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Indian Statistical Institute MSQE past year papers ∗†
Econschool
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- ∗Source: ISI MSQE 2004-2015 and ISI MSQE 2015-
- †Updated June 24,
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- Contents Index Page
B. ∑kj=0(−1)j+
k j
f (x + j)
C. ∑kj=0(−1)j
k j
f (x + k − j)
D. ∑kj=0(−1)j+
k j
f (x + k − j)
- Let In = ∫^0 ∞ xne−xdx, where n is some positive integer. Then In equals A. n! − nIn− 1 B. n! + nIn− 1 C. nIn− 1 D. none of these.
- If x^3 = 1, then ∆ =
a b c b c a c a b
∣∣ equals
A. (cx^2 + bx + a)
1 b c x c a x^2 a b
B. (cx^2 + bx + a)
x b c 1 c a x^2 a b
C. (cx^2 + bx + a)
x^2 b c x c a 1 a b
D. (cx^2 + bx + a)
1 b c x^2 c a x a b
- Consider any integer I = m^2 + n^2 , 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.
- 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 statements 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
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 ∫^ x^2 x (^4) +2^3 +1x 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) = (ab++xx^ )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
(a b
) (a b
)a+b
C. 2 log (ab^ )^ + b^2 − aba^2 D.
(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∑n f 1 (x) = ∑ni=1 xri , and f 2 (x) = i=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; D. none of the above.
- If h(x) = (^1) −^1 x , then h(h(h(x)) equals A. (^1) −^1 x B. x C. (^1) x D. 1 − x
- The function x|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.
∫ (^2) dx (x−2)(x−1)x equals
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.
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∫ (^) dx 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.
- The inverse of the function √−1 + x is A. √x^1 − 1 , B. x^2 + 1, C. √x − 1 , D. none of these.
- The domain of continuity of the function f (x) = √x + x x+1− 1 − (^) xx 2 +1+1 is A. [0, 1) B. (1, ∞) C. [0, 1) ∪ (1, ∞), D. none of these
- Consider the following linear programme: minimise x − 2 y subject to x + 3y ≥ 3 3 x + y ≥ 3 x + y ≤ 3 An optimal solution of the above programme is given by A. x = 34 , y = (^34) B. x = 0, y = 3 C. x = − 1 , y = 3 D. none of the above.
- Consider two functions f 1 : {a 1 , a 2 , a 3 } → {b 1 , b 2 , b 3 , b 4 } and f 2 : {b 1 , b 2 , b 3 , b 4 } → {c 1 , c 2 , c 3 }. The function f 1 is defined by f 1 (a 1 ) = b 1 , f 1 (a 2 ) = b 2 , f 1 (a 3 ) = b 3 and the function f 2 is defined by f 2 (b 1 ) = c 1 , f 2 (b 2 ) = c 2 , f 2 (b 3 ) = f 2 (b 4 ) = c 3. Then the mapping f 2 ◦ f 1 : {a 1 , a 2 , a 3 } → {c 1 , c 2 , c 3 } 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.
- 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,
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. K(n) is always divisible by 4 B. K(n) is always divisible by 5 C. K(n) is always divisible by 6 D. All Statements are incorrect.
- 25 books are placed at random on a shelf. The probability that a particular pair of books shall be always together is A. 252 B. 251 , C. 3001 D. 6001
- P (x) is a quadratic polynomial such that P (1) = −P (2). If -1 is a root of the equation, the other root is A. 45 , B. 85 , C. 65 ,
D. 35.
- The correlation coefficients between two variables X and Y obtained from the two equations 2 x + 3y − 1 = 0 and 5x − 2 y + 3 = 0 are A. equal but have opposite signs, B. −^23 and 25 , C. 12 and −^35 , D. Cannot say.
- If a, b, c, d are positive real numbers then ab + bc + cd + da is always
A. less than
B. less than 2 but greater than or equal to
C. less than 4 but greater than or equal 2 D. greater than or equal to 4.
- 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.
- The equation 5x^3 − 5 x^2 + 2x − 1 has 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.
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- An infinite geometric series has first term 1 and sum 4. It common ratio is A. (^12) B. (^34) C. 1 D. (^13)
- A continuous random variable X has a probability density function f (x) = 3x^2 with 0 ≤ x ≤ 1. If P (X ≤ a) = P (x > a), then a is: A. √^16 B. (^13 )
(^12)
C. (^12) D. (^12 )
(^13)
- If f (x) =
ex^ +
ex^ + √ex^ +... then f ′(x) equals to A. 2 ff^ ( (xx))+1−^1. B. f^22 (fx ()x−)f−^ ( 1 x) C. (^) f 22 (fx^ ()+x)+1f (x) D. (^2) ff (^ (xx)+1)
- limx→ 4
√x+5− 3 x− 4 is A. (^16) B. 0 C. (^14) D. not well defined
- If X = 2^65 and Y = 2^64 + 2^63 +... + 2^1 + 2^0 , then A. Y = X + 2^64. B. X = Y. C. Y = X + 1 D. Y = X − 1
0 e
x ex+1 dx^ = A. log(1 + e) B. log2. 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
C. 2
D. 4
- Let
y =
f (x) g(x) h(x) l m n a b c
where l, m, n, a, b, c are non-zero numbers. Then dydx equals A. (^) ∣∣ ∣∣ ∣∣
f ′(x) g′(x) h′(x) 0 0 0 0 0 0
B. ∣∣
f ′(x) g′(x) h′(x) 0 0 0 a b c
C. ∣∣
f ′(x) g′(x) h′(x) l m n a b c
D. ∣∣
f ′(x) g′(x) h′(x) l − a m − b n − c 1 1 1
- If f (x) = |x − 1 | + |x − 2 | + |x − 3 |, then f (x) is differentiable at A. 0 B. 1 C. 2 D. 3
- If (x − a)^2 + (y − b)^2 = c^2 , then 1 +
[ (^) dy dx
] 2
is independent of A. a B. b C. c D. Both b and c
- 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 D. (^) (m!)(nmnm!−m)! − 1
- Determine all values of the constants A and B such that the following function is continuous for all values of x. f (x) =
Ax − B if x ≤ − 1 2 x^2 + 3Ax + B if − 1 < x ≤ 1 4 if x > 1
A. A = B = 0 B. A = 34 , B = −^14 C. A = 14 , B = (^34) D. A = 12 , B = (^12)
- The value of limx→∞ (3x^ + 3^2 x) 1 x^ is A. 0 B. 1 C. e D. 9
- 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 = 650 , ∑^ Y = 100, ∑^ Y 2 = 460, ∑^ XY = 508. It was later discovered that at the time of 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. A. (^45) B. (^23) C. 1 D. (^56)
- The point on the curve y = x^2 − 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
- 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 A. (^12) B. 1 C. (^15) D. (^34)
- 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
