formed by the lines and y-axis. Or Find the solution of pair of linear equation with the help of graph.2x + y = 6, 4x 2y = 4 Q.26 In an equilateral triangle ABC, D is a point on side BC such that 4BD = BC. Prove that Q.27 If prove that Q.28 In the given figure, three circle each of radius 3.5 cm are drawn in such a way that each of them touches the other two. Find the area of shaded region enclosed between these three circles. [Use Material downloaded from mycbseguide.com. 4 / 26

58 Q.29 Water is flowing at the rate of 15 km/hour through a pipe of diameter 14 cm into a cuboidial pond which is 50 m long and 44 m wide. In what time will the level of water in pond rise by 21 cm? Q. 30 Find the mean, mode and median of the following frequency distribution : Class Frequency Or The following distribution shows the daily pocket allowance of children of a locality. The mean pocket allowance is Rs18. Find the missing frequency k. Daily Pocekt Allowence Number of Children k 5 4 Material downloaded from mycbseguide.com. 5 / 26

59 CBSE SAMPLE PAPER 03 CLASS X Mathematics Solutions Solution 1: Since point lies on line Then Solution 2: AP is and so on Common difference Next term Solution 3: Solution 4: We know that, If two triangle are similar then, Material downloaded from mycbseguide.com. 6 / 26

60 Solution 5: CP = CQ = 11 cm [Tangents from same external point] CQ = BC + BQ ButBQ = BR 11 = 7 + BR [as BC = 7 cm ] BR = 4 cm Solution 6: Number of ways t0o select a ball = 10 Number of ways to select a black ball = 6 Probability of getting a black ball = Solution 7 Let, If possible, is a rational number. From, we notice LHS is an irrational number and RHS is rational number, which is not possible. Hence, our supposition is wrong. Hence, Is an irrational number. Solution 8: Adding (i) and (ii) we get Material downloaded from mycbseguide.com. 7 / 26

61 Subtract (i) from (ii) we get Solving and we get Solution 9: Here, For equal roots,d=0 D 0 0 But m [ In quadratic equation, q ] Solution 10: Solution 11: Let h be the height of the tower In Material downloaded from mycbseguide.com. 8 / 26

62 Solution 12: Volume of solid sphere Volume of 1 cone Number of cones formed Solution 13: Let original duration of flight be Distance km Usual speed When time is increased, then time taken And new speed A.T.Q., hours. hrs Original duration of the flight hours Solution 14: Let the 4 th term Material downloaded from mycbseguide.com. 9 / 26

63 And 8 th term Then their sum Similarly On subtracting equation from we get [From ] Therefore, AP is Solution 15: DE = EB = 1 : 3 Material downloaded from mycbseguide.com. 10 / 26

64 In (alt. angles) (V.O.A.) Solution 16: Area of Similarly, area of Area of quadrilateral ABCD Area of Area of Or Let the height of parallelogram taking AB as base be h. Now AB = Area of = Material downloaded from mycbseguide.com. 11 / 26

65 Now, Solution 17: one aeroplane flying at point Another aeroplane flying at point C. Both are in the same line. In right In right From Since m Vertical distance between two aeroplane = m Material downloaded from mycbseguide.com. 12 / 26

66 Or In As BP = GR then From (i) and (ii) Now, TR = TP + PR = ( ) m. Height of tower =TR = 75 m. Distance between building and tower = GR = Material downloaded from mycbseguide.com. 13 / 26

67 Solution 18: Sides of parallelogram ABCD, touch circle at P, Q, R, S AP = AS, BP = BQ, CQ =CR, DR = DS [Length of the tangents from a point outside the circle is equal ] Consider, AB + CD AP + PB + CR + DR AS + BQ +CQ +DS (AS +SD ) + (BQ +QC) AD + BC AB + AB AD + AD [ Opposite sides of a parallelogram are equal ] 2 AB 2AD AB AD AS Adjacent sides of a parallelogram are equal. Hence parallelogram is a rhombus Solution 19: Material downloaded from mycbseguide.com. 14 / 26

68 Steps of construction: 1. Draw a line segment BC = 8 cm. 2. Then construct B = at B. 3. Then construct C = at C. 4. Line segment from the angles B and C, when produced, meet at A. 5. ABC is the constructed triangle. 6. Draw an acute angle CBX below BC. 7. Take points at BX, such that 8. Join 9. Draw parallel to meeting BC at C. 10. Draw C A parallel to CA, meeting BA at A. Or Draw ABC in which BC = 7 cm, and hence C = 30. Construction of similar triangle A'BC' as shown below: A BC is the required triangle similar to ABC where each side is of the side of ABC Material downloaded from mycbseguide.com. 15 / 26

69 Solution 20: AB BC Solution 21: Class interval Frequency cf x 4+x x y 9+x+y x+y 20 Median Median class Median Material downloaded from mycbseguide.com. 16 / 26

70 14.4 Or We have Classes Mid value (x i ) f i f i u i c.f Let assumed mean a = 45, Here h = 10 We know that mean Mean Since maximum frequency = 12 Modal class = Material downloaded from mycbseguide.com. 17 / 26

71 Here, Now Mode Mode = 45 Now Median class is Now median Here Median = Thus, Median = 42.5 Solution 22: Elementary Events are Material downloaded from mycbseguide.com. 18 / 26

72 Total number of elementary events =36 a) Let A=5 will not come up on either of them. Total number of elementary events favorable to A=25 b) Let B=5 will come up at least one die. Total number of elementary events favorable to B=11 c) Let C=5 will come up at both dice. Total number of elementary events favorable to C=1 Solution 23: Let n be any positive integer. When Material downloaded from mycbseguide.com. 19 / 26

73 When N (Where ) When (Where Square of any positive integer is either of the form Solution 24: P ( ) ( ) Material downloaded from mycbseguide.com. 20 / 26

74 Equating the coefficients, we get Also Using From and we get Solution 25: (i) Both lines intersect at point (5,0) on x- axis Material downloaded from mycbseguide.com. 21 / 26

75 (ii) Required portion is shaded in the graph. Or 2x + y = 6, 4x 2y = 4 For equation 2x + y 6 = 0, we have following points which lie on the line x 0 3 y 6 0 For equation 4x 2y 4 = 0, we have following points which lie on the line x 0 1 y -2 0 We can clearly see that lines are intersecting at (2, 2) which is the solution. Hence x = 2 and y = 2 and lines are consistent. Solution 26: In equilateral 4BD=BC Material downloaded from mycbseguide.com. 22 / 26

76 Construction: Draw In right AED, In right [using (i)] Solution 27: Now, LHS Solution 28: Material downloaded from mycbseguide.com. 23 / 26

77 ABC is an equilateral triangle each of whose side is of length = = 7 cm ar( ABC) = = = Area of 3 sectors = Area of shaded region = = Solution 29: let the level of water rises in the tank in hours Length of the water flow in hours = metres Diameter of the pipe = 14 cm Radius Volume of water Volume of water flow in hours in the pond Material downloaded from mycbseguide.com. 24 / 26

78 Volume of cuboidal pond with water level of height 21 cm ATQ, Hence, the level of water in the pond will rise by 21 cm in 2 hours. Solution 30: Table for mean, mode and median : C.I Mean For Median : Median Class : Median Material downloaded from mycbseguide.com. 25 / 26

79 For Mode: Maximum frequency Modal Class is Mode Or Daily Pocket Allowance Number of Children(f i ) Mid-point (x i ) f i u i k 20 1 k Material downloaded from mycbseguide.com. 26 / 26

80 SAMPLE QUESTION PAPER 04 Class-X ( ) Mathematics Time allowed: 3 Hours Max. Marks: 80 General Instructions: (i) All questions are compulsory. (ii) The question paper consists of 30 questions divided into four sections A, B, C and D. (iii) Section A contains 6 questions of 1 mark each. Section B contains 6 questions of 2 marks each. Section C contains 10 questions of 3 marks each. Section D contains 8 questions of 4 marks each. (iv) There is no overall choice. However, an internal choice has been provided in four questions of 3 marks each and three questions of 4 marks each. You have to attempt only one of the alternatives in all such questions. (v) Use of calculators is not permitted. SECTION - A 1. Find the zeroes of the quadratic polynomial. 2. Write the sum of the first fifteen natural numbers. 3. Find the coordinate of the mid-point of the line segment joining the points whose coordinates are and. 4. In fig-1, centre of the circle is O. From outside point P, two tangents PA and PB are drawn to touch the circle. Given ÐAPB = 60 o. Find ÐAOB. 5. A ladder of length 4 m. makes an angle of 45 o with the ground when placed against an Material downloaded from mycbseguide.com. 1 / 21

81 electric post. Determine the distance between the feet of the ladder and the electric post. 6. Captain of Indian cricket team tosses two different coins, one of 1 and other of 2 simultaneously. What is the probability that he gets at least one head? SECTION - B 7. Find the discriminant of the equation and hence find the nature its roots. 8. If the 10 th term of an A.P. is 47 and its 1 st term is 2, find the sum of its first 15 terms. 9. Find a relation between x and y such that the point (x, y) is equidistant from the points (7, 1) and (3, 5). 10. In fig-2, and. Prove that, is isosceles. 11. Two concentric circles are of radius 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle. 12. The height and base diameter of a solid cylinder are 210 m and 24 cm respectively. Find the volume of the cylinder. SECTION - C 13. The taxi charges in a city consist of a fixed charge together with the charge for the distance covered. For a distance of 10 km, the charge paid is 200 and for a journey of 15 km, the charge paid is 275. What are the fixed charges and the charge per km? How much a person has to pay for travelling a distance of 25 km? Material downloaded from mycbseguide.com. 2 / 21

82 14. Show that is irrational. 15. Show that any positive odd integer is of the form 4q + 1 or 4q + 3, where q is some integer. 16. The 4 th term of an A.P. is equal to three times the 1 st term & the 7 th term exceeds twice the 3 rd term by 1. Find the 1 st term and common difference. Or The first and the last terms of an AP are 17 and 350 respectively. If the common difference is 9, how many terms are there and what is their sum? 17. If A(-5, 7), B(- 4, -5), C(-1, -6) and D(4, 5) are the vertices of a quadrilateral, find the area. Or Find the coordinates of the points which divide the line-segment joining the points (-4, 0) and (0, 6) in four equal parts. 18. Prove that the sum of the squares of the sides of a rhombus is equal to sum of the squares on its diagonals 19. Prove that, the lengths of tangents drawn from an external point to a circle are equal. Or In fig-3, PQ and RS are two parallel tangents to a circle with centre O and another tangent EF with point of contact C intersecting PQ at E and RS at F. Prove that EOF = Evaluate: Material downloaded from mycbseguide.com. 3 / 21

83 21. A box contains 90 discs which are numbered from 1 to 90. If one disc is drawn at random from the box, find the probability that it bears a perfect square number. Or Two horses are considered for a race. The probability of selection of the first horse is 1/4 and that of second is 1/3. What is the probability that : (a) both of them will be selected. (b) only one of them will be selected. (c) none of them will be selected. 22. A box contains 5 red marbles, 8 white marbles and 4 green marbles. One marble is taken out of the box at random. What is the probability that the marble taken out will be i) white ii) not green? SECTION - D 23. Two water taps together can fill a tank in hours. The tap of larger diameter takes 10 hrs less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank. Or A train, travelling at a uniform speed for 360 km, would have taken 48 minutes less to travel the same distance if its speed were 5 km/hr more. Find the original speed of the train. 24. Find the sum of all three-digit numbers which leaves the remainder 3 when divided by Construct a triangle similar to triangle ABC with its side equal to of the corresponding sides of the triangle ABC. Or Construct an isosceles triangle whose base is 7cm and altitude 5 cm and then construct another triangle whose sides are times the corresponding sides of the isosceles triangle. 26. Prove that:. Material downloaded from mycbseguide.com. 4 / 21

84 27. In fig-3, of ABC is a right angle and AB = 3 units, AC = 4 units. Semicircles are drawn on AB, AC and BC as diameters. Also a circle circumscribing the ABC is drawn. Find the area of the shaded region. 28. Two ships are sailing in the sea on either side of a light-house. The angles of depression of two ships as observed from the top of the light-house are 60 o and 45 o respectively. If the distance between the ships is meters, find the height of the light-house. 29. A person connects a pipe of internal diameter 20m from a canal into a empty cylindrical tank in his field which is 10 m in diameter and 2 m deep. If the rate of flow of water through the pipe is 6km/hr, then after how much time the flow of water should be stopped to avoid over flow of water from the tank? What value we have from the problem? 30. The median of the following frequency distribution is 35. Find the value of x. Class Interval Frequency x 3 2 Or Compute the mode for the following frequency distribution. Size of items: Frequency: Material downloaded from mycbseguide.com. 5 / 21

85 CBSE SAMPLE PAPER 04 CLASS X Mathematics Solution SECTION- A 1. We have =(x+5)(x+2) So, the zeroes of are - 5 and The sum of the first fifteen natural numbers = 3. Coordinates of the mid-point are Possible outcomes are (H, H), (H, T), (T, H), (T, T) which are equally likely. Outcomes favorable to the event are (H, H), (H, T), (T, H).Hence the required Probability = SECTION- B 7. Here,. Discriminant Material downloaded from mycbseguide.com. 6 / 21

86 Hence the given quadratic equation has two equal real roots. 8. Let the common difference of the A.P be d. Given, the first term (a) = 2 and the 10 th term ( ) = 47 As, n th term, So, i.e., 9d = 47-2 i.e., As, the sum of the first n-terms of an AP, Therefore, the sum of the first 15-terms of the AP 9. Since the point P(x, y) is equidistant from the points A(7, 1) and B(3, 5) So, PA = PB i.e., i.e., i.e., i.e., i.e.,, This is the required relqtion. 10. Hence, (corresponding angles)... (i) Material downloaded from mycbseguide.com. 7 / 21

87 ... (ii) Therefore, PQ = PR [sides opposite to equal angles ] Hence, is isosceles. 11. Let C is the centre of two concentric circles of radii 5cm and 3 cm. Let AB, a chord of the bigger circle touches the smaller circle at M. CM = 3 cm, CA = 5 cm [radius through point of contact is to tangent ] [by Pythagoras Theorem ] i.e., Therefore, AB=2AM = 2*4 = 8 cm. [Line segment, drawn from centre of a circle perpendicular to any chord, bisects the chord ] Hence, the length of the chord is 8 cm. 12. Let V, h and r denote the volume, height and base radius of the cylinder respectively. Here, h=210 m, Then, Hence the volume of the cylinder is Material downloaded from mycbseguide.com. 8 / 21

88 SECTION - C 13. Let the fixed charge = x and the charge per km = y According to the first condition, x + 10y = 200 (i) According to the second condition, x + 15y = 275 (ii) Now subtracting (i) from (ii), we get, 5y = 75 i.e., y = 15 From (i), x+ 10*15 = 200 (putting value of y ] i.e., x = i.e., x = 50 Hence, the fixed charge = 50 and the charge per km = 15 Now for travelling a distance of 25 km, the person has to pay = ( *15) = let us assume that, is rational. i.e., let for two coprime numbers a and b ( a, b are integers, b 0), i.e., Since, is rational so, is rational. This is a contradiction, because is irrational. This contradiction is only for our incorrect assumption, that, is rational. Material downloaded from mycbseguide.com. 9 / 21

89 Hence, the conclusion is that, is irrational. 15. According to Euclid s division lemma, Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 r < b. Let us start with taking a, where a is a positive odd integer. We apply the division algorithm with a and b = 4. Since, 0 r < 4, the possible remainders are 0, 1, 2 and 3. i.e., a can be 4q or 4q + 1 or 4q + 2, or 4q + 3, where q is the quotient. However, since a is odd, a cannot be 4q or 4q + 2 (since they are both divisible by 2) Therefore, any odd integer is of the form 4q + 1 or 4q Let, a, d and be the first term, common difference and the term of the AP. As, According to the first condition, i.e., a + (4-1)d = 3a i.e., 3d = 2a. (i) Also, according to the 2 nd condition, i.e., a + (7-1)d - [a + (3-1)d ] = 1 i.e., a + 6d - a - 2d = 1 i.e., 4d = 1 i.e., Now from (i), i.e., Hence the first term = and common difference = Material downloaded from mycbseguide.com. 10 / 21

90 17. By joining A to C, we get and As, area of a triangle with vertices at points is (i) So, using formula (i) So the area of is. Again, using formula (i), So, the area of quadrilateral ABCD = 18. Let ABCD is rhombus. Its diagonals AC and BD intersect at point O. To prove that, Material downloaded from mycbseguide.com. 11 / 21

91 Since diagonals of a rhombus bisects each other perpendicularly, so, AO = OC = ; BO = OD= and AC is perpendicular to BD at point O. and are all right angled triangle. Using Pythagoras Theorem, From, From, From, From, Adding (i), (ii), (iii), (iv), we get, [ since AO = CO and BO = DO ] i.e., Hence proved 19. Let from an external point P two tangents PQ and PR are drawn to the circle with centre at O. The tangents touch the circle at points Q and R. We have to prove that, PQ = PR. Let us draw the line segments OQ, OR and OP. Now in DPOQ and DPOR, Material downloaded from mycbseguide.com. 12 / 21

92 [each = 90 o as OQ and OR are radii through points of contacts] OQ = OR [radii of same circle] OP = OP [ Common side] Therefore, [RHS] Hence, PQ = PR [corresponding part of congruent triangles] 20. We know,,, = 21. One disc is drawn at random from the box means that all the discs are equally likely to be drawn. Let the event of drawing one disc bearing a perfect square number be E. Material downloaded from mycbseguide.com. 13 / 21

93 Given that total number of discs in the box = 90 Therefore, total number of all possible outcomes = 90 1, 4, 9, 16, 25, 36, 49, 64, 81 are perfect square numbers between 1 to Therefore, outcomes favorable to the event E = 9 So, 22. One marble is taken out of the box at random means, all the marbles are equally likely to be taken out. Therefore, the total number of possible outcomes = = 17 Let the event of taking out of one white marble be W and also let the event of taking out of one green marble be G Then number of outcomes favorable to the event W = 8 Therefore, [answer of (i) ] Again the number of outcomes favourable to the event G = 4 So, Therefore, P( not green) = 1 - P(G) [answer of (ii) ] SECTION - D 23. Let the tap of smaller diameter can fill the tank separately in x hrs Then the tap of bigger diameter can fill the tank separately in (x - 10) hrs. According to the question, two water taps together can fill a tank in hours i.e., in hrs. Then, in 1 hr smaller diameter tap can fill part of the tank, Material downloaded from mycbseguide.com. 14 / 21

94 in 1 hr bigger diameter tap can fill and 1 hr two taps together can fill part of the tank, part of the tank, So, i.e., i.e., i.e., i.e., i.e., Now, using the quadratic formula, we get, Since, so, x = 25 Therefore, tap of smaller diameter can fill the tank separately in 25 hrs, and the tap of bigger diameter can fill the tank separately in 15 hrs. 24. The first three-digit number which leaves remainder 3 when divided by 5 is 103.[as 103 = 5* ] Last three-digit number which leaves remainder 3 when divided by 5 is 998. [ as 998 = 5* ] Now the three-digit numbers which leaves remainder 3 when divided by 5 are 103, 108, 111,, 998, which form an AP. First term, a = 103, common difference, d = = 5 Material downloaded from mycbseguide.com. 15 / 21

95 Let, 998 be the term ( ) of the series. As, So, i.e., i.e., Therefore, 998 is the 180 th term of the series. Now sum of the terms of the series = = Steps of construction: Draw any ray BX making an acute angle with BC on the side opposite to the vertex A. Locate 4 (the greater of 3 and 4 in ) points on BX so that Material downloaded from mycbseguide.com. 16 / 21

96 Join B 4 C and draw a line through B 3 (the 3rd point, 3 being smaller of 3 and 4 in ) parallel to B 4 C to intersect BC at C. Draw a line through C parallel to the line CA to intersect BA at A. Thus, A BC is the required triangle 26. Hence proved. 27. In ABC,, AB=3 units, AC = 4 units So, by Pythagoras Theorem, Material downloaded from mycbseguide.com. 17 / 21

97 Area of the semi-circle on BC (as diameter) i.e., area of BECB = Area of the semi-circle on AC (as diameter) i.e., area of ADCA = Area of the semi-circle on AB (as diameter) i.e., area of AFBA = Also from the fig. (area of AHBA + area of AGCA ) = area of semi-circle BHGCB - area of ABC = - = - = Area of AFBHA + area of AGCDA = area of AFBA + area of ADCA - (area of AHBA + area of AGCA ) = + - sq units. Material downloaded from mycbseguide.com. 18 / 21

98 Hence, area of the shaded part i.e., area of area of BECB +area of AGCDA + area of AFBHA = sq. units =+ + - sq. units = 28. Let h meters be the height of the light-house AB. Also let two ships be at C and D (in fig) By question, CD = meters. Angle of depression of two ships as observed from the top (A) of the light-house are 60 o and 45 o respectively. In the fig., Therefore, (alternate angle) Now in, Again, in, Material downloaded from mycbseguide.com. 19 / 21

99 [ by (i) ] Hence, the height of the light-house is meters 29. Radius of the cylindrical tank =, its depth is 2 m. Internal radius of the pipe = So volume of the tank = Flow of water through the pipe = 3 km/hr Volume of water will flow through the pipe per minute = Therefore, time taken to fill the tank So, flow of water should be stopped after 1 hr 40 min to avoid over flow from the tank. Value: Water is most important substance in the earth for lives ( human, animals, plants) so Material downloaded from mycbseguide.com. 20 / 21

100 we should use our water wisely and not waste it. 30. Class Interval Frequency Cumulative Frequency x 16+x x x Given median = 35. This lies in the class So, l = 30, f = 6, cf = the cumulative frequency of the class preceding = 10 We know, Here, Hence value of x is 5. Material downloaded from mycbseguide.com. 21 / 21

101 SAMPLE QUESTION PAPER 05 Class-X ( ) Mathematics Time allowed: 3 Hours Max. Marks: 80 General Instructions: (i) All questions are compulsory. (ii) The question paper consists of 30 questions divided into four sections A, B, C and D. (iii) Section A contains 6 questions of 1 mark each. Section B contains 6 questions of 2 marks each. Section C contains 10 questions of 3 marks each. Section D contains 8 questions of 4 marks each. (iv) There is no overall choice. However, an internal choice has been provided in four questions of 3 marks each and three questions of 4 marks each. You have to attempt only one of the alternatives in all such questions. (v) Use of calculators is not permitted. SECTION A 1. Which term of the progression 4, 9, 14, 19, is 109? 2. Find the sum of the zeroes of the quadratic polynomial. 3. If and, then what is the value of cot a? 4. A die is thrown once. What is the probability of getting a composite number? 5. Find the coordinates of a point A, where AB is the diameter of a circle whose centre is (2, 3) and B is (1, 4). 6. In fig-1, AQ, AR and BC are the tangents. If AQ = 10 cm, find the perimeter of DABC. Material downloaded from mycbseguide.com. 1 / 20

102 SECTION B 7. Determine an A.P whose 3 rd term is 16 and difference of 7 th term and 5 th term is Find the coordinates of the points of trisection of the line segment joining points (4, 1) and B( 2, 3). 9. Find two consecutive positive integers, sum of whose squares is In Fig 2, and AB = 6 cm. Find the length of DC. 11. In fig-3, from an outside point P, PA is a tangent to a circle whose centre is at C. A is the point of contact. If PC = 10 cm and PA= 8 cm. Find the diameter of the circle cubes each of volume are joined end to end. Find the surface area of the resulting cuboid. SECTION C Material downloaded from mycbseguide.com. 2 / 20

103 13. Find two consecutive odd positive integers, sum of whose squares is How many terms of the sequence 18, 16, 14, should be taken so that their sum is zero? 15. In fig-4, DE BH and EF HC. Show that, DF BC. 16. Show that the points (1, 7), (4, 2), ( 1, 1) and ( 4, 4) are the vertices of a square. Or Find the area of the rhombus, if its vertices are (3,0), (4,5), (-1,4) and (-2,-1) taken in order. 17. The ages of workers in a factory are given below : Age (in years) Number of workers Find the modal age of workers. Or Compute the Median for the given data. Class interval Frequency Two dice one blue and one grey, are thrown at the same time. What is the probability that the sum of the two numbers appearing on the top of the dice is 6? 19. Show that is irrational. Material downloaded from mycbseguide.com. 3 / 20

104 Or Use Euclid s division algorithm to find the HCF of 4052 and Prove that, 21. In fig 5 ABCD is a quadrant of a circle with radius 28 cm and a semicircle BEDB is drawn with BD as diameter. Find the area of the shaded region. Or The radius of a radius of a circle is 20cm Three more concentric circles are drawn inside it in such a manner that it is divided into four parts of equal area. Find the radius of the largest of the three concentric circle 22. Find the HCF of 96 and 404 by the prime factorization method. Hence, find their LCM. SECTION D 23. Construct a tangent to a circle of radius 4 cm from a point on the concentric circle of radius 6 cm. 24. Five years hence, the age of Anubhab will be three times that of his son. Five years ago, Anubhab s age was seven times that of his son. What are their present ages? 25. Prove that:. Or If a cosθ b sin θ = c prove that 26. The king, queen and jack of clubs are removed from a deck of 52 playing cards and the remaining cards are shuffled. A card is drawn from the remaining cards. Find the Material downloaded from mycbseguide.com. 4 / 20

105 probability of getting a card of (i) heart (ii) king (iii) club. 27. In fig-6, AB and CD are two diameters of a circle with center at O.. OD is the diameter of the smaller circle.given that, OA = 7 cm, find the area of the shaded region. 28. From the top of a building 100 m. high, the angles of depression of the top and bottom of a tower are observed to be 45 o and 60 o respectively. Find the height of the tower. Also find the distance between the foot of the building and the bottom of the tower. 29. A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in fig-7. If the height of the cylinder is 10 cm and its base is of radius 3. 5 cm, find the total surface area of the article. 30. The sum of first six terms of an A.P is 42. The ratio of its 10 th term to its 30 th term is 1 : 3. Find the 1 st term and the 13 th term of the A.P. Or For what values of n nth term of the series 3, 10, 17...And 63, 65, 67..Are equal. Material downloaded from mycbseguide.com. 5 / 20

106 CBSE SAMPLE PAPER 05 CLASS X - Mathematics Marking Scheme SECTION A 1. 4+(n 1).5 = 109 (n 1).5 = 105 n = =22 2. are the two zeroes. So, sum the zeroes = 3. Given = 4. From 1 to 6 there are two composite numbers 4 and 6. P(getting a composite number) 5. Let (x, y) be the coordinates of point A. Material downloaded from mycbseguide.com. 6 / 20

107 Coordinates of A are (3, 10). 6. Perimeter of ABC =AB+BC+AC = (AQ QB)+(BS + SC)+(AR RC) =AQ BS + BS + RC +AQ RC = 2AQ = 20 cm. [Since The lengths of tangents drawn from an external point to a circle are equal. SECTION B 7. n-th term,, a = 1 st term and d = common difference of the AP Now d = 6 Also, a+ (3 1)d = 16 or, a + 12 = 16 or, a = 4. Therefore the terms of the AP are 4, 10, 16, 22, 28, 8. Let point P(x, y) trisects the line segment joining points (4, 1) and B( 2, 3) [P is nearer to the first point (4, 1) ] Then, coordinates of P are Again let point Q(x, y) trisects the line segment joining points (4, 1) and B( 2, 3) Material downloaded from mycbseguide.com. 7 / 20

108 [Q is nearer to the second point ( 2, 3)] Then, coordinates of Q are 9. Let the consecutive positive numbers be x and (x+1) According to the question, x = 13, since x is positive, Hence the required numbers are 13 and We know that, if one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional, then the two triangles are similar. Here, Given, [vertically opposite] Material downloaded from mycbseguide.com. 8 / 20

109 Therefore, ~. i.e., Hence, i.e., DC = 12 cm. 11. We know that, The tangent at any point of a circle is perpendicular to the radius through the point of contact. So,. By Pythagoras theorem, Hence the diameter of the circle = 2*6 cm = 12 cm. 12. Length of each side of the cube of volume is 4 cm. If 2 cubes each of volume are joined end to end, then a cuboid will be formed. The dimensions of the cuboid formed are (4 + 4) cm, 4 cm, 4 cm i.e., 8 cm, 4 cm and 4 cm. The surface area of the resulting cuboid SECTION C 13. Let the consecutive odd positive integers be (2x 1) and (2x + 1). (x is positive integer 1] According to question, i.e., Material downloaded from mycbseguide.com. 9 / 20

110 Hence the required numbers are (2 11 1) and ( ) i.e., 21 and 23. ALTERNATE PROCESS: Let the consecutive odd positive integers be x and (x + 2) [ x 1 ] According to question, [since, x is odd positive integer] x = 21. Hence the required numbers are 21 and In the given sequence, first term, a = 18,common difference, d = =14 16 = 2 = constant Therefore, the sequence is an AP. Let the sum of the first n terms is zero. We know,. i.e., Hence, the sum of 19 terms of the sequence is zero. Material downloaded from mycbseguide.com. 10 / 20

111 15. By question DE BH. Therefore in ABH, [Since, any line, drawn parallel to one side of a triangle, divides the other two sides in the same ratio.] Also by question, EF HC. So, [by same reason ] Therefore by (i) and (ii), We know that, if a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side. DF BC [ by (iii)] Hence proved. 16. Let (1, 7), (4, 2), ( 1, 1) and ( 4, 4) are the coordinates of the points A, B, C and D respectively. Length of, Length of Length of Length of Material downloaded from mycbseguide.com. 11 / 20

112 Now, Since, the lengths of the sides are equal, so ABCD is either a square or a rhombus. Length of Length of Again, i.e., lengths of the diagonals are equal. So ABCD is a square. Hence, the given points are the vertices of a square. Or Let (3, 0), (4, 5), ( 1, 4) and ( 2, 1) are the vertices A, B, C, D of a rhombus ABCD 17. From the given data, we se that the maximum frequency is 104 and the corresponding class is Material downloaded from mycbseguide.com. 12 / 20

113 Therefore, the modal class is Here, l = 50, h = 10, f = 104, and Therefore, i.e., Hence the modal age of workers is years. 18. When the blue die shows 1, the grey die can show any one of the numbers 1, 2, 3, 4, 5, 6. The same is true when the blue die shows 2, 3, 4, 5, 6. So, the number of possible outcomes = 6 6 = 36. Let the event that, the sum of the two numbers appearing on the top of the dice is 6 be denoted by E. Thereforem the outcomes favourable to the event E are: (1, 5), (2, 4), (3, 3), (4, 2), (5, 1) So number of outcomes favourable to the event E = 5 Therefore, 19. If possible let us assume, that, is a rational number. i.e.,, where p and q are coprime numbers and i.e., As, 5, p, q are all integers so is rational. Now from (i) we get that is also a rational. Material downloaded from mycbseguide.com. 13 / 20

114 But this contradicts the fact that is always an irrational number. Therefore our assumption is wrong. Hence we conclude that, is irrational Hence proved. 21. Since ABCD is a quadrant of a circle with radius 28 cm, so, AB = AD = 28 cm. Area of the quadrant ABCD = sq. cm area(bdcb) = area(quadrant ABCD) area ( ABD) = sq. cm By Pythagoras Theorem, Now, area of the shaded region = area (semi-circle BEDB) area(bdcb) Material downloaded from mycbseguide.com. 14 / 20

115 = = sq. cm 22. By prime factorization, of 96 and 404 can be written as and Therefore, the HCF of 96 and 404 is Since Product of two numbers = (their HCF) (their LCM) Hence LCM of 96 and 404 = SECTION D 23. Steps of construction : Material downloaded from mycbseguide.com. 15 / 20

116 Draw a circle of 4 cm radius with centre as O. Draw a circle of 6 cm radius taking O as its centre. Locate a point P on this circle and join OP. Bisect OP. Let M be the mid-point of OP. Take M as its centre and MO as its radius, draw a circle. Let it intersect the given circle at the points Q and R. Join PQ and PR. PQ and PR are the required tangents. 24. Let the present age of Anubhab and his son be x years and y years respectively. Five years hence, age of Anubhab will be (x+5) years and five years hence his son s age will be (y+5) years. By question, (x+5) = 3(y+5) i.e., x 3y = 15 5 i.e., x 3y = 10 (i) Five years ago, Anubhab s age was (x 5) years and that of his son was (y 5) years. By question, (x 5)=7(y 5) i.e., x 7y = 5 35 i.e., x 7y = 30 (ii) Now subtracting (ii) from (i), From (i), x= = 40 Hence, the present ages of Anubhab and his son are 40 years and 10 years. 25. [ Multiplying numerator and denominator by (coseca cota) ] Material downloaded from mycbseguide.com. 16 / 20

117 [multiplying numerator and denominator by (1-cosA)] [ Dividing numerator and denominator by sina ] Hence proved. 26. After removing king, queen and jack of clubs from a deck of 52 playing cards there are 49 cards left in the deck. Out of these well shuffled 49 cards one card can be chosen in 49 different ways. Therefore number of total outcomes = 49 i) There are 13 heart cards in the deck of well shuffled 49 cards out of which one heart card can be chosen in 13 different ways. Therefore, number of outcomes favourable to the event of getting heart = 13 P(getting one heart card ) = ii) There are 3 king cards in the deck of well shuffled 49 cards out of which one king can be chosen in 3 different ways. P(getting one king ) = iii) There are (13 3) i.e., 10 club cards in the deck of well shuffled 49 cards out of which one club can be chosen in 10 different ways. P(getting one club card ) = 27. According to question, OA= 7 cm. Therefore, OA = OB = OD = 7 cm. Material downloaded from mycbseguide.com. 17 / 20

118 Area of the circle with OD as diameter = Area of the semi-circle with OA as radius = Area of ABC = Hence the area of the shaded area = Area (circle with OD as diameter) + area(semi-circle with OA as radius) ar( ABC) = 28. Let AB and CD are the building and the tower respectively. By question, AB = 100 m Also by question, angles of depression of the top and bottom of a tower are observed from the top of the building to be 45 o and 60 o respectively. In the figure,. Let, the height of the tower CD be h meter. is drawn. EB = CD = h meter and AE = (100 h) m Material downloaded from mycbseguide.com. 18 / 20

119 In DAEC, Then, in DAEC, (alternate angle) i.e., EC = AE = (100 h) m BD= EC = (100 h) m Now in DABD, (al) then, i.e., i.e., i.e., Hence the height of the tower = m (approx.) The distance between the foot of the building and the bottom of the tower = = m 29. Let r and h are the base radius and height of the cylinder respectively. Given, h = 10 cm and r = 3.5 cm. Then radius of each of the hemisphere = 3.5 cm. We know that, curved surface are of a cylinder = sq. units and surface area of the hemisphere = sq. units Material downloaded from mycbseguide.com. 19 / 20

120 Total surface area of the article = Curved surface area of the cylinder + 2 one hemisphere) (surface area of = sq. units sq. units 30. We know that, if a and d are the first term and the common difference of an AP, then, term, and sum of first n terms, According to the question, i.e., i.e., Also by question, i.e., i.e., i.e., i.e., i.e., Now from (i), 7d = 14, i.e., d = 2 a = d = 2 Now 13 th term of the AP = 2+ (13 1)2 = 26 Hence the first term = 2 and the 13 th term = 26. Material downloaded from mycbseguide.com. 20 / 20

121 SAMPLE QUESTION PAPER 06 Class-X ( ) Mathematics Time allowed: 3 Hours Max. Marks: 80 General Instructions: (i) All questions are compulsory. (ii) The question paper consists of 30 questions divided into four sections A, B, C and D. (iii) Section A contains 6 questions of 1 mark each. Section B contains 6 questions of 2 marks each. Section C contains 10 questions of 3 marks each. Section D contains 8 questions of 4 marks each. (iv) There is no overall choice. However, an internal choice has been provided in four questions of 3 marks each and three questions of 4 marks each. You have to attempt only one of the alternatives in all such questions. (v) Use of calculators is not permitted. SECTION - A 1. Find the coefficient of x 0 in x 2 + 3x + 2 = 0 2. What is the common difference of the A.P. in which a 18 -a 14 =32? 3. If the points A(x, 2), B(-3,-4) and C(7, -5) are colinear, then find the value of x. 4. If PA and PB are tangents from an outside point P such that PA = 15 cm and APB = 60. Find the length of chord AB. 5. Find the length of the shadow of a 20m tall pole, on the ground when the sun s elevation is If the probability of winning a game is 0.995, then find the probability of losing a game. SECTION - B 7. Solve:. Material downloaded from mycbseguide.com. 1 / 19

122 8. Find the number of terms in the following series: -5 + (-8) + (-11) + + (-230) 9. Find the area (in sq.units) of the triangle formed by the points A(a, 0), O(0, 0) and B(0, b). 10. In two similar triangles ABC and PQR, if their corresponding altitudes AD and PS are in the ratio 4 : 9, find the ratio of the areas of triangles ABC and PQR. 11. In the adjoining figure, O is the centre of the circle. AP and AQ two tangents drawn to the circle. B is a point on the tangent QA and PAB = 125, Find POQ. 12. The dimensions of a metallic cuboid are: 100 cm, 80 cm, 64 cm. It is melted and recast into a cube. Find the surface area of the cube. SECTION - C 13. The two forces are acting on a body such that their maximum and minimum value of resultant is 27 N and 13 N respectively. Find the values of forces. 14. Prove that is irrational. 15. A sweet seller has 420 Kaju burfis and 130 Badam barfis. She wants to stack them in such a way that each stack has the same number, and they take up the least area of the tray. What is the maximum number of burfis that can be placed in each stack for this purpose? (Use Euclid division algorithm) 16. Find the sum of all multiples of 9 lying between 100 and 500. Or Material downloaded from mycbseguide.com. 2 / 19

123 Check whether 301 is a term of the list of numbers 5, 11, 17, 23, Find a point on y axis which is equidistant from (2, 2) and (9, 9). Or Find a relation between x and y if the points (2, 1), (x, y) and (7, 5) are collinear 18. The hypotenuse of a right triangle is 6 m more than the twice of the shortest side. If the third side is 2 m less than the hypotenuse, then find the sides of the triangle. 19. Prove that the parallelogram circumscribing a circle is a rhombus. Or An umbrella has 8 ribs, which are equally spaced, a ssuming umbrella to be a flat circle of radius 45 cm. Find the area between two c onsecutive ribs of the umbrella. 20. If 7 cosec A - 3cot A = 7, prove that 7cot A - 3cosec A = A card is drawn from a well- shuffled pack of 52 playing cards. What is the probability that the card drawn is (a) either a red or a king (b) a black face card (c) a red queen card. Or At a car park there are 100 vehicles, 60 of which are cars, 30 are vans and the remainder are lorries. If every vehicle is equally likely to leave, find the probability of: (a) van leaving first. (b) lorry leaving first. (c) car leaving second if either a lorry or van had left first 22. From a bag containing 5 red, 6 black and 7 yellow balls, a ball is drawn at random. Find the probability that it is: (a) not a yellow ball (b) neither a black nor a red ball Material downloaded from mycbseguide.com. 3 / 19

124 (c) either a black or a yellow ball SECTION - D 23. The sum of the areas of two squares is 640 m 2. If the difference in their perimeters be 64m find the sides of the two squares. Or In a class test, the sum of marks obtained by Puneet in Mathematics and Science is 28. Had he got 3 more marks in Mathematics and 4 marks less in Science, the product of marks obtained in the two subjects would have been 180? Find the marks obtained in two subjects separately. 24. A girl walking uphill covers a distance of 20 m during the first minute, 18m during the second minute and 16m during the third minute and so on. How much distance will she cover in (i) 10 th minute (ii) 10 minutes? 25. Draw a circle of radius 8 cm. From a point 12 cm away from its centre, construct the pair of tangents to the circle and measure their lengths. Or Draw a line segment AB of length 8 cm. Taking A as centre, draw a circle of radius 4 cm. and constant the pair of tangents of the circle from point B and measure their lengths. 26. If sec A+ tan A= p, then prove that sin A = (p 2 +1)/(p 2-1) 27. In the adjoining figure, PQ=24 cm, PR=7 cm and O is the centre of the circle. Find the area of the shaded region. Material downloaded from mycbseguide.com. 4 / 19

125 28. From the top of a hill, the angles of depression of two consecutive kilometre stones due east are found to be 30 and 45. Find the height of the hill glass spheres each of radius 2 cm are packed in a cuboidal box of internal dimensions 16 cm x 8 cm x 8 cm and then the box is filled with water. Find the volume of water filled in the box. 30. The following frequency distribution gives the monthly consumption of electricity of 68 consumers of a locality. Find the mean, median and mode of the data. Monthly Consumption Number of Consumers Or The median of the following dats is 525. Find the value of x and y if the total frequencry is 100. Classes Number of Consumers 2 5 X Y Material downloaded from mycbseguide.com. 5 / 19

126 CBSE SAMPLE PAPER 06 CLASS X - Mathematics Solutions Section A d= AB=15cm 5. 20cm Section B a = -5 ; d = -3 a n = = -5 + (n 1) (-3) => -225 = (n-1)(-3) => 75 = n-1 => n = 76 Material downloaded from mycbseguide.com. 6 / 19

127 9. Area of ABC = = - sq.units = sq.units ( as area cannot be negative) 10. Since the areas of two similar triangles are in the ratio of the squares of corresponding altitudes, Thus, the ratio of areas of triangles ABC and PQR = 16 : PAB+ PAQ =180 [Linear Pair of angles] PAQ o = 180 o PAQ = 180 o o PAQ = 55 o In Quadrilateral APOQ, PAQ+ APO+ AQO+ POQ = POQ=360 [Radius is perpendicular with tangent at their point of contact] POQ= POQ = Volume of cube = Volume of cuboid (side) 3 = Length Breadth Height (side) 3 = Material downloaded from mycbseguide.com. 7 / 19

128 Side = = 80 cm. Surface Area of the Cube = 6a 2 = = cm 2 Section C 13. Let the two forces in Newton(N) are x and y respectively. Then, x + y = (1) [ For maximum force] x - y = (2) [ For minimum force] Solving (1) and (2) x = 20 ; y = 7 Required forces are 20N & 7N 14. Suppose is rational. p and 1 are coprime But is irrational while is rational and an irrational can never be equal to a rational number. Thus our assumption is wrong and hence is irrational. 15. The number of stacks will be least if the number of burfis in each stack is equal to the HCF of 420 and130. Now, let us use Euclid s algorithm to find their HCF. 420 = = Material downloaded from mycbseguide.com. 8 / 19

129 30 = So, the HCF of 420 and 130 is 10. Hence, the sweet seller can make stacks of 10m burfis of each kind to cover the least area of the tray. 16. a = 108, a n = l = 495 ; d = 9 a n = a + (n 1) d 495 = (n -1) 9 n = 44 S 44 = (a + l) = ( ) = 13,266 Or We have : a 2 a 1 = 11 5 = 6, a 3 a 2 = = 6, a 4 a 3 = = 6 As a k + 1 a k is the same for k = 1, 2, 3, etc., the given list of numbers is an AP. Now, a = 5 and d = 6. Let 301 be a term, say, the nth term of the this AP. We know that a n = a + (n 1) d So, 301 = 5 + (n 1) 6 i.e., 301 = 6n 1 So, n = But n should be a positive integer. So, 301 is not a term of the given list of numbers. Material downloaded from mycbseguide.com. 9 / 19

130 17. PA = PB Squaring on both the sides, Point is (0,11) Or If the given points A(2,1), B(x,y) and C(7,5) are collinear, then the area formed by these points will be 0. This is the requried relation between x and y. 18. Let the shortest side be x meters in length. Then, hypotenuse = (2x+16)m And the third side = (2x+4)m Material downloaded from mycbseguide.com. 10 / 19

131 By using Pythagoras theorem, we have (2x + 6) 2 = x 2 + (2x + 4) 2 (x 10)(x + 2) = 0 x = 10 or x = Let ABCD be parallelogram circumscribing circle with centre 0. AS = AP (Tangents from A) BP = BQ (Tangents from B) CQ = CR (Tangents from C) DS = DR (Tangents from D) Now, AP + BP + CR + DR = AS + BQ + CQ + DS ( AP + BP) + ( CR + DR) = (AS + DS ) + ( BQ + CQ ) AB + CD = AD + BC 2 AB = 2 BC AB = BC AB = BC = CD = DA ABCD is a rhombus 20. 7cosecA 3 cota = 7 Material downloaded from mycbseguide.com. 11 / 19

132 7cosecA 7 = 3 cota 7(CosecA 1) = 3 cota Multiplying by (coseca + 1) both sides, we get 7(cosecA 1) (coseca + 1) = 3 cota (coseca + 1) 7(cosec 2 A-1) = 3 cota (coseca + 1) 7cot 2 A=3 cota (coseca + 1) 7cotA = 3(cosecA + 1) 7cotA coseca=3 21. n(s)= 52 (a). Let A be the event of drawing either a red or a king card n(a) = 28 P(A) = (b) Let B be the event of drawing a black face card. n(b) = 6 P(B) = (c) Let C be the event of drawing a red queen card. n(c) = 2 P(C) = 22. (a) let A be the event of drawing a ball other than yellow in colour. n(a) = 11 P(A) = Material downloaded from mycbseguide.com. 12 / 19

133 (b). Let B be the event of drawing a ball which is yellow in colour. n(b) = 7 P(B) = (c). Let C be the event of drawing a ball which is either black or yellow. n(c) = 13 P(C) = Section D 23. Let the areas of two squares be a 1 2, a2 2 a a2 2 = (1) 4 a 1-4a 2 =64 a 1 a 2 = 16 a 1 = 16 +a 2 Substitute the value of a 1 in (i) we get (16 +a 2 ) 2 + a 2 2 = 640 a a2-192 = 0 Material downloaded from mycbseguide.com. 13 / 19

134 = 8, -24 But length cannot be negative. a 2 = 8 a 1 = 24 Therefore the sides are 24 & The required sequence is 20,18,16,14,... a = 20, d = -2, n = 10 (i). 10 th minute a n =a + (n -1) d a 10 =20 + 9(-2) = 2 m She covers a distance of 2 m during 10 th minute. (ii) 10 minutes = 5(22) = 110 After 10 minutes she will cover 110 m. Material downloaded from mycbseguide.com. 14 / 19

135 Steps of Construction: Step 1: Draw a circle with centre O and radius 8 cm. Step 2: Take a point P outside the circle such that OP = 12 cm. Join OP. Step 3: Draw a perpendicular bisector of line segment OP. It intersects OP at O. Step 4: Take O as centre and O O as radius and draw a circle which intersects the previous circle at points A and B. Step 5: Join P to A and P to B. PA & PB are the required tangents from point P to circle with centre O. Length of PA = PB = 14.4 cm 26. Let p (sec A + tan A) By componendo and dividend method. Material downloaded from mycbseguide.com. 15 / 19

136 sina = LHS 27. PQ = 24 cm, PR = 7 cm Since P is a right angle, By Pythagoras theorem, we have QR 2 = PQ 2 + PR 2 = = = 625 QR = 25 Since QR is a diameter of a circle, radius of the circle r = 25/2 cm. Area of the shaded region = Area of semi circle Area of right triangle PQR 28. Let h = height of the hill C and D are position of two stones. Material downloaded from mycbseguide.com. 16 / 19

137 In CAB, we have In DAB, we have 29. Given, Radius of 1 glass sphere. R = 2cm Therefore, Volume of such 21 glass spheres Given, Dimensions of the cuboidal box = 16 cm 8 cm 8 cm Therefore, Volume of the cuboidal box = Volume of the water added in cuboid box = Material downloaded from mycbseguide.com. 17 / 19

138 Volume of the cuboidal box volume of the 21 glass sphere = 320 cm Monthly consumption (in units) No. of consumers (f i ) Cumulative frequency(cf) Class mark d i =x i -135 u i f i u i (f 0 ) (f 1 ) (f 2 ) TOTAL 68 7 From the table, n = 68, n/2 = 34, h = 20 Mean = Material downloaded from mycbseguide.com. 18 / 19

139 Thus, the mean monthly consumption of electricity is units. Now, is the median class because its frequency 42 is just greater than 34. Median = = = 137 units Model class is because it has highest frequency Mode = = = Material downloaded from mycbseguide.com. 19 / 19

140 SAMPLE QUESTION PAPER 07 Class-X ( ) Mathematics Time allowed: 3 Hours Max. Marks: 80 General Instructions: (i) All questions are compulsory. (ii) The question paper consists of 30 questions divided into four sections A, B, C and D. (iii) Section A contains 6 questions of 1 mark each. Section B contains 6 questions of 2 marks each. Section C contains 10 questions of 3 marks each. Section D contains 8 questions of 4 marks each. (iv) There is no overall choice. However, an internal choice has been provided in four questions of 3 marks each and three questions of 4 marks each. You have to attempt only one of the alternatives in all such questions. (v) Use of calculators is not permitted. Section A Q.1 Is x = 2. y = 3 a solution of linear equation? Q.2 The n th term of an AP is 7-4n. Find the common difference Q.3 Find the value of. Q.4 A ladder is 10 meter long reaches a window 8 meter above the ground. Find the distance of the foot of the ladder from the base of the wall. Q.5 What is the relation between the lengths of two tangents from an external point to a circle? Q.6 One card is drawn from well shuffled pack of 52 cards. Find the probability that the card will be ace. Section B Material downloaded from mycbseguide.com. 1 / 32

141 Q.7 Find the highest positive integer by which dividing the numbers 396, 436, 542 reminders 5, 11, and 15 respectively. Q.8 For what value of p, the following system of equation have a unique solution : Q.9 Find two numbers whose sum is and product is Q.10 If then evaluate Q.11 The shadow of tower standing on a plane ground is found to be 40 m longer when the Sun s altitude reduces to from. Find the height of the tower. Q.12 A hemispherical tank full of water id emptied by a pipe at the rate of litre per second, How much time will take to empty the full tank if it is 3m in diameter. Section C Q.13 A pole has to be erected at a point on this boundary of a circular part of diameter 13 meters in such a way that the difference of its distances from two diametrically opposite fixed gates A and B on the boundary is 7 meters. Is it possible to do so? If yes, at what distance from the two gates should the pole be erected. Q.14 Find the sum of number between 1 to 100 divisible by 6. Q.15 In the given figure, the line segment XY is parallel to side AC of and it divides the triangle into two parts of equal areas. Find the ratio. Material downloaded from mycbseguide.com. 2 / 32

142 Q.16 Find the area of triangle formed by joining the mid points of the sides of the triangle ABC whose vertices are A(0,-1), B(2,1) and C(0,3). Find the ratio of this area and the area of ABC. Or Find the cirumcentre of the triangle whose vertices are (-2, -3), (-1, 0), (7, -6). Q.17 From a point on a bridge across a river the angles of depression of the banks on opposite sides of the river are and respectively. If the bridge is at a height of 4m from the banks, find the width of the river. Or If the angle of elevation of a cloud from a point 'h' metres above a lake is α and the angle of depression of its reflection in the lake is β, prove that the height of the cloud is Q.18 Two concentric circles are if radii 25 cm. and 7 cm. respectively. Find the length of the chord of the bigger circle which touches the smaller circle. Q.19 Draw a line segment AB of length 8 cm. Taking A as centre, draw a circle of radius 4 cm. and constant the pair of tangents of the circle from point B and measure their lengths. Or Draw a circle of radius 8 cm. From a point 12 cm away from its centre, construct the pair of tangents to the circle and measure their lengths. Q.20 Find the co- ordinate of the circumcentre of the triangle whose vertices are (8,6), (8,-2) and (2, -2). Also its circum radius. Q.21 The distribution below shows the pocket money during a days of 60 employees in an office. Find the median pocket money of the employees. Pocket Money (in Rupees) Number of Employees Material downloaded from mycbseguide.com. 3 / 32

143 Or A survey conducted on 20 households in a locality by a group of students resulted in the following frequency table for the number of family members in a household. Find the mode. Family size No. of families Q.22 A box contains 30 discs which are number 1 to 30. If one disc is drawn at random from the box. Find the probability that it hears (i) A two digit number (ii) A perfect square number. Section D Q.23 Prove that is an irrational number. Or Use Euclid s division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8. Q.24 Divide by and verify the division algorithm. Q.25 Coach of a cricket team buys one bat and 2 balls for Rs. 300.Later he buys another 2 bates and 3 balls of the same kind for Rs Represent this situation algebraically and solve it by graphical method. Also find out that how much money coach will pay for the purchase of one bat and one ball. Or The sum of two numbers is 16 and the sum of their reciprocals is. Find the numbers. Q.26 BE and CF are medians of triangle ABC right angled at A. Prove that 4(BE 2 + CF 2 ) = 5BC 2 Q.27 Prove that Material downloaded from mycbseguide.com. 4 / 32

144 Q.28 Two circular flower beds lie on two sides AB and CD of a square lawn ABCD of side 56 m. If the centre of each circular flower bed is point of intersection O of the diagonals of square lawn. Find the sum of the areas of the lawn and flower beds. Q.29 A tent is the form of a cylinder of diameter 4.2 m and height 4 m, surrounding by a cone of equal base and height 2.8 m. find the capacity of the tent and the cost of canvas for making the tent at 100 per sq. m. Q.30 The median of the following dats is 525. Find the value of x and y if the total frequencry is 100. Classes Number of Consumers 2 5 X Y Or The distribution below gives the marks of 100 students of a class. Marks No. of Students Draw a less than type and a more than type ogive from the given data. Hence, obtain the median marks from the graph. Material downloaded from mycbseguide.com. 5 / 32

145 CBSE SAMPLE PAPER 07 CLASS X - Mathematics Solutions Sol.1 The given equation is Putting in given equation, we have = = = Hence, is a solution of given equation. Sol.2 Here, a n =7-4n If n=1 then a 1 =7-4x1=3 If n=2 then a 2 =7-4x2=-1 Common difference d= a 2 - a 1 =-1-3=-4 Sol.3.(putting the value of trigonometric ratios) Sol.4 According to the figure ABC is a right angled triangle in which Now, By the Pythagoras Theorem Material downloaded from mycbseguide.com. 6 / 32

146 Sol.5 Both tangent lines are equal. Sol.6 There are 4 aces in a pack The number of Favorable outcomes=4 The total number of outcomes=52 P (card is ace) Sol.7 It is given that on dividing 396 by the required number, there is remainder of 5; this means that = 391 is exactly divisible by the required number. Similarly = 425 is also exactly divisible by the required number and = 527 is also exactly divisible by required number. Also the required number is the largest number satisfying the above property. Therefore it is HCF of 391, 425 and 527. Let us now find the HCF of 391, 491 and = = = HCF (391, 425, 527) = 17 Sol.8 For unique solution Or Sol.9 Let one number be x Material downloaded from mycbseguide.com. 7 / 32

147 Anotthe number will be (27-x ) [ Sum of two number = 27] Their product According to the question So the required two numbers are 13 and 14 Ans. Sol.10 Sol.11 Let AB be height of the tower and BC m Material downloaded from mycbseguide.com. 8 / 32

148 DB m In ABC, we have In ABD, we have Substituting obtained from equation in equation We get m Material downloaded from mycbseguide.com. 9 / 32

149 Sol.12 radius of hemispherical tank m Volume of the tank So, the volume of water to be emptied litres Litres Since, litres of water is emptied in 1 second litres of water will be emptied in second second Or minutes Minutes. Sol.13 Let P the required location of the pole from the gate B be m, i.e. BP m. Now the difference of the distance of the pole from the two gates AP-BP (or BP-AP) 7 m. Therefore, AP = m. Now, AB 13 m, an since AB is a diameter (By Pythagoras theorem) Material downloaded from mycbseguide.com. 10 / 32

150 i.e. i.e. i.e. So, the distance of the pole from gate B satisfies the equation So, it would be possible to place the pole if this equation has real roots. To see if this is so or not, let us consider it s discriminant. This discriminant is So, the given quadratic equation has two real roots, and it is possible to erect the pole on the boundary of the park. Solving the quadratic equation by the quadratic formula, we get Therefore, or Since is the distance between the pole and the gate B, it must be positive. Therefore, will have to be ignored. So. Thus the pole has to be erected on the boundary of the park at a distance of 5 m from the gate B and 12 m from the gate A Sol.14 The numbers between 1 to 100 divisible by 6, 96, Or, Or Material downloaded from mycbseguide.com. 11 / 32

151 Or Or Or Or Sol.15 We have XY AC (Given) So, BXY A and (Corresponding angles) Therefore (AA Similarity Criterion) So, (Theorem 6.6) Also (Given) So, Material downloaded from mycbseguide.com. 12 / 32

152 Therefore, from and i.e., or or or or Sol.16 Let the vertices of ABC is A(0,-1), B(2,1) and C(0,3) D,E,F are mid of AB, BC & CA. Midpoint formula Co-ordinate of D Co-ordinate of E Co-ordinate of F Vertices of DEF is D(1,0), E(1,2), F(0,1) Here Material downloaded from mycbseguide.com. 13 / 32

153 Area of DEF In ABC Ratio Or Let the centre of the circumcircle be O(x, y). The points are A(-2, -3), B(-1, 0) and C(7, -6). OA = OB = Radius Also, OB = OC = Radius Material downloaded from mycbseguide.com. 14 / 32

154 adding (i) and (ii), we get 5x = 15 => x = 3 put value of x in (i) we get, => 3y = -9 => y = -3 So the circumcenter is (3,-3) Sol.17 In fig. A and B represent points on the bank on opposite sides of the river, so that AB is the width of the river. P is a point on the bridge at a height of 4m, i.e. DP = 4m. we are interested to determine the width of the river, which is the length of the side AB of the APB. Now, AB = AD + DB In right APD, So, Material downloaded from mycbseguide.com. 15 / 32

155 Also in right So, DB = 4m AB = AD + DB m m Therefore, the width of the river is m Or Let AN be the surface of the lake and O be the point of observation such that OA = h meters Let P be the position of the cloud and P' be its reflection in the lake. Then PN = P'N Let Material downloaded from mycbseguide.com. 16 / 32

156 From (i) and (ii), we get Sol.18 Let O be the centre of the concentric circles of radii 25 cm. and 7 cm. respectively. Let AB be a chord of the larger circle touching the smaller circle at P. Then Material downloaded from mycbseguide.com. 17 / 32

157 AP = PB and OP AB Applying Pythagoras theorem in OPA, We have cm. cm. Sol.19 Steps of Construction: (1) Draw a line segment AB = 8 cm. (2) Taking A as centre draw a circle of radius = 4cm. (3) Draw a perpendicular bisect of AB and let it intersect AB in P. (4) With P as centre and PA or PB as radius, draw another circle intersecting the given circle in Q & R. (5) Join BQ and BR. Thus BQ and BR are the required tangents from point P to the circle. Lengths BQ =BR = 6.9 Material downloaded from mycbseguide.com. 18 / 32

158 cm. Sol.20 Let A (8,6), B (8, 2) and C ( 2, 2) be the vertices of given the triangle and Let P( ) be the circumference of the triangle Then PA = PB = PC Taking Again, taking Material downloaded from mycbseguide.com. 19 / 32

159 => x = 5 Coordinates of P are (5, 2) Radius PA Sol.21 Pocket Money (In Rupees) Number of Employees (f) Cumulative Frequency This lies in class Median group in Now using formula where Material downloaded from mycbseguide.com. 20 / 32

160 Rs. Or Since the maximum frequency = 8 and it corresponds to the class 3-5 Modal class = 3-5 Here, We know that mode M o is given by Sol.22 There are 30 discs in the box out of which one disc can be drawn in 30 ways. Total number of elementary events 30 (i) From number 1 to 30, there are 21 two digit numbers, namely 10,11,12.30 out of these 21 two digit numbered disc can be chosen in 21 ways. Favourable number of elementary events 21 Hence P (getting a to digit numbered disc) (ii) Those numbers from 1 to 30 which are perfect square 1,4,9,16,25 i.e. squares of 1, 2,3,4,5 Material downloaded from mycbseguide.com. 21 / 32

161 respectively. Therefore, there are 5 disc marked with the number which are perfect squares. Favourable number of elementary events 5 Hence, P (Getting a disc marked with a number which is perfect square) Sol.23 Proof : Let us assume, to the contrary that is rational. So, we can find integers r and such that Suppose r and s have a common factor other than 1. Then we divide by the common factor to get, where a and b are coprime. So, we can write for some integer c. Substituting for a we get, that is,. This mean that 2 divides and so 2 divides b (again using theorem 1.3 with p =2 ). Therefore, a and b have at least 2 as a common factor. But this contradict the fact that a and b have no common factor other than 1. This contradiction has arisen because of our incorrect assumption that is rational. So, we conclude that is irrational. Or According to Euclid s division lemma, Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 r < b. Let a be any positive integer and b = 3. Then, let a = 3q + r, where q > 0 and 0< r < 3 a = 3q or a = 3q + 1 or a = 3q +2 We have three cases : i) when a = 3q, then, a3 = (3q) 3 = 9(3q 3 ) = 9k [where k = 3q 3 is an integer] ii) when a = 3q + 1, then, a 3 = (3q + 1) 3 = 9(3q 3 ) + 9(3q 2 ) + 9q + 1 Material downloaded from mycbseguide.com. 22 / 32

162 = 9(3q 3 +3q 2 +3q) + 1 = 9k1 + 1 [where k 1 = 3q 3 + 3q 2 + 3q is an integer] iii) When a = 3q + 2 then, a 3 = (3q + 2) 3 = 9(3q 3 ) + 9(6q 2 ) + 9(4q) + 8 = 9(3q 3 + 6q 2 + 4q) + 8 = 9k [where k 2 = 3q 3 + 6q 2 + 4q is an integer] Therefore, the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8. Sol.24 Dividend and Divisor So quotient Remainder Now Divisor quotient + remainder Dividend Material downloaded from mycbseguide.com. 23 / 32

163 In this way, the division algorithm is verified. Sol.25 Let the cost of one bat Rs. And cost of one ball Rs. Algebraically form Solution by graphical method: Table x y Material downloaded from mycbseguide.com. 24 / 32

164 x y Plot the points and draw the lines passing through them to represent the equations as shown in figure. The two lines intersect at point ( 150, 75) So is the required solution of the pair of linear equation Cost of one bat = Rs. 150 & Cost of one ball = Rs. 75 Cost of one bat & one ball = Rs. 225 Or Let the required numbers be x and y. Then, x + y = 16 Material downloaded from mycbseguide.com. 25 / 32

165 And, xy = 48 We can write x y = = = x + y = 16 (i) x y = 8 (ii) Or, x + y = 16 (iii) x y = 8 (iv) On solving (i) and (ii), we get x = 12 and y = 4 On solving (iii) and (iv), we get x = 4 and y = 12 Thus, the required numbers are 12 and 4. Sol.26 Material downloaded from mycbseguide.com. 26 / 32

166 BE and CF are medians of the ABC in which From ABC, (Pythagoras Theorem ) From ABE, or (E is midpoint of AC ) or or From FAC, or ( F is the mid-point of AB) or or Adding and we have Material downloaded from mycbseguide.com. 27 / 32

167 i.e. [from (i)] Sol.27 L.H.S. R.H.S. Sol.28 OE AB BE AB Material downloaded from mycbseguide.com. 28 / 32

168 m Area of segment Area of two segment Area of square lawn Sum of the areas of the lawn and the flower beds Ans. Sol.29 Radius of cylinder m. Height of cone m and height of cylinder m Material downloaded from mycbseguide.com. 29 / 32

169 Capacity of the tent = Vol. of cone + Vol. of cylinder Curved surface area of tent = Curved surface area of cone + Curved surface area of cylinder Cost of canvas per Rs. 100 Cost of canvas 75.9 Ans. Material downloaded from mycbseguide.com. 30 / 32

170 Sol.30 Class - interval Frequency (f) c.f x 7 + x x x x y 56 + x+ y x +y x + y x + y Given n = 100 or Median is 525 which lies interval Median 525 = 500 Or Or Or Material downloaded from mycbseguide.com. 31 / 32

171 From equation Or Marks Cf Marks Cf Less than 5 4 More than Less than10 10 More than 5 96 Less than15 20 More than Less than More than Less than More than Less than More than Less than More than Less than More than 35 5 Material downloaded from mycbseguide.com. 32 / 32

172 SAMPLE QUESTION PAPER 08 Class-X ( ) Mathematics Time allowed: 3 Hours Max. Marks: 80 General Instructions: (i) All questions are compulsory. (ii) The question paper consists of 30 questions divided into four sections A, B, C and D. (iii) Section A contains 6 questions of 1 mark each. Section B contains 6 questions of 2 marks each. Section C contains 10 questions of 3 marks each. Section D contains 8 questions of 4 marks each. (iv) There is no overall choice. However, an internal choice has been provided in four questions of 3 marks each and three questions of 4 marks each. You have to attempt only one of the alternatives in all such questions. (v) Use of calculators is not permitted. SECTION A 1. The sum and product of zeros of a quadratic polynomial are and 7 respectively. Write the polynomial? 2. Can two positive integers have their H.C.F and L.C.M as 12 and 512 respectively? Justify. 3. If, then determine the value of. 4. Write the relation between Mean, Mode and Median. 5. If the straight line joining two points P (5, 8) and Q (8, k) is parallel to x-axis, then write the value of k. 6. A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ = 12 cm. Write the length of PQ. SECTION B Material downloaded from mycbseguide.com. 1 / 21

173 7. The 7 th term of an A.P. is 4 and its 13 th term is 16. Find the sum of its first 19 terms. 8. If the points (4, 3) and (x, 5) lie on the circumference of the circle whose centre is (2, 3), then find the value of x. 9. Show that is irrational. 10. In Fig-1, if EF BC and FG CD, prove that,. 11. A quadrilateral ABCD is drawn to circumscribe a circle (fig-2). Prove that, AB + CD = AD + BC. 12. From a solid cylinder whose height is 2.4 cm and diameter 1.4 cm, a conical cavity of the same height and same diameter is hollowed out (fig-3). Find the total surface area of the remaining solid. SECTION C 13. Find the roots of the equation 3x 2 7x 2 = 0 by the method of completing the square. Material downloaded from mycbseguide.com. 2 / 21

174 14. Solve the pair of linear equations 8x + 5y = 9 and 3x + 2y = 4 by cross-.multiplication method. 15. Poved that if in two triangles, sides of one triangle are in the same ratio of the sides of the other triangle, then their corresponding angles are equal. 16. Prove that the points A( 5, 4), B( 1, 2) and C(5, 2) are the vertices of an isosceles rightangled triangle. Or The vertices of a triangle are A (-1, 3), B (1, -1) and C (5, 1). Find the length of the median through the vertex C. 17. Cards marked with numbers 3, 4, 5,, 50 are placed in a box and mixed thoroughly. One card is drawn at random from the box. Find the probability that number on the drawn card is a two digit number which is a perfect square. Or A die is thrown once. Find the probability of getting (i) an even number (ii) a number greater than 3 (iii) a composiite number 18. A die is thrown once. Find the probability of getting (i) a prime number; (ii) an odd number. 19. Solve for x : Or If the roots of the equation (b c)x 2 + (c a)x + (a b) = 0 are equal, then prove that 2b = a + c. 20. If, then prove that, 21. Two tangents PA and PB are drawn to a circle with centre O from an external point P. Prove that APB = 2 OAB. Material downloaded from mycbseguide.com. 3 / 21

175 Or A circle with centre O, diameter AB and a chord AD is drawn. Another circle is drawn with AO as diameter to cut AD at C. Prove that BD = 2OC. 22. State the Fundamental Theorem of Arithmetic. Use Euclid s division algorithm to find the HCF of 196 and Hence find the LCM of these numbers. SECTION D 23. Draw a pair of tangents to a circle of radius 5 cm which are inclined to each other at an angle of 60. Or Draw a line segment AB of length 8 cm. Taking A as centre, draw a circle of radius 4 cm. and constant the pair of tangents of the circle from point B and measure their lengths. 24. Check graphically whether the pair of equations x + y = 8 and x 2y = 2 is consistent. If so, solve them graphically. Also find the coordinates of the points where the two lines meet the y-axis. 25. If and show that, Or If prove that 26. If the median of the distribution given below is 28.5, find the values of x and y.the sum of all frequency is 60 Material downloaded from mycbseguide.com. 4 / 21

176 Class-interval Frequencry 5 x y Find the area of the shaded region in fig-5, where ABCD is a square of side 20cm. 28. The angle of elevation of a cloud from a point 60 m above a lake is 30 o and the angle of depression of the reflection of the cloud in the lake is 60 o. Find the height of the cloud. 29. A metallic right circular cone 20 cm high and whose vertical angle is 60 o is cut into two parts at the middle of its height by a plane parallel to its base. Find the volume of the frustum so obtained. 30. The term and the sum of first n terms of an A.P are respectively are and and. Prove that,. Or Find the sum of first 40 positive integers divisible by 6. Also find the sum of first 20 positive integers divisible by 5 or 7. Material downloaded from mycbseguide.com. 5 / 21

177 CBSE SAMPLE PAPER 08 CLASS X - Mathematics Solutions SECTION A 1. Sum of zeroes = Product of zeroes = Coefficient of x 2 = 5 and coefficient of x = 1 and the term free from x = 35 The polynomial is 2. No. There can not exist two numbers satisfying the given condition, because here L.C.M (= 512) is not divisible by H.C.F ( = 12). 3. i.e., 4. 3 Median = Mode + 2 Mean 5. k = 8 6. SECTION B 7. 7 th term,. By question, a + 6d = 4 (i) 13 th term,. By question, a + 12d = 16 (ii) Material downloaded from mycbseguide.com. 6 / 21

178 Now sum of first 19 terms = [ Adding (i) and (ii), we get, 2a+ 18d = 16 ] 8. Length of radius = i.e., i.e., (x 2) 2 = 0 i.e., x = 2 Hence, value of x = If possible, let us assume that is rational and equals to i.e.,, where a and b are positive integers prime to each other and b >1 i.e.,.. (i) From (i), we see that, is not an integer, as a and b are prime to each other, so are also prime to each other, but 3b is an integer i.e in (i), a fraction equals to an integer, which contradicts our initial assumption. Hence, is irrational Since, EF BC, Since, FG CD, Material downloaded from mycbseguide.com. 7 / 21

179 By (i) and (ii), 11. Since, the lengths of tangents drawn from an external point to a circle are equal. AP = AS (i) BP = BQ (ii) CQ = CR (iii) DR = DS (iv) Now, AB + CD = AP + PB + CR + RD = AS + BQ + CQ + DS = (AS + DS) + (BQ + CQ) = AD + BC Hence proved. 12. Height of the solid cylinder (h)= 2.4 cm Diameter of its base (2r)= 1.4 cm. Material downloaded from mycbseguide.com. 8 / 21

180 Therefore its base radius (r)= 0.7 cm Height and diameter of the conical cavity are equal to those of the cylinder. Remaining surface area = (Curved surface area( outside) of cylinder) + (surface area of its bottom) + ( curved surface area of the conical cavity) = 1 SECTION C 13. i.e., [multiplying both sides by 3 ] i.e., i.e., i.e., i.e., Material downloaded from mycbseguide.com. 9 / 21

181 i.e., i.e., Therefore, the roots are and 14. Solving equations (i) and (ii) by cross-multiplication method, we get, i.e., i.e., i.e., Hence the solutions are x = 2, y = Let ABC and DEF be two triangles such that,. To prove that, From AB and AC cutting AP = DE, AQ = DF let us join P and Q Material downloaded from mycbseguide.com. 10 / 21

182 so, PQ BC [If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.] So, (common angle) Therefore, DABC and DDEF are equiangular and so their corresponding sides are in the same ratio. Hence, i.e., So i.e., PQ = EF DEF APQ (S-S-S) So Hence the corresponding angles of the triangles are equal. 16. Length of units, Length of units, Length of units, Material downloaded from mycbseguide.com. 11 / 21

183 Here we get, Hence the triangle is an isosceles triangle. Also, we observe that Therefore, by Pythagoras theorem, DABC is a right-angled triangle (right angle at B) Hence the triangle is an isosceles right-angled triangle. 17. According to the question, cards are mixed thoroughly and one card is drawn at random from the box, so the event of drawing a card is equally and likely. Since cards are marked with numbers 3, 4, 5,, 50, So there are 48 cards Here the total number of possible outcomes = 48. Let the event of drawing a card at random bearing two digit perfect square number be E. Then the number of outcomes favourable to the event E = 4 (here two digit perfect numbers are 16, 25, 36, 49 ) Therefore, 18. When a die is thrown once, then the number of total outcomes = 6 Let the E be the event of getting one prime number. Here prime numbers are 2, 3, 5 Then the number outcomes favourable to E = 3 Hence, Again let F be the event of getting an odd number. Here odd numbers are 1, 3, 5 Material downloaded from mycbseguide.com. 12 / 21

184 Hence, 19. (x 1, 2, 3) i.e., ( given x 1, 2, 3) i.e., i.e., i.e., i.e., x = 0 or x = 4 Hence the solutions are x = 0 and x = Given, i.e., i.e., [multiplying both sides by ] i.e., i.e., i.e., Hence proved. 21. According to the question, from an outside point P two tangents PA and PB are drawn to a circle with centre O (fig-4). To prove that, APB = 2 OAB. Material downloaded from mycbseguide.com. 13 / 21

185 Since the lengths of tangents drawn from an external point to a circle are equal. So, PA = PB. i.e., PAB is isosceles. [since, radius through point of contact is perpendicular to the tangent at the point of contact] Hence proved. 22. Fundamental Theorem of Arithmetic : Every composite number can be expressed ( factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur. Since 867 > 255, we apply the division lemma to 867 and 255, to get 867 = = = The remainder is 0 (zero) and divisor is 51. Therefore the HCF of 867 and 255 is 51. Hence LCM of 867 and 255 = SECTION D 23. Construct a circle of radius 5 cm. Let its centre be O. Material downloaded from mycbseguide.com. 14 / 21

186 Now construct one radius (say OP) of the circle. At O, draw an angle of 60, and draw a perpendicular to OP at P, 90. Extend these lines to meet at T (say). Now, taking T as centre and a radius equal to TP draw an arc which cuts the circle at Q. Join T, Q. Hence TP and TQ are the two tangents to the given circle of radius 5 cm which are inclined to each other at an angle of 60 [For justification of the construction: i.e., ] 24. x y = 8 x Three solutions for equation (i)are given in the table : Three solutions for equation (ii) are given in the table : x Material downloaded from mycbseguide.com. 15 / 21

187 1 0 3 Drawing Line AC Drawing Line PR Plotting points A(0, 8), B(4, 4) and C(8, 0) on graph paper the straight line AC is obtained as graph of the equation (i) Plotting points P(0, 1), Q(2, 0) and R(8, 3) on graph paper the straight line PR is obtained as graph of the equation (ii) From the graph, it is clear that a point M(6, 2) common to both the lines AC and PR. So the pair of equations is consistent and the solutions of the equations are x = 6 and y = 2. From the graph it is seen that the coordinates of the points where the lines AC and PR meets the y-axis are (0, 8) and (0, 1) respectively. 25. Given, (i) and (ii) adding (i) and (ii), we get, and subtracting (ii) from (i), we get, Material downloaded from mycbseguide.com. 16 / 21

188 . Hence proved. 26. Class interval Frequency Cumulative Frequency x 5+x x x y 40+x+y x+y Total 60 It is given that, n = 60 i.e., 45 + x + y = 60 i.e., x + y = 15 The median is 28.5, which lies in the class So, l = 20, f = 20, cf = 5+x, h = 10 We know, Material downloaded from mycbseguide.com. 17 / 21

189 Here, Therefore y = 15 8 = Let the square be ABCD of side 20 cm. Area of the square ABCD =. Diameter of each circle (in fig-5) = Therefore radius of each circle = 5 cm. So area of each circle = Total area of four squares = Hence area of the shaded region in the fig-5 = Let in the adjacent figure EC be the surface of water in the lake. A is the position of the observer. AE = 60 m. Also let B is the position of cloud and D be its image for fig. Material downloaded from mycbseguide.com. 18 / 21

190 In the lake and BF = h metre So, BC = CD = (h + 60)m (see fig) FC = 60 m. By question, In ABF, In AFD, [ since DF = DC + CF] By (i) and (ii), Hence height of the cloud from the water surface of the lake =BC = m = 120 m 29. Let ADH be a metallic right circular cone, whose height is 20 cm. is cut into two parts at Material downloaded from mycbseguide.com. 19 / 21

191 the middle of its height by a plane parallel to its base. The frustum is EBFHCDE Given that,, AC = 20 cm 1 According to question, AB = BC = 10 cm. In ABF, Again in In ACH, Therefore volume of the frustum EBFHCDE = = [here H = AC = 20 cm AB = 10 cm] = Hence the required volume = 30. Let first term and the common difference of the AP be a and d respectively. Then Material downloaded from mycbseguide.com. 20 / 21

192 According to question, This is an identity. Now putting, n = 2n 1 and m = 2m 1, we get, Hence proved. Material downloaded from mycbseguide.com. 21 / 21

193 SAMPLE QUESTION PAPER 09 Class-X ( ) Mathematics Time allowed: 3 Hours Max. Marks: 80 General Instructions: (i) All questions are compulsory. (ii) The question paper consists of 30 questions divided into four sections A, B, C and D. (iii) Section A contains 6 questions of 1 mark each. Section B contains 6 questions of 2 marks each. Section C contains 10 questions of 3 marks each. Section D contains 8 questions of 4 marks each. (iv) There is no overall choice. However, an internal choice has been provided in four questions of 3 marks each and three questions of 4 marks each. You have to attempt only one of the alternatives in all such questions. (v) Use of calculators is not permitted. SECTION A 1. Find the zeroes of the quadratic polynomial. 2. A ladder of length H metre makes an angle of 60 o with the ground when placed against a wall. If the distance between the feet of the ladder and the wall be 5 m, then determine H. 3. Captain of a team tosses two different coins, one golden colour and other of silver colour simultaneously. What is the probability that he gets atleast one head? 4. Write the sum of the first 100 natural numbers. 5. Find the coordinate of the mid-point of the line segment joining the points whose coordinates are and. 6. From an outside point A, two tangents AB and AC are drawn to touch the circle with centre at O (in fig-1). Given BAC = 30 o. Find AOB. Material downloaded from mycbseguide.com. 1 / 21

194 SECTION B 7. Find the values of y for which the distance between the points P(2, 3) and Q(10, y) is 10 units. 8. Find the discriminant of the equation and hence find the nature of its roots. 9. The sum of first 30 and 40 terms of an AP are respectively 2265 and 4020, then find the common difference of the AP. 10. The height and base diameter of a solid cylinder are 16 m and 2r m respectively. The cylinder is melted and recast to 12 solid spheres of same base diameter. Find r. 11. Two concentric circles are of radius 7 cm and cm. Find the length of the chord of the bigger circle, which touches the smaller circle. 12. If ABC and DEF are two similar triangles and their areas are 81 cm 2 and 144 cm 2 respectively. The bases of the triangles are respectively BC and EF. If EF = 15 cm, find BC. SECTION C 13. The denominator of a fraction is one more than thrice of the numerator. If the sum of the fraction and its reciprocal is, then find the fraction. 14. Use Euclid s division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m An army contingent of 616 members is to march behind an army band of 32 members in Material downloaded from mycbseguide.com. 2 / 21

195 a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march? 16. The ratio of the sum of n-terms of two APs is (2n + 4) : (5n +2), find the ratio of their 12 th terms. Or Find the sum of all 3 digit numbers which leave remainder 3 when divided by The vertices of a ABC are A(4, 9), B and C(9, 4). A line is drawn to intersect sides AB and AC at P and Q respectively, such that. Find the area of APQ. Or The vertices of a ABC is (1, 2), (3, 1) and (2, 5). Point D divides AB in the ratio 2:1 and P is the mid-point of CD. Find the coordinates of the point P. 18. In fig-2, ABC and ABD are on the same base AB and on opposite sides of AB. If CD intersects AB at O, then show that,. 19. In fig-3, PQ and RS are two parallel tangents to a circle with centre O and another tangent EF with point of contact C intersecting PQ at E and RS at F. Prove that EOF = 90. Material downloaded from mycbseguide.com. 3 / 21

196 Or Prove that, the lengths of tangents drawn from an external point to a circle are equal. 20. Prove that:. 21. A bag contains 5 red ball and some blue balls. If the probability of drawing a blue ball is double that of a red ball, determine the number of blue balls in the bag. Or A bag contains 8 red balls and x blue balls. the odd against drawing a blue ball are 2:5. what is the value of x? 22. One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting : a) a king of red colour (b) a spade (c) a face card SECTION D 23. If we add 1 to the numerator and subtract 1 from the denominator, a fraction reduces to 1. It becomes if we only add 1 to the denominator. What is the fraction? Or The sum of a 2 digit number and number obtained by reversing the order of the digits is 99. If the digits of the number differ by 3. Find the number. 24. Find the common difference of an A.P. whose 1 st term is 100 and the sum of whose first six terms is 5 times the sum of the next six terms. Material downloaded from mycbseguide.com. 4 / 21

197 25. Draw a circle with the help of a bangle. Take a point outside the circle. Construct the pair of tangents from this point to the circle. Or Let ABC be a right triangle in which AB = 6 cm, BC = 8 cm and B = 90. BD is the perpendicular from B on AC. The circle through B, C, D is drawn. Construct the tangents from A to this circle. 26. Prove that, 27. Find the area of the segment APB shown in fig-4, if radius of the circle is 14 cm, and (use ) 28. There are two poles either on each bank of a river, just opposite to each other. One pole is 60 metre high. From the top of this pole, the angles of depression of the top and the foot of the other pole are 30 o and 45 o respectively. Find the width of the river and the height of the other pole. 29. In fig-5, a toy is in the form of a cone mounted on a hemisphere of common base of radius 7 cm. The total height of the toy is 31 cm. Find the total surface area of the toy. [ ] Material downloaded from mycbseguide.com. 5 / 21

198 30. The following frequency distribution gives the monthly consumption of electricity of 68 consumers of a locality. Find the median and mode of the data. Monthly Consumption Number of Consumer Or If the mean of the following distribution is 27, find the value of p. Also find the median and mode. Class-interval No of workers 8 p Material downloaded from mycbseguide.com. 6 / 21

199 CBSE SAMPLE PAPER 09 CLASS X - Mathematics Solutions SECTION- A 1. 3x 2 + 7x + 2 So the zeroes are. 2. Therefore, H = 10 m. 3. Total outcomes = 2 2 = 4 4. Sum of the first 100 natural numbers = 5. Coordinates of the required mid-point = i.e., 6. Since tangents drawn from an external point to a circle subtend equal angles at the centre of the circle. Material downloaded from mycbseguide.com. 7 / 21

200 = SECTION- B 7. By question, i.e., i.e., i.e., i.e., Hence values of y are 3, 9 8. Discriminant of is = = 0 Therefore roots of the given quadratic equation are real and equal. 9. We know that the sum of first n-terms of an AP (whose first term is and common difference is d) is By question, Again by question, Subtracting (i) from (ii), we get, 10d = 50 i.e., d = 5 Hence the common difference of the AP is The volume of the solid cylinder = Volume of each sphere = [since its base diameter is equal to that of the cylinder] By question,. Hence r = 1 m Material downloaded from mycbseguide.com. 8 / 21

201 11. Let C is the centre of two concentric circles of radii 5cm and 3 cm. Let AB, a chord of the bigger circle which touches the smaller circle at M. CM = cm, CA = 7 cm [radius through point of contact is to tangent] [by Pythagoras Theorem ] i.e., Therefore, AB = 2AM = 2 6 = 12 cm. [Line segment, drawn from centre of a circle perpendicular to any chord, bisects the chord ] Hence, the length of the chord is 12 cm. 12. We know that, the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Therefore for the given problem, Hence, BC = cm SECTION C 13. Let the fraction be According to first condition, y = 3x + 1. So the fraction becomes Again by the 2 nd condition, Material downloaded from mycbseguide.com. 9 / 21

202 Therefore. x = 5 (since, x >0) Hence the required fraction = 14. According to Euclid s division lemma, Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 r < b. Let a be any positive integer and b = 3. Then, let a = 3q + r, where q > 0 and 0< r < 3 a = 3q or a = 3q + 1 or a = 3q +2 We have three cases : i) when a = 3q, then, a3 = (3q) 3 = 9(3q 3 ) = 9k [where k = 3q 3 is an integer] ii) when a = 3q + 1, then, a 3 = (3q + 1) 3 = 9(3q 3 ) + 9(3q 2 ) + 9q + 1 = 9(3q 3 +3q 2 +3q) + 1 = 9k1 + 1 [where k 1 = 3q 3 + 3q 2 + 3q is an integer] iii) When a = 3q + 2 then, a 3 = (3q + 2) 3 = 9(3q 3 ) + 9(6q 2 ) + 9(4q) + 8 = 9(3q 3 + 6q 2 + 4q) + 8 = 9k [where k 2 = 3q 3 + 6q 2 + 4q is an integer] Therefore, the cube of any positive integer is of the form 9m, 9m + 1 or 9m To find the maximum number of columns in which members can march, we have to calculate the HCF of 32 and 616. Using Euclid s algorithm, 616 = , 32 = Therefore the HCF of 32 and 616 is 8 Hence they can march in 8 columns each. 16. Let, be the first terms of the two APs also let be their common differences respectively. According to question, This is an identity. Material downloaded from mycbseguide.com. 10 / 21

203 Putting n = 23, we get, i.e., i.e., i.e., i.e., Hence ratio of their 12 th terms is 50 : Since P divides AB internally in the ratio 2 : 3, So, abscissa i.e., x-coordinate of P = and ordinate i.e., y-coordinate of P = Therefore coordinates of P are (3, 5) Since, Q divides AC internally in the ratio 2 : 3 Again, abscissa i.e., x-coordinate of Q = and ordinate i.e., y-coordinate of Q = Therefore coordinates of Q are (6, 7) Using, formula :, area of APQ = 18. Given that, ABC and ABD are on the same base AB but on opposite sides of AB. Material downloaded from mycbseguide.com. 11 / 21

204 To prove that, Construction :Let CM AB and DN AB are drawn. Proof: In CMO and DNO, CMO = DNO = 90 o COM = DON [ vertically opposite ] CMO ~ DNO [by A-A property ] So, (ii) and (iii) From (ii) and (iii), Hence proved. 19. Given, two tangents PQ and RS, which touch the circle, with centre O, at points A and B. Also, PQ RS. Another tangent EF touches the circle at C and meets the lines PQ and RS at E and F respectively. To prove,. Material downloaded from mycbseguide.com. 12 / 21

205 Construction : O, C joined. Proof :. [Since, radius through point of contact is perpendicular to the tangent] In OA = OC [radius of same circle] EA = EC [Tangents drawn from outside point to a circle are equal] i.e., Similarly, it can be shown that, Since PQ RC and EF is a transversal, So,. Hence proved. 20. Material downloaded from mycbseguide.com. 13 / 21

206 = RHS Hence proved. 21. Let the number of blue balls in the bag be x. Therefore total number of balls in the bag = 5 + x Let probability of drawing a blue ball be P(blue). Let probability of drawing a red ball be P(red). Then, and According to the question, P(blue) = 2 P(red) i.e., i.e., x=10 [since x can not be negative ] Hence number of blue balls in the bag is 10. Material downloaded from mycbseguide.com. 14 / 21

207 22. Total number of favourable outcomes = 52. a) We know that, cards of diamond and heart are of red colour. There exist one king of Diamond and one king of heart. Number of outcomes favourable to getting a king of red colour = 2 P(getting a spade) b) A well-shuffled deck of 52 cards contains 13 cards of spade. So, number of outcomes favourable to getting a spade = 13. P(getting a king of red colour) c) A well-shuffled deck of 52 cards contains 12 face cards. So, number of outcomes favourable to getting a face card = 12. P(getting a face card) SECTION - D 23. Let the numerator and the denominator be x and y respectively. i.e., the fraction is According to question we get, and Material downloaded from mycbseguide.com. 15 / 21

208 Now, and Substituting y in (ii) with the help of (i), 2x = x i.e., x = 3 1 Now from (i), y = 3 +2 =5. Hence the fraction is. 24. Let the 1 st term and common difference of the AP be a and d respectively. Here a = 100. We know that sum of first n-terms of an AP,. Now sum of first 6-terms of the AP is. and sum of next 6-terms of the AP = S 12 - S 6 = 6( d) - 3( d) = d d = d According to the question, d = 5(600+51d) i.e., d = d i.e., = 255d 15d i.e., 240d = 2400 d = 10 Hence the common difference of the AP is ( 10). Material downloaded from mycbseguide.com. 16 / 21

209 25. Steps of construction : Draw a circle with help of a bangle. Take two non parallel chords AB and CD Draw perpendicular bisectors of these chords. Mark point O where two bisectors intersect. Point O is center of the circle. Take a point P outside the circle and join PO. Bisect PO. Let M be mid-point of PO. Taking M as centre and MO as radius, draw a circle. Let it intersect the given circle at the points Q and R. Join PQ and PR. 26. Hence proved. 27. Area of the segment APB = Area of the sector OAPBO area of OAB. Material downloaded from mycbseguide.com. 17 / 21

210 Here area of the sector OAPBO = Let, OM AB is drawn. Given, Since, perpendicular drawn from centre of a circle to a chord bisects the chord, So, here, Now Therefore area of OAB = Hence, Area of the segment APB = Area of the sector OAPBO area of OAB = 28. Let two poles AB(= 60 m) and PQ be either on each bank of a river. Material downloaded from mycbseguide.com. 18 / 21

211 Let PQ = h m. and width of the river i.e., BQ = x m. In figure, PR AB. AR = AB BR = AB PQ = (60 h) m Acc to question, Angle of depression of P from A = XAP = APR (alternate angle) = and angle of depression of Q from A = XAQ = AQB (alternate angle) = Now in AQB, i.e., i.e., x =60 Now in APR, i.e., i.e., [since x = 60] i.e., i.e., Material downloaded from mycbseguide.com. 19 / 21

212 Hence the width of the river is 60 m and height of the other pole is m. 29. Total height of the toy is 31 cm i.e., in fig-5, AD = 31 cm. Radius of the common base is 7 cm. i.e., OC = OB = OD = 7 cm. AO = AB OD = 31 7 = 24 cm. Surface area of the toy = slant surface area of the cone + curved surface area of the hemisphere Here, slant surface area of the cone = sq. unit Curved surface area of the hemisphere = Hence the total surface area of the toy =( ) cm 2 = 858 cm For table = 1 Monthly consumption (in units) Number of consumer (fi) Cumulative frequency Material downloaded from mycbseguide.com. 20 / 21

213 Total N=68 For Median. Here,, then, which lies in the interval Median class is Here, l = 125, n = 68, f = 20, cf = 22 and h = 20. For Mode : In the given data, maximum frequency is 20 and it corresponds to the class interval Modal class = and = 125, f 1 = 20, f 0 = 13, f 2 = 14 and h = 20 Now, Hence Median= 137 units, Mode = units. Material downloaded from mycbseguide.com. 21 / 21

214 SAMPLE QUESTION PAPER 10 Class-X ( ) Mathematics Time allowed: 3 Hours Max. Marks: 80 General Instructions: (i) All questions are compulsory. (ii) The question paper consists of 30 questions divided into four sections A, B, C and D. (iii) Section A contains 6 questions of 1 mark each. Section B contains 6 questions of 2 marks each. Section C contains 10 questions of 3 marks each. Section D contains 8 questions of 4 marks each. (iv) There is no overall choice. However, an internal choice has been provided in four questions of 3 marks each and three questions of 4 marks each. You have to attempt only one of the alternatives in all such questions. (v) Use of calculators is not permitted. SECTION - A 1. The decimal expansion of the rational number will terminate after how many places of decimal? 2. Find the nature of the roots of the equation. 3. A boy walks 12m due east and 5 m due south. How far is he from the starting point? 4. If the point C(k, 4) divides the join of points A(2, 6) and B(5, 1) in the ratio 2 : 3, then find the value of k. 5. Find the area of annulus whose inner and outer radii are 6 cm and 8 cm. 6. In a family of three children, find the probability of having at least one boy. SECTION - B 7. Check whether on simplification gives a rational or irrational number. Material downloaded from mycbseguide.com. 1 / 21

215 8. Gunal saved Rs.10 in the first week of a month and then increased his weekly savings by Rs If in the nth week, his savings become Rs.59.50, find n. 9. Without using trigonometric tables prove that: tan 1 tan 11 tan 21 tan 69 tan 79 tan 89 = Find the third vertex of the triangle ABC. If two of its vertices are at A(-3, 1) and B(0, 2) and the mid-point of BC is at D 11. D and E are points on the sides AB and AC respectively of a ABC. If AD = 5.7 cm, DB = 9.5 cm, AE = 4.8 cm and EC = 8 cm, then determine whether DE BC or not. 12. Three cubes each of side 15 cm are joined end to end. Find the total surface area of the resulting cuboid. SECTION - C 13. Show that any positive odd integer is of the form (4m + 1 ) or ( 4m + 3), where m is some integer. 14. If two zeroes of the polynomial x 4 + 3x 3-20x 2-6x + 36 are and -, then find the other zeroes of the polynomial. 15. Solve: 16. A class consists of a number of boys whose ages are in A.P., the common difference being 4 months. If the youngest boy is just eight years old and if sum of the ages is 168 years, then find the number of boys in the class. OR Find for the AP in -9, -14, -19, Find the coordinates of the centroid of a triangle whose vertices are A( ), OR Material downloaded from mycbseguide.com. 2 / 21

216 In figure, ABC is a right triangle right-angled at B. Medians AD and CE are of respective lengths 5 cm and, find length of AC. 18. If then show that 19. Prove that: OR If, show that 20. A square field and an equilateral triangular park have equal perimeters. If the cost of ploughing the field at rate of Rs 5 perm 2 is Rs 720, find the cost of maintaining the park at therate of Rs 10 per m The following table shows the marks obtained by 100 students of class X in a school during a particular academic session. Find the mode of this distribution. Marks Less than 10 Less than 20 Less than 30 Less than 40 Less than 50 Less than 60 Less than70 Less than 80 No of students OR The percentage of marks obtained by 100 students in an examination are given below: Marks Frequency Material downloaded from mycbseguide.com. 3 / 21

217 Determine the median percentage of marks. 22. A bag contains 6 red balls and some blue balls. If the probability of drawing a blue ball from the bag is double that of a red ball, find the number of blue balls in the bag. SECTION - D 23. Solve the following system of linear equations graphically: 3x + y = 12 and x - 3y = -6. Shade the region bounded by these lines and the x-axis. Also find the ratio of areas of triangles formed by given lines with x-axis and the y-axis. OR A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours it can go 40 km upstream and 55 km down stream. Determine the speed of the stream and that of the boat in still water. 24. A plane left 30 minutes late than its scheduled time and in order to reach the destination 1500 km away in time, it had to increase the speed by 250 km/h from the usual speed. Find its usual speed. 25. Prove that the ratio of areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Use the above theorem, in the following. The areas of two similar triangles are 81 cm 2 and 144 cm 2. If the largest side of the smaller triangle is 27 cm, find the largest side of the larger triangle. OR In a triangle if the square of one side is equal to the sum of the squares on the other two sides. Prove that the angle apposite to the first side is a right angle. Use the above theorem to find the measure of in figure given below. Material downloaded from mycbseguide.com. 4 / 21

218 26. Prove that the intercept of a tangent between two parallel tangents to a circle subtends a right angle at the centre of the circle. 27. Construct a triangle similar to given ABC in which AB = 4 cm, BC = 6 cm and ABC = 60, such that each side of the new triangle is of given ABC. 28. The angle of elevation of the top a tower at a point on the level ground is 30. After walking a distance of 100m towards the foot of the tower along the horizontal line through the foot of the tower on the same level ground, the angle of elevation of the top of the tower is 60. Find the height of the tower. 29. A container (open at the top) made up of a metal sheet is in the form of a frustum of a cone of height 16 cm with radii of its lower and upper ends as 8 cm and 20 cm respectively. Find (i) the cost of milk when it is completely filled with milk at the rate of Rs 15 per litre. (ii) the cost of metal sheet used, if it costs Rs 5 per 100 cm The median of the following data is Find the missing frequencies x and y, if the total frequency is 100. Class-Interval Frequency 7 10 x 13 y OR In the following distribution, locate the median mean and mode. Monthly consumption of electricity No. of Consumers Material downloaded from mycbseguide.com. 5 / 21

219 SAMPLE PAPER 10 (CLASS X - MATHEMATICS ) Marking Scheme 1. The decimal expansion of will terminate after 4 places of decimal 2. D = 108 > 0 Therefore the roots are real and distinct. 3 Let the starting point be B. Then BC =12 m & CA = 5m By using Pythagoras theorem, The distance between the starting point and end point = 13 m. 4 Material downloaded from mycbseguide.com. 6 / 21

220 By using section formula, 5. Area of Annulus = 6 Sample Space = { BBB, BBG, BGB, BGG, GBB, GBG, GGB,GGG} n(s) = 8 P(having at least one boy) = Section B Question numbers 7 to 12 carry two marks each 7. = = ( a rational number) 8. a = 10 ; d = 2.75 a n = = 10 + (n 1) (2.75) n = 19 OR a = -9 d = (-9) = = -5 a 30 - a 20 = (a + 29d) - (a + 19d) Material downloaded from mycbseguide.com. 7 / 21

221 = 10d = 10 x (-5) = LHS = tan (90-89 ) tan (90-79 ) tan (90-69 ) tan 69 tan 79 tan 89 = cot 89 cot 79 cot 69 tan 69 tan 79 tan 89 = cot 89 cot 79 cot 69 = Let the third vertex be (x, y) 11. AD = 5.7 cm, DB = 9.5 cm, AE = 4.8 cm and EC = 8 cm Since D and E are the points on AB and AC respectively. Check whether or not. 0.6 = 0.6 Therefore, (each equal to 0.6) Hence, by the converse of Thales theorem DE BC 12. Resulting solid is a cuboid with length = 3(15) = 45 cm ; breadth = 15 cm ; Height = 15 cm TSA of the cuboid = 2 (lb + bh + hl) = 2 ( ) = 3150 cm 2 Section C Question numbers 13 to 22 carry three marks each 13. Let n be any arbitrary positive odd integer. On dividing n by 4, let m be the Quotient and r be the remainder. So, by Euclid s division lemma, we have Material downloaded from mycbseguide.com. 8 / 21

222 n = 4m + r, where m 0 and r < 4. As m 0 and r < 4 and r is an integer, r can take values 0, 1, 2, 3. n = 4m or n = 4m + 1 or n = 4m + 2 or n = 4m + 3 n = 4m + 1 or n = 4m + 3( since n is odd) Thus, any positive odd integer is of the form (4m + 1) or (4m + 3), where m is some integer. 14. Since and - are two zeroes of the polynomial (x - ) (x + ) is a factor of the polynomial. By long division method x 4 + 3x 3-20x 2-6x + 36 = (x 2-2) (x 2 + 3x - 18) = (x 2-2) (x + 6) (x - 3) The other zeroes of the Polynomial are -6, (x 2-5x + 4) +(x 2-5x + 6)= (x 2-6x + 8) 6x 2 30x +30 = 10x 2-60x +80 4x 2 30x +50=0 2x 2 15x +25 = 0 (x-5)(2x-5) = 0 x=5; x=5/ ,100, 104,... a = 96, S n = = 2016 ; d = 4 S n = ( (n 1) 4)=2016 n 2 +47n-1008=0 (n+63)(n-16) =0 n=-63 is rejected; n =16 Therefore no of boys in the class = The vertices of triangle are A(2, ), Material downloaded from mycbseguide.com. 9 / 21

223 Therefore the coordinates of centroid are = RHS OR L.H.S Material downloaded from mycbseguide.com. 10 / 21

224 20. Let the side of the park be a meter. 5 x a 2 = 720 a = 12m. Perimeter of square = 48 m. Perimeter of triangle = 48m. Side of triangle = 16m. Now Area of triangle = = 64 m 2. Cost of maintaining the park = Rs. (10 x 64 ) = Rs Marks No.of students(f i ) (f 0 ) (f 1 ) (f 2 ) TOTAL 68 Modal class is because it has highest frequency Mode = Material downloaded from mycbseguide.com. 11 / 21

225 OR Marks (class) No. of students (Frequency) Cumulative Frequency Here n = 100 Therefore, which lies in the class l 1 (The lower limit of the median class) = 45 c (The cumulative frequency of the class preceding the median class) = 48 f (The frequency of the Median class)= 23 h (The class size) = 5 Median So, the median percentage of marks is Let the number of blue balls be x. Thus the sum of the possible outcomes = 6 +x Material downloaded from mycbseguide.com. 12 / 21

226 Now the sum of the favourable outcomes if the red balls are drawn = 6 P(E) = Again the sum of the favourable outcomes if the blue balls are drawn = x P(E) = =2 x=12 Hence the number of blue balls = 12. Section D 23 Since the lines intersect at (3, 3), there is a unique solution given by x=3, y = 3 Area of triangle ABC formed by lines with x - axis = ½ x 10 x 3 = 15sq. units Area of triangle BDE formed by lines with y - a x is = ½ x 10 x 3 = 15 sq units Ratio of these areas = 1 :1 OR Material downloaded from mycbseguide.com. 13 / 21

227 Let the speed of boat is }x\text{ km/h in still water and stream y km/h According to question, on solving eq. (i) and (ii) we get, on solving eq. (iii) and (iv) we get, x = 8km/h y = 3km/h 24. Let the usual speed of plane be x km/hour Time taken = hrs. with usual speed Time taken after increasing speed = hrs Given that x x = 0 (x ) ( x -750 ) = 0 x = 750 or (Rejected) Usual speed of plane = 750km/h. 25 Given: ABC ~ DEF To Prove: Construction: Draw AG BC and DH EF Proof : Material downloaded from mycbseguide.com. 14 / 21

228 Now in ABG and DEH, B = E AGB = DHE(Each 90 ) AGB ~ DHE But, From (i) and (ii), we get, Similarly, we can prove Let the largest side of the larger triangle be x cm, then x = 36 cm (Using the theorem) OR Given: A such that To prove: Triangle ABC is right angled at B Construction: Construct a triangle DEF such that Material downloaded from mycbseguide.com. 15 / 21

229 and Proof: is a right angledtriangle right angled at E [construction] By Pythagoras theorem, we have Thus, in and we have [By Construction and (i)] Hence, In is a right triangle. Now in is right angled at K 26 Material downloaded from mycbseguide.com. 16 / 21

230 Since tangent is perpendicular to the radius of the circle SPO = SRO = OQT = 90 In right triangles OPS and ORS OS = OS (common) OP = OR (radii of circle) OPS ORS(RHS congruence) 1 = 2 Similarly 3= 4 Now = =90 SOT = DABC in which AB = 4 cm, BC = 6 cm and ABC = 60 DA BC is the required similar triangle. Material downloaded from mycbseguide.com. 17 / 21

231 28 In right BAC, AB = (100 + AD) x ---(i) In right DBAD, AB = AD x ---(ii) From (i) and (ii) we get AD = 3 AD AD = 50 m From (ii) AB = 50 m = 50 x 1.732m or, AB = 86.6 m 29. The Container is a frustum of cone h = 16cm, r = 8cm, R = 20cm Volume of the container = x ph ( R 2 + Rr + r 2 ) = x 3.14 x 16 ((20) (8) + (8) 2 ) cm 3 = x 3.14 x 16 ( ) cm 3 = ( x 3.14 x 16 x 624 ) cm 3 = (3.14 x 3328) cm 3 = cm 3 = litres Cost of milk = Rs (10.45 x 15) = Rs Material downloaded from mycbseguide.com. 18 / 21

232 Now, slant height of the frustum of cone = l Total surface area of the container = pl ( R+r) +p r 2 = 3.14 x 20 (20 + 8) (8) 2 cm 2 = 3.14 [ 20 x ] cm 2 = 3.14 x 624 = cm 2 Cost of metal Used = Rs x = Rs x 5 = Rs = Rs 98 (Approx.) 30 CLASS INTERVAL FREQUENCY CUMULATIVE FREQUENCY x 17+x x y 30+x+y x+y x+y x+y Given n(total frequency ) = = 63 + x + y x + y = (1) Material downloaded from mycbseguide.com. 19 / 21

233 The median is which lies in the class So, median class is l = 20 f = y c.f = 30 + x h = 5 Using formula, Median = l = y = x 20x + 3y = (2) Solving (1) and (2), we get x = 17 y = 20 OR Monthly consumption of electricity No. of consumers (F) C.F Class Mark (X) FX ƒx=9320 Now Median class = and this is in class Material downloaded from mycbseguide.com. 20 / 21

234 Here, We know that Hence, Median = 137 Again Mean For mode, since the maximum frequency is 20 and this corresponds to the class Here, Thus, Median = 137, Mean = and Mode = The three measures are approximately the same in the class. Material downloaded from mycbseguide.com. 21 / 21

235 SAMPLE QUESTION PAPER 11 Class-X ( ) Mathematics GENERAL INSTRUCTIONS (i) All questions are compulsory. (ii) The question paper consists of 30 questions divided into four sections A, B,C and D. (iii) Section A contains 6 questions of 1 mark each, Section B contains 6 questions of 2 marks each, Section C contains 10 questions of 3 marks each and Section D contains 8 questions of 4 marks each. (iv) Use of Calculator and log tables is not permitted. Q.1 Find the value of k for which the pair of linear equations and has no solution Q.2. for what value of p, are 2p + 1, 13, 5p-3 three consecutive terms of an A.P? Q.3 without using trigonometry table, evaluate Q.4 In figure,, BC = 7.5 cm, AM = 4cm and MC = 2 cm. Find the length BN Q.5 The two tangents from an external point P to a circle with centre O and PA and PB. If what is the value of? Q.6 A pair of dice is thrown once. Find the probability of getting the same number on each dice. Q. 7 Prove that is an irrational number. Material downloaded from mycbseguide.com. 1 / 27

236 Q.8 Solve the equations graphically: What is the area of the triangle formed by the two lines and the line y = 0? Q.9 Find the roots of the following equation Q.10 Prove that : Q.11 The angle of elevation of the top of a building from the foot of a tower is and angle of elevation of top of the tower from the foot of the building is. If the tower is 50 m high. Find the height of the building. Q.12 The dimensions of a metallic cuboid are 100 cm 80 cm 64 cm. It is melted and recast into a cube. Find the surface area of the cube. Q.13 A person on tour has Rs for his expense. If he extends his tour for 3 days, he has to cut down his daily expense by RS. 70. Find the duration of the tour. Q.14 The first and last term of an AP are 8 and 350 respectively. If its common difference is 9, how many terms are there and what is their sum? Q.15 Prove that the ratio of areas of two similar triangles is equal to the ratio of the square of their corresponding sides. Q.16 Point P divides the line segment joining the points A (2,1) and B (5, -8) such that If P lies on the line, find the value of k. OR Show that the points and are the vertices of a square. Q.17 From the top of a 7m high building, the angle of elevation of the top of a tower is and angle of depression of the foot of the tower is Find the height of the tower. OR A girl who is 1.2 m tall, spots a balloon moving with the wind in a horizontal line at a height of 88.2 m from the ground. The angle of elevation of the balloon from the eye of the girl at Material downloaded from mycbseguide.com. 2 / 27

237 any instant is. After sometime, the angle of elevation reduces to. Find the distance travelled by the balloon during the interval. Q.18 In figure, a triangle ABC is drawn to circumscribe a circle of radius 2 cm such that the segment BD and DC into which BC is divided by the point of contact D are the lengths 4cm and 3cm respectively. If area of ABC = 21, then find the lengths of sides AB and AC. Q.19 Draw a triangle ABC with BC = 7cm, and. then construct another triangle whose sides are times the corresponding sides of ABC. OR Draw a circle of radius 2.3 cm take a point P on it. Without using the centre of the circle. Draw tangent to it at P. Q.20 The mid points of sides of a triangle are (3,4), (4,1) and (2,0). Find the coordinate of vertices of the triangle. Q.21 Weekly income of 600 families is given below. Income in Rs. Frequency Material downloaded from mycbseguide.com. 3 / 27

238 Find the median. OR The following tables gives production yield per hectare of wheat of 100 farms of village: Production Yeild (in hr.) No. of Farms Change the distribution to a more than type distribution and draw its Ogive. Q.22 All kings, queens and aces are removed from a pack of 52 cards. The remaining cards are well shuffled and then a card id drawn from it. Find the probability that the drawn card is (i) a black face card (ii) a red card. Q. 23 Show that only one of the number n, n+2 and n+4 is divisible by 3. OR Use Euclid Division Lemma to show that cube of any positive integer is either of the form 9m, (9m + 1) or (9m + 8). Q. 24 Obtain all other zeroes of the polynomial, if two of its zeroes are and. Material downloaded from mycbseguide.com. 4 / 27

239 Q.25 Solve the following pair of equations for and ; OR If in a rectangle the length is increased and breadth is decreased by 2 units each, the area is reduced by 28 square units, and if the length is reduced by 1 unit and breadth is increased by 2 units, the area increased by 33 square units. Find the dimensions of the rectangle. Q 26. In figure AB units, CD units and units, prove that Q.27 Prove that Q.28 ABC is a right triangle, right angled at A. Find the area of shaded region if AB = 6cm, BC = 10cm and O is the centre of incircle of ABC. (take ) Q.29 A gulab jamun, when ready for eating, contains sugar syrup of about 30% of its volume. Find approximately how much syrup would be found in 45 such gulab jamuns, each shaped Material downloaded from mycbseguide.com. 5 / 27

240 like a cylinder with two hemispherical ends, if the complete length of each of them is 5cm and its diameter is 2.8 cm. Q.30 Draw less than Ogive and more than Ogive for the following distribution and hence find its median. Class Frequency OR Find the mean, mode and median for the following data. Classes Frequency Material downloaded from mycbseguide.com. 6 / 27

241 SAMPLE PAPER 11 (CLASS X - MATHEMATICS ) Marking Scheme 1. Since pair of equations has no solution Then, i.e. 2. 2p+1, 13,5p-3 are consecutive terms of an A.P 13-2p - 1 = 5p p = 5p = 5p + 2p 28 = 7p p = In MN AB [ Let x = BN] 3x = 15 x = 5 Hence BN = 5 cm. 5. PA and PB are tangents to the circle Material downloaded from mycbseguide.com. 7 / 27

242 [ Tangent makes an angle of with the radius at the point of contact ] In quadrilateral OAPB, [Angle sum property of a quadrilateral] 6. Total number of outcomes when a pair of dice is thrown is 36 Same number on each dice i.e. (1,1), (2,2), (3,3), (4,4), (5,5), (6,6) Number of ways of getting the same number on each dice = 6. Required probability = 7. Let if possible is a rational number. From we notice LHS is an irrational number and RHS is rational number, which is not possible. Hence, our supposition is wrong. Hence, is an irrational number. 8. Material downloaded from mycbseguide.com. 8 / 27

243 From From From graph, we observe that solution of equation is (0,2) Area = 5 square units 9. Material downloaded from mycbseguide.com. 9 / 27

244 x 2-3x - 28 = -30 x 2-3x + 2 = 0 (x - 2)(x - 1)=0 x = 2,1 Required roots are 2, Taking LHS = =RHS 11. Solution : Let TP be height of tower of 50m, Material downloaded from mycbseguide.com. 10 / 27

245 Let distance AP between building and tower be. To find : AB, height of building In right In right AB AB = 16.67m Hence height of the building is m 12. Dimensions of the metallic cuboid are 100 cm 80 cm 64 cm (Where a is side of cube) a = 80 cm Surface area of cube 13. Number of days for a tour = x Daily expense = y A.T.Q. If tour be extended for 3 days New number of days = x + 3 Daily expense = y - 70 Now (x + 3)(y - 70) = 4200 xy - 70x + 3y -210 = x + 3y = 0 [From (i)] Put Material downloaded from mycbseguide.com. 11 / 27

246 x x = 0 x 2 + 3x = 0 (x + 15)(x - 12) = 0 x = 12, x = -15 (rejected) Duration of Tours = 12 Hours 14. Here a = 8, l =350, d=9 From formula, l= a n = a + (n - 1) d, we get a + ( n-1) d= (n - 1)9 = 350 (n - 1)9 = (n - 1)9 = 342 n - 1 = n - 1 = 38 n = From formula, we get 15. Given : To prove : Construction : Draw AM BC and DK EF Proof : ABC Material downloaded from mycbseguide.com. 12 / 27

247 Also ABM and DEK (given) ( angles of similar s ) (Construction) By AA Rule Equating (i) and (ii) So, Hence proved 16. P is the point of intersection of line segment AB and line 2x - y + k = 0. Here, 3 AP = AP + PB 2 AP = PB AP : PB = 1 : 2 P divides the line segment joining A (2,1) and B(5, -8) in ratio 1:2 Coordinates of point P are i.e. p(3, -2) As point P lies on the line 2x - y + k = k = 0 6 = -8 OR Material downloaded from mycbseguide.com. 13 / 27

248 Diagonal Diagonal Hence proved. 17. AB is a building of height 7 m and CD is tower of height h m. AB = ED = 7m and CE = (h-7)m Let BD = AE = x m In right AED, In right AEC, Height of the tower is 28 m. OR In right Material downloaded from mycbseguide.com. 14 / 27

249 In right 18. Let AE = AF = x Length of tangents from an external point is equal. ar BOC = ar AOB = ar AOC = Material downloaded from mycbseguide.com. 15 / 27

250 [from (i)] 21x + 7x - 49 = 0 2x 2 + 7x - 49 = 0 2x x - 7x -49 = 0 2x(x + 7) - 7(x + 7) = 0 (2x - 7)(x + 7) = 0 [rejected] length of side AB = = 7.5 cm and AC = = 6.5 cm 19. OR Material downloaded from mycbseguide.com. 16 / 27

251 Steps of construction: 1. Draw a circle of radius 2.3 cm and take a point P on it. 2. Draw chord PQ. 3. Mark a point R in the major arc QP. 4. Join PR and RQ. 5. Draw 6. Produce to as shown in figure, then is the required tangent at the point P. 20. Let A(x 1, y 1 ), B(x 2, y 2 ) and C(x 3, y 3 ) are the vertices of ABC. (3,4), (4,1),(2,0) are mid points of sides AB,BC,CA As (3,4) is mid point of AB x 1 + x 2 = 6...(i) y 1 + y 2 = 8...(ii) As (4,1) are mid points of BC x 2 + x 3 = 4...(iii) y2 + y3 = 2... (iv) As (2,0) are mid points of AC x 1 + x 3 = 4...(v) y 1 + y 3 = 0..(vi) Adding (i), (iii), (v), we get 2 (x 1 + x 2 + x 3 ) = 18 x 1 + x 2 + x 3 = 9 as x 1 + x 2 = 6 [from (i)] x 3 = 3 x 1 = 1, x 2 = 5 Material downloaded from mycbseguide.com. 17 / 27

252 Similarly, y 1 = 3, y 2 = 5, y 3 = -3 Coordinate of vertices of are A(1, 3), B(5, 5), C(3, -3) 21. Income in Rs. Number of families (f) c. f n 600 = 300 Median class = l = 1000, c.f.=250, f=190, h=1000 Median OR More than type Ogive Production yield (Kg/ha) C.F More than or equal to More than or equal to More than or equal to More than or equal to Material downloaded from mycbseguide.com. 18 / 27

253 More than or equal to More than or equal to Now, draw the Ogive by plotting the points (50, 100), (55, 98), (60, 90), (65, 78), (70, 54), (75, 16) 22. Number of kings = 4, number of queens =4, number of aces = 4 After removing all kings, queens and aces, number of cards left = = 40 (i) Number of black face cards in the remaining cards ( 2 jacks) = 2 Probability of black face card (ii) Out of 12 cards removed, 6 are of red colour. Number of red- coloured cards left 26 6 = 20 Number of ways of drawing a red card = 20 Probability of getting a red card 23. Let n = 3k ; 3k+1 or 3k +2 (i) When n = 3k n is divisible by 3 n+ 2 = 3k +2 n + 2 is not divisible by 3 and n + 4 = 3k + 4 = 3k = 3(k +1) + 1 n + 4 is not divisible by 3. Material downloaded from mycbseguide.com. 19 / 27

254 (ii) When n = 3k + 1 n is not divisible by 3 n + 2 = (3k + 1) + 2 = 3k + 3 = 3 (k + 1) n + 2 is divisible by 3 n + 4 = 3k = 3k + 5 = 3k = 3(k +1)+2 n + 4 is not divisible by 3 (iii) When n = 3k + 2, n is not divisible by 3 n + 2 = (3k +2) +2 = 3k + 4 = 3(k +1) +1 n + 2 is not divisible by 3 n + 4 = (3k +2) + 4 = 3k + 6 = 3(k+2) n + 4 is divisible by 3. Only one of the numbers n, n + 2 and n + 4 is divisible by 3. OR Let a = 3q + r a = 3q; then a 3 = 27q 3 =9m ; where m=3q 3 when a=3q+1 ; then a 3 =27q 3 +27q 2 +9q+1 =9 (3q 3 +3q 2 +q)+1 = 9m+8 (where m = 3q 3 + 3q 2 +q) when a=3q+2 ; then a 3 = (3q+2) 2 = 27q 3 +54q 2 +36q+8 = 9(3q 3 +6q 2 +4q)+8 = 9m+8 (where m = 3q 3 +6q 2 +4q) Hence, cubes of any positive integer is either of the form9m,( 9m+1 ) or( 9m+8 ). 24. Solution : are the zeroes of Material downloaded from mycbseguide.com. 20 / 27

255 is a factor of is factor of For other factor other factor of p(x) is x 2-3x + 2=0 For other zeroes, x 2-3x + 2=0 (x - 2)(x - 1) x = 2, x = 1 Other zeroes are 1 and Put and also ; put and We have Equation (i) - 2 x equation (ii) Material downloaded from mycbseguide.com. 21 / 27

256 Put x - y = 5..(iii) Solving (iii) and (iv) for x and y We have OR Let the length and breadth of a rectangle be x and y meters. According to question, Area = xy (x + 2) (y - 2) = xy - 28 or 2x - 2y = 24 or x - y = 12...(i) and (x - 1) (y + 2) = xy x - y = 33...(ii) on subtracting eq (ii) - (i), we get, x = 21 and 21 - y = 12 so y = y = 9 length =21m, breadth = 9m 26. Let BQ = units, DQ b units Material downloaded from mycbseguide.com. 22 / 27

257 and Similarly Also From (i) and (ii) (Hence Proved ) 27. Taking LHS = Material downloaded from mycbseguide.com. 23 / 27

258 = = = = = = = = = RHS 28. Since and OP, OQ are radius through contact point must be on tangents lines. Therefore, OPAQ is a square. Let OP AP = AQ = OQ Now in ABC, i.e AC 2 = 100 AC 2 = 64 Ac = 8 BR = PB = 6 - x CQ = 8 - x = CR [ tangent from an external point ] Now again BC = CR + BR 10 = 8 - x x 10 = 14-2x 2x = 4 x = 2 Material downloaded from mycbseguide.com. 24 / 27

259 Area of shaded portion = Area of triangle - Area of circle 29. Radius of the hemispherical part = 1.4 cm Length of the cylindrical part = 5 - (2 x 1.4) = 2.2cm Volume of 1 gulab jamun = volume of two hemispherical ends + Volume of cylindrical part Volume of 45 gulab jamuns Volume of syrup found in 45 gulab jamuns 30. Solution : table for less than Ogive and more than Ogive C.I. For less than Ogive For more than Ogive C.I. (less than) c.f. Point C.I. (more than) c.f. Point (30,10) (20,100) (40,18) (30,90) (50,30) (40,82) (60,54) (50,70) Material downloaded from mycbseguide.com. 25 / 27

260 (70,60) (60,46) (80,85) (70,40) (90,100) (80,15) Less than Ogive and more than Ogive. We notice both the curves intersect at (58.3, 50) Median = 58.3 OR We have, class Mid- value(x i ) (f i ) f i u i c.f Material downloaded from mycbseguide.com. 26 / 27