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PREBOARD EXAMINATION ISC 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. SECTION A Question 1. Each question carries 3 marks. [10X3 = 30] (i) If A is a square matrix of order 3, and adj. A 36, find A . 4 1 3 (ii) Simplify: cos cos x sin x 5 5 (iii) If the line y 2 x k touches the ellipse 3 x 2 5 y 2 15 , find the value of k. 1 1 (iv) Evaluate: Lim x 1 log x x 1 d 2 y b 4 x2 y 2 (v) If 2 2 1, prove that dx 2 a 2 y 3 a b (vi) Evaluate: 1 e x 1 2 dx (vii) A speaks the truth in 55 percent cases and B speaks the truth in 75 percent cases. Find the probability that they will contradict each other in stating the same fact. (viii) Given n 10, x 55, y 55, xy 350, x 2 385. Predict the value of y when x 6. (ix)If 1, , 2 are the cube roots of unity, show that 2 5 2 2 729. 6 (x) Solve the differential equation: dy sin x y sin x y . dx -2Question 2. a 2 b2 2 (a) Using properties of determinants, prove that a b2 c2 b2 c2 b2 c2 a 2 4a 2b 2 c 2 c 2 a 2 [5] 1 1 1 (b) If A 2 1 3 , find A 1 , and use it to solve the system of equations: x 2 y z 4, x y z 0, 1 1 1 x 3 y z 2. [5] Question 3. (a) Use Lagrange s Mean Value theorem to find a point on the curve y x 2 4 defined in 2, 4 where the tangent is parallel to the chord joining the end points on the curve. [5] (b) Find the equation of the hyperbola whose foci are 0, 10 and which passes through 2,3 . [5] Question 4. (a) If sin 1 x sin 1 y sin 1 z , prove that x 2 y 2 z 2 2 yz 1 x 2 . =0. [5] (b) P, Q and R represent three switches in an ON position and P ', Q ' and R ' represent the three switches in an OFF position. Construct the switching circuit representing the polynomial Q Q P R Q ' R Using the Boolean laws , simplify the expression and construct the equivalent switching circuit. Question 5. 2 (a) If y 3cos log x 4sin log x , then show that x y2 xy1 y 0. [5] [5] (b) Find the dimensions of the rectangle of greatest area that can be inscribed in a semi-circle of radius r. Question 6. 4 (a) Evaluate: log 1 tan x dx [5] (b) Find the area between the curve y x 4 x and the x axis from x 0 to x 5. [5] 0 Question 7. (a) Ten students got the following percentage of marks in mathematics and physics: Mathematics 56 64 75 85 85 87 91 95 97 98 Physics 66 72 56 66 74 78 74 88 90 89 Calculate Spearman s coefficient of rank correlation and comment on r. -3- [5] (b) For the data given below, find the regression equation of X on Y. Using the equation, calculate the value of X when Y=15. [5] X 20 25 30 35 40 45 Y 12 14 16 20 22 25 Question 8. (a) Two cards are drawn randomly from a well shuffled pack of 52 cards one after another without replacement. Find the probability that one of these is an ace and the other is a queen of the opposite colour. [5] (b) There are 2 bags. One bag contains 6 green and 3 red balls. The second bag contains 5 green and 4 red balls. One ball is transferred from the first bag to the second bag. Then one ball is drawn from the second bag. Find the probability that it is a red ball. [5] Question 9. (a) Solve the differential equation: sin x dy y cos x x sin x. dx (b) Using De Moivre s theorem, find the value of 1 i 3 1 i 3 2 [5] 4 [5] SECTION B Question 10. a b c . [5] sin A sin B sin C (b) Find the volume of the parallelepiped whose three coterminous edges are 2i 3 j 4k , i 2 j k and [5] 3i j 2k . (a) Using vector method, prove that in triangle ABC, Question 11. (a) Find the equation of the line passing through the point 1,3, 8 and perpendicular to the lines x 2 y 5 z 1 x 2 y 1 z 3 . and [5] 1 2 3 3 2 5 (b) Find the equation of the plane passing through the point 2i 2 j 2k and parallel to the plane r r . 2i 4 j 2k 5. [5] Question 12. (a) The overall percentage of passes in a certain examination is 75. If 5 candidates from a certain town appear in the examination, what is the probability that at least four pass the examination? [5] (b) A consulting firm rents cars from 3 agencies such that 20% of the cars are rented from agency A, 30% from agency B and 50% from agency C. It is known that 70% of the cars from agency A, 80% of the cars from B and 90% of the cars from C are in good condition. If a car taken on rent is in good condition, what is the probability that is from agency B? [5] -4- SECTION C Question 13. (a) A furniture firm manufactures chairs and tables, each requiring the use of 3 machines A, B and C. Production of one chair requires 2 hours on machine A, 1 hour on machine B and 1 hour on machine C. Each table requires 1 hour each on machine A and B and 3 hours on machine C. the profit obtained by selling one chair is Rs 30 while that obtained by selling one table is Rs 60. The total time available per week on machine A is 70 hours, that on machine B is 40 hours and that on machine C is 90 hours. How many chairs and tables should be made per week so as to maximize the profit? Formulate the problem as LPP and solve it graphically. [5] (b) A man borrowed some money and paid it back in three equal quarterly installments of Rs 9261 each. If the first installment is to be paid one year after the date of borrowing and the rate of interest charged was 20% per annum, compounded quarterly, find the sum he borrowed& the total interest charged. [5] Question 14. (a) A bill of Rs 56,100 is drawn on 22.01.1990 at 11 months and is discounted on 13.10.1990 at the rate of 10%. Find the Banker s gain. x (b) A radio manufacturer finds that he can sell x radios per week at Rs p each, where p 2 100 .[5] 4 x2 His cost of production of x radios per week is Rs 120 . Find the value of x when the marginal 2 cost function and the marginal revenue function are equal. [5] Question 15. (a) Find the cost of living index number for the year 2005 with 2000 as the base year from the following data: [5] Commodity Weights Price in 2000 Price in 2005 Rice 3 15 20 Wheat 3 6 10 Pulses 2 32 40 Milk 1 12 18 Clothing 1 150 200 (b) Find the four quarterly moving averages for the following data: Year 1980 1981 1982 Jan March 45 39 46 April June 56 49 56 July Sept 39 45 49 [5] Oct Dec 30 41 40
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