ISI Admission Test 2018 Sample Paper JRF Physics, JRF QROR, MS(QMBA), M.Tech(QROR) MMA

FOR ISI ADMISSION TEST EXAM PREPARATION

ISI Admission Test 2018
Sample Paper · JRF
Physics, JRF QROR,
MS(QMBA),
M.Tech(QROR) MMA
EXAM YEAR TYPE

ISI Admission Test 2018 Sample Paper
SUBJECT

JRF Physics, JRF QROR, MS(QMBA), M.Tech(QROR) MMA

Notes · Sample Papers · Previous Year Papers · Mock Tests

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SAMPLE QUESTIONS (MCQ -TYPE)

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1. The number of isosceles (but not equilateral) triangles with integer

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sides and no side exceeding 10 is

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(A) 65 (B) 75 (C) 81 (D) 90.

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2. The number of squares in the following figure is

o m(C) 29
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(A) 25 (B) 26 (D) 30.

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3. The number of trailing zeros in 100!

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(A) 21 (B) 23 (C) 24 (D) 25.

4. The number of common terms in the two sequences {3, 7, 11, . . . , 407}
and {2, 9, 16, . . . , 709} is

(A) 13 (B) 14 (C) 15 (D) 16.

5. One needs to choose six real numbers x1 , x2 , . . . , x6 such that the

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product of any five of them is equal to other number. The number

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of such choices is

c. o s e m(D) 93.
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(A) 3 (B) 33 (C) 63

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6. The volume of the region S = {(x, y, z) : |x| + |y| + |z| ≤ 1} is

a (A) 1/6 (B) 1/3 (C) 2/3 (D) 4/3.

7. The greatest common divisor of all numbers of the form p2 − 1, where
p ≥ 7 is a prime, is

(A) 6 (B) 12 (C) 24 (D) 48.

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8. Let a and b be two positive integers such that

a = k1 b + r1 and b = k2 r1 + r2 ,

where k1 , k2 , r1 , r2 are positive integers with r2 < r1 < b. Then
gcd(a, b) is same as

(A) gcd(r1 , r2 ) (B) gcd(k1 , k2 ) (C) gcd(k1 , r2 ) (D) gcd(k2 , r1 ).

9. If α is a root of x2 − x + 1 = 0, then α2018 + α−2018 is

(A) -1 (B) 0 (C) 1 (D) 2.

10. A new flag of ISI club is to be designed with 5 vertical strips using
some or all of the four colours: green, maroon, red and yellow. In
how many ways this can be done so that no two adjacent strips have
the same colour ?

(A) 120 (B) 324 (C) 432 (D) 576.

11. The value of λ for which the system of linear equations 2x−y−z = 12,
x − 2y + z = −4, x + y + λz = 4 has no solution is

(A) 2 (B) -2 (C) 3 (D) -3.

12. The rank of the matrix
 
1 2 3 4
 
 5 6 7 8 
  is
 6 8 10 12 
 
151 262 373 484

(A) 1 (B) 2 (C) 3 (D) 4.
" #
2 i
13. If A = , the trace of A10 is
i 0

(A) 2 (B) 2(1+i) (C) 0 (D) 210 .

14. Let A be a 3 × 3 real matrix with all diagonal entries equal to 0. If 1 + i
is an eigenvalue of A, the determinant of A equals

(A) -4 (B) -2 (C) 2 (D) 4.

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15. Let G be a finite group of even order. Then which of the following

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statements is correct ?

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(A) The number of elements of order 2 in G is even

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e G has no subgroup of order 2
(B) The number of elements of order 2 in G is odd
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l (D) None of the above.
(C)
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16. Consider a large village, where only two newspapers P1 and P2 are
available to the families. It is known that the proportion of families
(i) not taking P1 is 0.48,
(ii) not taking P2 is 0.58,
(iii) taking only P2 is 0.30.

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The probability that a randomly chosen family from the village takes

m0.40 (D) cannot be determined.
only P1 is

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(A) 0.24 (B) 0.28
l a(C)

g of which have the same weight and the
17. There are eight coins,aseven
other one weighs more. In order to find the coin having more weight,
a person randomly chooses two coins and puts one coin on each side
of a common balance. If these two coins are found to have the same
weight, the person then randomly chooses two more coins from the
rest and follows the same method as before. The probability that the

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coin will be identified at the second draw is

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(A) 1/2 (B) 1/3 (C) 1/4 (D) 1/6.

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18. Let A1 = (0, 0), A2 = (1, 0), A3 = (1, 1) and A4 = (0, 1) be the four

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vertices of a square. A particle starts from the point A1 at time 0 and

a moves either to A2 or to A4 with equal probability. Similarly, in each of
the subsequent steps, it randomly chooses one of its adjacent vertices
and moves there. Let T be the minimum number of steps required to
cover all four vertices. The probability P (T = 4) is

(A) 0 (B) 1/16 (C) 1/8 (D) 1/4.

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19. Let X1 , X2 , . . . , Xn be independent and identically distributed with
P (Xi = 1) = P (Xi = −1) = p and P (Xi = 0) = 1 − 2p for all
i = 1, 2, . . . , n. Define
Y n  n
Y  n
Y 
an = P Xi = 1 , bn = P Xi = −1 and cn = P Xi = 0 .
i=1 i=1 i=1

Which of the following is true as n tends to infinity?

(A) an → 1/3, bn → 1/3, cn → 1/3
(B) an → p, bn → p, cn → 1 − 2p
(C) an → 1/2, bn → 1/2, cn → 0
(D) an → 0, bn → 0, cn → 1.

20. Consider the set of all functions from {1, 2, . . . , m} to {1, 2, . . . , n},
where n > m. If a function is chosen from this set at random, the
probability that it will be strictly increasing is
n
m n
n m+n−1
m m+n−1

(A) m /n (B) m /m (C) m−1 /n (D) m /mn .

21. The angle between the tangents drawn from the point (1, 4) to the
parabola y 2 = 4x is

(A) π/2 (B) π/3 (C) π/4 (D) π/6.

22. The x-axis divides the circle x2 + y 2 − 6x − 4y + 5 = 0 into two parts.
The area of the smaller part is

(A) 2π − 1 (B) 2(π − 1) (C) 2π − 3 (D) 2(π − 2).

23. For n ≥ 1, let
1 2 n
an = 2
+ 2 + ··· + and bn = c0 + c1 r + c2 r 2 + · · · + cn r n ,
2 3 (n + 1)2
where |ck | ≤ M for all integer k and |r| < 1. Then

(A) both {an } and {bn } are Cauchy sequences
(B) {an } is a Cauchy sequence, but {bn } is not a Cauchy sequence
(C) {an } is not a Cauchy sequence, but {bn } is a Cauchy sequence
(D) neither {an } nor {bn } is a Cauchy sequence.

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24. The sum of the infinite series

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2 6 10 14

(A) 2 om .
1+ + 2 + 3 + 4 + ··· is
3 3 3 3

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emsolution of the differential equation s
(B) 3 (C) 4 (D) 6.

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25. The

ag 2 2
(1 + x y )ydx + (x y − 1)xdy = 0 2 2
is

(A) xy = log x − log y + C
(B) xy = log y − log x + C
(C) x2 y 2 = 2(log x − log y) + C
(D) x2 y 2 = 2(log y − log x) + C.

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26. Let Ci (i = 0, 1, . . . , n) be the coefficient of xi in (1 + x)n . Then

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em
C0 C1 C2 Cn
− + − · · · + (−1)n

s
2 3 4 n+2
is equal to
g l a
1
(A) n+1 a
1
(B) n+2 1
(C) n(n+1) 1
(D) (n+1)(n+2) .

27. Number of real solutions of the equation x7 + 2x5 + 3x3 + 4x = 2018
is

(A) 1 (B) 3 (C) 5 (D) 7.

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28. Consider the following functions

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( (
1, |x| ≤ 1 1, |x| ≤ 2
f (x) = and g(x) =

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a following
0, |x| > 1 2, |x| > 2.

Define h (x) = f (g(x)) and h (x) = g(f (x)). Which oflthe
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g
1 2

a
statements is correct?

(A) h1 and h2 are continuous everywhere
(B) h1 is continuous everywhere and h2 has discontinuity at ±1
(C) h2 is continuous everywhere and h1 has discontinuity at ±2
(D) h1 has discontinuity at ±2 and h2 has discontinuity at ±1.

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29. Let f be a continuous function with f (1) = 1. Define
Z t2
F (t) = f (x)dx.
t

The value of F ′ (1) is

(A) -2 (B) -1 (C) 1 (D) 2.

30. Consider the function
x2 x3 xn
 
f (x) = 1 + x + + + ... + e−x ,
2! 3! n!

where n ≥ 4 is a positive integer. Which of the following statements
is correct?

(A) f has no local extremum
(B) For every n, f has a local maximum at x = 0
(C) f has no local extremum if n is odd and has a local maximum at
x = 0 when n is even
(D) f has no local extremum if n is even and has a local maximum at
x = 0 when n is odd.

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