# Question 1 ( 1.0 marks) places of decimals? Solution: Now, on dividing by 2, we obtain =

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1 Question 1 ( 1.0 marks) The decimal expansion of the rational number places of decimals? will terminate after how many The given expression i.e., can be rewritten as Now, on dividing by 2, we obtain = Thus, the given expression will terminate after 4 decimal places. Question 2 ( 1.0 marks) For what value of k, ( 4) is a zero of the polynomial x 2 x (2k + 2)? Let p(x) = x 2 x (2k + 2). If ( 4) is a zero of p(x), then p ( 4) = 0. p( 4) = 0 Þ ( 4) 2 ( 4) 2k 2 = 0 Þ k 2 = 0 Þ 18 2k = 0 Þ 2k = 18 Þ k = 9 Thus, the required value of k is 9.

2 Question 3 ( 1.0 marks) For what value of p, are 2p 1, 7 and 3p three consecutives terms of an A.P.? Let 2p 1, 7, and 3p be the three consecutive terms of an A.P. The difference between any two consecutive terms of an A.P. is equal. =>2 nd term 1 st term = 3 rd term 2 nd term => 7 (2p 1) = 3p 7 => 7 2p + 1 = 3p 7 => 8 2p = 3p 7 => 15 = 5p Thus, if p = 3, then 2p 1, 7, and 3p are the three consecutive terms of an A.P. Question 4 ( 1.0 marks) In Fig. 1, CP and CQ are tangents to a circle with centre O. ARB is another tangent touching the circle at R. If CP = 11 cm, and BC = 7 cm, then find the length of BR. Fig. 1 It is given that CP and CQ are the tangents from the same point C. We know that the lengths of tangents drawn from an external point to a circle are equal. CP = CQ = 11 cm [CP = 11 cm]

3 =>CQ = 11 cm => CB + BQ = 11 cm => 7 cm + BQ = 11 cm [BC = 7 cm] => BQ = 11 cm 7 cm = 4 cm According to the given figure, BQ and BR are the tangents from the same point B. =>BR = BQ = 4 cm. Thus, the length of BR is 4 cm. Question 5 ( 1.0 marks) In Fig. 2, M = N = 46. Express x in terms of a, b and c where a, b and c are lengths, of LM, MN and NK respectively. Fig. 2 In KPN and KLM, we have KNP = KML = 46 [Given] NKP = MKL [Common] Thus, KPN KLM [by AA similarity criterion of triangles]

4 Question 6 ( 1.0 marks) If, then find the value of (2cot 2 θ + 2). It is given that. Question 7 ( 1.0 marks) Find the value of a so that the point (3, a) lies on the line represented by 2x 3y = 5. If the point (3, a) lies on the line represented by 2x 3y = 5, then the point (3, a) satisfies the equation 2x 3y = 5. =>2 3 3 a = 5 => 6 3a = 5 => 6 5 = 3a => 1 = 3a

5 Thus, the required value of a is. Question 8 ( 1.0 marks) A cylinder and a cone are of same base radius and of same height. Find the ratio of the volume of cylinder to that of the cone. It is given that the cylinder and the cone are of same base radius and same height. Therefore, let Radius of cylinder = Radius of cone = r Height of cylinder = Height of cone = h =>Volume of cylinder = π r 2 h Volume of cone = Thus, the ratio of the volume of the cylinder to that of the cone is 3:1. Question 9 ( 1.0 marks) Find the distance between the points and. Let the given points be A and B. We know that the distance between two points (x 1, y 1 ) and (x 2, y 2 ) is

6 Thus, the distance between the given points is 2 units. Question 10 ( 1.0 marks) Write the median class of the following distribution: Classes Frequency The given data can be written in the tabular form as Class Frequency Cumulative frequency

7 Here, Median class is that class whose cumulative frequency is greater than and nearest to. Thus, the median class of the given distribution is Section B Question Number 11 to 15 carry 2 marks each. Question 11 ( 2.0 marks) If the polynomial is divided by another polynomial, the remainder comes out to be, find a and b. The given polynomial i.e., 6x 4 + 8x 3 +17x 2 +21x +7 can be divided by 3x 2 + 4x +1 as Thus, the remainder obtained is x + 2. It is given that the remainder is of the form ax + b. On comparing both of them, we obtain a = 1 and b = 2. Thus, the respective values of a and b are 1 and 2. Question 12 ( 2.0 marks)

8 Find the value(s) of k for which the pair of linear equations kx + 3y = k 2 and 12x + ky = k has no solution. The given equations are kx + 3y = k 2 and 12x + ky = k. The above equations can be rewritten as kx + 3y (k 2) = 0, and 12x + ky k = 0 The equations a 1 x + b 1 y + c 1 = 0 and a 2 x + b 2 y + c 2 = 0 have no solution if. On comparing the given equations, we obtain a 1 = k, a 2 = 12, b 1 = 3, b 2 = k, c 1 = (k 2), c 2 = k On considering, we obtain k 2 = 36, which gives k = ± 6 Thus, for k = 6 and 6, the given set of equations will have no solutions. Question 13 ( 2.0 marks) If S n, the sum of first n terms of an A.P. is given by S n =, then find its nth term. It is given that S n = 3n 2 4n. We know that.

9 Thus, the n th term of the given A.P. is (6n 7). Question 14 ( 2.0 marks) Two tangents PA and PB are drawn to a circle with centre O from an external point P. Prove that APB = 2 OAB. Fig. 3 OR Prove that the parallelogram circumscribing a circle is a rhombus. We start by joining OB. We know that the tangent at any point of a circle is perpendicular to the radius through the point of contact. => OAP = OBP = 90º Now,

10 OAP OBP + AOB = 360º [Angle sum property of quadrilaterals] => 90º º + AOB = 360º => AOB = 360º 180º 3 = 180º 3 (1) Now, in OAB, OA is equal to OB as both are radii. => 1 = 2 [In a triangle, angles opposite to equal sides are equal] Now, on applying angle sum property of triangles in AOB, we obtain AOB = 180º => 21 + AOB = 180º => 21 + (180º 3) = 180º [Using (1)] => 21 = 3 => APB = 2OAB Thus, the given result is proved. OR Consider that parallelogram ABCD circumscribes a circle. Sides AB, BC, CD, and AD touch the circle at points P, Q, R, and S respectively. It is known that the lengths of the tangents drawn from an external point to a circle are equal. => AP = AS (i) PB = BQ (ii) DR = DS (iii)

11 CR = CQ (iv) On adding all the equations, we obtain (AP + PB) + (DR + RC) = (AS + DS) + (BQ + CQ) => AB + CD = AD + BC => 2AB = 2BC [ ABCD is a gm, AB = CD and AD = BC] => AB = BC Thus, we can say that ABCD is a rhombus since ABCD is a parallelogram and its adjacent sides are equal (AB = BC). Question 15 ( 2.0 marks) Simplify: Thus, the value of the given expression is 1. Section C Question Number 16 to 25 carry 3 marks each. Question 16 ( 3.0 marks) Prove that is an irrational number.

12 is rational. Accordingly, there exist positive co- Let us assume on the contrary that primes a and b such that 5 is a factor of. This implies that 5 is a factor of. for some positive integer c Now, This means that b 2 is a multiple of 5, or b is a multiple of 5. Therefore, we see that both and b are multiples of 5 or 5 is a factor of both and b. This contradicts that a and b are co-primes, which has happened because of our incorrect assumption. Hence, is an irrational number. Question 17 ( 3.0 marks) Solve the following pair of equations: The given pair of linear equations is

13 Let and Accordingly, equations (1) and (2) become 5u + v = 2 (3) 6u 3v = 1 (4) On multiplying equation (3) with 3 and adding it to equation (4), we obtain On substituting the value of u in equation (3), we obtain Now and => x 1 = 3 and y 2 = 3

14 => x = 4 and y = 5 Thus, the solution of the given pair of equations is x = 4 and y = 5. Question 18 ( 3.0 marks) The sum of 4th and 8th terms of an A.P. is 24 and sum of 6th and 10th terms is 44. Find A.P. Let a be the first term and d be the common difference of the A.P. It is known that the n th term of an A.P. is given by. Now, we have On subtracting equation (1) from equation (2), we obtain On substituting the value of d in equation (1), we obtain

15 The A.P. is a, a + d, a + 2d => 13, , => 13, 8, 3 Thus, the A.P. is 13, 8, 3 Question 19 ( 3.0 marks) Construct a ABC in which BC = 6.5 cm, AB = 4.5 cm and ABC = 60. Construct a triangle similar to this triangle whose sides are triangle ABC. of the corresponding sides of the A triangle whose sides are following the given steps. of the corresponding sides of ABC can be drawn by Step 1 Draw a 6.5 cm long line segment BC. Step 2 Draw an arc of any radius while taking B as the centre. Let it intersect line BC at point O. Taking O as the centre, draw another arc to cut the previous arc at point O'. Join BO', which is the ray making 60 with line BC. Step 3 Taking B as the centre, draw an arc of radius 4.5 cm intersecting the extended line segment BO' at point A. Join AC. In ABC, AB = 4.5 cm, BC = 6.5 cm, and ABC = 60. Step 4 Draw a ray BX making an acute angle with BC on the opposite side of vertex A. Step 5

16 Locate 4 points (as 4 is greater in 3 and 4), B 1, B 2, B 3, and B 4 on line segment BX. Step 6 Join B 4 C and draw a line through B 3 such that it is parallel to B 4 C and intersecting BC at C'. Step 7 Draw a line through C' parallel to AC and intersecting AB at A'. A'BC' is the required triangle. Justification The construction can be justified by proving In A BC and ABC, A C B = ACB (Corresponding angles) A BC = ABC (Common) => A BC ABC (AA similarity criterion) In BB 3 C and BB 4 C, (1)

17 B 3 BC = B 4 BC (Common) BB 3 C = BB 4 C (Alternate interior angles) => BB 3 C BB 4 C (AA similarity criterion) From equations (1) and (2), we obtain => This justifies the construction. Question 20 ( 3.0 marks) In Fig. 4, ABC is right angled at C and DE AB. Prove that ABC ADE and hence find the lengths of AE and DE. Fig. 4 In Fig, 5, DEFG is a square and BAC = 90. Show that DE 2 = BD EC. OR

18 Fig. 5 In ABC and ADE: ACB = AED (= 90 ) BAC = DAE (Common) => ABC ADE (By AA similarity criterion) It is known that corresponding sides of similar triangles are proportional. On applying Pythagoras theorem to ABC, we obtain AB 2 = AC 2 + BC 2

19 => AB 2 = = = 169 => AB = 13 cm On substituting the value of AB in equation (1), we obtain Thus, the lengths of AE and DE are approximately 1.15 cm and 2.77 cm respectively. OR DEFG is a square. => EDG = FED = 90 Now, BDG + EDG = 180 [Linear pair] => BDG + 90 = 180 => BDG = 90 Similarly, CEF = 90

20 Let DBG = α By angle sum property in BDG: DGB + DBG + BDG = 180 => DGB = 90 α By angle sum property in ABC: ECF + ABC + BAC = 180 [ECF = BCA] => ECF = 90 α By angle sum property in FEC: CEF + ECF + CFE = 180 => CFE = α Now, in BDG and FEC: BDG = FEC (= 90 ) DBG = EFC (= α) By AA similarity criterion, BDG FEC We know that corresponding sides of similar triangles are proportional. [In square DEFG, DE = DG = EF = GF] => DE 2 = BD EC Thus, the given result is proved. Question 21 ( 3.0 marks) Find the value of sin 30 geometrically. OR

21 Without using trigonometrical tables, evaluate: Consider an equilateral ABC. The measure of each interior angle of an equilateral triangle is 60. =>BAC = ACB = CBA = 60 Now, draw a perpendicular AD BC. In an equilateral triangle, the perpendicular drawn from any vertex to the opposite side bisects the angle at that vertex and also bisects the other side. => BAD = CAD = 30 => BD = DC Now, ABD is a right triangle, right-angled at D with BAD = 30º and ABD = 60º Let the length of side AB be 2a. Accordingly, Now, we have sin30º =

22 Thus, the value of sin 30º is. OR Question 22 ( 3.0 marks) Find the point on y-axis which is equidistant from the points (5, 2) and ( 3, 2) OR The line segment joining the points A (2, 1) and B (5, 8) is trisected at the points P and Q such that P is nearer to A. If P also lies on the line given by 2x y + k = 0, find the value of k. The required point is equidistant from points (5, 2) and ( 3, 2). Since the point lies on y-axis, its x-coordinate will be 0. Thus, let its coordinates be (0, y).

23 The distance between two points (x 1, y 1 ) and (x 2, y 2 ) is given by. =>Distance between (0, y) and (5, 2) = Similarly, distance between (0, y) and ( 3, 2) = According to the given condition, Thus, the required point is (0, 2). OR P (x 1, y 1 ) and Q (x 2, y 2 ) are the points of trisection of line segment AB, i.e., AP = PQ = QB Therefore, point P divides AB internally in the ratio 1:2. Thus, by section formula, It is also given that point P (3, 2) lies on line 2x y + k = 0. Thus, the value of k is 8. Question 23 ( 3.0 marks)

24 If P (x, y) is any point on the line joining the points A (a, 0) and B (0, b), then show that. Since the three points A (a, 0), P (x, y), and C (0, b) lie on the same line, the area of the triangle formed by these three points is zero. Hence, proved. Question 24 ( 3.0 marks) In Fig. 6, PQ = 24 cm, PR = 7 cm and O is the centre of the circle. Find the area of shaded region (take π = 3.14) Fig. 6 We know that the angle in a semicircle is a right angle. =>RPQ = 90

25 Thus, PQR represents a right-angled triangle where PR = 7 cm and PQ = 24 cm. On applying Pythagoras theorem to right PQR, we obtain QR 2 = PQ 2 + PR 2 = (24 cm) 2 + (7 cm) 2 = ( ) cm 2 = 625 cm 2 => QR = = 25 cm Area of semicircle =, where Area of semicircle = cm 2 Area of PQR = = 7 12 cm 2 = 84 cm 2 Thus, area of the shaded region = Area of semicircle Area of PQR = = cm 2 Question 25 ( 3.0 marks) 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 probability of getting a card of (i) heart (ii) queen (iii) clubs. After removing the king, queen, and jack of clubs, the number of remaining cards is 49, out of which 13 are hearts, 13 are diamonds, 13 are spades, and 10 are clubs. (i) Probability of drawing a heart (ii) Probability of drawing a queen

26 (iii) Probability of drawing a club card Section D Question Number 26 to 30 carry 6 marks each. Question 26 ( 6.0 marks) The sum of the squares of two consecutive odd numbers is 394. Find the numbers. OR Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds, they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars? Let the two odd numbers be (2n 1) and (2n + 1), where n is a natural number. It is given that (2n 1) 2 + (2n + 1) 2 = 394 => 4n n + 4n n = 394 => 8n = 394 => 8n 2 = 392 => n 2 = 49 => n = 7 => 2n 1 = 2 (7) 1 = 13 => 2n + 1 = 2 (7) + 1 = 15 Hence, the odd numbers are 13 and 15. OR Let the speed of car A be u and that of car B be v.

27 First condition (when both cars travel in the same direction) Let both the cars meet at point C in 5 hours. Car A travels distance AC, whereas car B travels distance BC. Distance = time speed => AC = 5 u BC = 5 v AC BC = 100 => 5u 5v = 100 => u v = 20 (1) Second condition (when both cars travel in opposite directions) Let both cars meet at point D. Car A will travel distance AD, whereas car B will travel distance BD. Distance = time speed => AD = 1 u => BD = 1 v AD + BD = 100 => u + v = 100 (2) On adding equations (1) and (2), we obtain

28 2u = 120 => u = 60 From equation (2), we obtain 60 + v = 100 => v = 40 Thus, the speed of car A is 60 km/h and that of car B is 40 km/h. Question 27 ( 6.0 marks) Prove that, if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio. Using the above result, do the following: In Fig. 7, DE BC and BD = CE. Prove that ABC is an isosceles triangle. Fig. 7 Let us consider a PQR. A line ST parallel to its base (i.e. QR) is drawn intersecting the sides PQ and PR at points S and T.

29 Now, line STdivides PQ into two parts i.e., PS and SQ. Line ST similarlydivides PR into two parts i.e., PT and TR.This information is sufficient to prove this theorem if we can prove that. To do this, let usjoin QT and RS and draw TU PS and SV PT. We know that area of triangle = => Taking PT as base and SV as height, we can write Similarly, And, Now, => (1)

30 Now, => (2) QST and RST are on the same base i.e., ST and between the same parallel lines i.e., ST and QR. =>ar ( QST) = ar ( RST) (3) From equations (1), (2), and (3), we obtain Hence, proved. Now, consider ABC. Here, a line DE parallel to its base (i.e., BC) is drawn such that it intersects sides AB and BC at points D and E respectively. By using the above result, we may find that It is given that BD = CE (4) =>AD = AE (5) On adding equations (4) and (5), we obtain BD + AD = CE + AE

31 => AB = AC Thus, ABC is an isosceles triangle. Question 28 ( 6.0 marks) A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of 30, which is approaching the foot of the tower with a uniform speed. Six seconds later the angle of depression of the car is found to be 60. Find the time taken by the car to reach the foot of the tower from this point. Let AB be a tower. Initially, the car was at point C. After 6 s, it reaches point D. Clearly, CD is the distance that the car travels in 6 s. Here, XAC is the angle of depression, when the car is at C, while XAD is the angle of depression, when the car is at D. ACB = XAC = 30 (Alternate interior angles) And, ADB = XAD = 60 (Alternate interior angles) Now, in ABD, In ABC,

32 On dividing equation (1) by equation (2), we obtain The car is approaching the foot of the tower with uniform speed (i.e., constant speed). Speed = Hence, ratio will be constant each time. Let the car take t seconds to reach the foot of the tower from this point. => t = 3 s Thus, the car will reach the foot of the tower in 3 seconds. Question 29 ( 6.0 marks) From a solid cylinder whose height is 8 cm and radius 6 cm, a conical cavity of height 8 cm and of base radius 6 cm, is hollowed out. Find the volume of the remaining solid

33 correct to two places of decimals. Also find the total surface area of the remaining solid. (take π = ) OR In Fig. 8, ABC is a right triangle right angled at A. Find the area of shaded region if AB = 6 cm, BC = 10 cm and O is the centre of the incircle of ABC. (take π = 3.14) Volume of the remaining solid = Volume of cylinder Volume of conical cavity Thus, the volume of the remaining solid up to two decimal places is cm 3. Now, the total surface area of the remaining solid is Curved surface area of cylinder + area of top face + curved surface area of cone = 2πrh + πr 2 + πrl

34 = πr (2h + r + l) The slant height (l) of the cone is given by. => Slant height of conical cavity = Therefore, total surface area of the solid = (6 cm) (2 8 cm + 6 cm + 10 cm) = (6 cm) (16 cm + 6 cm + 10 cm) = cm 32 cm = cm 2 Thus, the total surface area of the remaining solid up to two decimal places is cm 2. OR By Pythagoras theorem, in ABC, we obtain BC 2 = AC 2 + AB 2 => AC 2 = BC 2 AB 2 = (10 cm) 2 (6 cm) 2 => AC 2 = 64 cm 2 => AC = 8 cm Let the radius of the incircle be r. Let the circle touch side AB at P, side AC at Q, and side BC at R. We know that the radius from the centre of the tangent is perpendicular to the tangent through the point of contact. => OPAB, OQAC, and ORBC

35 Also, the tangents drawn from an external point to the circle are equal. => AP = AQ (Tangents from point A) BP = BR (Tangents from point B) CR = CQ (Tangents from point C) Now, in quadrilateral OPAQ, AQ = AP and AQO = APO = PAQ = 90 Therefore, OPAQ is a square. => OP = AQ = AP = OQ = r => AB = 6 cm r => BR = 6 cm r CQ = 8 cm r => CR = 8 r Now, BC = BR + CR => 10 cm = 6 cm r + 8 cm r => 10 cm = 14 cm 2r => r = 2 cm Thus, area of shaded region = Area of ABC Area of circle = AB AC πr 2 = (8 cm) (6 cm) π (2 cm) 2 = 24 cm cm 2 = 24 cm cm 2 = cm 2 Thus, the area of the shaded region is cm 2.

36 Question 30 ( 6.0 marks) The following table gives the daily income of 50 workers of a factory: Daily income (in Rs.) Number of workers Find the Mean, Mode and Median of the above data. Let the assumed mean a be 150. The table for the given data can be drawn as Class interval Number of workers (f i ) Class mark (x i ) d i = x i 150 f i d i Cumulative frequency (cf) Total Mean is given by Thus, the mean of the given data is It can be seen in the data table that the maximum frequency is 14. The class corresponding to this frequency is => Modal class =

37 Lower limit of modal class (l) = 120 Class size (h) = = 20 Frequency of modal class (f 1 ) = 14 Frequency of class preceding the modal class (f 1 ) = 12 Frequency of class succeeding the modal class (f 1 ) = 8 Mode is given by Thus, the mode of the given data is 125. Here, number of observations (n) = 50 This observation lies in class interval Therefore, the median class is Lower limit of median class (l) = 120 Cumulative frequency of class preceding the median class (cf) = 12 Frequency of median class (f) = 14

38 Median of the data is given by Thus, the median of the given data is

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