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Indian Statistical Institute MSQE past year papers
(PEA) ∗†
Econschool
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
∗Source †Updated: July 30, 2021
1
1 ISI PEA 2006
- If f (x) = log (1+ 1 −xx^ )^ , 0 < x < 1 , then f (^ 1+^2 xx 2 )^ equals A. 2 f (x) B. f^ ( 2 x) C. (f (x))^2 D. none of these
- If u = ϕ(x − y, y − z, z − x), then ∂u∂x + ∂u∂y + ∂u∂z equals A. 0 B. 1 C. u D. none of these
- 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 A.
m i
B.
n i
C.
m k − i
n i
D.
m i
n k − i
- The number of disjoint intervals over which the function f (x) = | 0. 5 x^2 −| x | | is decreas- ing is A. one B. two C. three D. none of these
- For a set of real numbers x 1 , x 2 ,...... , xn, the root mean square (RMS) defined as RMS = {^ N^1 ∑ni=1 x^2 i^ }^1 /^2 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
- 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 A. (^4772) B. (^2572) C. 15355 D. 15398.
- Let f (x) = log(1+^
xp )−log( 1 − xq ) x , x^6 = 0.^ If^ f^ is continuous at^ x^ = 0,^ then the value of^ f^ (0) is A. (^1) p − (^1) q B. p + q C. (^1) p + (^1) q D. none of these.
- Consider four positive numbers x 1 , x 2 , y 1 , y 2 such that y 1 , y 2 > x 1 x 2 Consider the number S = (x 1 y 2 + x 2 y 1 ) − 2 x 1 x 2. The number S is A. always a negative integer B. can be a negative fraction C. always a positive number D. none of these.
- 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.
- The number of positive pairs of integral values of (x, y) that solves 2xy − 4 x^2 +12x− 5 y = 11 is A. 4 B. 1 C. 2 D. none of these.
- 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] 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
D. none of these.
- If f (x) =
x +
x +
x + √x +.. ., then f ′(x) is A. (^2) f (xx)− 1 B. (^2) f (x^1 )− 1. C. (^) x√^1 f (x) ; D. (^2) f (x^1 )+.
- If P = logx(xy) and Q = logy(xy), then P + Q equals A. P Q B. PQ C. QP D. P Q 2
- The solution to
∫ (^2) x (^3) + x^4 +2x dx^ is A. x 44 x+2 (^3) +2x + constant; B. log x^4 + log 2x+ constant; C. 12 log |x^4 + 2x| + constant; D.
∣x 44 x+2 (^3) +2x
∣ + constant.
- The set of all values of x for which x^2 − 3 x + 2 > 0 is A. (−∞, 1) B. (2, ∞) C. (−∞, 2) ∩ (1, ∞) D. (−∞, 1) ∪ (2, ∞)
- Consider the functions f 1 (x) = x^2 and f 2 (x) = 4x^3 + 7 defined on the real line. Then A. f 1 is one-to-one and onto, but not f 2 B. f 2 is one-to-one and onto, but not f 1 C. both f 1 and f 2 are one-to-one and onto; D. none of the above.
- If f (x) =
(a+x b+x
)a+b+2x , a > 0 , b > 0 , then f ′(0) equals A.
(b (^2) −a 2 b^2
) (a b
)a+b− 1
B.
2 log (ab^ )^ + b^2 ab−a^2
) (a b
)a+b
C. 2 log
(a b
(b (^2) −a 2 ba
- The linear programming problem
maxx,y z = 0. 5 x + 1. 5 y subject to: x + y ≤ 6 3 x + 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.
- 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 B. f (x; θ) is a p.d.f. only for 0 < θ < (^12) C. f (x; θ) is a p.d.f. only for 12 ≤ θ < 1 D. f (x; θ) is not a p.d.f. for any value of θ.
- The correlation coefficient r for the following five pairs of observations satisfies
x 5 1 4 3 2 y 0 4 2 0 − 1
A. r > 0; B. r < − 0 .5; C. − 0. 5 < r < 0; D. r = 0.
- 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 f 1 (x) = ∑ni=1 xri , and f 2 (x) = ∑ni=1 aixi + b, where xi > 0 for all i, 0 < r < 1 , ai > 0 and b are constants. Then A. f 1 is not homothetic but f 2 is; B. f 2 is not homothetic but f 1 is; C. both f 1 and f 2 are homothetic;
A. divisible by 6 but not always divisible by 12 B. divisible by 12 but not always divisible by 24 C. divisible by 24 but not always divisible by 120 D. divisible by 120 but not always divisible by 720.
- Two varieties of mango, A and B, are available at prices Rs. p 1 and Rs. p 2 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. p 1 = p 2 B. p 2 = 34 p 1 C. p 1 = p 2 or p 2 = 34 p 1 D. 34 ≤ p p^21 < 1
- For a given bivariate data set (xi, yi; i = 1, 2 ,... , n) , the squared correlation coefficient (r^2 ) between x^2 and y is found to be 1. Which of the following statements is the most appropriate? A. In the (x, y) scatter diagram, all points lie on a straight line. B. In the (x, y) scatter diagram, all points lie on the curve y = x^2. C. In the (x, y) scatter diagram, all points lie on the curve y = a+bx^2 , a > 0 , b > 0 D. In the (x, y) scatter diagram, all points lie on the curve y = a+bx^2 , a, b any real numbers.
- 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.
- 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)
- Suppose a discrete random variable X takes on the values 0, 1 , 2 ,... , n with frequencies
proportional to binomial coefficients
( (^) n 0
( (^) n 1
( (^) n n
respectively. Then the mean (μ) and the variance (σ^2 ) of the distribution are A. μ = n 2 and σ^2 = n 2 B. μ = n 4 and σ^2 = n 4 C. μ = n 2 and σ^2 = n 4 D. μ = n 4 and σ^2 = n 2
- 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
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
D. Statements (a), (b) and (c) are all incorrect.
- Given that f is a real-valued differentiable function such that f (x)f ′(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.
- Let p, q, r, s be four arbitrary positive numbers. Then the value of (p^2 +p+1)(q^2 +q+1)(r^2 +r+1)(s^2 +s+1) pqrs is at least as large as A. 81 B. 91 C. 101. D. None of these.
- If x = t t−^11 and y = t t−t^1 , t > 0 , t 6 = 1 then the relation between x and y is A. yx^ = x 1 y^ , B. x 1 y^ = y 1 x^ , C. xy^ = yx, D. xy^ = y 1 x^.
- 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.
- The value of
0 [x]nf^ ′(x)dx,^ where [x] stands for the integral part of^ x, n^ is a positive integer and f ′^ is the derivative of the function f, is A. (n + 2n) (f (2) − f (0)), B. (1 + 2n) (f (2) − f (1)) C. 2nf (2) − (2n^ − 1) f (1) − f (0), D. none of these.
- 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.
- 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.
- 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?
A. 1/ 4 ,
B. 1/ 2 ,
C. 2 / 3
D. 3/ 4
- 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′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 ′s method: Divide the set into 2000 and 3000 numbers and follow the procedure of A. Z′^ s method: Calculate the mean of 4500 numbers (which are 6 = 1 ) and then add 1. Then A. all methods are correct, B. X′s method is correct, but Y and Z′s methods are wrong, C. X′s and Y ′s methods are correct but Z s′ methods is wrong, D. none is correct.
- 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
- 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.
- The function f (x) = x^4 − 4 x^3 + 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.
- Consider the number K(n) = (n + 3) (n^2 + 6n + 8) defined for integers n. Which of the following statements is correct?
A. all roots between 1 and 2 , B. all negative roots, C. a root between 0 and 1 , D. all roots greater than 2.
- The probability density of a random variable is
f (x) = ax^2 e−kx^ (k > 0 , 0 ≤ x ≤ ∞) Then, a equals A. k 23 , B. k 2 , C. k 22 , D. k
- Let( x = r be the mode of the distribution with probability mass function p(x) = n x
px(1 − p)n−x. Then which of the following inequalities hold. A. (n + 1)p − 1 < r < (n + 1)p, B. r < (n + 1)p − 1 C. r > (n + 1)p D. r < np.
- Let y = (y 1 ,... , yn) be a set of n observations with y 1 ≤ y 2 ≤... ≤ yn. Let y′^ = (y 1 , y 2 ,... , yj + δ,... , yk − δ,... , yn) where yk − δ > yk− 1 >... > yj+1 > yj + δ δ > 0. Let σ : standard deviation of y and σ′^ : standard deviation of y′. Then A. σ < σ′, B. σ′^ < σ, C. σ′^ = σ, D. nothing can be said.
- Let x be a r.v. with pdf f (x) and let F (x) be the distribution function. Let r(x) = 1 xf−F^ ( x(x)).
Then for x < eμ^ and f (x) = e−^
(log x 2 −μ)^2 x√ 2 π ,^ the function^ r(x) is A. increasing in x, B. decreasing in x C. constant, D. none of the above.
- 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×nxn× 1 where elements of y are some rearrangements of the elements of x. Then
A. P is bistochastic with diagonal elements 1 , B. P cannot be bistochastic, C. P is bistochastic with elements 0 and 1 , D. P is a unit matrix.
- Let f 1 (x) = (^) xx+1. Define fn(x) = f 1 (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.
- limn→∞ (^1) 1+−xx−−^22 nn , x > 0 equals A. 1 B. - C. 0 D. The limit does not exist.
- Consider the function f (x 1 , x 2 ) = max { 6 − x 1 , 7 − x 2 }. The solution (x∗ 1 , x∗ 2 ) to the optimization problem minimize f (x 1 , x 2 ) subject to x 1 + x 2 = 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.
C. log 1+ 2 e. D. 2 log(1 + e)
- 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? A. 19. B. (^12) C. (^29) D. (^13)
- 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 12.
- Let f, g and h be real valued functions defined as follows: f (x) = x(1 − x); g(x) = 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
- 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
- If f (x) is a real valued function such that
2 f (x) + 3f (−x) = 55 − 7 x for every x ∈ R, then f (3) equals A. 40 B. 32
C. 26
D. 10
- 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. A. (^1516) B. 167 C. 245 D. 17522
- If x 1 , x 2 , x 3 are positive real numbers, then x 1 x 2 +^
x 2 x 3 +^
x 3 x 1 is always A. ≤ 3 B. ≤ 313 C. ≥ 3 D. 3
- limn→∞ 12 +2^2 n+ 3 ... +n^2 equals A. 0 B. (^13) C. (^16) D. 1.
- Suppose b is an odd integer and the following two polynomial equations have a common root. x^2 − 7 x + 12 = 0 and x^2 − 8 x + b = 0 The root of x^2 − 8 x + b = 0 that is not a root of x^2 − 7 x + 12 = 0 is A. 2 B. 3 C. 4 D. 5
- Suppose n ≥ 9 is an integer. Let μ = n^12 + n^13 + n^14. Then, which of the following relationships between n and μ is correct? A. n = μ B. n > μ
