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Class 12 ISC Prelims 2017 : Mathematics

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MATHEMATICS (Three hours) (Candidates are allowed additional 15 minutes for only reading the paper. They must NOT start writing during this time) Section A Answer Question 1 (compulsory) and five other questions. Section B and Section C Answer two questions from either Section B or Section C. All working, including rough work, should be done on the same sheet as, and adjacent to, the rest of the answer. The intended marks for questions or parts of questions are given in brackets [ ]. Mathematical tables and graph papers are provided. Slide rule may be used. ..... SECTION A Question 1 [10 3] (iv) 1 2 3 1 1 x 1 4 5 6 2 0 3 2 5 3 Find the value of x if 2 2 Find the value of k so that the line 2 x y k 0 may touch the ellipse x 2 y 2 1 1 2 tan 1 tan 1 3 7 4 Prove that: 1 lim cot x x 0 x Using L Hospital s rule, evaluate: (v) Evaluate: (i) (ii) (iii) x tan 2 x dx 2 cos x dx (vi) Evaluate: (vii) For 5 observations of pairs of 0 x, y of variables x and y the following results are 2 2 obtained: x 15 , y 25 , x 55 , y 135 , xy 83 . Find: regression coefficients bxy byx and (viii) (ix) (x) . A bag contains 5 red, 6 white and 7 black balls. Two balls are drawn at random, what is the probability that both balls are red or both are black? z 2i z 3 If z x iy and , show that 6 x 4 y 5 0 Solve the differential equation: x y 1 dy 1 dx This Paper consists of 5 printed pages and 1 blank page. Sample paper Turn over Question 2 (a) (b) Using properties of determinants, prove that: a b c c b c a b c a b a a b c [5] 2(a b)(b c)(c a) Using Martin s rule, solve the following system of equations: [5] 2 3 4 3 x y z 5 4 6 4 x y z 3 2 2 6 x y z Question 3 (a) (b) x 1 1 x 1 tan 1 tan x 2 x 2 4 Solve the equation for x : A and B represent switches in on position and A/ and B / represent in off [5] [5] position. Construct a switching circuit representing the Boolean function A / B/ A / B A B . Using laws of Boolean Algebra, simplify the function and draw the simplified circuit. Question 4 (a) (b) f ( x ) x x 3 2 Verify Rolle s Theorem for the function in the interval [0, 3]. Find the equation of the hyperbola whose foci are (8, 3), (0, 3) and eccentricity e [5] [5] 4 3. Question 5 (a) (b) d2y 2 x a sin , y a 1 cos , If find dx at Show that a cylinder of a given volume, which is open at the top, has minimum total surface area, provided its height is equal to the radius of its base. [5] [5] Question 6 x 2 x 2 dx 2x 3 (a) Evaluate: (b) Find the area bounded by the curve y 2 4 x 1 [5] and the lines x 1 and y 4 . [5] 2 Sample paper Turn over Question 7 (a) Calculate Karl Pearson s correlation coefficient between the marks in History and Political Science obtained by 10 students. Marks in History Marks in Pol. Science (b) 10 12 25 22 13 16 25 14 22 17 17 18 12 17 25 23 [5] 21 24 20 17 If the two regression lines of a bivariate distribution are 40 x 18 y 5 and 8 x 10 y 6 0 , find: (i) Mean of x and y (ii) (iii) (iv) [5] Regression coefficients Coefficient of correlation Value of x when y = 12 Question 8 (a) (b) 3 5 and of student B The probability of student A passing an examination is [5] 4 passing is 5 . Assuming the two events A passes , B passes as independent, find the probability of (i) Both students passing the examination. (ii) Only A passing the examination. (iii) Only one of them passing the examination. (iv) None of them passing the examination. A bag contains 6 red and 5 blue balls; another bag contains 5 red and 8 blue balls. [5] A ball is drawn from the first bag and without noticing colour is put in the second bag. A ball is then drawn from the second bag. Find the probability that the ball drawn is blue in colour. Question 9 (a) (b) 2 2i Using De-Moivre s theorem, find the values of x Solve the differential equation 3 4 . dy y x2 y2 dx [5] [5] 3 Sample paper Turn over SECTION B Question 10 (a) Using vector method find the area of the triangle with vertices A (1, 1, 2), B (2, 3, 5) and C (1, 5, 5). (b) If a , b , c are three vectors, show that [5] a b [5] c a 2 a b c b c Question 11 (a) (b) Find the equation of a line passing through the point (1, 2, 4) x 5 y 13 z 7 16 7 and and perpendicular to the lines 3 x 1 y 2 z 3 3 8 5 . Find the equation of the plane passing through the intersection of the planes r . i j k 6 and r . 2i 3 j 4k 5 [5] [5] and the point (1, 1, 1) . Question 12 (a) (b) A Company has two plants to manufacture motorcycles. 70% motorcycles are [5] manufactured at the first plant, while 30% are manufactured at the second plant. At the first plant, 20% motorcycles are defective, while at the second plant 10% are defective. A motorcycle, randomly picked up, is found to be standard quality. Find the probability that it has come out from the second plant. 4 If the mean and variance of a binomial distribution are 4 and 3 respectively, find the probability of getting 4 successes. [5] SECTION C Question 13 (a) A bill for 3235 was drawn on March 3,2015 at 6 months and discounted on April 3, 2015 at 8% p.a. Find the discounted value of bill. [5] 4 Sample paper Turn over (b) A dealer wishes to purchase a number of geysers and fans. He has only 57, 600 to invest and has space for at most 20 items. A geyser costs him 3600 and a fan 2400. His expectation is that he can sell a geyser at a profit of 220 and a fan at a profit of 180. Assuming that he can sell all the items he can buy, how should he invest his money to maximize the profit? Solve graphically and find the maximum profit. [5] Question 14 (a) Find the amount of an annuity of 2000 payable at the end of every month for 5 years if money is worth 6% p.a. compounded monthly? [5] (b) A company sells its product at the rate of 10 per unit. The variable costs are estimated to run 25% of the total revenue received. If the fixed costs for the product are 4,500, find: (i) The total revenue function (ii) The total cost function (iii) The profit function (iv) The break-even point (v) The numbers of units the company must sell to cover its fixed cost. [5] Question 15 (a) Calculate the index number for the year 2016 with 2015 as the base year by the simple averages of price relatives method from the following data. Commodity Weight Price (Rs. Per unit) Year 2015 Price (Rs. Per unit) Year 2016 (b) [5] A 40 32.00 B 25 80.00 C 5 1.00 D 20 10.24 E 10 4.00 40.00 120.00 1.00 15.36 3.00 Daily absence from a school during 3 weeks is recorded as follows: [5] Mon Tues. Wed. Thu. Fri. Week 1 23 23 21 33 40 Week 2 38 52 43 58 63 Week 3 52 54 61 51 51 Draw a graph illustrating these figures. Calculate 6 days moving averages and plot them on the same graph. 5 Sample paper Turn over

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