ISI PEA Past Year Papers (2006-2024) PDF

Indian Statistical Institute MSQE (PEA) Past Year Papers (2006-2024) by Econschool ([email protected]) Updated: May 20, 2024 Contents 1 ISI PEA 2006 2 2 ISI PEA 2007 6 3 ISI PEA 2008 12 4 ISI PEA 2009 18 5 ISI PEA 2010 25 6 ISI PEA 2011 31 7 ISI PEA 2012 37 8 ISI PEA 2013 43 9 ISI PEA 2014 49 10 ISI PEA 2015 56 11 ISI PEA 2016 62 12 ISI PEA 2017 69 13 ISI PEA 2018 76 14 ISI PEA 2019 84 15 ISI PEA 2020 92 16 ISI PEA 2021 100 17 ISI PEA 2022 107 18 ISI PEA 2023 114 19 ISI PEA 2024 123 Econschool econschool.in [email protected] 1 ISI PEA 2006 1+x 2x 1. If f (x) = log 1−x , 0 < x < 1, then f 1+x2 equals A. 2f (x) f (x) B. 2 C. (f (x))2 D. none of these ∂u ∂u ∂u 2. If u = φ(x − y, y − z, z − x), then ∂x + ∂y + ∂z equals A. 0 B. 1 C. u D. none of these 3. Let A and B be disjoint sets containing m and n elements, respectively, and let C = A ∪ B. The number of subsets S of C that contain k elements and that also have the property that S ∩ A contains i elements is ! m A. i ! n B. i ! ! m n C. k−i i ! ! m n D. i k−i 4. The number of disjoint intervals over which the function f (x) = |0.5x2 −| x | | is decreas- ing is A. one B. two C. three D. none of these 5. For a set of real numbers x1 , x2 ,...... , xn , the root mean square (RMS) defined as n o1/2 RMS = N1 ni=1 x2i P is a measure of central tendency. If AM denotes the arithmetic mean of the set of numbers, then which of the following statements is correct? A. RMS < AM always B. RMS > AM always C. RMS < AM when the numbers are not all equal ISI PEA 2006 Index Page 2 Econschool econschool.in [email protected] D. RMS > AM when numbers are not all equal 6. Let f (x) be a function of real variable and let ∆f be the function ∆f (x) = f (x+1)−f (x). For k > 1, put ∆ f = ∆ ∆ f. Then ∆k f (x) equals k k−1 ! Pk j k A. j=0 (−1) f (x + j) j ! Pk j+1 k B. j=0 (−1) f (x + j) j ! Pk j k C. j=0 (−1) f (x + k − j) j ! Pk j+1 k D. j=0 (−1) f (x + k − j) j R ∞ n −x 7. Let In = 0 x e dx, where n is some positive integer. Then In equals A. n! − nIn−1 B. n! + nIn−1 C. nIn−1 D. none of these. a b c 3 8. If x = 1, then ∆ = b c a equals c a b 1 b c A. (cx2 + bx + a) x c a x2 a b x b c 2 B. (cx + bx + a) 1 c a x2 a b x2 b c C. (cx2 + bx + a) x c a 1 a b 1 b c D. (cx2 + bx + a) x2 c a x a b 9. Consider any integer I = m2 + n2 , where m and n are any two odd integers. Then A. I is never divisible by 2 B. I is never divisible by 4 C. I is never divisible by 6 D. none of these. ISI PEA 2006 Index Page 3 Econschool econschool.in [email protected] 10. A box has 10 red balls and 5 black balls. A ball is selected from the box. If the ball is red, it is returned to the box. If the ball is black, it and 2 additional black balls are added to the box. The probability that a second ball selected from the box will be red is 47 A. 72 25 B. 72 55 C. 153 98 D. 153. log(1+ x p) −log(1− xq ) 11. Let f (x) = x , x 6= 0. If f is continuous at x = 0, then the value of f (0) is 1 1 A. p − q B. p + q 1 1 C. p + q D. none of these. 12. Consider four positive numbers x1 , x2 , y1 , y2 such that y1 , y2 > x1 x2 Consider the number S = (x1 y2 + x2 y1 ) − 2x1 x2. The number S is A. always a negative integer B. can be a negative fraction C. always a positive number D. none of these. 13. Given x ≥ y ≥ z, and x + y + z = 12, the maximum value of x + 3y + 5z is A. 36 B. 42 C. 38 D. 32. 14. The number of positive pairs of integral values of (x, y) that solves 2xy −4x2 +12x−5y = 11 is A. 4 B. 1 C. 2 D. none of these. 15. Consider any continuous function f : [0, 1] → [0, 1]. Which one of the following state- ments is incorrect? A. f always has at least one maximum in the interval [0,1] B. f always has at least one minimum in the interval [0,1] ISI PEA 2006 Index Page 4 Econschool econschool.in [email protected] C. ∃x ∈ [0, 1] such that f (x) = x D. the function f must always have the property that f (0) ∈ {0, 1} f (1) ∈ {0, 1} and f (0) + f (1) = I ISI PEA 2006 Index Page 5 Econschool econschool.in [email protected] 2 ISI PEA 2007 (1+x)a −1 1. Let α and β be any two positive real numbers. Then limx→0 (1+x)β −1 equals α A. β α+1 B. β+1 α−1 C. β−1 D. 1 2. Suppose the number X is odd. Then X 2 − 1 is A. odd; B. not prime; C. necessarily positive; D. none of the above. 3. The value of k for which the function f (x) = kekx is a probability density function on the interval [0, 1] is A. k = log 2; B. k = 2 log 2; C. k = 3 log 3; D. k = 3 log 4 4. p and q are positive integers such that p2 − q 2 is a prime number. Then, p − q is A. a prime number; B. an even number greater than 2 C. an odd number greater than 1 but not prime; D. none of these. 5. Any non-decreasing function defined on the interval [a, b] A. is differentiable on (a, b) B. is continuous in [a, b] but not differentiable; C. has a continuous inverse; D. none of these. x 3 4 6. The equation 1 2 1 = 0 is satisfied by 1 8 1 A. x = 1; B. x = 3 C. x = 4 ISI PEA 2007 Index Page 6 Econschool econschool.in [email protected] D. none of these. s √ r q 7. If f (x) = x+ x+ x+ x +..., then f 0 (x) is x A. 2f (x)−1 1 B. 2f (x)−1. C. √1 ; x f (x) 1 D. 2f (x)+1. 8. If P = logx (xy) and Q = logy (xy), then P + Q equals A. P Q P B. Q Q C. P PQ D. 2 R 2x3 +1 9. The solution to dx x4 +2x is x4 +2x A. 4x3 +2 + constant; 4 B. log x + log 2x+ constant; 1 C. 2 log |x4 + 2x| + constant; x4 +2x D. 4x3 +2 + constant. 10. The set of all values of x for which x2 − 3x + 2 > 0 is A. (−∞, 1) B. (2, ∞) C. (−∞, 2) ∩ (1, ∞) D. (−∞, 1) ∪ (2, ∞) 11. Consider the functions f1 (x) = x2 and f2 (x) = 4x3 + 7 defined on the real line. Then A. f1 is one-to-one and onto, but not f2 B. f2 is one-to-one and onto, but not f1 C. both f1 and f2 are one-to-one and onto; D. none of the above. a+b+2x 12. If f (x) = a+x b+x , a > 0, b > 0, then f 0 (0) equals a+b−1 b2 −a2 a A. b2 b a+b a b2 −a2 a B. 2 log b + ab b ISI PEA 2007 Index Page 7 Econschool econschool.in [email protected] a b2 −a2 C. 2 log b + ab b2 −a2 D. ba. 13. The linear programming problem maxx,y z = 0.5x + 1.5y subject to: x + y ≤ 6 3x + y ≤ 15 x + 3y ≤ 15 x, y ≥ 0 has A. no solution; B. a unique non-degenerate solution; C. a corner solution; D. infinitely many solutions. 14. Let f (x; θ) = θf (x; 1)+(1−θ)f (x; 0), where θ is a constant satisfying 0 < θ < 1 Further, both f (x; 1) and f (x; 0) are probability density functions (p · d, f.). Then A. f (x; θ) is a p.d.f. for all values of θ 1 B. f (x; θ) is a p.d.f. only for 0 < θ < 2 1 C. f (x; θ) is a p.d.f. only for 2 ≤θ 0; B. r < −0.5; C. −0.5 < r < 0; D. r = 0. 16. An n -coordinated function f is called homothetic if it can be expressed as an increasing transformation of a homogeneous function of degree one. Let f1 (x) = ni=1 xri , and P f2 (x) = ni=1 ai xi + b, where xi > 0 for all i, 0 < r < 1, ai > 0 and b are constants. Then P A. f1 is not homothetic but f2 is; B. f2 is not homothetic but f1 is; C. both f1 and f2 are homothetic; D. none of the above. ISI PEA 2007 Index Page 8 Econschool econschool.in [email protected] 1 17. If h(x) = 1−x , then h(h(h(x)) equals 1 A. 1−x B. x 1 C. x D. 1 − x |x| 3 18. The function x|x| + x is A. continuous but not differentiable at x = 0 B. differentiable at x = 0 C. not continuous at x = 0; D. continuously differentiable at x = 0. R 2dx 19. (x−2)(x−1)x equals x(x−2) A. log (x−1)2 + constant; (x−2) B. log x(x−1)2 + constant; x2 C. log (x−1)(x−2) + constant; (x−2)2 D. log x(x−1) + constant. 20. Experience shows that 20% of the people reserving tables at a certain restaurant never show up. If the restaurant has 50 tables and takes 52 reservations, then the probability that it will be able to accommodate everyone is 209 A. 1 − 552 52 4 B. 1 − 14 × 5 50 4 C. 5 50 1 D. 5 21. For any real number x, define [x] as the highest integer value not greater than x. For R 3/2 example, [0.5] = 0, = 1 and [1.5] = 1. Let I = 0 [x] + [x2 ] dx. Then I equals A. 1 √ 5−2 2 B. 2 √ C. 2 2 D. none of these. 22. Every integer of the form (n3 − n) (n2 − 4) ( for n = 3, 4,...) is A. divisible by 6 but not always divisible by 12 B. divisible by 12 but not always divisible by 24 ISI PEA 2007 Index Page 9 Econschool econschool.in [email protected] C. divisible by 24 but not always divisible by 120 D. divisible by 120 but not always divisible by 720. 23. Two varieties of mango, A and B, are available at prices Rs. p1 and Rs. p2 per kg, respectively. One buyer buys 5 kg. of A and 10 kg. of B and another buyer spends Rs 100 on A and Rs. 150 on B. If the average expenditure per mango (irrespective of variety) is the same for the two buyers, then which of the following statements is the most appropriate? A. p1 = p2 B. p2 = 34 p1 C. p1 = p2 or p2 = 34 p1 3 p2 D. 4 ≤ p1 0, b > 0 D. In the (x, y) scatter diagram, all points lie on the curve y = a + bx2 , a, b any real numbers. 25. The number of possible permutations of the integers 1 to 7 such that the numbers 1 and 2 always precede the number 3 and the numbers 6 and 7 always succeed the number 3 is A. 720 B. 168 C. 84 D. none of these. 26. Suppose the real valued continuous function f defined on the set of non-negative real numbers satisfies the condition f (x) = xf (x), then f (2) equals A. 1 B. 2 C. 3 D. f (1) 27. Suppose a discrete random variable X takes ! on the! values 0, 1,!2,... , n with frequencies n n n proportional to binomial coefficients , ,..., respectively. Then the 0 1 n mean (µ) and the variance (σ 2 ) of the distribution are ISI PEA 2007 Index Page 10 Econschool econschool.in [email protected] n n A. µ = 2 and σ 2 = 2 n n B. µ = 4 and σ 2 = 4 n n C. µ = 2 and σ 2 = 4 n n D. µ = 4 and σ 2 = 2 28. Consider a square that has sides of length 2 units. Five points are placed anywhere inside this square. Which of the following statements is incorrect? √ A. There cannot be any two points whose distance is more than 2 2. B. The square can be partitioned into four squares of side 1 unit each such that at least one unit square has two points that lies on or inside it. √ C. At least two points can be found whose distance is less than 2. D. Statements (a), (b) and (c) are all incorrect. 29. Given that f is a real-valued differentiable function such that f (x)f 0 (x) < 0 for all real x, it follows that A. f (x) is an increasing function; B. f (x) is a decreasing function; C. |f (x)| is an increasing function; D. |f (x)| is a decreasing function. (p2 +p+1)(q2 +q+1)(r2 +r+1)(s2 +s+1) 30. Let p, q, r, s be four arbitrary positive numbers. Then the value of pqrs is at least as large as A. 81 B. 91 C. 101. D. None of these. ISI PEA 2007 Index Page 11 Econschool econschool.in [email protected] 3 ISI PEA 2008 R dx 1. x+x log x equals A. log |x + x log x|+ constant B. log |1 + x log x|+ constant C. log | log x|+ constant D. log |1 + log x|+ constant. √ 2. The inverse of the function −1 + x is A. √1 , x−1 B. x2 + 1, √ C. x − 1, D. none of these. √ x+1 x+1 3. The domain of continuity of the function f (x) = x+ x−1 − x2 +1 is A. [0, 1) B. (1, ∞) C. [0, 1) ∪ (1, ∞), D. none of these 4. Consider the following linear programme: minimise x − 2y subject to x + 3y ≥ 3 3x + y ≥ 3 x+y ≤3 An optimal solution of the above programme is given by A. x = 43 , y = 3 4 B. x = 0, y = 3 C. x = −1, y = 3 D. none of the above. 5. Consider two functions f1 : {a1 , a2 , a3 } → {b1 , b2 , b3 , b4 } and f2 : {b1 , b2 , b3 , b4 } → {c1 , c2 , c3 }. The function f1 is defined by f1 (a1 ) = b1 , f1 (a2 ) = b2 , f1 (a3 ) = b3 and the function f2 is defined by f2 (b1 ) = c1 , f2 (b2 ) = c2 , f2 (b3 ) = f2 (b4 ) = c3. Then the mapping f2 ◦ f1 : {a1 , a2 , a3 } → {c1 , c2 , c3 } is A. a composite and one − to − one function but not an onto function. B. a composite and onto function but not a one − to − one function. C. a composite, one − to − one and onto function. D. not a function. ISI PEA 2008 Index Page 12 Econschool econschool.in [email protected] 1 t 6. If x = t t−1 and y = t t−1 , t > 0, t 6= 1 then the relation between x and y is 1 A. y x = x y , 1 1 B. x y = y x , C. xy = y x , 1 D. xy = y x. 7. The maximum value of T = 2xB + 3xS subject to the constraint 20xB + 15xS ≤ 900 where xB ≥ 0 and xS ≥ 0, is A. 150, B. 180, C. 200, D. none of these. 8. The value of 02 [x]n f 0 (x)dx, where [x] stands for the integral part of x, n is a positive R integer and f 0 is the derivative of the function f, is A. (n + 2n ) (f (2) − f (0)), B. (1 + 2n ) (f (2) − f (1)) C. 2n f (2) − (2n − 1) f (1) − f (0), D. none of these. 9. A surveyor found that in a society of 10,000 adult literates 21% completed college edu- cation, 42% completed university education and remaining 37% completed only school education. Of those who went to college 61% reads newspapers regularly, 35% of those who went to the university and 70% of those who completed only school education are regular readers of newspapers. Then the percentage of those who read newspapers regularly completed only school education is A. 40%, B. 52%, C. 35%, D. none of these. 10. The function f (x) = x|x|e−x defined on the real line is A. continuous but not differentiable at zero, B. differentiable only at zero, C. differentiable everywhere, D. differentiable only at finitely many points. 11. Let X be the set of positive integers denoting the number of tries it takes the Indian cricket team to win the World Cup. The team has equal odds for winning or losing any match. What is the probability that they will win in odd number of matches? ISI PEA 2008 Index Page 13 Econschool econschool.in [email protected] A. 1/4, B. 1/2, C. 2/3 D. 3/4 12. Three persons X, Y, Z were asked to find the mean of 5000 numbers, of which 500 are unities. Each one did his own simplification. X 0 s method: Divide the set of number into 5 equal parts, calculate the mean for each part and then take the mean of these. Y 0 s method: Divide the set into 2000 and 3000 numbers and follow the procedure of A. Z 0 s method: Calculate the mean of 4500 numbers (which are 6= 1 ) and then add 1. Then A. all methods are correct, B. X 0 s method is correct, but Y and Z 0 s methods are wrong, C. X 0 s and Y 0 s methods are correct but Zs0 methods is wrong, D. none is correct. 13. The number of ways in which six letters can be placed in six directed envelopes such that exactly four letters are placed in correct envelopes and exactly two letters are placed in wrong envelopes is A. 1 B. 15 C. 135 D. None of these 14. The set of all values of x for which the inequality |x − 3| + |x + 2| < 11 holds is A. (−3, 2), B. (−5, 2), C. (−5, 6), D. none of these. 15. The function f (x) = x4 − 4x3 + 16x has A. a unique maximum but no minimum, B. a unique minimum but no maximum, C. a unique maximum and a unique minimum, D. neither a maximum nor a minimum. 16. Consider the number K(n) = (n + 3) (n2 + 6n + 8) defined for integers n. Which of the following statements is correct? A. K(n) is always divisible by 4 B. K(n) is always divisible by 5 ISI PEA 2008 Index Page 14 Econschool econschool.in [email protected] C. K(n) is always divisible by 6 D. All Statements are incorrect. 17. 25 books are placed at random on a shelf. The probability that a particular pair of books shall be always together is 2 A. 25 1 B. 25 , 1 C. 300 1 D. 600 18. P (x) is a quadratic polynomial such that P (1) = −P (2). If -1 is a root of the equation, the other root is 4 A. 5 , 8 B. 5 , 6 C. 5 , 3 D. 5. 19. The correlation coefficients between two variables X and Y obtained from the two equa- tions 2x + 3y − 1 = 0 and 5x − 2y + 3 = 0 are A. equal but have opposite signs, B. − 23 and 25 , 1 C. 2 and − 35 , D. Cannot say. a 20. If a, b, c, d are positive real numbers then b + cb + c d + d a is always √ A. less than 2. √ B. less than 2 but greater than or equal to 2, C. less than 4 but greater than or equal 2 D. greater than or equal to 4. 21. The range of value of x for which the inequality log(2−x) (x − 3) ≥ −1 holds is A. 2 < x < 3, B. x > 3, C. x < 2, D. no such x exists. 22. The equation 5x3 − 5x2 + 2x − 1 has A. all roots between 1 and 2 , B. all negative roots, ISI PEA 2008 Index Page 15 Econschool econschool.in [email protected] C. a root between 0 and 1 , D. all roots greater than 2. 23. The probability density of a random variable is f (x) = ax2 e−kx (k > 0, 0 ≤ x ≤ ∞) Then, a equals k3 A. 2 , k B. 2 , k2 C. 2 , D. k 24. Let x! = r be the mode of the distribution with probability mass function p(x) = n px (1 − p)n−x. Then which of the following inequalities hold. x A. (n + 1)p − 1 < r < (n + 1)p, B. r < (n + 1)p − 1 C. r > (n + 1)p D. r < np. 25. Let y = (y1 ,... , yn ) be a set of n observations with y1 ≤ y2 ≤... ≤ yn. Let y 0 = (y1 , y2 ,... , yj + δ,... , yk − δ,... , yn ) where yk − δ > yk−1 >... > yj+1 > yj + δ δ > 0. Let σ : standard deviation of y and σ 0 : standard deviation of y 0. Then A. σ < σ 0 , B. σ 0 < σ, C. σ 0 = σ, D. nothing can be said. xf (x) 26. Let x be a r.v. with pdf f (x) and let F (x) be the distribution function. Let r(x) = 1−F (x). (log x−µ)2 µ e− 2 Then for x < e and f (x) = √ x 2π , the function r(x) is A. increasing in x, B. decreasing in x C. constant, D. none of the above. 27. A square matrix of order n is said to be a bistochastic matrix if all of its entries are non-negative and each of its rows and columns sum to 1. Let yn×1 = Pn×n xn×1 where elements of y are some rearrangements of the elements of x. Then A. P is bistochastic with diagonal elements 1 , ISI PEA 2008 Index Page 16 Econschool econschool.in [email protected] B. P cannot be bistochastic, C. P is bistochastic with elements 0 and 1 , D. P is a unit matrix. x 28. Let f1 (x) = x+1. Define fn (x) = f1 (fn−1 (x)) , where n ≥ 2. Then fn (x) is A. decreasing in n, B. increasing in n, C. initially decreasing in n and then increasing in n, D. initially increasing in n and then decreasing n. 1−x−2n 29. limn→∞ 1+x−2n ,x > 0 equals A. 1 B. -1 C. 0 D. The limit does not exist. 30. Consider the function f (x1 , x2 ) = max {6 − x1 , 7 − x2 }. The solution (x∗1 , x∗2 ) to the optimization problem minimize f (x1 , x2 ) subject to x1 + x2 = 21 is A. (x∗1 = 10.5, x∗2 = 10.5), B. (x∗1 = 11, x∗2 = 10) C. (x∗1 = 10, x∗2 = 11), D. None of these. ISI PEA 2008 Index Page 17 Econschool econschool.in [email protected] 4 ISI PEA 2009 1. An infinite geometric series has first term 1 and sum 4. It common ratio is 1 A. 2 3 B. 4 C. 1 1 D. 3 2. A continuous random variable X has a probability density function f (x) = 3x2 with 0 ≤ x ≤ 1. If P (X ≤ a) = P (x > a), then a is: A. √1 6 1 1 2 B. 3 1 C. 2 1 1 3 D. 2 √ r q 3. If f (x) = ex + ex + ex +... then f 0 (x) equals to f (x)−1 A. 2f (x)+1. f 2 (x)−f (x) B. 2f (x)−1 2f (x)+1 C. f 2 (x)+f (x) f (x) D. 2f (x)+1 √ x+5−3 4. limx→4 x−4 is A. 16 B. 0 1 C. 4 D. not well defined 5. If X = 265 and Y = 264 + 263 +... + 21 + 20 , then A. Y = X + 264. B. X = Y. C. Y = X + 1 D. Y = X − 1 R 1 ex 6. 0 ex +1 dx = A. log(1 + e) B. log2. ISI PEA 2009 Index Page 18 Econschool econschool.in [email protected] C. log 1+e 2. D. 2 log(1 + e) 7. There is a box with ten balls. Each ball has a number between 1 and 10 written on it. No two balls have the same number. Two balls are drawn (simultaneously) at random from the box. What is the probability of choosing two balls with odd numbers? 1 A. 9. 1 B. 2 2 C. 9 1 D. 3 8. A box contains 100 balls. Some of them are white and the remaining are red. Let X and Y denote the number of white and red balls respectively. The correlation between X and Y is A. 0. B. 1. C. -1. D. some real number between − 12 and 21. x 9. Let f, g and h be real valued functions defined as follows: f (x) = x(1 − x); g(x) = 2 and h(x) = min{f (x), g(x)} with 0 ≤ x ≤ 1. Then h is A. continuous and differentiable B. differentiable but not continuous C. continuous but not differentiable D. neither continuous nor differentiable 10. In how many ways can three persons, each throwing a single die once, make a score of 8? A. 5 B. 15 C. 21 D. 30 11. If f (x) is a real valued function such that 2f (x) + 3f (−x) = 55 − 7x for every x ∈ R, then f (3) equals A. 40 B. 32 ISI PEA 2009 Index Page 19 Econschool econschool.in [email protected] C. 26 D. 10 12. Two persons, A and B, make an appointment to meet at the train station between 4 P.M. and 5 P.M.. They agree that each is to wait not more than 15 minutes for the other. Assuming that each is independently equally likely to arrive at any point during the hour, find the probability that they meet. 15 A. 16 7 B. 16 5 C. 24 22 D. 175 13. If x1 , x2 , x3 are positive real numbers, then x1 x2 x3 + + x2 x3 x1 is always A. ≤ 3 1 B. ≤ 3 3 C. ≥ 3 D. 3 12 +22 +...+n2 14. limn→∞ n3 equals A. 0 1 B. 3 1 C. 6 D. 1. 15. Suppose b is an odd integer and the following two polynomial equations have a common root. x2 − 7x + 12 = 0 and x2 − 8x + b = 0 The root of x2 − 8x + b = 0 that is not a root of x2 − 7x + 12 = 0 is A. 2 B. 3 C. 4 D. 5 1 1 1 16. Suppose n ≥ 9 is an integer. Let µ = n 2 + n 3 + n 4. Then, which of the following relationships between n and µ is correct? A. n = µ B. n > µ ISI PEA 2009 Index Page 20 Econschool econschool.in [email protected] C. n < µ D. None of the above. 17. Which of the following functions f : R → R satisfies the relation f (x+y) = f (x)+f (y)? A. f (z) = z 2 B. f (z) = az for some real number a C. f (z) = log z D. f (z) = ez 18. For what value of a does the following equation have a unique solution? x a 2 2 x 0 =0 2 1 1 A. 0 B. 1 C. 2 D. 4 19. Let f (x) g(x) h(x) y= l m n a b c dy where l, m, n, a, b, c are non-zero numbers. Then dx equals A. f 0 (x) g 0 (x) h0 (x) 0 0 0 0 0 0 B. f 0 (x) g 0 (x) h0 (x) 0 0 0 a b c C. f 0 (x) g 0 (x) h0 (x) l m n a b c D. f 0 (x) g 0 (x) h0 (x) l−a m−b n−c 1 1 1 ISI PEA 2009 Index Page 21 Econschool econschool.in [email protected] 20. If f (x) = |x − 1| + |x − 2| + |x − 3|, then f (x) is differentiable at A. 0 B. 1 C. 2 D. 3 dy 2 h i 21. If (x − a)2 + (y − b)2 = c2 , then 1 + dx is independent of A. a B. b C. c D. Both b and c 22. A student is browsing in a second-hand bookshop and finds n books of interest. The shop has m copies of each of these n bools. Assuming he never wants duplicate copies of any book, and that he selects at least one book, how many ways can he make a selection? For example, if there is one book of interest with two copies, then he can make a selection in 2 ways. A. (m + 1)n − 1 B. nm C. 2nm − 1 nm! D. (m!)(nm−m)! −1 23. Determine all values of the constants A and B such that the following function is con- tinuous for all values of x.    Ax − B if x ≤ −1 f (x) =  2x2 + 3Ax + B if − 1 < x ≤ 1  4 if x > 1 A. A = B = 0 B. A = 43 , B = − 14 C. A = 41 , B = 3 4 D. A = 21 , B = 1 2 1 24. The value of limx→∞ (3x + 32x ) x is A. 0 B. 1 C. e D. 9 ISI PEA 2009 Index Page 22 Econschool econschool.in [email protected] 25. A computer while calculating correlation coefficient between two random variables X and Y from 25 pairs of observations obtained the following results: X = 125, X 2 = P P 650, Y = 100, Y 2 = 460, XY = 508. It was later discovered that at the time of P P P inputing, the pair (X = 8, Y = 12) had been wrongly input as (X = 6, Y = 14) and the pair (X = 6, Y = 8) had been wrongly input as (X = 8, Y = 6). Calculate the value of the correlation coefficient with the correct data. 4 A. 5 2 B. 3 C. 1 5 D. 6 26. The point on the curve y = x2 − 1 which is nearest to the point (2, −0.5) is A. (1, 0) B. (2, 3) C. (0, −1) D. None of the above 27. If a probability density function of a random variable X is given by f (x) = kx(2−x), 0 ≤ x ≤ 2, then mean of X is 1 A. 2 B. 1 1 C. 5 3 D. 4 28. Suppose X is the set of all integers greater than or equal to 8. Let f : X → R. and f (x + y) = f (xy) for all x, y ≥ 4. If f (8) = 9, then f (9) = A. 8 B. 9 C. 64 D. 81 29. Let f : R → R be defined by f (x) = (x − 1)(x − 2)(x − 3). Which of the following is true about f ? h 1 1 i A. It decreases on the interval 2 − 3− 2 , 2 + 3− 2 h 1 1 i B. It increases on the interval 2 − 3− 2 , 2 + 3− 2 1 i C. It decreases on the interval −∞, 2 − 3− 2 D. It decreases on the interval [2,3] ISI PEA 2009 Index Page 23 Econschool econschool.in [email protected] 30. A box with no top is to be made from a rectangular sheet of cardboard measuring 8 metres by 5 metres by cutting squares of side x metres out of each corner and folding up the sides. The largest possible volume in cubic metres of such a box is A. 15 B. 12 C. 20 D. 18 ISI PEA 2009 Index Page 24 Econschool econschool.in [email protected] 5 ISI PEA 2010 h i 1 1 1 1 1. The value of 100 1.2 + 2.3 + 3.4 +... + 99.100 A. is 99 , B. is 100 C. is 101 (100)2 D. is 99. √ √ 2. The function f (x) = x( x + x + 9) is A. continuously differentiable at x = 0, B. continuous but not differentiable at x = 0, C. differentiable but the derivative is not continuous at x = 0, D. not differentiable at x = 0. 3. Consider a GP series whose first term is 1 and the common ratio is a positive integer r(> 1). Consider an AP series whose first term is 1 and whose (r + 2)th term coincides with the third term of the GP series. Then the common difference of the AP series is A. r − 1 B. r C. r + 1 D. r + 2 4. The first three terms of the binomial expansion (1+x)n are 1, −9, 297 8 respectively. What is the value of n? A. 5 B. 8 C. 10 D. 12 5. Given logp x = α and logq x = β, the value of log pq x equals αβ A. β−α β−α B. αβ α−β C. αβ , αβ D. α−β 6. Let P = {1, 2, 3, 4, 5} and Q = {1, 2}. The total number of subsets X of P such that X ∩ Q = {2} is A. 6 ISI PEA 2010 Index Page 25 Econschool econschool.in [email protected] B. 7 C. 8 D. 9. 7. An unbiased coin is tossed until a head appears. The expected number of tosses required is A. 1 B. 2 C. 4 D. ∞. 8. Let X be a random variable with probability density function ( c x2 if x ≥ c f (x) = 0 x 0, is A. (1 + x) 1+x B. x 1−x C. x , x D. 1+x 17. Let Xi , i = 1, 2,... , n be identically distributed with variance σ 2. Let cov (Xi , Xj ) = ρ for all i 6= j. Define X̄n = n1 Xi and let an = Var X̄n Then limn→∞ an equals P A. 0 , B. ρ, C. σ 2 + ρ ISI PEA 2010 Index Page 27 Econschool econschool.in [email protected] D. σ 2 + ρ2. 18. Let X be a Normally distributed random variable with mean 0 and variance 1. Let Φ(.) be the cumulative distribution function of the variable X. Then the expectation of Φ(X) is A. − 12 , B. 0 , 1 C. 2 , D. 1.   Pr loge ( xk ) 19. Consider any finite integer r ≥ 2. Then limx→0  P∞ k=0 xk  equals k=1 k! A. 0 , B. 1 C. e, D. loge 2. 20. Consider 5 boxes, each containing 6 balls labelled 1,2,3,4,5,6. Suppose one ball is drawn from each of the boxes. Denote by bi , the label of the ball drawn from the i -th box, i = 1, 2, 3, 4, 5. Then the number of ways in which the balls can be chosen such that b1 < b2 < b3 < b4 < b5 is A. 1 B. 2 C. 5 D. 6 Pm n+r 21. The sum r=0 r equals ! n+m+1 A. n+m ! n+m B. (n + m + 1) n+1 ! n+m+1 C. n ! n+m+1 D. n+1 22. Consider the following 2 -variable linear regression where the error i ’s are independently and identically distributed with mean 0 and variance 1; yi = α + β (xi − x̄) + i , i = 1, 2,... , n Let α̂ and β̂ be ordinary least squares estimates of α and β respectively. Then the correlation coefficient between α̂ and β̂ is ISI PEA 2010 Index Page 28 Econschool econschool.in [email protected] A. 1 B. 0 C. -1 1 D. 2 23. Let f be a real valued continuous function on [0, 3]. Suppose that f (x) takes only rational values and f (1) = 1. Then f (2) equals A. 2 B. 4 C. 8 D. None of these √ x2 +x22 e−(w /(x1 +x2 )) dw with the property that 2 2 2 24. Consider the function f (x1 , x2 ) = 0 1 R f (0, 0) = 0. Then the function f (x1 , x2 ) is A. homogeneous of degree -1 B. homogeneous of degree 21 , C. homogeneous of degree 1 D. None of these. 25. If f (1) = 0, f 0 (x) > f (x) for all x > 1, then f (x) is A. positive valued for all x > 1, B. negative valued for all x > 1, C. positive valued on (1,2) but negative valued on [2, ∞) D. None of these. 26. Consider the constrained optimization problem max (ax + by) subject to (cx + dy) ≤ 100 x≥0,y≥0 (c+d) where a, b, c, d are positive real numbers such that d b > (a+b). The unique solution (x∗ , y ∗ ) to this constrained optimization problem is A. x∗ = 100 a , y∗ =0 B. x∗ = 100 c , y∗ =0 C. x∗ = 0, y ∗ = 100 b , D. x∗ = 0, y ∗ = 100 d. 27. For any real number x, let [x] be qthe largest integer not exceeding x. The domain of definition of the function f (x) = ( |[|x| − 2]| − 3)−1 is ISI PEA 2010 Index Page 29 Econschool econschool.in [email protected] A. [-6,6] B. (−∞, −6) ∪ (+6, ∞) C. (−∞, −6] ∪ [+6, ∞) D. None of these. 28. Let f : R → R and g : R → R be defined as  1   −1 if x < − 2      1 1   f (x) = − if − ≤x 0  and g(x) = 1 + x − [x], where [x] is the largest integer not exceeding x. Then f (g(x)) equals A. -1 B. − 21 , C. 0 , D. 1. function and a1 f (x) + a2 f (−x) = b1 − b2 x for all x with a1 6= a2 and 29. If f is a real valued b2 6= 0. Then f bb21 equals A. 0, 2a2 b1 B. − a21 −a22 2a2 b1 C. a21 −a22 b1 D. More information is required to find the exact value of f b2. 30. For all x, y ∈ (0, ∞), a function f : (0, ∞) → R satisfies the inequality |f (x) − f (y)| ≤ |x − y|3 Then f is A. an increasing function, B. a decreasing function, C. a constant function. D. None of these. ISI PEA 2010 Index Page 30 Econschool econschool.in [email protected] 6 ISI PEA 2011 r q r q 1. The expression 13 + 3 23/3 + 13 − 3 23/3 is A. A natural number, B. A rational number but not a natural number, C. An irrational number not exceeding 6 , D. An irrational number exceeding 6. √ (x+3) 2. The domain of definition of the function f (x) = (x2 +5x+4) is A. (−∞, ∞)\{−1, −4} B. (−0, ∞)\{−1, −4} C. (−1, ∞)\{−4} D. None of these. 3. The value of log4 2 − log8 2 + log16 2 −...... A. loge 2, B. 1 − loge 2, C. loge 2 − 1, D. None of these. 4. The function max {1, x, x2 } , where x is any real number, has A. Discontinuity at one point only, B. Discontinuity at two points only, C. Discontinuity at three points only, D. No point of discontinuity. x−y 5. If x, y, z > 0 are in HP, then y−z equals x A. y , y B. z x C. z D. None of these. x 6. The function f (x) = 1+|x| , where x is any real number is, A. Everywhere differentiable but the derivative has a point of discontinuity. B. Everywhere differentiable except at 0. C. Everywhere continuously differentiable. D. Everywhere differentiable but the derivative has 2 points of discontinuity. ISI PEA 2011 Index Page 31 Econschool econschool.in [email protected] 7. Let the function f : R++ → R++ be such that f (1) = 3 and f 0 (1) = 9, where R++ is the f (1+x) 1/x positive part of the real line. Then limx→0 f (1) equals A. 3 , B. e2 , C. 2, D. e3. 8. Let f, g : [0, ∞) → [0, ∞) be decreasing and increasing respectively. Define h(x) = f (g(x)). If h(0) = 0, then h(x) − h(1) is A. Nonpositive for x ≥ 1, positive otherwise, B. Always negative, C. Always positive, D. Positive for x ≥ 1, nonpositive otherwise. 9. A committee consisting of 3 men and 2 women is to be formed out of 6 men and 4 women. In how many ways this can be done if Mr. X and Mrs. Y are not to be included together? A. 120 B. 140 C. 90 D. 60 10. The number of continuous functions f satisfying xf (y) + yf (x) = (x + y)f (x)f (y), where x and y are any real numbers, is A. 1 , B. 2, C. 3 D. None of these. 11. If the positive numbers x1 ,... , xn are in AP, then 1 1 1 √ √ +√ √ +....... + √ √ equals x 1 + x2 x2 + x3 xn−1 + xn A. √ n √ , x1 + xn 1√ B. √ x1 + xn , 2n√ C. √ x1 + xn , D. None of these. ISI PEA 2011 Index Page 32 Econschool econschool.in [email protected] 12. If x, y, z are any real numbers, then which of the following is always true? A. max{x, y} < max{x, y, z} B. max{x, y} > max{x, y, z} x+y+|x−y| C. max{x, y} = 2 D. None of these. P4 13. If x1 , x2 , x3 , x4 > 0 and i=1 xi = 2, then P = (x1 + x2 ) (x3 + x4 ) is A. Bounded between zero and one, B. Bounded between one and two, C. Bounded between two and three, D. Bounded between three and four. 14. Everybody in a room shakes hand with everybody else. Total number of handshakes is 91. Then the number of persons in the room is A. 11 B. 12 C. 13 , D. 14 15. The number of ways in which 6 pencils can be distributed between two boys such that each boy gets at least one pencil is A. 30 B. 60 C. 62 D. 64 16. Number of continuous functions characterized by the equation xf (x) + 2f (−x) = −1, where x is any real number, is A. 1 B. 2 C. 3 D. None of these R1 17. The value of the function f (x) = x + 0 (xy 2 + x2 y) f (y)dy is px + qx2 , where A. p = 80, q = 180 B. p = 40, q = 140 C. p = 50, q = 150 D. None of these. ISI PEA 2011 Index Page 33 Econschool econschool.in [email protected] 18. If x and y are real numbers such that x2 + y 2 = 1, then the maximum value of |x| + |y| is 1 A. 2 , √ B. 2 C. √1 2 D. 2. 19. The number of onto functions from A = {p, q, r, s} to B = {p, r} is A. 16 B. 2, C. 8, D. 14 20. If the coefficients of (2r + 5) th and (r − 6) th terms in the expansion of (1 + x)39 are equal, then r C12 equals A. 45 B. 91 C. 63 D. None of these. " # C 2 21. If X = and |X 7 | = 128, then the value of C is 1 C A. ±5, B. ±1 C. ±2 D. None of these. 22. Let f (x) = Ax2 +Bx+C, where A, B, C are real numbers. If f (x) is an integer whenever x is an integer, then A. 2A and A + B are integers, but C is not an integer. B. A + B and C are integers, but 2A is not an integer. C. 2A, A + B and C are all integers. D. None of these. 23. Four persons board a lift on the ground floor of a seven-storey building. The number of ways in which they leave the lift, such that each of them gets down at different floors, is A. 360 B. 60 C. 120 ISI PEA 2011 Index Page 34 Econschool econschool.in [email protected] D. 240 24. The number of vectors (x, x1 , x2 ) , where x, x1 , x2 > 0, for which x x |log (xx1 )| + |log (xx2 )| + log + log = |log x1 + log x2 | x1 x2 holds, is A. One B. Two C. Three D. None of these. 25. In a sample of households actually invaded by small pox, 70% of the inhabitants are attacked and 85% had been vaccinated. The minimum percentage of households (among those vaccinated) that must have been attacked [Numbers expressed as nearest integer value] is A. 55 B. 65 C. 30 D. 15 26. In an analysis of bivariate data (X and Y) the following results were obtained. Variance of X (σx2 ) = 9, product of the regression coefficient of Y on X and X on Y is 0.36, and the regression coefficient from the regression of Y on X (βyx ) is 0.8. The variance of Y is A. 16 B. 4 C. 1.69 D. 3 27. For comparing the wear and tear quality of two brands of automobile tyres, two sam- ples of 50 customers using two types of tyres under similar conditions were selected. The number of kilometers x1 and x2 until the tyres became worn out, was obtained from each of them for the tyres used by them. The sample results were as follows: x̄1 = 13, 200 km, x̄2 = 13, 650 km Sx1 = 300 km, Sx2 = 400 km. What would you conclude about the two brands of tyres (at 5% level of significance) as far as the wear and tear quality is concerned? A. The two brands are alike B. The two brands are not the same, C. Nothing can be concluded, D. The given data are inadequate to perform a test. ISI PEA 2011 Index Page 35 Econschool econschool.in [email protected] 28. A continuous random variable x has the following probability density function: f (x) = α+1 α x0 x0 x for x > x0 , α > 1 The distribution function and the mean of x are given respectively by α x A. 1 − x0 , α−1 α x0 −α x B. 1 − x0 , α−1 α x0 −α x αx0 C. 1 − x0 , α−1 α x αx0 D. 1 − x0 , α−1 29. Suppose a discrete random variable X takes ! on the! values 0, 1,!2,... , n with frequencies n n n proportional to binomial coefficients , ,..., respectively. Then the 0 1 n mean (µ) and the variance (σ 2 ) of the distribution are n n A. µ = 2 and σ 2 = 2 n n B. µ = 4 and σ 2 = 4 n n C. µ = 2 and σ 2 = 4 n n D. µ = 4 and σ 2 = 2. 30. Let {Xi } be a sequence of i.i.d random variables such that Xi = 1 with probability p and Xi = 0 with probability 1 − p. Define ( 1 if ni=1 Xi = 100 P y= 0 otherwise Then E (y 2 ) is A. ∞, ! n B. p100 (1 − p)n−100 , 100 C. np, D. (np)2. ISI PEA 2011 Index Page 36 Econschool econschool.in [email protected] 7 ISI PEA 2012 1. Kupamonduk, the frog, lives in a well 14 feet deep. One fine morning she has an urge to see the world, and starts to climb out of her well. Every day she climbs up by 5 feet when there is light, but slides back by 3 feet in the dark. How many days will she take to climb out of the well? A. 3 B. 8 C. 6 D. None of the above 2. The derivative of f (x) = |x|2 at x = 0 is, A. -1 B. Non-existent C. 0 D. 1/2 3. Let N = {1, 2, 3,...} be the set of natural numbers. For each n ∈ N define An = {(n + 1)k : k ∈ N }. Then A1 ∩ A2 equals A. A3 B. A4 C. A5 D. A6. 4. Let S = {a, b, c} be a set such that a, b and c are distinct real numbers. Then min{max{a, b}, max{b, c}, m is always A. the highest number in S, B. the second highest number in S, C. the lowest number in S, D. the arithmetic mean of the three numbers in S. 5. The sequence < −4−n >, n = 1, 2, · · · , is A. Unbounded and monotone increasing, B. Unbounded and monotone decreasing, C. Bounded and convergent, D. Bounded but not convergent. R x 6. 7x2 +2 dx equals 1 A. 14 ln (7x2 + 2) + constant ISI PEA 2012 Index Page 37 Econschool econschool.in [email protected] B. 7x2 + 2 C. ln x+ constant, D. None of the above. 7. The number of real roots of the equation 2(x − 1)2 = (x − 3)2 + (x + 1)2 − 8 is A. Zero, B. One, C. Two, D. None of the above. 8. The three vectors [0,1],[1,0] and [1000,1000] are A. Dependent, B. Independent, C. Pairwise orthogonal, D. None of the above. 9. The function f (.) is increasing over [a, b]. Then [f (.)]n , where n is an odd integer greater than 1, is necessarily A. Increasing over [a, b] B. Decreasing over [a, b] C. Increasing over [a, b] if and only if f (.) is positive over [a, b] D. None of the above. 1 2 3 10. The determinant of the matrix 4 5 6 is 7 8 9 A. 21 B. -16 C. 0 D. 14 11. In what ratio should a given line be divided into two parts, so that the area of the rectangle formed by the two parts as the sides is the maximum possible? A. 1 is to 1 , B. 1 is to 4 , C. 3 is to 2 , ISI PEA 2012 Index Page 38 Econschool econschool.in [email protected] D. None of the above. 12. Suppose (x∗ , y ∗ ) solves: Minimize ax + by subject to xα + y α = M and x, y ≥ 0, where a > b > 0, M > 0 and α > 1. Then, the solution is x∗α−1 a A. α−1 = y∗ b 1 B. x∗ = 0, y ∗ = M α 1 C. y ∗ = 0, x∗ = M α D. None of the above. 13. Three boys and two girls are to be seated in a row for a photograph. It is desired that no two girls sit together. The number of ways in which they can be so arranged is A. 4P2 × 3! B. 3P2 × 2! C. 2! × 3! D. None of the above √ √ √ 14. The domain of x for which x + 3 − x + x2 − 4x is real is, A. [0,3] B. (0,3) C. {0} D. None of the above 15. P(x) is a quadratic polynomial such that P(1) = P(−1). Then A. The two roots sum to zero, B. The two roots sum to 1 , C. One root is twice the other, D. None of the above. q √ q √ 16. The expression 11 + 6 2 + 11 − 6 2 is A. Positive and an even integer, B. Positive and an odd integer, C. Positive and irrational, D. None of the above. 17. What is the maximum value of a(1 − a)b(1 − b)c(1 − c), where a, b, c vary over all positive fractional values? ISI PEA 2012 Index Page 39 Econschool econschool.in [email protected] A. 1 1 B. 8 1 C. 27 1 D. 64. 18. There are four modes of transportation in Delhi: ( A ) Auto-rickshaw, (B) Bus, (C) Car, and (D) Delhi-Metro. The probability of using transports A, B, C, D by an indi- vidual is 19 , 29 , 49 , 29 respectively. The probability that he arrives late at work if he uses transportation A, B, C, D is 75 , 47 , 67 , and 67 respectively. What is the probability that he used transport A if he reached office on time? 1 A. 9 , 1 B. 7 3 C. 7 2 D. 9 19. What is the least (strictly) positive value of the expression a3 +b3 +c3 − 3abc, where a, b, c vary over all strictly positive integers? You may use the identity a3 + b3 + c3 − 3abc = 1 2 (a + b + c) ((a − b)2 + (b − c)2 + (c − a)2 ) A. 2 B. 3 C. 4 D. 8 20. If a2 + b2 + c2 = 1, then ab + bc + ca is, A. −0.75 B. Belongs to the interval [−1, −0.5] C. Belongs to the interval [0.5, 1] D. None of the above. 21. Consider the following linear programming problem: Maximize a+b subject to a+2b ≤ 4 a + 6b ≤ 6 a − 2b ≤ 2 a, b ≥ 0 An optimal solution is: A. a = 4, b = 0 B. a = 0, b = 1 C. a = 3, b = 1/2 D. None of the above. R −1 1 22. The value of −4 x dx equals, A. ln 4 B. Undefined, ISI PEA 2012 Index Page 40 Econschool econschool.in [email protected] C. ln(−4) − ln(−1) D. None of the above. 23. Given x ≥ y ≥ z, and x + y + z = 9, the maximum value of x + 3y+ 5z is A. 27 B. 42 C. 21 D. 18 24. A car with six sparkplugs is known to have two malfunctioning ones. If two plugs are pulled out at random, what is the probability of getting at least one malfunctioning plug. A. 1/15 B. 7/15 C. 8/15 D. 9/15. 25. Suppose there is a multiple choice test which has 20 questions. Each question has two possible responses - true or false. Moreover, only one of them is correct. Suppose a student answers each of them randomly. Which one of the following statements is correct? A. The probability of getting 15 correct answers is less than the probability of getting 5 correct answers, B. The probability of getting 15 correct answers is more than the probability of getting 5 correct answers, C. The probability of getting 15 correct answers is equal to the probability of getting 5 correct answers, D. The answer depends on such things as the order of the questions. 26. From a group of 6 men and 5 women, how many different committees consisting of three men and two women can be formed when it is known that 2 of the men do not want to be on the committee together? A. 160 B. 80 C. 120 D. 200 27. Consider any two consecutive integers a and b that are both greater than 1. The sum (a2 + b2 ) is A. Always even, B. Always a prime number, ISI PEA 2012 Index Page 41 Econschool econschool.in [email protected] C. Never a prime number, D. None of the above statements is correct. 28. The number of real non-negative roots of the equation x2 − 3|x| − 10 = 0 is, A. 2 B. 1 C. 0 D. 3 29. Let < an > and < bn >, n = 1, 2, · · · , be two different sequences, where < an > is convergent and < bn > is divergent. Then the sequence < an + bn > is A. Convergent, B. Divergent, C. Undefined, D. None of the above. 30. Consider the function |x| f (x) = 1 + |x| This function is, A. Increasing in x when x ≥ 0, B. Decreasing in x, C. Increasing in x for all real x, D. None of the above. ISI PEA 2012 Index Page 42 Econschool econschool.in [email protected] 8 ISI PEA 2013 1−x 1 1. Let f (x) = 1+x ,x 6= −1. Then f f x , x 6= 0 and x 6= −1, is A. 1 B. x C. x2 1 D. x 1.2+2.3+...+n(n+1) 2. The limiting value of n3 as n → ∞ is, A. 0 B. 1 C. 1/3 D. 1/2 3. Suppose a1 , a2 ,... , an are n positive real numbers with a1 a2... an = 1. Then the mini- mum value of (1 + a1 ) (1 + a2 )... (1 + an ) is A. 2n B. 22n C. 1 D. None of the above. 4. Let the random variable X follow a Binomial distribution with parameters n and p where n(> 1) is an integer and 0 < p < 1. Suppose further that the probability of X = 0 is the same as the probability of X = 1 Then the value of p is 1 A. n , 1 B. n+1 n C. n+1 n−1 D. n+1. 5. Let X be a random variable such that E (X 2 ) = E(X) = 1. Then E (X 100 ) is A. 1 , B. 2100 , C. 0, D. None of the above. 6. If α and β are the roots of the equation x2 − ax + b = 0, then the quadratic equation whose roots are α + β + αβ and αβ − α − β is A. x2 − 2ax + a2 − b2 = 0 B. x2 − 2ax − a2 + b2 = 0 ISI PEA 2013 Index Page 43 Econschool econschool.in [email protected] C. x2 − 2bx − a2 + b2 = 0 D. x2 − 2bx + a2 − b2 = 0 7. Suppose f (x) = 2 x2 + x12 −3 x + x1 −1 where x is real and x 6= 0. Then the solutions of f (x) = 0 are such that their product is A. 1 B. 2 C. -1 D. -2 8. Toss a fair coin 43 times. What is the number of cases where number of Head is > number of Tail? A. 243 , B. 243 − 43 C. 242 D. None of the above. 9. The minimum number of real roots of f (x) = |x|3 + a|x|2 + b|x| + c where a, b and c are real, is A. 0 B. 2 C. 3 D. 6 10. Suppose f (x, y) where x and y are real, is a differentiable function satisfying the following properties: (i) f (x + k, y) = f (x, y) + ky (ii) f (x, y + k) = f (x, y) + kx; and (iii) f (x, 0) = m, where m is a constant. Then f (x, y) is given by A. m + xy B. m + x + y C. mxy D. None of the above. 11. Let I = 2343 {x − [x]}2 dx where [x] denotes the largest integer less than or equal to x. R Then the value of I is 343 A. 3 , 343 B. 2 , 341 C. 3 , D. None of the above. ISI PEA 2013 Index Page 44 Econschool econschool.in [email protected] 12. The coefficients of three consecutive terms in the expression of (1 + x)n are 165,330 and 462. Then the value of n is A. 10 B. 11 C. 12 D. 13 13. If a2 + b2 + c2 = 1, then ab + bc + ca lies in h i 1 A. 2 ,1 B. [−1, 1] h i C. − 21 , 21 h i D. − 12 , 1 14. Let the function f (x) be defined as f (x) = |x − 4| + |x − 5|. Then which of the following statements is true? A. f (x) is differentiable at all points, B. f (x) is differentiable at x = 4, but not at x = 5 C. f (x) is differentiable at x = 5 but not at x = 4, D. None of the above. R 1 R x 2 xy 15. The value of the integral 0 0x e dxdy is A. e e B. 2 , 1 C. 2 (e − 1) 1 D. 2 (e − 2) 16. Let N = {1, 2,...} be a set of natural numbers. For each x ∈ N , define An = {(n + 1)k, k ∈ N }. Then A1 ∩ A2 equals A. A2 B. A4 C. A5 D. A6 n √ o 1 17. limx→0 x 1 + x + x2 − 1 is A. 0 B. 1 1 C. 2 , D. Non-existent. ISI PEA 2013 Index Page 45 Econschool econschool.in [email protected] ! ! ! ! n n n n 18. The value of +2 +3 +... + (n + 1) equals 0 1 2 n A. 2n + n2n−1 B. 2n − n2n−1 C. 2n D. 2n+2 1 1 0 0    0 0 1 1  19. The rank of the matrix   is 1 0 1 0     0 1 0 1 A. 1 B. 2 C. 3 D. 4 20. Suppose an odd positive integer 2n + 1 is written as a sum of two integers so that their product is maximum. Then the integers are A. 2n and 1 , B. n + 2 and n − 1 C. 2n − 1 and 2 D. None of the above. 21. If |a| < 1, |b| < 1, then the series a(a + b) + a2 (a2 + b2 ) + a3 (a3 + b3 ) +...... converges to a2 b2 A. 1−a2 + 1−b2 a(a+b) B. 1−a(a+b) a2 ab C. 1−a2 + 1−ab , a2 ab D. 1−a2 − 1−ab. 22. Suppose f (x) = x3 − 6x2 + 24x. Then which of the following statements is true? A. f (x) has a maxima but no minima, B. f (x) has a minima but no maxima, C. f (x) has a maxima and a minima, D. f (x) has neither a maxima nor a minima. 23. An urn contains 5 red balls, 4 black balls and 2 white balls. A player draws 2 balls one after another with replacement. Then the probability of getting at least one red ball or at least one white ball is ISI PEA 2013 Index Page 46 Econschool econschool.in [email protected] 105 A. 121 67 B. 121 20 C. 121 D. None of the above. 1 t 24. If logt x = t−1 and logt y = t−1 , where logt x stand for logarithm of x to the base t. Then the relation between x and y is A. y x = x1/y B. x1/y = y 1/x C. xy = y x , D. xy = y 1/x f 00 (x) R2 25. Suppose f 0 (x) = 1 for all x. Also, f (0) = e2 and f (1) = e3. Then −2 f (x)dx equals A. 2e2 B. e2 − e−2 C. e4 − 1 D. None of the above. 26. The minimum value of the objective function z = 5x + 7y, where x ≥ 0 and y ≥ 0, subject to the constraints 2x + 3y ≥ 6, 3x − y ≤ 15, −x + y ≤ 4, and 2x + 5y ≤ 27 is A. 14 B. 15 C. 25 D. 28 ! 2 5 27. Suppose A is a 2 × 2 matrix given as. Then the matrix A2 − 3A − 13I, where 3 1 I is the 2 × 2 identity matrix, equals A. I ! 0 0 B. 0 0 ! 1 5 C. 3 0 D. None of the above. 28. The number of permutations of the letters a, b, c, and d such that b does not follow a, c does not follow b, and d does not follow c is A. 14 B. 13 ISI PEA 2013 Index Page 47 Econschool econschool.in [email protected] C. 12 D. 11 29. Given n observations x1 , x2 ,... , xn , which of the following statements is true? A. The mean deviation about arithmetic mean can exceed the standard deviation B. The mean deviation about arithmetic mean cannot exceed the standard devi- ation C. The root mean square deviation about a point A is least when A is the median D. The mean deviation about a point A is minimum when A is the arithmetic mean 30. Consider the following classical linear regression of y on x, yi = βxi + ui , i = 1, 2,... ,... , n where E (ui ) = 0, V (ui ) = σ 2 for all i, and u0i s are homoscedastic and non-autocorrelated. Now, let ûi be the ordinary least square estimate of ui. Then which of the following statements is true? Pn A. i=1 ûi = 0 Pn Pn B. i=1 ûi = 0, and i=1 xi ûi = 0 Pn Pn C. i=1 ûi = 0, and i=1 xi ûi 6= 0 Pn D. i=1 xi ûi = 0 ISI PEA 2013 Index Page 48 Econschool econschool.in [email protected] 9 ISI PEA 2014 1−x 1 1. Let f (x) = 1+x ,x 6= −1. Then f f x , x 6= 0 and x 6= −1, is A. 1 B. x C. x2 1 D. x 2. What is the value of the following definite integral? Z π 2 2 ex cos(x)dx 0 π A. e 2 x B. e 2 − 1 C. 0 π D. e 2 + 1 3. Let f : R → R be a function defined as follows: f (x) = |x − 1| + (x − 1) Which of the following is not true for f ? A. f (x) = f (x0 ) for all x, x0 < 1 B. f (x) = 2f (1) for all x > 1 C. f is not differentiable at 1 D. The derivative of f at x = 2 is 2 4. Population of a city is 40% male and 60% female. Suppose also that 50% of males and 30% of females in the city smoke. The probability that a smoker in the city is male is closest to A. 0.5 B. 0.46 C. 0.53 D. 0.7 5. A blue and a red die are thrown simultaneously. We define three events as follows: - Event E : the sum of the numbers on the two dice is 7 - Event F : the number on the blue die equals 4 - Event G : the number on the red die equals 3 Which of the following statements is true? A. E and F are disjoint events. B. E and F are independent events. ISI PEA 2014 Index Page 49 Econschool econschool.in [email protected] C. E and F are not independent events. D. Probability of E is more than the probability of F. 6. Let p > 2 be a prime number. Consider the following set containing 2 × 2 matrices of integers: ( " # ) 0 a Tp = A = : a, b ∈ {0, 1,... , p − 1} b 0 A matrix A ∈ Tp is p -special if determinant of A is not divisible by p. How many matrices in Tp are p -special? A. (p − 1)2 B. 2p − 1 C. p2 D. p2 − p + 1 7. A ”good” word is any seven letter word consisting of letters from {A, B, C} (some letters may be absent and some letter can be present more than once), with the restriction that A cannot be followed by B, B cannot be followed by C, and C cannot be followed by A. How many good words are there? A. 192 B. 128 C. 96 D. 64 8. Let n be a positive integer and 0 < a < b < ∞. The total number of real roots of the equation (x − a)2n+1 + (x − b)2n+1 = 0 is A. 1 B. 3 C. 2n − 1 D. 2n + 1 9. Consider the optimization problem below: maxx,y x + y subject to 2x + y ≤ 14 −x + 2y ≤ 8 2x − y ≤ 10 x, y ≥ 0 The value of the objective function at optimal solution of this optimization problem: A. does not exist B. is 8 ISI PEA 2014 Index Page 50 Econschool econschool.in [email protected] C. is 10 D. is unbounded 10. A random variable X is distributed in [0, 1]. Mr. Fox believes that X follows a distribu- tion with cumulative density function (cdf) F : [0,1]→ [0, 1] and Mr. Goat believes that X follows a distribution with cdf G : [0, 1] → [0, 1]. Assume F and G are differentiable, F 6= G and F (x) ≤ G(x) for all x ∈ [0, 1]. Let EF [X] and EG [X] be the expected values of X for Mr. Fox and Mr. Goat respectively. Which of the following is true? A. EF [X] ≤ EG [X] B. EF [X] ≥ EG [X] C. EF [X] = EG [X] D. None of the above. 11. Let f : [0, 2] → [0, 1] be a function defined as follows: ( x if x ≤ α f (x) = 1 2 if x ∈ (α, 2] where α ∈ (0, 2). Suppose X is a random variable distributed in [0, 2] with probability density function f. What is the probability that the realized value of X is greater than 1? A. 1 B. 0 1 C. 2 3 D. 4 12. The value of the expression 100 Z 1 X xk dx k=1 0 k is 100 A. 101 1 B. 99 C. 1 99 D. 100 13. Consider the following system of inequalities. x1 − x 2 ≤3 x2 − x 3 ≤ −2 x3 − x 4 ≤ 10 x4 − x 2 ≤α x4 − x 3 ≤ −4 where α is a real number. A value of α for which this system has a solution is ISI PEA 2014 Index Page 51 Econschool econschool.in [email protected] A. -16 B. -12 C. -10 D. None of the above 14. A fair coin is tossed infinite number of times. The probability that a head turns up for the first time after even number of tosses is 1 A. 3 1 B. 2 2 C. 3 3 D. 4 15. An entrance examination has 10 ”true-false” questions. A student answers all the ques- tions randomly and his probability of choosing the correct answer is 0.5. Each correct answer fetches a score of 1 to the student, while each incorrect answer fetches a score of zero. What is the probability that the student gets the mean score? 1 A. 4 63 B. 256 1 C. 2 1 D. 8 16. For any positive integer k, let Sk denote the sum of the infinite geometric progression whose first term is (k−1) and common ratio is k1. The value of the expression ∞ P k! k=1 Sk is A. e. B. 1 + e C. 2 + e D. e2. 17. Let G(x) = 0x tet dt for all non-negative real number x. What is the value of limx→0 x1 G0 (x), R where G0 (x) is the derivative of G at x A. 0 B. 1 C. e D. None of the above 18. Let α ∈ (0, 1) and f (x) = xα + (1 − x)α for all x ∈ [0, 1]. Then the maximum value of f is A. 1 B. greater than 2 ISI PEA 2014 Index Page 52 Econschool econschool.in [email protected] C. in (1,2) D. 2 19. Let n be a positive integer. What is the value of the expression n X kC(n, k) k=1 where C(n, k) denotes the number of ways to choose k out of n objects? A. n2n−1 B. n2n−2 C. 2n D. n2n 20. The first term of an arithmetic progression is a and common difference is d ∈ (0, 1). Suppose the k -th term of this arithmetic progression equals the sum of the infinite geometric progression whose first term is a and common ratio is d. If a > 2 is a prime number, then which of the following is a possible value of d? 1 A. 2 1 B. 3 1 C. 5 1 D. 9 21. In period 1, a chicken gives birth to 2 chickens (so, there are three chickens after period 1). In period 2, each chicken born in period 1 either gives birth to 2 chickens or does not give birth to any chicken. If a chicken does not give birth to any chicken in a period, it does not give birth in any other subsequent periods. Continuing in this manner, in period (k + 1), a chicken born in period k either gives birth to 2 chickens or does not give birth to any chicken. This process is repeated for T periods - assume no chicken dies. After T periods, there are in total 31 chickens. The maximum and the minimum possible values of T are respectively A. 12 and 4 B. 15 and 4 C. 15 and 5 D. 12 and 5 22. Let a and p be positive integers. Consider the following matrix   p 1 1 A= 0 p a    0 a 2 If determinant of A is 0, then a possible value of p is ISI PEA 2014 Index Page 53 Econschool econschool.in [email protected] A. 1 B. 2 C. 4 D. None of the above 23. For what value of α does the equation (x − 1) (x2 − 7x + α) = 0 have exactly two unique roots? A. 6 B. 10 C. 12 D. None of the above 24. What is the value of the following infinite series? ∞ (−1)k−1 loge 3k X k 2 k=1 A. loge 2 B. loge 2 loge 3 C. loge 6 D. loge 5 25. There

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