Booklet Number: 2016 TEST CODE: DST. Objective type: 30 Questions Time: 2 hours

Transcription

1 Booklet Number: 016 TEST CODE: DST Forenoon Objective type: 30 Questions Time: hours You are being given a separate Answer Sheet for answering all the questions. Write your Name, Registration Number, Test Centre, Test Code and the Number of this Question Booklet etc. in the appropriate places on the Answer Sheet. This test contains 30 questions in all. For each of the 30 questions there are four suggested answers of which only one is correct. For each question, indicate your choice of the correct answer by darkening the appropriate oval completely on the Answer Sheet. For each question: * 4 marks are allotted for each correct answer, * 0 mark for each incorrect answer, and * 1 mark for each unattempted question. ALL ROUGH WORK MUST BE DONE IN THE SPACE AVAILABLE IN THIS QUESTION BOOKLET ONLY. YOU ARE NOT ALLOWED TO USE CALCULATORS IN ANY FORM. STOP! WAIT FOR THE SIGNAL TO START

2 1. Let f(x) be a real quadratic polynomial such that (i) the sum of its zeros is, (ii) the product of its zeros is 3 and (iii) f( ) = 3. Then f(x) is equal to (A) x + x + 3. (B) x + x 3. (C) x x 3. (D) x x Let y = f(x) be a real valued differentiable function with f(0) = 1, f(1) = 3. Then there must be a real number ξ, 0 < ξ < 1 such that (A) f (ξ) = 3. (B) f (ξ) =. (C) f (ξ) = 1. (D) f (ξ) If A, B, C are the interior angles of a triangle ABC, then which of the following is true? (A) cot ( ) A+B+C = (B) cot ( A+B+C (C) cot ( A+B C (D) cot ( A+B C ) = 1 ) = tan C ) = cot C. 4. If P (x(t), y(t)) moves in the x-y plane tracing a curve C in such a way that at any point t, the slope of the tangent is x(t) y(t), then the curve C is (A)a hyperbola. (B) a parabola. (C) an ellipse. (D) a circle. 5. The maximum area of a rectangle inscribed in a circle of radius is (A). (B). (C). (D) none of these. 6. A pair of linear equations 3x + y = c, 5x + y = d, where c and d are real constants, (A) has no solution. (B) has a unique solution. (C) has an infinite number of solutions. (D) may or may not have a solution. 1

3 7. The base b of a triangle is increasing at a rate of 3 inches per hour while its height h is decreasing at a rate of 3 inches per hour. Which of the following must be true about the area A of the triangle? (A) A remains constant. (B) A is always increasing. (C) A is decreasing only when b > h. (D) A is decreasing only when b < h. 8. Let x, y, z be three real numbers such that x + y = 5 and y + z = 8. Then x + z (A) must be greater than 5 but less than 8. (B) must be 8. (C) must be 5. (D) can be any real number. 9. If f is a continuous function defined for all real x such that its maximum value is 5 and its minimum value is 7, then which of the following must be true? (A) f(x) takes only strictly positive values. (B) f( x ) attains every value between 5 and 7. (C) f(x) attains every value between 5 and 7. (D) f( x ) cannot assume the value Among all the 4-digit positive integers that can be formed with the digits, 3, 5, 7 and 9, no digit being repeated, how many are divisible by 5? (A) 4 (B) 10 (C) 15 (D) If log x + log 4 x + log 16 x = 1 4, the value of x is (A) 8. (B) 7. (C) 4. (D). 1. Let f(x) be a real-valued differentiable function. If f(x) = 0 for exactly five distinct points in (0, 1), then the number of points at which f (x) = 0 in the same interval must be (A) 5. (B) at least 5. (C) 4. (D) at least 4.

4 13. A man borrows Rs and agrees to repay the same in 1 monthly instalments, each instalment being less than the preceding one by Rs His first instalment will be equal to (A) Rs (B) Rs (C) Rs (D) Rs In the expansion of (1 + x) 11 in increasing powers of x, where x > 0, if the value of the term involving x 4 is 4 times the value of the term involving x, then the value of x is (A) 4. (B) 3. (C). (D) The number of ways in which a coin can be tossed 17 times to produce more heads than tails is (A) 9. (B) 8. (C) 16. (D) The maximum value of the function x 3 3x 9x 10 in the interval [3, 8] is (A) 44. (B) 38. (C) 5. (D) The value of 1 + 1! + 1 4! + 1 6! + 1 8! +... is (A) between 7 6 and 7 4. (B) less than 7 4. (C) greater than 7 6. (D) not a finite number. 18. The area of the region bounded by the curves y = 8x and y = x is (A) (B) (C) 1 4. (D) Suppose that A is a 3 3 real matrix such that for each 3 1 real vector x, x Ax = 0 where x stands for the transpose of x. Which of the following must be true? (A) A = A (B) A = A (C) det(a) 0 (D) trace(a) Let f(x) = a+bx+cx be a quadratic polynomial with real coefficients. If f(1) = f(3) = 5 and f() < 5, then which of the following cannot be a value for f(10)? (A) 4 (B) 6 (C) 7 (D)

5 1. A linear system of equations consisting of m equations in n unknowns is known to have a unique solution. If the coefficient matrix of this system is A, then (A) n > m. (B) rank(a) = m. (C) rank(a) = n. (D) A must be a square matrix.. If f : R R is a bounded odd function, then which of the following must be true about lim x 0 f(x)? (A) It cannot exist. (B) It must be 0. (C) It must be 0, if it exists. (D) It must exist, but may be nonzero. 3. Which of the following is a possible domain of definition of the realvalued function f(x) =? 1 x x (A) ( 1, 1) (B) < x < 0 (C) 0 < x < (D) None of the above. 4. If P (a, b), Q (c, d) are any two distinct given points in R then the set {(ta + (1 t)c, tb + (1 t)d) : t R} represents (A) the line segment joining P and Q excluding the end-points. (B) the line segment joining P and Q including the end-points. (C) the entire straight line passing through P and Q. (D) the space spanned by the vectors (a, b) and (c, d). 5. If the line y = 1 4 x + b is tangent to the curve y = 1 x (a, 1 a ), where a > 0, then the value of b is at the point (A) 1. (B) 1. (C). (D). 4

6 6. Let S be the set of all possible triples (x 1, x, x 3 ), where each x i is either 0 or 1. For each pair (i, j), i j, i, j {1,, 3}, let A ij be the subset of S consisting of all triples with x i = x j. Then A 1, A 3, A 31 are (A) disjoint but their union is not equal to S. (B) not disjoint and their union is not equal to S. (C) disjoint and their union is equal to S. (D) not disjoint but their union is equal to S. (1 + x) 1/x e 7. lim equals x 0 x (A) e. (B) e. (C) e. (D) e. 8. The function f : R R, defined as f(x) = x a, a > 0, is differentiable everywhere for (A) a (1, ). (B) a ( 1 4, 3]. (C) a ( 1, 1]. (D) a [1, ]. 9. The equation x 5 + y 16 1 λ(x 1) = 0 represents a circle if (A) λ = 1. (B) λ = (C) λ = (D) λ = A ball is thrown up into the air. Its position at a time t is given by s(t) = t 16t where t is in seconds. It will attain its greatest height at (A) 4 seconds. (B) 3 seconds. (C) seconds. (D) 1 second. 5