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Important Questions for ISC Mathematics 2017

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Arijit Saha
The Bhawanipur Gujarati Education Society School (BGES), Kolkata

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ISC MATHEMATICS 2017 TEST PAPER Answer Section A and Either Section B or C SECTION A [10x3+10x5 = 80 marks] Question 1( Answer Q1 and any other five) 2 3 1 2 -1 -1 and B = 1 2 , Find (AB) 1 0 1. a) If A= b) Determine the value of k so that the line y=2x+k may touch the ellipse 3x 2 5 y 2 15. [ 23 ] [0] c) Prove that 4(cot 1 3 cos ec 5) 1 d) Evaluate Lt log sin 2 x sin x using L Hospital s Rule. [1] x 0 e) Evaluate tan x sin x cos x dx [2 tan x C ] f) Given mx-y+10=0 and -2x+5y=14 are the lines of regression of x on y and y on x respectively. Find the value of m if the coefficient of correlation between x and y is 1/ 10. [4] g) In a hostel 60% of the students read Hindi newspaper, 40% read English newspaper and 20% read both the newspapers. A student is selected at random. Find the probability that she reads neither Hindi nor English newspaper. [0.2] 1 h) Evaluate 5x 3 dx [13/10] 0 i) If p and q are complex cube roots of unity, then prove that (1+p)(1+q)(1+p2)(1+q2)=1. j) Solve the differential equation: 1 y 2 1 log x dx xdy 0. [(1+logx)2=-2tan-1y+C] Question 2 a b ax by a) Prove using the properties of determinants that: b c bx cy b 2 ac ax 2 2bxy cy 2 . ax by bx cy 0 5 1 0 3 , find the inverse of A and hence solve the following system of linear equations: b) If A= 2 0 0 3 1 5x +2y = - 1, 3z-x = 4, 3y - z = 5. [-1,2,1] Question 3 a) Solve tan 1 ( x 1) tan 1 ( x 1) tan 1 8 . 31 [1/4] b) Draw the circuit: (x+y )(x +y)(x +y ). Simplify it by the laws of Boolean. Construct the simplified circuit. [x y ] Question 4 a) Find the equation of the hyperbola whose directrix is 2x+y=1, focus at (1,1) & e = 3. [7x2-2y2+12xy-2x+4y-7=0] b) Verify Lagrange s Mean Value Theorem for the function f ( x) tan 1 x in [0, 1] . 4 Question 5 a) If y log( x 1 x 2 ) prove that (1 x 2 ) y 2 xy1 0. b) An open tank with square base of side x metres and vertical height h metres is to be constructed so as to contain C cubic metres of water. Show that the expenses on lining the inside of the tank with lead would be least if h=x/2. Question 6 a) Evaluate cos x sin x 2 sin x 3 2 log (sin x 1) sin 2 x 2 sin x 3 C dx b) Find the area of the region bounded by the curve x2=4y and the straight line x = 4y 2. [9/8] Question 7 a) Calculate Spearman s rank correlation coefficient between production and sales of medicines: Production 79 96 96 69 59 79 68 62 Sales 125 137 156 125 107 136 125 107 (0.92) a) Find the most likely price in Mumbai(x) corresponding to price of Rs 70 in Kolkata(y) from the followings: x 67, y 65, x 3.5, y 2.5, xy 0.8 . [ 72.6] Question 8 a) A problem in mathematics is given to three students whose chances of solving it are 1/2, 1/3, 1/4. What is the probability that the problem is solved by exactly one? [11/24] b) Bag A contains 5 red and 3 green and bag B contains 3 red and 4 green balls. A ball is transferred from bag A to B and then a ball is drawn from bag B. What is the probability that the ball drawn is green. [35/64] Question 9 a) Use De Moivre s theorem to prove 1 3i b) Solve: x sin y dy tan ydx 0 Given x=0 when y= /2 . 1 3i 5 5 32 . [xlogsiny=siny(logsiny-1)+1] SECTION B (Any two) [2x10 =20 marks] Question 10 a) In any ABC, prove by vector method that cosB= (c2+a2-b2)/2ca b) If the vectors i 2 j k , 3 i 2 j 7 k and i 6 j 5 k are coplanar, find the value of . [5] Question 11 a) Find the shortest distance between the lines r i 2 j 4k 2i 3 j 6k and r 3i 3 j 5k 2i 3 j 8k . [Ans: 14/ 241] b) Find the vector equation of the plane passing through the point 3i j 2k and parallel to the lines r j 3k 2i 5 j k and r i 3 j k 5 j 4k . [Ans: r.(25i 8 j 10k ) 87 ] Question 12 a) The probability that, on joining a professional college, a student will successfully complete the course of studies is 3/5. Determine the probability that out of five students joining (i) none and (ii) at least two will successfully complete the course. [32/3125,2853/3125] b) A company has two plants to manufacture bicycles. The first and second plants manufacture 60% and 40% bicycles respectively. 80% and 90% of bicycles are rated as standard quality at first and second plants respectively. A bicycle of standard quality was found. Find the probability that it come from second plant. [3/7] SECTION C (Any two) [2x10 =20 marks] Question 13 a) A man invests a certain sum of money half yearly with a certain bank which pays interest at 10% p.a. compounded half yearly. If he allows his deposits to accumulate with the company, an amount of Rs 19098 accumulates to his credit after he has made his investments for the 8th time. Find the yearly deposit made by the man. [4000] b) A dealer wishes to purchases a number of fans and sewing machines. He has only Rs. 5760 to invest and has space for at the most 20 items. A fan costs him Rs. 360 and a sewing machine Rs. 240. He expects to sell a fan at a profit of Rs. 22 and a sewing machine for a profit of Rs. 18. Assuming that he can sell all the items that he buys, how should he invest his money to maximize his profit? Solve it graphically. [8,12] Question 14 a) A bill for Rs 7650 was drawn on 8 March, 2003 at 7 months. It was discounted on 18 May, 2003 and the holder of the bill received Rs 7497. What rate of interest did the banker charge? [5%] b) A bicycle manufacturer produces x units per week at a fixed cost of Rs 2500 and variable cost of Rs (x+78) per unit. He is a monopolist and the demand function for his product is x= (600-p)/8, where the price is Rs p per unit. Show that the maximum profit is obtained when 29 units are produced per week. Also find the monopoly price. [Rs 368] Question 15 a) Calculate the price index by Arithmetic Mean Method: Commodity A B C D Weight 22 48 17 13 Base price 40 15 20 30 Current price 52 24 22 16 [131.03] b) Calculate a four monthly moving averages from the following data: 1989 1989 1990 Jan 18 July 31 Jan 24 Feb 16 Aug 29 Feb 28 March 23 Sept 35 March 29 Plot the moving averages graphically. April 27 Oct 27 April 30 May 28 Nov 29 May 29 June 19 Dec 24 June 22

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