[Class-X] MATHEMATICS SESSION:

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1 [Class-X] MTHEMTICS SESSION: Time allowed: 3 hrs. Maximum Marks : 80 General Instructions : (i) ll questions are compulsory. (ii) This question paper consists of 30 questions divided into four sections, B, C and D (iii) Section contains 6 questions of 1 mark each. Section B contains 6 questions of 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 3 questions of 4 marks each. You have to attempt only one of the alternatives in all such questions (v) Use of calculator is not permitted. SECTION- Question numbers 1 to 6 carry 1 mark each. Q.1 If x = 3 is one root of the quadratic equation x kx 6 = 0, then find the value of k. Q. What is the HCF of smallest prime number and the smallest composite number? Q.3 Find the distance of a point P(x, y) from the origin. Q.4 In an P, if the common difference (d) = 4, and the seventh term (a7) is 4, then find the first term. Q.5 What is the value of (cos 67º sin 3º)? B 1 Q.6 Given BC ~ PQR, if = then find PQ 3 arbc arpqr SECTION-B Question numbers 7 to 1 carry mark each. Q.7 Given that is irrational, prove that (5 + 3 ) is an irrational number. Topper nswers Page - 1

2 Q.8 In Fig. 1.BCD is a rectangle. Find the values of x and y. D x + y C 14 cm x y 30 cm B Q.9 Find the sum of first 8 multiples of 3. Q.10 Find the ratio in which P(4, m) divides the line segment joining the points (,3) and B(6, 3). Hence find m. Q.11 Two different dice are tossed together. Find the probability: (i) of getting a doublet (ii) of getting a sum 10, of the numbers on the two dice. Q.1 n integer is chosen at random between 1 and 100. Find the probability that it is : (i) divisible by 8. (ii) not divisible by 8. SECTION-C Question numbers 3 to carry 3 mark each. Q.13 Find HCF and LCM of 404 and 96 and verify that HCF LCM = Product of the two given numbers. Q.14 Find all zeroes of the polynomial (x 4 9x 3 + 5x + 3x 1) if two of its zeroes are ( + 3 ) and ( 3 ). Q.15 If (,1), B(a, 0), C(4, b) and D(1, ) are the vertices of a parallelogram BCD, find the values of a and b. Hence find the lengths of its sides. If ( 5,7), B( 4, 5), C( 1, 6) and D(4,5) are the vertices of a quadrilateral, find the area of the quadrilateral BCD. Q.16 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 its speed by 100 km./h from the usual speed. Find its usual speed. Page - CBSE Question Bank

3 Q.17 Prove that the area of an equilateral triangle described on one side of the square is equal to half the area of the equilateral triangle described on one of its diagonal. If the area of two similar triangles are equal, prove that they are congruent. Q.18 Prove that the lengths of tangents drawn from an external point to a circle are equal. Q.19 If 4 tan = 3, evaluate 4sin cos1 4sin cos1 If tan = cot ( 18º), where is an acute angle, Find the value of. Q.0 Find the area of the shaded region in Fig,, where arcs drawn with centres, B, C and D intersect in pairs at mid-points P, Q, R and S of the sides B, BC, CD, and D respectively of a square BCD of side 1 cm. [Use = 3.14] P P S Q D R Q.1 wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in Fig. 3. 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 C heap of rice is in the form of a cone of base diameter 4 m and height 3.5 m. find the volume of the rice. How much canvas cloth is required to just cover the heap? Topper nswers Page - 3

4 Q. The table below shows the salaries of 80 persons : Salary (In thousand Rs.) No. of Persons Calculate the median salary of the data. SECTION-D Question numbers 3 to 30 carry 4 mark each. Q.3 motor boat whose speed is 18 km/hr in still water takes 1 hr more to go 4 km upstream than to return downstream to the same spot. Find the speed of the stream. train travels at a certain average speed for a distance of 63 km and then travels at a distance of 7 km at an average speed of 6 km/hr more than its original speed. If it takes 3 hours to complete total journey, what is the original average speed? Q.4 The sum of four consecutive numbers in an P is 3 and the ratio of the product of the first and the last term to the product of two middle terms is 7 : 15 Find the numbers. Q.5 In an equilateral BC, D is a point on side BC such that BD = 3 1 BC, Prove that 9(D) = 7(B) Prove that, in a right triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides. Q.6 Draw a triangle BC with BC = 6 cm, B = 5 cm and BC = 60º. Then construct a triangle whose sides are 4 3 of the corresponding sides of the BC. 4 Page - 4 CBSE Question Bank

5 Q.7 Prove that : sin sin 3 cos cos = tan Q.8 The diameters of the lower and upper ends of a bucket in the form of a frustum of a cone are 10 cm and 30 cm respectively. If its height is 4 cm, find : (i) The area of the metal sheet used to make the bucket. (ii) Why we should avoid the bucket made by ordinary plastic? [Use = 3.14] Q.9 s observed from the top of a 100 m high light house from the sea-level, the angles of depression of two ships are 30º and 45º. If one ship is exactly behind the other on the same side of the light house, Find the distance between the two ships. [Use 3 = 1.73] Q.30 The mean of the following distribution is 18. Find the frequency f of the class Class Frequency f 5 4 The following distribution gives the daily income of 50 workers of a factory : Dally Income (in Rs.) Number of workers Convert the distribution above to a less than type cumulative frequency distribution and draw its ogive. Topper nswers Page - 5

6 1. x = 3 is one root of the equation 9 6k 6 = 0 k = 1 SOLUTIONS. The required numbers are and 4. HCF of and 4 is. 3. OP = x y 4. a + 6( 4) = 4 a = 8 5. cos 67 = sin 3º cos 67 º sin 3 = 0 6. ar BC = ar PQR 1 1 = 3 9 B PQ 7. Let us assume is a rational number. p = where q 0 and p q are integers. q = p 5q 3q is a rational number as RHS is rational This contradicts the given fact that is irrational. Hence 5 + is an irrational number 8. B = DC and BC = D x y 30 and x y 14 Solving to get x = and y = 8 9. S = = 3( ) 8 9 = 3 = Page - 6 CBSE Question Bank

7 10. Let P : PB = K : 1 6k = 4 k 1 k = 1, ratio is 1 : Hence m = = Total number of possible outcomes = 36 (i) Doublets are (1, 1) (, ) (3, 3) (4, 4) (5, 5) (6, 6) Total number of doublets = Prob (getting a doublet) = or 36 6 (ii) Favourable outcomes are (4, 6) (5, 5) (6, 4) i.e., Prob (getting a sum 10) = or Total number of outcomes = 98 (i) Favourable outcomes are 8, 16, 4,..., 96 i.e., Prob (integer is divisible by 8) = or (ii) Prob (integer is not divisible by 8) = = = 101 = = 3 = 5 3 HCF of 404 and 96 = = 4 LCM of 404 and 96 = = 9696 HCF LCM = = lso = Hence HCF LCM = Product of 404 and p(x) = x 4 9x 3 + 5x + 3x and 3 are zeroes of p(x) p(x) = (x 3 ) (x + 3 ) g(x) = ( x 4x + 1) g(x) (x 4 9x 3 + 5x + 3x 1) (x 4x + 1) = x x 1 g(x) = x x 1 = (x + 1) (x 1) Therefore other zeroes are x = 1 and = 1 Therefore all zeroes are + 3, 3, 1 and 1 Topper nswers Page - 7

8 15. BCD is a parallelogram diagonals C and BD bisect each other therefore (1,) (4,b) D C O (o,p) B (,1) Mid point of BD is same as mid point of C a 1 4 b 1, =, a 1 b 1 = 1 and = 1 a = 1, b = 1 therefore length of sides are rea of quad BCD = r BD + r BCD C 4,9 ( 1, 6) D 10 units each. 5,7 ( 4, 5) B rea of BD = 1 ( 5) ( 5 5) + ( 4) (5 7) + (4) (7 + 5) = 53 sq units rea of BCD = 1 ( 4) ( 6 5) + ( 1) (5 + 5) + 4 ( 5 + 6) = 19 sq units Hence area of qad. BCD = = 7 sq unit 16. Let the usual speed of the plane be x km/hr = x x x + 100x = 0 x + 600x 500x = 0 (x + 600) (x 500) = 0 x 600, x = 500 Speed of plane = 500 km/hr 8 Page - 8 CBSE Question Bank

9 17. Let the side of the square be 'a' units C = a + a = a C = a units E a rea of equilateral BCF = D a a a 3 a sq.u 4 C B a a a F rea of equilateral CE = rea BCF = 1 r CE 3 ( a ) = 4 3 a sq. u. Let BC ~ PQR. ar BC B BC C = = = ar PQR PQ QR PR Given ar BC = ar PQR B BC C = 1 = = PQ QR PR B = PQ, BC = QR, C = PR Therefore BC PQR (sss congruence rule) 18. Given : Two tangents P and Q are drawn from a point to a circle with centre O. P O To prove : P = Q Construction : Join OP, OQ and O. Proof : P is a tangent at P and OP is the radius through P. OP P. Similarly, OQ Q. In the right triangle OP and OQ, we have OP = OQ [radii of the same circle] O = O [common] OP ~ OQ [by RHS congruence] Hence, P = Q Topper nswers Page - 9 Q

10 19. 4 tan = 3 tan = 4 3 sin = 5 3 and cos = sin cos 1 = 4 sin cos tan = cot ( 18º) 90º = 18º 3 = 108º 36º 0. Radius of each arc drawn = 6 cm 36 rea of one quadrant = (3.14) 4 rea of four quadrants = = cm rea of square BCD = 1 1 = 114 cm Hence rea of shaded region = = cm 1. Total surface rea of article = CS of cylinder + CS of hemispheres CS of cylinder = rh = = 0 cm surface rea of two hemispherical scoops = = 154 cm Total surface rea of article = = 374 cm Radius of conical heap = 1 m 1 Volume of rice = m 3 = 58 m 3 7 rea of canvas cloth required = r = 1 (3.5) = 1.5 m = 471.4m 10 Page - 10 CBSE Question Bank

11 . Salary (in thousand Rs.) No. of persons (f) cf N 80 = = 140 Median class is h N Median = + C f 5 = 10 + (140 49) = = 13.4 Median salary is Rs 13.4 thousand or Rs 1340 (approx) 3. Let the speed of stream be x km/hr. The speed of the boat upstream = (18 x) km/hr and Speed of the boat downstream = (18 + x) km/hr s given in the question x 4 = 1 18 x x + 48x 34 = 0 (x + 54) (x 6) = 0 x 54, x = 6 Speed of the stream 6 km/hr. Let the original average speed of train be x km/hr Therefore + = 3 x x 6 x 39x 16 = 0 (x 4) (x + 3) = 0 x 3 x = 4 Original speed of train is 4km/hr. Topper nswers Page - 11

12 4. Let the four consecutive terms of the.p. be a 3d, a d, a + d, a + 3d. By given conditions (a 3d) + (a d) + (a + d) + (a + 3d) = 3 4a = 3 a = 8 (a 3d)(a and (a d)(a 8a = 18d d = 4 d = ± 3d) d) 7 = 15 Numbers are, 6, 10, 14, or 14, 10, 6,. 5. Draw E BC EB EC (RHS congruence rule) BE = EC = 1 BC = 1 B Let B = BC = C = x Now BE = x and DE = BE BD = = 6 x x x 3 Now B = E + BE...(1) and D = E + DE...() From (1) and () B D = BE DE x x D = D = x 8 D = x 36 9D = 7B x 6 x x Given : right-angled triangle BC in which B = 90º. To Prove : (Hypotenuse) = (Base) + (Perpendicular). i.e., C = B + BC 1 Page - 1 CBSE Question Bank

13 Construction : From B draw BD C. B D C Proof : In triangle DB and BC, we have DB = BC [Each equal to 90º] and, = [Common] So, by -similarity criterion, we have DB ~ BC D B = B C [ In similar triangles corresponding sides are proportional] B = D C...(i) In triangles BDC and BC, we have CDB = BC [Each equal to 90º] and, C = C [Common] So, by -similarity criterion, we have BDC ~ BC DC BC = BC C In similar triangles corresponding sides are proportional BC = C DC dding equation (i) and (ii), we get B + BC = D C + C DC B + BC = C (D + DC) B + BC = C C...(ii) B + BC = C Hence, C = B + BC Topper nswers Page - 13

14 6. ' 5cm B 60º O B 1 6cm B C' B 3 C B 4 Steps of constructions X (1) Draw a BC with side BC = 6cm, B = 5 cm and BC = 60º () Draw a ray BX making an acute angle with BC on the opposite side of vertex (3) Mark 4 points B1, B B3. B4 (as 4 is greater in 3 and 4) on line BX such that BB1 = B1B = BB3 = B3B4 (4) Join B4C and draw a line through B3, parallel to B4C intersecting BC at C' (5) Draw a line through C' parallel to C intersecting B at '. 'BC' is the required triangle 7. LHS = = sin sin cos 3 sin (1 sin cos ( cos 3 cos ) 1) = sin [1 (1 cos cos (cos )] 1) = tan = RHS 8. Here r1 = 15 cm, r = 5 cm and h = 4 cm (i) rea of metal sheet = CS of the bucket + area of lower end = (r1 + r) + r where = 4 (15 5) = 6 cm Surface area of metal sheet = 3.14 ( ) cm = cm we should avoid use of plastic because it is non-degradable or similar value 14 Page - 14 CBSE Question Bank

15 No. of workers 9. Let B be the tower and ships are at points C and D. B tan 45º = BC B = 1 BC B = BC lso tan 30º = 1 B = 3 BC CD 1 B = 3 BC CD B + CD = 3 B CD = B ( 3 1 ) = 100 (1.73 1) = 73. m 30. Class x r fx f 0f For x f 704 0f 704 0f Mean = 18 = 40 f f = f f = 8 f fx Coumulative frequency distribution table of less than type is Daily income Cumulative frequency Less than Less than 10 1 Less than (00, 50) 50 Less than (180, 40) Less than Less than (140, 6) (160, 34) 0 10 (10, 1) Daily income Topper nswers Page - 15

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