ICSE QUESTION PAPER Class X MATHS (2017)

Transcription

1 ICSE QUESTION PAPER Class X MATHS (017) (Two and a half hours) Answers to this Paper must be written on the paper provided separately. You will not be allowed to write during the first 15 minutes. This time is to be spent in reading the Question Paper. The time given at the head of this Paper is the time allowed for writing the answers. Attempt all questions from Section A and any four questions from Section B. All working, including rough work, must be clearly shown and must be done on the same sheet as the rest of the answer. Omission of essential working will result in loss of marks. The intended marks for questions or parts of questions are given in brackets [ ]. Mathematical tables are provided. Question 1 SECTION A (40 Marks) Attempt all questions from this Section. (a) If b is the mean proportion between a and c, show that [] a a b b a b b c c c (b) Solve the equation 4x 5x = 0 and give your answer correct to two decimal places. [] (c) AB and CD are two parallel chords of a circle such that AB = 4 cm and CD = 10 cm. If the radius of the circle is 1 cm. find the distance between the two chords. [4]

2 Question (a) Evaluate without using trigonometric tables, [] 1 sin 8 sin 6 tan 8 cot 5 sec 0 4 (b) [] 1 1 If A = and B= and A 5B =5C. Find matrix C, where C is a by matrix. 4 (c) Jaya borrowed Rs. 50,000 for years. The rates of interest for two successive years are 1% and 15% respectively. She repays,000 at the end of the first year. Find the amount she must pay at the end of the second year to clear her debt. [4] Question (a) The catalogue price of a computer set is Rs. 4,000. The shopkeeper gives a discount of 10% on the listed price. He further gives an off-season discount of 5% on the discounted price. However, sales tax at 8% is charged on the remaining price after the two successive discounts. Find [] (i) the amount of sales tax a customer has to pay (ii) the total price to be paid by the customer for the computer set. (b) P(1, -) is a point on the line segment A(, -6) and B(x, y) such that AP : PB is equal to :. Find the coordinates of B. [] (c) The marks of 10 students of a class in an examination arranged in ascending order is as follows: [4] 1, 5, 4, x, x + 4, 55, 61, 71, 80 If the median marks is 48, find the value of x. Hence find the mode of the given data.

3 Question 4 (a) What must be subtracted from 16x 8x + 4x + 7 so that the resulting expression has x + 1 as a factor? [] (b) In the given figure ABCD is a rectangle. It consists of a circle and two semi-circles each of which are of radius 5 cm. Find the area of the shaded region. Give your answer correct to three significant figures. [] (c) Solve the following inequation and represent the solution set on a number line x 7, xl [4] Question 5 SECTION B (40 Marks) Attempt any four questions from this section (a) a 5 Hence solve for a and b given X =. b 50 Given matrix B =, find the matrix X if, X = B 4B. [] (b) How much should a man invest in Rs. 50 shares selling at Rs. 60 to obtain an income of Rs. 450, if the rate of dividend declared is 10%. Also find his yield percent, to the nearest whole number. [] (c) Sixteen cards are labeled as a, b, c,. m, n, o, p. They are put in a box and shuffled. A boy is asked to draw a card from the box. What is the probability that the card drawn is: [4] (a) a vowel (b) a consonant (c) none of the letters of the word median

4 Question 6 (a) Using a ruler and a compass construct a triangle ABC in which AB = 7 cm, CAB = 60 o and AC = 5 cm. Construct the locus of [] (i) points equidistant from AB and AC (ii) points equidistant from BA and BC Hence construct a circle touching the three sides of the triangle internally. (b) A conical tent is to accommodate 77 persons. Each person must have 16 m of air to breathe. Given the radius of the tent as 7 m, find the height of the tent and also its curved surface area. [] (c) 7m n 5 If, use properties of proportion to find 7m n (i) m : n m (ii) m n n [4] Question 7 (a) A page from a savings bank account passbook is given below: [4] Date Particulars Amount withdrawn (Rs.) Amount Deposited (Rs.) Balance (Rs.) Jan 7, 016 B/F, Jan 10, 016 By Cheque Feb 8, 016 To Self Apr 6, 016 By Cheque May 4, 016 By Cash May 7, 016 By Cheque (i) Calculate the interest for the 6 months from January to June 016, at 6% per annum. (ii) If the account is closed on 1 st July 016, find the amount received by the account holder. (b) Use a graph paper for this question (Take cms = 1 unit on both x and y axis) [6] (i) Plot the following points: A(0, 4), B(, ), C(1, 1) and D(, 0) (ii) Reflect points B, C, D on the y-axis and write down their coordinates. Name the images as B, C, D respectively. (iii) Join the points A, B, C, D, D, C, B and A in order, so as to form a closed figure. Write down the equation of the line of symmetry of the figure formed.

5 Question 8 (a) Calculate the mean of the following distribution using step deviation method. [] Marks Number of students (b) In the given figure PQ is a tangent to the circle at A, AB and AD are bisectors of CAQ and PAC. If BAQ = 0 o, prove that: [] (i) BD is a diameter of the circle (ii) ABC is an isosceles triangle (c) The printed price of an air conditioner is Rs /-. The wholesaler allows a discount of 10% to the shopkeeper. The shopkeeper sells the article to the customer at a discount of 5% of the marked price. Sales tax (under VAT) is charged at the rate of 1% at every stage. Find: [4] (i) VAT paid by the shopkeeper to the government (ii) The total amount paid by the customer inclusive of tax. Question 9 (a) In the figure given, O is the centre of the circle. DAE = 70 o, Find giving suitable reasons the measure of: [] (i) BCD (ii) BOD (iii) OBD (b) A(-1, ), B(4, ) and C(, -) are the vertices of a triangle. [] (i) Find the coordinates of the centroid G of the triangle (ii) Find the equation of the line through G and parallel to AC

6 (c) Prove that [4] sin θsin θ = tan θ cos θ cos θ Question 10 (a) The sum of the ages of Vivek and his younger brother Amit is 47 years. The product of their ages in years is 550. Find their ages. [4] (b) The daily wages of 80 workers in a project are given below. [6] Wages (in Rs.) No. of Workers Use a graph paper to draw an ogive for the above distribution. (Use a scale of cm = Rs. 50 on x-axis and cm = 10 workers on y-axis). Use your ogive to estimate: (i) the median wage of the workers (ii) the lower quartile wage of workers (iii) the numbers of workers who earn more than Rs. 65 daily Question 11 (a) The angles of depression of two ships A and B as observed from the top of a light house 60 m high are 60 o and 45 o respectively. If the two ships are on the opposite sides of the light house, find the distance between the two ships, Give your answer correct to the nearest whole number. [] (b) PQR is a triangle. S is a point on the side QR of PQR such that PSR = QPR. Given QP = 8 cm, PR = 6 cm and SR = cm [] (i) Prove PQR SPR (ii) Find the length of QR and PS area of ΔPQR (iii) area of ΔSPR (c) Mr. Richard has a recurring deposit account in a bank for years at 7.5% p. a. simple interest. If he gets Rs. 85 as interest at the time of maturity, find [4] (i) The monthly deposit (ii) The maturity value

7 Class X Mathematics Board Paper 017(Solution) SECTION A 1. (a) Cost price of an article = Rs.,450 (i) Markedprice of the article = Cost price + 16% of Cost price 16 = = = Rs.400 (ii) Price paid by the customer = Marked price + Sales Tax 10 = = =Rs (b) 1x 515x 4 7x 1, xr Take 1x 5 15x 4 15x 4 7x 1 9 x 1 i.e. 4.5 x 1 1x 15x 9 15x 7x 8 0 x 9 8x 8 9 x x 1 9 x x 1 Solution set {x : 4.5 x 1, xr} The solution on the number line is as follows:

8 . sin65 cos (c) sin8.sec6 cosec 0 cos5 sin58 sin(90 5 ) cos(90 58 ) 1 1 sin8 cos5 sin58 cos(90 8 ) sin 0 (a) cos5 sin cos5 sin58 sin sin8 5 x y Given : A, B and A B x x 9 x x 9 4x Now, A A A We have A B Two matrices are equal if each and every corresponding element is equal. 9 4x 9 16 Thus, y 4x 16 and 1 y x 4 and y 1 r (b) Population after n years Present population Present population,00, After first year, population,00, ,00,000 10,0, Population after two years,0, ,5,000 Thus, the population after two years is,5, n

9 (c) Three vertices of a parallelogram taken in order are A(, 6), B(5, 10) and C(, ) (i) We need to find the co-ordinates of D. We know that the diagonals of a parallelogram bisect each other. Let x, y be the co-ordinates of D. 6 Mid point of diagonal AC,, 4 5x 10 y And, mid point of diagonal BD, Thus, we have 5x 10 y and 4 5 x 6 and 10 y 8 x 1 and y D (1, ) (ii) Length of diagonal BD (iii) A(, 6) (x, y ) and B(5, 10) (x, y ) Slope of line AB m 1 1 y y x x 5 1 Equation of line AB is given by y y m(x x ) y 6 (x ) y 6 x 6 x y 0 x y

10 . (a) Area of one semi-circle = Area of both semi-circles = Area of one triangle = Area of both triangles = 1 Area of shaded portion cm

11 (b) Marks (x) Total No. of n = 0 Students (f) fx fx = 90 c.f fx 90 Mean n 0 Number of observations 0 (even) Median th th n n observation 1 observation th 0 0 observation 1 observation 15 observation 16 observation th th th Mode The number (marks) with highest frequency (c) In the given figure, TS SP, mtsr = mosp = 90 o In TSR, mtsr + mtrs +mrts = x = 180 x = x = 5 Now, y = x [Angle subtended at the centre is double that of the angle subtended by the arc at the same centre] y = 5 y = 50 In OSP, mosp + mspo + mpos = z + 50 = 180 z = z = 40

12 4. (a) Given, P = Rs n = years = 4 months r = 6% n(n 1) r (i) Interest P Thus, the interest earned in years is Rs (ii) Sum deposited in two years ,000 Maturity value = Total sum depositedin two years + Interest = 4, ,500 = 5,500 Thus, the maturity value is Rs. 5,500. (b) (K + )x Kx + 6 = 0.(1) Substituting x = in equation (1), we get (K + )() K() + 6 = 0 9(K + ) K + 6 = 0 9k + 18 k + 6 = 0 6k + 4 = 0 K = 4 Now, substituting K = 4 in equation (1), we get ( 4 + )x ( 4)x + 6 = 0 x + 4x + 6 = 0 x x = 0 x x + x = 0 x(x ) + 1(x ) = 0 (x + 1)(x ) = 0 So, the roots are x = 1 and x =. Thus, the other root of the equation is x = 1.

13 (c) Each int erior angle of the regular hexagon n n Steps of construction: i. Construct the regular hexagon ABCDEF with each side equal to 5 cm. ii. Draw the perpendicular bisectors of sides AB and AF and make them intersect each other at point O. iii. With O as the centre and OA as the radius draw a circle which will pass through all the vertices of the regular hexagon ABCDEF. SECTION B 5. (a) (i) (ii) (iii) (iv) (v) See the figure. Reflection of points on the y-axis will result in the change of the x-coordinate Points will be B (, 5), C ( 5, ), D ( 5, ), E (, 5). The figure BCDEE D C B is a hexagon. The lines of symmetry is x-axis or y-axis.

14 (b) Principal for the month of April = Rs. 0 Principal for the month of May = Rs Principal for the month of June = Rs Principal for the month of July = Rs Total Principal for one month = Rs. 18,50 1 Time = year, Rate = 4% 1 Interest earned = Money received on closing the account on 1 st August, 010 = Last Balance + Interest earned = Rs. ( ) = Rs (a) Given that a, b and c are in continued proportion. a b b ac b c L.H.S. (a b c)(a b c) a(a b c) b(a b c) c(a b c) a ab ac ab b bc ac bc c a ac b ac c a b b b c b ac a b c R.H.S. (b) i. The line intersects the x axis where, y = 0. Thus, the co-ordinates of A are (4, 0). ii. Length of AB= units Length of AC units iii. Let Q divides AC in the ratio m1 : m. Thus, the co-ordinates of Q are (0, y) m1x mx1 Since x m m 1 m ( ) m (4) 1 0 m1 4m m1 m m1 m m1 m 1 Required ratio is : 1.

15 iv. A(4, 0) = A(x1, y1) and B(, 4) = b(x, y) Slope of AC = = = 4 6 Equation of line AC is given by y y = m(x x ) y 0 = (x 4) y = x 8 x y (c) Consider the following distribution: Class Interval Frequency (f) Class mark (x) fx Total n = f = 100 fx = 600 fx 600 Mean 6 n (a) Radius of small sphere = r = cm Radius of big sphere = R = 4 cm 4 4 Volume of small sphere r cm Volume of big sphere R 4 cm Volume of both the spheres cm Weneed to find R. h 8 cm (Given) 1 1 Volume of the cone R 1 8 Volume of the conevolume of both the sphere 1 88 R1 8 R R1 R1 6 8 R 6 cm

16 (b) The given polynomials are ax + x 9 and x + 4x + a. Let p(x) = ax + x 9 and q(x) = x + 4x + a Given that p(x) and q(x) leave the same remainder when divided by (x + ), Thus by Remainder Theorem, we have p( ) = q( ) a( ) + ( ) 9 = ( ) + 4( ) + a 7a = a 7a + 18 = 66 + a 7a a = a = a = 8 a (c) sin cos L.H.S. 1 cot 1 tan sin cos cos sin 1 1 sin cos sin cos sin cos cos sin sin cos sin cos sin cos sin cos sin cos sin cos sin cos sin cos sin cos R.H.S.

17 8. (a) Construction: Join AD and CB. In APD and CPB A = C..(Angles in the same segment) D = B..(Angles in the same segment) APD CPB.(By AA Postulate) AP PD CP PB...(Corresponding sides of similar triangles) AP PB = CP PD (b) Total number of balls = = 0 (i) Number of green balls 9 Numberof favourable cases Number of favourable cases 9 P(Green ball) Total number of balls 0 (ii) Number of white balls 5, Number of red balls 6 Number of favourable cases Number of favourable cases 11 P(White ball or Red ball) Total number of balls 0 (iii) P(Neither green ball nor white ball) P(Red ball) Number of Red balls Total number of balls 6 0

18 (c) (i) 100 shares at Rs. 0 premium means: Nominal value of the share is Rs. 100 Market value of each share = = Rs. 10 Investment = Rs Investment 9600 Number of shares 80 Market Value of each Share 10 (ii) Sale price of each share = Rs. 160 The sale proceeds = = Rs. 1,800 (iii) New investment = Rs. 1,800 Market Value of each share = Rs. 40 Investment 1800 Number of shares 0 Market Value of each share 40 (iv) Dividend in the 1 st investment: Number of shares Rate of dividendn.v. of each share 80 8% Rs. 640 nd Dividend in the investment Number of shares Rate of dividendn.v. of each share 010% Rs Thus, change in two dividends = = Rs. 960

19 9. (a) Consider the following figure: In AEC, AE tan0 EC 1 AE AE AE 69.8 m In BEC, EB tan4 EC EB EB 5.47 m Thus, height of first tower, AB AE EB m 1 m (correct to significant figures) And, height of second tower, CD EB 5.47 m 5.4 (correct to significant figures)

20 (b) The cumulative frequency table of the given distribution table is as follows: The ogive is as follows: y Weight in Kg Number of workers Cumulative frequency (110, 47) (100, 4) C B (90, 6) (10, 50) 5 0 (80, ) A (70, 11) (60, 4) x Number of workers = th (i) Upper quartile (Q) term 7.5 term 9 4 th 50 th Lower quartile (Q 1) term 1.5 term th (ii) Through mark of 95 on the x axis, draw a vertical line which meets the graph at point C. Then through point C, draw a horizontal line which meets the y-axis at the mark of 9. Thus, number of workers weighing 95 kg and above =

21 10. (a) (i) Selling price of the manufacturer = Rs Marked price of the wholesaler Rs. 0,000 (ii) For retailer, C.P. Marked price Discount Rs % of Rs Rs Rs Rs Rs. 000 Rs. 7,000 Now, C.P.inclusive of tax Rs % of Rs (iii) For wholesaler, C.P. Rs S.P. Rs =Rs Rs =Rs Rs. 160 =Rs. 9,160 Profit S.P. C.P. Rs. ( ) Rs VAT paid by wholesaler Rs. 000 Rs (b) AB C AB 5C

22 (c) (i)consider ADE and ACB. A A Common mb me 90 Thus by Angle-Angle similarity, triangles, ACB (ii)since ADE ACB, their sides are proportional. ADE. AE AD DE...(1) AB AC BC In ABC, by Pythagoras Theorem, we have AB BC AC AB 5 1 AB 1 cm From equation (1), we have, 4 AD DE AD 1 1 AD cm 4 DE Also DE cm 1 (iii) We need to find the area of ADE and quadrilateral BCED Area of ADE = AE DE 4 cm Area of quad.bced = Area of ABC Area of ADE 1 10 BC AB cm 10 1 Thus ratio of areas of ADE to quadrilateral BCED= 80 8

23 11. (a) Let the two natural numbers be x and (8 x). Then, we have 1 1 x 8 x 15 8 x x x 8 x 15 (b) 8 x x 8 x 15 4 x 1 x 8 x x x 8 x 60 15x 8x x x 15x 8x 60 0 x x 60 0 x 0x x 60 0 x x 0 0 x 0 or x 0 0 x or x 0 Since sum of two natural numbers is 8, x cannot be equal to 0 x = and 8 x 8 5 Hence, required natural numbers are and 5. x 1x y 7y 6x 8 9y 7 x 1x 6x 8 y 7y 9y 7 (Using componendo-dividendo) x 1x 6x 8 y 7y 9y 7 x y x y x y x y x y x y x y (Using componendo-dividendo) 4 6 x y x x : y = : y

24 (c) 1. Draw a line segment AB of length 5.5 cm.. Make an angle mbax = 105 using a protractor.. Draw an arc AC with radius AC = 6 cm on AX with centre at A. 4. Join BC. Thus ABC is the required triangle. (i) Draw BR, the bisector of ABC, which is the locus of points equidistant from BA and BC. (ii) Draw MN, the perpendicular bisector of BC, which is the locus of points equidistant from B and C. (iii) The angle bisector of ABC and perpendicular bisector of BC meet at point P. Thus, P satisfies the above two loci. Length of PC = 4.8 cm