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ISC Board Specimen 2011 : Mathematics

5 pages, 39 questions, 39 questions with responses, 100 total responses,    0    0
ISC 12th
Indian School Certificate Examination (ISC), New Delhi
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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 x 1 3 (i) If is a symmetric matrix then find the value of x . 2x + 3 x + 2 (ii) If cos(2 sin 1 x ) = 1 find the values of x 9 (iii) Find the value of k if the line y = 4 x + k [3] touches the parabola y 2 = 16 x . (iv) Evaluate: Lim (1 + sin x )cot x x 0 (v) Evaluate: si n [3] xdx [3] [3] [3] 2 (vi) Evaluate: 3 + 5 cos x log[ 3 + 5 sin x ]dx [3] 0 (vii) Five books on Mathematics and 3 books on Physics are placed at random on a bookshelf. What is the probability that books on Mathematics are placed side by side on the bookshelf? [3] (viii) If the two lines of regression are 4 x + y = 225 and x + 4 y = 150 , find the value of Karl Pearson s coefficient of correlation for the data represented by x and y . [3] (ix) If ( 3 + i )6 = x + iy find the values ' x ' and y . (x) Solve log dy = 4x 2 y 2 . dx [3] [3] ------------------------------------------------------------------------------------------------------------------------------1 ISC Specimen Question Paper Question 2 (a) Prove that: bc b 2 + bc a 2 + ac ac a 2 + ab b 2 + ab (b) c 2 + bc 3 c 2 + ac =(ab+bc+ca) [5 ] ab 3 6 1 1 2 1 If A = 2 5 1 and B = 0 5 5 find AB hence or otherwise solve the 2 4 1 2 24 27 following linear equations by matrix method: 3 x + 6 y z = 7 2 x + 5 y z = 14 2 x + 4 y z = 11 [5] [5] Question 3 (a) Verify Lagrange s Mean Value Theorem for the given function: f ( x ) = x ( x + 3)( x 2) on [ 1, 4] (b) Find the foci and equation of the directrices of the hyperbola 9 x 2 16 y 2 + 18 x + 64 y = 199 . Question 4 2 + 3 cos x 1 x ] = 2 tan 1 ( tan ) . 3 + 2 cos x 2 5 (a) Prove that cos 1 [ (b) (i) Write the Boolean expression corresponding to the switching circuit given below: [5] [5] (ii) Simplify the expression and construct the switching circuit for the simplified expression. ------------------------------------------------------------------------------------------------------------------------------2 ISC Specimen Question Paper Question 5 (a) (b) d2 y dy If y = sin(2 sin x ) show that (1 x ) 2 x + 4y = 0. dx dx 1 2 [5] Find the dimensions of the rectangle of greatest area that can be inscribed in a semi-circle of radius 10 2 meters. [5] Question 6 x2 + 1 x4 + 1dx . (a) Evaluate [5] (b) Find the area of the region bounded between the curves x = y 2 and x = 3 2 y 2 [5] Question 7 (a) Ten students got the following percentage of marks in Mathematics and Physics: Marks in Mathematics 56 64 75 85 85 87 91 95 97 98 Marks in Physics 66 72 56 66 74 78 74 88 90 89 Calculate Spearman s coefficient of rank correlation and comment on r. (b) [5] 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) (b) Two numbers are selected randomly from the first fifteen natural numbers. If the sum of the numbers selected is even, find the probability that both numbers are even. [5] In bag A there are 6 white and 4 red balls, while in bag B there are 7 white and 8 red balls and in the bag C , there are 4 white and 3 red balls. One ball is taken out at random from each bag. Find the probability that all the three balls are of the same colour. [5] Question 9 (a) Solve : (1+y 2 )dx = (tan 1 y x )dy . (b) Find the locus of z if Am p[ z 1 ]= . z +1 4 [5] [5] ------------------------------------------------------------------------------------------------------------------------------3 ISC Specimen Question Paper SECTION B Question 10 (a) With usual notations using vector method prove that in a triangle ABC a b c = = sin A sin B sin C (b) [5] Find the volume of the parallelepiped whose three coterminus edges are 2i-3j+4k, i+2j-k and 3i-j+2k . [5] Question 11 (a) (b) Determine the equations of the line passing through the point (1,3,-8) and perpendicular to x 2 y 5 z 1 x + 2 y 1 z + 3 the lines and . = = = = 1 2 3 2 5 3 Find the equation of the plane passing through the point 2i 2 j + 2k and parallel to the plane r .( 2 i 4 j + 2 k ) = 5 [5] [5] Question 12 (a) (b) The overall percentage of passes in a certain examination is 75. If five candidates from a certain town appear in the examination, what is the probability that at least four pass the examination? [5] A consulting firm rents cars from three 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 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 probability that it is from agency B? [5] SECTION C Question 13 (a) (b) A furniture firm manufactures chairs and tables, each requiring the use of three 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 by selling one table the profit is Rs. 60. The total time available per week on machine A is 70 hours, on machine B is 40 hours and on machine C is 90 hours. How many chairs and tables should be made per week so as to maximize profit? Formulate the problem as L.P.P. and solve it graphically. [5] 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 rate of interest charged was 20%pa compounded quarterly, find the sum he borrowed. Find also total interest charged. [5] ------------------------------------------------------------------------------------------------------------------------------4 ISC Specimen Question Paper Question 14 (a) (b) A bill for Rs.56,100 is drawn on 22-1-90 at 11months and is discounted on 13.10.90 at the rate of 10% . Find the Bankers gain. [5] A radio manufacturer finds that he can sell x radios per week at Rs p each, x x2 where p = 2(100 ) . His cost of production of x radios per week is Rs. 120 x + . 4 2 Find the value of x when marginal cost function and marginal revenue function are equal. [5] Question 15 (a) Find the cost of living index number for the year 2005 with 2000 as base from the following data, using weighted average of price relatives: Commodity Weights Rice Wheat Pulses Milk Clothing (b) 3 3 2 1 1 Price(inRs) 2000 15 6 32 12 150 Price(inRs) 2005 20 10 40 18 200 Find four quarterly moving average: Year 1980 1981 1982 Jan-March 45 39 46 April - June 56 49 56 [5] [5] July-Sept. 39 45 49 Oct - Dec. 30 41 40 ------------------------------------------------------------------------------------------------------------------------------5 ISC Specimen Question Paper

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