ISI Admission Test 2020 Question Paper JRF Physics, JRF QROR, MS(QMBA), M.Tech(QROR) MMA

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ISI Admission Test 2020
Question Paper · JRF
Physics, JRF QROR,
MS(QMBA),
M.Tech(QROR) MMA
EXAM YEAR TYPE

ISI Admission Test 2020 Question Paper
SUBJECT

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

Notes · Sample Papers · Previous Year Papers · Mock Tests

co m .
. s e
em l a

as ag

1. Let A = (1, −1), B = (−2, 0), C = (1, 2) and D be the vertices of a

m
parallelogram in the X − Y plane listed clockwise. Then the point D is

m . c o
co(1−t )+i 1 − t be a complex number where t is a real number
(A) (4, 1) (B) (−2, −3) (C) (3, 0) (D) (−2, 1)

2. Let z .=
e m
s
√

emthat |t| < 1. Then the locus of z in the complex plane is a
2 2

s g l
la (A) an ellipse a
such

ag (B) a hyperbola

(C) a parabola (D) a pair of straight lines

R 2 x2 R e4 p
3. Let 1 e dx = a. Then the value of e loge x dx is

(A) e4 − a (B) 2e4 − a

.c om
(C) e4 − e − 4a (D) 2e4 − e − a

(B) e + 2se
m
4. The area bounded by the curves y = ex , y = xex and the y-axis is
(A) e − 2
l a
g the inequality
(C) e − 1 (D) 2e − 3

5. The set of all solutionsaof
1 1
>
2x − 1 1 − 2x−1
is
   
(A) 1, ∞ (B) 0, log2 ( 34 )

m
m c. o
     

c. o
(C) 0, log2 ( 43 ) ∪ 1, ∞ (D) − 1, ∞

e m
em la s
s
x
6. If lim ae −bx cos x = 5, then

la ag
x→0

ag
(A) a and b are uniquely determined
(B) a is uniquely determined, but not b
(C) b is uniquely determined, but not a
(D) neither a nor b is uniquely determined

m . c
1

c. o se m
m a
se
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g l
Page 1 of 6

7. Consider four events P, Q, R and S such that if any of P and Q occurs,
then either R occurs or S doesn’t occur. If exactly one of R and S
always occurs, which of the following statements is necessarily true?
(The notation E c denotes the complement of the event E)
(A) R =⇒ P
(B) R =⇒ P c
(C) Rc =⇒ Qc
(D) Rc =⇒ Q

8. The particular solution of
 dy 
loge = 5x + 7y, y(0) = 0
dx
is
(A) e5x + 5e−7y = 7 (B) 7e5x − 5e−7y = 5

(C) 5e5x + 7e7y = 12 (D) 7e5x + 5e−7y = 12

9. Define Aj = ni=1 ij , j = 0, 1, 2, 3. Then
P

A1 A2
lim
n→∞ A0 A3

is
(A) 0 (B) 21 (C) 23 (D) 1

10. Let p, q, r ∈ R. If f (x) = px2 + qx + r be such that p + q + r = 3 and
f (x + y) = f (x) + f (y) + xy, for all x, y ∈ R. Then the value of f (5) is
(A) 25 (B) 30 (C) 35 (D) 40

11. If n C0 ,n C1 ,n C2 , ...,n Cn denote the binomial coefficients in the expansion
of (1 + x)n , p > 0 is a real number and q = 1 − p, then
n
X
r2 n Cr pn−r q r
r=0

is
(A) np2 q 2 (B) n2 p2 q 2 (C) npq + n2 p2 (D) npq + n2 q 2

12. If |z+3−2i| = 8 and the maximum and the minimum values of |2z+9−8i|
are α and β, respectively, then the value of α + β is
(A) 10 (B) 21 (C) 32 (D) 27

2

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co m .
. s e
em l a

as a g

13. Consider the cubic equation x3 = 2x + 5. Which of the following state-

m
co
ments about the above equation is true?

o mtwo positive real roots and one negative real root .
(A) All its roots are real and positive

c
. has two negative real roots and one positive real root e m
m s
(B) It has

s e It has one real root and a pair of complex roots
(C) It
g l a
g l a (D)
a
a 14. Consider
3
two
3
real-valued sequences {x
n } and {y
n } satisfying the condi-
tion xn − yn → 0 as n → ∞. Then, as n → ∞,
(A) xn − yn → 0 always
(B) xn − yn → 0 only if {xn } converges
(C) xn − yn → 0 only if {|xn | − |yn |} converges
(D) xn − yn → 0 only if {|x2n + xn yn + yn2 |} converges

m
c. o
d sin x R2 3
15. Let dx P (x) = e x , x > 0. If 1 x3 esin x dx = P (k) − P (1), then which

m
of the following is a possible value of k?

s e
a
(A) 2 (B) 4 (C) 8 (D) 16

g l
measured along a line a
16. The distance of the point (1, −2, 3) from the plane x − y + z = 11
x y z
parallel to = = is 2 3 6
(A) 5 (B) 6 (C) 7 (D) 8

17. The number of words that can be constructed using 10 letters of the
English alphabet such that all five vowels appear exactly once in the
word is

m
(A) 21 C5 10! (B) 21 C5 (5!)2

m c. o
c. o e m
(C) 10 P5 21 P5 (D) 10 P5 (21)5

em 18. Let f : [0, ∞) → R be a differentiable function with a
l s
las ag bounded by
f (0) = 1 and

g
0

a
f (x)f (x) > 0, for all x. Let A(n) be the area of the region
x-axis, y-axis, graph of f and the line x = n. Then
(A) {A(n)}n≥1 is a convergent sequence
(B) {A(n)}n≥1 is an oscillatory sequence
(C) the function A : N → R is increasing
(D) none of the above statements is true

m . c
3

c. o s e m
m a
se
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g l Page 3 of 6

19. Let x, y, z be three natural numbers. Then the number of triplets (x, y, z)
such that xyz = 100 is
(A) 36 (B) 25 (C) 72 (D) 18

20. How many distinct straight lines can one form that are given by an
equation ax + by = 0, where a and b are numbers from the set
{0, 1, 2, 3, 4, 5, 6, 7}?
(A) 63 (B) 57 (C) 37 (D) 49

21. Consider three non-zero matrices A, B and C such that ABB 0 = CBB 0
where B 0 is the transpose of B. Which of the following statements is
necessarily true?
(A) r(A) = r(C)
(B) non-zero eigenvalues of A and C are identical
(C) AB = CB
(D) none of the above

22. Let m and n be nonzero integers. Define
n o
Am,n = x ∈ R : n2 x3 + 2020x2 + mx = 0 .

Then the number of pairs (m, n) for which Am,n has exactly two points
is
(A) 0 (B) 10 (C) 16 (D) ∞

23. Consider two independent events with the same probability p (0 < p <
1). Then the probability of occurrence of at least one of the two events
is
(A) the same for all p
(B) linearly increasing in p
(C) strictly convex in p
(D) strictly concave in p

24. Let S be the set of all 3 × 3 real matrices A = ((aij )) such that the
matrix ((a3ij )) has rank one. Define a set R = {rank(A) : A ∈ S}. Then
R is equal to
(A) {1} (B) {1, 2} (C) {1, 3} (D) {1, 2, 3}

4

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co m .
. s e
em l a

as a g

25. The function f : R → R is defined by

m
co
(

om .
1
e− x , x>0

m
f (x) =

.c e
0, x≤0.

e m l a s
a s g
Then

g l (B) f is continuous, but not differentiable everywhere a
a
(A) f is not continuous

(C) f is differentiable but f 0 is not continuous
(D) f is differentiable and f 0 is continuous

26. For a cyclic group G of order 12, the number of subgroups of G is
(A) 2 (B) 6 (C) 8 (D) 11

m
co
27. Let f : R → R be a continuously differentiable function and f (1) = 4.

.
Then the value of

em
Z f (x)
2t

a s
lim dt

l
x→1 4 x−1
is
(A) 8f 0 (1) a g
(B) 2f 0 (1) (C) 4f 0 (1) (D) f 0 (1)

28. The series
2x 4x3 8x7
+ + + ...
1 + x2 1 + x4 1 + x8

m
(A) is uniformly convergent for all x

m c. o
(B) is convergent for all x, but the convergence is not uniform

c. o e m
(C) is convergent only for |x| ≤ 12 , but the convergence is not uniform

em
(D) is uniformly convergent on [− 12 , 12 ]
la s
las ag
g
29. Let S and T be two non-empty sets and f : S → T be a function such

a that f (A ∩ B) = f (A) ∩ f (B) for all subsets A and B of S. Then
(A) there exist A ⊂ S such that f −1 f (A) 6= A
(B) f is one-to-one
(C) there exist disjoint subsets A, B of S such that f (A) ∩ f (B) 6= φ
(D) none of the above statements is necessarily true

m . c
5

c. o s e m
m a
se
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30. Let x1 , x2 , ..., xn ∈ R be distinct reals. Define the set
n
o
A = f1 (t), f2 (t), ..., fn (t) : t ∈ R ,

where for 1 ≤ k ≤ n,
(
1, if xk ≤ t
fk (t) =
0, otherwise.

Then A contains
(A) exactly n distinct elements
(B) exactly (n + 1) distinct elements
(C) exactly 2n distinct elements
(D) infinitely many distinct elements

6

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