PAPER : IIT-JAM 2020 1

PAPER : IIT-JAM
MATHEMATICS MA–2020

SECTION-A
[Multiple Choice Questions (MCQ)]
Q.1 – Q.10 carry ONE mark each.2018

1. Let f  x   2 x 3  9 x 2  7 . Which of the following is true ?

(a) f is one-one in the interval [-1,1] (b) f is one-one in the interval [2,4]
(c) f is not one-one in the interval [-4,0] (d) f is not one-one in the interval [0,4]
2. Let T :  2   2 be the linear transformation given by T  x, y     x, y  . Then
(a) T 2k  T for all k  1
(b) T 2 k 1  T for all k  1
(c) The range of T2 is a proper subspace of the range of T
(d) The range of T2 is equal to the range of T
dv u
3. If u  x 3 and v  y 2 tansfrom the differential equation 3x5 dx  y  y 2  x3  dy  0 to du  2 u  v ,
 
then  is
(a) 4 (b) 2 (c) –2 (d) –4
4. Which of the following is False?
x 1 sin x cos x
(a) lim
x  e x
0 (b) lim 1/ x  0
x  0 xe
(c) lim
x 0 1  2 x
 0 (d) lim
x 0 1  2 x
0
 n2
 n2 n
5. The radius of convergence of the power series    x is
n 1  n 
1 1 1
(a) e 2 (b) (c) (d)
e e e2
6. Let g :    be a twice differentiable function. If f  x, y   g  y   xg   y  , then
f 2 f f f 2 f f f 2 f f f 2 f f
(a)  y  (b)  y  (c)  x  (d)  x 
x xdy y y xdy x x xdy y y xdy x
n

7. Let sn  1 
 1
, n   . Then the sequence sn  is
n
(a) monotonically increasing and is convergent to 1
(b) monotonically decreasing and is convergent to 1
(c) neither monotonically increasing nor monotonically decreasing but is convergent to 1
(d) divergent
8. Consider the following group under matrix multiplication
 1 p q  
  
H   0 1 r : p, qr   
 
 0 0 1  
  
Then the center of the group is isomorphic to
(a)   \ 0 , (b)  ,   (c)   2 ,   (d)  ,      \ 0 ,  
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
9. If the directional derivative of the function z  y 2 e2 x at (2, –1) along the unit vactor b  iˆ  ˆj is zero,
then    equals.
1 1
(a) (b) (c) 2 (d) 2 2
2 2 2
10. If the equation of the tangent plane to the surface z  16  x 2  y 2 at the point P 1,3,6  is
ax  by  cz  d  0 . Then the value of d is
(a) 16 (b) 26 (c) 36 (d) 46
11. Let M be a 4×3 real matrix and let e1 , e2 , e3  be the standard basis of 3 . which of the following is true ?

(a) If rank (M) = 1, then Me1 , Me2  is a linearly independent set

(b) If rank (M) = 2, then Me1 , Me2  is a linearly independent set

(c) If rank (M) = 2, then Me1 , Me3  is a linearly independent set

(d) If rank (M) = 3, then Me1 , Me3  is a linearly independent set

12. Let S 1  z   : z  1 be the circle group under multiplication and i  1 . Then the set

   : 
ei 2  is infinite is
(a) empty (b) non- empty and finite
(c) countably infinite (d) uncountable
1  sn2
13. Define s1    0 and sn 1  , n  1 . Which of the following is true ?
1 
2 1 1
(a) If sn  , then sn  is monotonically increasing and lim sn 
 n  
2 1 1
(b) If sn  , then sn  is monotonically decreasing and lim sn 
 n  
2 1 1
(c) If sn  , then sn  is monotonically increasing and lim sn 
 n  
2 1 1
(d) If sn  , then sn  is monotonically decreasing and lim sn 
 n  
14. Let M be a real 6×6 matrix. Let 2 and –1 be two eigenvalues of M. If M 5  aI  bM , where a, b   ,
then
(a) a  10, b  11 (b) a  11, b  10 (c) a  10, b  11 (d) a  10, b  11
1 1
15. Let f : 0,1   be a continuous function such that f     and
2 2
1


f  x   f  y    x  y   sin x  y
2
 for all x, y 0,1 . Then  f  x dx is
0

1 1 1 1
(a)  (b)  (c) (d)
2 4 4 2
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 2 1 2 1
 x sin x  y sin y , xy  0

 
 x 2 sin , x  0, y  0
16. Let f  x, y    x
 1
 y 2 sin , y  0, x  0
 y
 0, x y0

Which of the following is true at (0, 0) ?

(a) f is not continuous
f f
(b) is continuous but is not continuous
x y
(c) f is not differentiable
f f
(d) f is differentiable but both and are not continuous
x y
 

17. Suppose that S is the sum of a convergent series  an . Define tn  an  an1  an 2 . Then the series  tn
n 1 n 1

(a) diverges (b) converges to 3S – a1– a2

(c) converges to 3S – a1– 2a2 (d) converges to 3S – 2a1– a2
18. Let a  iˆ  ˆj  kˆ and r  xiˆ  yjˆ  zkˆ, x, y, z   . Which of the following is False ?
    
(a)   a  r   a (b)   a  r   0
       
(c)   a  r   a (d)     a  r  r   4  a  r 

19. Let F    : 2020  1 . Consider the groups

  z    1 z  
G    :  F , z    and H    : z  
 0 1    0 1  
under matrix multiplication. Then the number of cosets of H in G is
(a) 1010 (b) 2019 (c) 2020 (d) infinite
20. Let f  x, y , z   x3  y 3  z 3  3 xyz ,. A point at which the gradient of the function f is equal to zero is
(a) (–1,1,–1) (b) (–1,–1,–1) (c) (–1,1,1) (d) (1,–1,1)
 x  a 2 ,x 0
21. Let a   . If f  x    3
 x  a  ,x 0
then
d2 f d2 f
(a) does not exist at x = 0 for any value of a (b) 2 exists at x = 0 for exactly one value of a
dx 2 dx
d2 f d2 f
(c) exists at x = 0 for exactly two values of a (d) exists at x = 0 for infinitely many values of a
dx 2 dx 2
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d2y dy
22. A solution of the differential equation 2 x 2 2
 3 x  y  0, x  0 that passes through the point (1, 1) is
dx dx
1 1 1 1
(a) y  (b) y  (c) y  (d) y 
x x2 x x3/2
23.  
Consider the differential equation L  y   y  y 2 dx  xdy  0 . The function f  x, y  is said to be an
1
integrating factor of the equation if f  x, y  L  y   0 becomes exact. If f  x, y   2 2 , then
x y
(a) f is an integrating factor and y  1  kxy , k   is NOT its general solution
(b) f is an integrating factor and y  1  kxy , k   is its general solution
(c) f is an integrating factor and y  1  kxy , k   is NOT its general solution
(d) f is NOT an integrating factor and y  1  kxy, k   is its general solution
24. Let M be an n  n  n  2  non-zero real matrix with M 2  0 and let    \ 0 . Then
(a)  is only eigenvalue of (M +I) and (M–I)
(b)  is only eigenvalue of (M +I) and (I–M)
(c) – is the only eigenvalue of (M +I) and (M–I)
(d) – is only eigenvalue of (M +I) and (I–M)
a
25. Let {an} be a sequence of positive real numbers. Suppose that l  lim n1 .which of the following is true ?
n  a
n

(a) If l  1, then lim an  1

n 
(b) If l  1, then lim a  0 n
n 

(c) If l  1, then lim an  1 (d) If l  1, then lim an  0

n  n 

26. Let D   x, y    : x  y  1 and f : D   be a non- constant continuous function. Which of the

2

following is TRUE ?
(a) The range of f is unbounded
(b) The range of f is a union of open intervals
(c) The range of f is a closed interval
(d) the range of f is a union of at least two disjoint closed intervals
27. The area bounded by the curves x 2  y 2  2 x and x 2  y 2  4 x , and the straight lines y = x and y = 0 is
 1  1   1  1
(a) 3    (b) 3    (c) 2    (d) 2   
 2 4  4 2  4 3 3 4
28. Let S be the surface of the portion of the sphere with centre at the origin and radius 4, above the xy-plane.
 

Let F  yiˆ  xjˆ  yx 3 kˆ . if n̂ is the unit outward normal to S, Then    F  ndS 
ˆ equals
S

(a) –32 (b) –16 (c) 16 (d) 32 

29. The value of the triple integral   x y  1dxdydz, where V is the region given by x 2  y 2  1, 0  z  2 is
2

V
(a)  (b) 2 (c)3 (d) 4
2 2 2
30. Let S be the part of the cone z  x  y between the planes z = 0 and z = 1. Then the value of the surface
integral   x  y  dS is
2 2

S
  
(a)  (b) (c) (d)
2 3 2
 PAPER : IIT-JAM 2020 5

SECTION-B
[Multiple Select Questions (MSQ)]
Q.31 – Q.40 carry TWO marks each.
 1 2  n  1  b  lim  1  1  ..  1 
31 Let a  lim  2  2  ..   and   which of the following is/are true?
n  n
 n n2  n n  1
 n2 nn
a
(a) a > b (b) a < b (c) ab  ln 2 (d)  ln 2
b
d2y dy
Let L  y   x
2
32. 2
 px  qy , where p,q are real constants Let y1  x  and y2  x  be two solution
dx dx
of L  y   0, x  0 that satisfy y1  x0   1 , y1  x0   0 , y2  x0   0 and y2  x0   1 for some x0  0 . Then,
(a) y1(x) is not a constant multiple of y2  x 

(b) y1(x) is constant multiple of y2  x 

(c) 1, ln x are solutions of L[y] = 0 when p = 1, q = 0
(d) x, ln x are solutions of L[y] = 0 when p + q  0
33. Cosider the following system of linear equations x  y  5 z  3, x  2 y  mz = 5 and x  2 y  4 z  k . The
system is consistent if
(a) m  4 (b) k  5 (c) m  4 (d) k  5
34. Let a , b   and a  b . Which of the following statement(s) is/are true ?
(a) There exists a continuous function f :  a, b   a, b  such that f is one-one
(b) There exists a continuous function f :  a, b   a, b  such that f is onto
(c) There exists a continuous function f :  a, b    a, b  such that f is one-one
(d) There exists a continuous function f :  a, b    a, b  such that f is onto
35. Let a, b, c   such that a<b<c. Which of the following is/are true for any continuous function f :   
satisfying f  a   b, f  b   c and f (c) = a ?
(a) There exist    a , c  such that f     
(b) There exist    a, b  such that f     
(c) There exists    a, b  such that  f  f     
(d) There exists    a, c  such that  f  f  f     
36. Let V be a non-zero vector space over a field F. Let S  V be a non-empty set. Consider the following
properties of S:
(I) For any vector space W over F, any map f : S  W extends to a linear map from V to W..
(II) For any vector space W over F and any two lienar maps f , g : V  W satisfying f (s)= g(s) for all
s  S we have f  v   g  v  for all v V ,
(III) S is linearly independent
(IV) The span of S is V
Which of the following statement (s) is/are True ?
(a) (I) implies (IV) (b) (I) implies (III) (c) (II) implies (III) (d) (II) implies (IV)
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n n

37. If sn 
 1 and tn 
 1 , n  0,1, 2,..., then
2n  3 4n  1
 
(a) s
n 0
n is absolutely convergent (b) t n is absolutely convergent
n 0

 
(c) s
n 0
n is conditionally convergent (d) t n is conditionally convergent
n 0

n
38. Let f be a real valued function of a real variable, such that f  0   K for all n   , where K> 0. Which
of the following is/are true ?
1

f
n
 0 n
(a)  0 as n  
n!
1
 n
f  0 n
  as n  
(b)
n!

(c) f 
n
 x  exists for all x   and for all n  
f    0  n

(d) The series  is absolutely convergent

n 1  n  1 !

39. Let G be a group with identity e. Let H be an abelian non-trivial proper subgroup of G with the property that
H  gHg 1  e for all g  H . K   g  G; gh  hgh  H  , then
(a) K is a proper subgroup of H
(b) H is a proper subgroup of K
(c) K = H
(d) There exists no abelian subgroup L  G such that K is a proper subgroup of L
40. Let S be that part of the surface of the paraboloid z =16 –x2–y2 which is above the plane z = 0 and D be
its projection on the xy- plane. Then the area of S equals

(a)  1  4  x 2  y 2 dxdy (b)  1  2  x 2  y 2 dxdy

D D

2 4 2 4

(c)  1  4r 2 drd  (d)  1  4r 2 rdrd 

0 0 0 0
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SECTION-C
[Numerical Answer Type (NAT)]
Q.41 – Q.50 carry ONE mark each.
41. Let  : S3  S 1 be a non-trivial non- injective group homomorphism. Then the number of elements in the
kernel of  is___________
42. Let f :    be such that f , f , f  are continuous functions with f  0, f   0 and f   0 . Then
f  x  f  x
lim is ___________
x 2
1  1
43. Let S   : n    and f : S   be defined by f  x   , then
n  x

 1 1 
max  : x     f  x   f    1 is ___________(upto two decimal places)
 3 3 

44. Let f  x, y   0 be a solution of the homogeneous differential equation  2 x  5 y  dx   x  3 y  dy  0

If f  x  , y  3  0 is a solution of the differential equation  2 x  5 y  1 dx   2  x  3 y  dy  0 then

the value of  is ___________
 n 
Consider the real vector space P2020   ai x ; ai   and 0  n  2020  .Let W be the subspace given by
i
45.
 i 0 

 n 
W    ai x i  P2020 ; ai  0 for all odd i  , Then the dimension of W is ___________
 i 0 
 2 2 2
46. Let F  xiˆ  yjˆ  zkˆ and S be the sphere given by  x  2    y  2    z  2   4. If n̂ is the unit

1 
ˆ is ___________
F  ndS
 
outward normal to S, then
S

1 2
x2
47. If e dxdy  k  e 4  1 , then k equals___________
0 2y
1
Let xn  n n and yn  e n , n   . Then the value of lim yn is ___________
1 x
48.
n 

dy
49. Consider the differential equation  10 y  f  x  , x  0 .Where f  x  is a continuous function such
dx
that lim f  x   1 . Then the value of lim y  x  is ___________
x 
x

df
50. Let f  x, y   e x sin y, x  t 3  1 and y  t 4  t . Then at t = 0 is__________ (upto two decimal
dt
places)
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Q.51 – Q.60 carry TWO marks each.
51. Let T :  7   7 be a linear tranformation with nullity (T) = 2. Then, the minimum possible value for Rank
(T2) is ___________

9 2 7
1
 1  . Then, the value of det((8I – M)3) is ___________
52. Let M   0 7 2

0 0 11 6
 
0 0 5 0

3 1
53. Consider the expansion of the function f  x   in powers of x, that is valid in x  . Then
1  x 1  2 x  2
the coefficient of x4 is___________
54. Suppose that G is a group of order 57 which is NOT cyclic. If G contains a unique subgroup H of order 19,
then for any g  H , o  g  is ___________
55. The minimum value of the function f  x, y   x 2  xy  y 2  3x  6 y  11 is___________
56. Let C be the boundary of the square with vertices (0, 0), (1, 0) ,(1, 1) and (0, 1) oriented in the counter clockwise
sense. Then the value of the line integral  x y dx   x  y  dy is ___________ (upto twodecimal places)
2 2 2 2

2
57. Let f  x   x  x, x  0 and g  x   a0  a1  x  1  a2  x  1 be the sum of first three terms of the
Taylor series of f (x) around x = 1. If g (3) = 3, then  is___________
58. If x 2  xy 2  c where c   , is the general solution of the exact differential equation
M  x, y  dx  2 xydy  0 then M(1, 1) is ___________

59. Let f :    be a differentiable function with f '  x   f  x  for all x, Suppose that f  x  and f  x  are

d2y dy
two non-zero solution of the differential equation 4 2
 p  3 y  0 satisfying f  x  f x   f  2 x 
dx dx
and f  x  f  x   f  x  then the value of p is ___________

1 1 1
60. The sum of the series 2  2 2  1  3  32  1  4  4 2  1  .... is ___________

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