1. IIT JAM MATHEMATICS Practice Test Paper
JD INSTITUTE IIT JEE and IIT JAM
Instructor: Prem Chandra Patel(IITB)
1. ay"
+ by'
+ cy = 0, where a, b, c R, then Eqn
i. Non. Homogenous of second order
ii. Homogenous of second order
iii. Linear homogenous of first order
iv. None of these
2. Let y1 & y2 be the some solution of y"
+ ay'
+ cy = 0 then general solution of D.E. always
i. c1y1 +y2 ii. c1y1 + c2y2 iii. y1 + c2y2 iv. None of these
3. If Eigen vector of matrix of A order 2 is (1, 2) with respect to Eigen value λ = 2, then matrix An
has one
i. Eigen Vector (1, 2n
) & Eigen value λ = 2n
ii. Eigen Vector (2n
, 1) & Eigen value 2n
iii. Eigen vector (1, 2) & Eigen value λ = 2n
iv. None of these
4. If K is compact set & f :K R be continuous function than f(K) has attains
i. only minima ii. only maxima iii. Maxima & minima iv. Neither maxima nor minima
5. Let {xn} be convergent seqn
of real number. If a1 >√7 +√2& an+1 = √7 + √ 𝑎 𝑛 − √7 𝑛 ≥ 1, then which one
of the following is the limit of the seen
i. √7 + 1 ii. √7 + √2 iii. √7 iv. √7 + √√7
6. The volume of the portion of the solid cylinder x2
+ y2
≤1 bounded above by the surface Z = x2
+ y2
&
bounded below by the xy-plane is
i.
𝜋
2
ii. 𝜋 iii.
3𝜋
2
iv. None of these
7. Let S = {(x, y) ∈R2
xy= 1} then
i. S is not connected but compact
ii. S is neither connected but not compact
iii. S is bounded but not connected
iv. S is unbounded but connected
8. Let if possible 𝛼 = lim
(𝑥,𝑦)−(0,0)
sin(𝑥2+𝑦2)
𝑥2+𝑦2 , 𝛽 = lim
(𝑥,𝑦)−(0,0)
𝑥2.𝑦2
𝑥2+𝑦2 then
i. 𝛼 exists but 𝛽 does not
ii. 𝛼 does not exists but 𝛽 exists
iii. 𝛼, 𝛽 do not exist
iv. Both 𝛼, 𝛽 exist.
9. Let Q is irrational number & S = R – Q the complement set of R then set of limit point of S
i. Countable set finite set
ii. Countable infinite set
iii. Uncountable set
iv. Empty set
10. Let A be 3x3 complex matrix s.t. A3
= I (I is 3x3 Identity matrix
i. A is diagonalizable
ii. A is not diagonalizable
iii. The minimal polynomial of A has reapeted root
iv. All eigenvalues of A are real.

2. IIT JAM MATHEMATICS Practice Test Paper
JD INSTITUTE IIT JEE and IIT JAM
Instructor: Prem Chandra Patel(IITB)
Q-11 to 30 carry two marks :-
11. Suppose that the dependent variable z & w are functions of the independent variable u & v dependent by the
eq. f(u, v, z, w) = 0 & g (u, v, z, w)= 0 where fzgw - fwgz = ? Which one of the following correct.
i. zu = fwgv – fwgz ii. zu = fvgw - fwgv
iii. zv = fzgu - fugz iv. zu = fzgw – fzgv
12. If lim
𝑥→𝑎
[
𝑓(𝑥)
𝑔(𝑥)
] exists then
i. Both lim
𝑥→𝑎
𝑓(𝑥) & lim
𝑥→𝑎
𝑔(𝑥) must exist
ii. lim
𝑥→𝑎
𝑓(𝑥) need not exist but lim
𝑥→𝑎
𝑔(𝑥) exist
iii. Neither lim
𝑥→𝑎
𝑓(𝑥) nor lim
𝑥→𝑎
𝑔(𝑥) may exist
iv. lim
𝑥→𝑎
𝑓(𝑥) exist but lim
𝑥→𝑎
𝑔(𝑥) need not exist
13. f:R →R is not f(o)=0 & |
𝑑𝑡
𝑑𝑥
| ≤ 4 we can conclude that f(1) is an
i. (5, 6) ii. (-4, 4) iii. [-4, 4] iv. (−∞, −4) ∪ (4, ∞)
14. Consider the sets of seqn
X={(xn) : xn ∈ {0, 1}, n ∈ N}
Y= {(xn) ∈ 𝑥 : xn = 1 for at most finitely value n} then
i. x is countable, y is finite
ii. x is uncountable, y is countable
iii. x is countable, y is countable
iv. x is uncountable, y is uncountable
15. Which of following sets of function f : R→R is a vector space over R
A1 = {f; lim
𝑥→2
𝑓(𝑥) = 0)}, A2= {𝑔; lim
𝑥→2
𝑔(𝑥) = 1)}, A3 = {h ;lim
𝑥→2
ℎ(𝑥)exist)
i. Only A1 ii. Only A2 iii. A1 & A3 but not A2 iv. All of the above.
16. f :R2
→R2
be a given by f (x, y) = (x, y + sin x). Then the derivative of f at (x, y) is linear transformation given
by
i. (
1 0
cos 𝑥 1
) ii. (
1 0
1 cos 𝑥
) iii. (
1 cos 𝑥
1 0
) iv. (
1 cos 𝑥
1 0
)
17. A function 𝑓 ∶ 𝑅2
→R is defined by f(x, y) =x2
y Let v=(1, 2) & a= (a1, a2) be two defined of R2
. The directional
derivative of f in the direction of v at a is :
i. a1 + a2 ii. a2 + 4a1 iii.
𝑎2
2
+ a1 iv. None of these
18. Which of the following subset of R4
is a basis of R4
?
B1 = { (1, 0, 0, 0), (1, 1, 0, 0), (1, 1, 1, 0), (1, 1, 1, 1)}
B2 = { (1, 0, 0, 0), (1, 2, 0, 0), (1, 2, 3, 0), (1, 2, 3, 4)}
B3 = { (1, 2, 0, 0), (0, 0, 1, 1), (2, 1, 0, 0), (-5, 5, 0, 0)} then
i. B1 & B2 but not B3
ii. B1 B2 & B3
iii. B1 & B3 but not B2
iv. Only B1
19. If y(x) is soon of D.E. y''
+ ay = 3e-x
then lim
𝑡→∞
𝑒−𝑡
𝑦(𝑡) is equal to
i. 1/3 ii. 3/10 iii. 0 iv. none of these

3. IIT JAM MATHEMATICS Practice Test Paper
JD INSTITUTE IIT JEE and IIT JAM
Instructor: Prem Chandra Patel(IITB)
20. Given a natural number n>1, s.t, (n-1)!= -1modn We can conclude that
i. n = pk
where P is prime k>1
ii. n=pq where p & q are district prime
iii. n=pqr where p & q, r are district prime
iv. n = p where p is prime
21. If ∑ 𝑎𝑛∞
𝑛=1 is a divergent series of positive number & <bn> of positive number which converges to zero but
for which ∑ 𝑏𝑛𝑎𝑛∞
𝑛=1
i. Converges ii. divergent
iii. may or may not converges iv. none of these
22. Let f : R→ R is real valued function & F is closed subset of R then f-1
(F) is
i. closed ii. open
iii. Neither open nor closed iv. none of these
23. If f is a non- constant real valued continuous function on R, then the range of f
i. is countable ii. is finite iii. is not countable iv. none of these
24. Let f be a continuous real valued function on the closed bounded interval [a, b]. If the maximum value for f is
attained at ‘c’ whose a<c<b, then
i. f1
(c)=0 ii. f1
(c)=1
iii. f1
(c) does not exist iv. f1
(c)=0 if f1
(c) exist
25. Soln
of y1
+ y cot x = cos3
x (1+y2
sin2
x) at y(0)=0 is given by
i. y cos x = tan (
1
4
−
1
4
cos4
x)
ii. y cos x = tan (−
1
4
𝑐𝑜𝑠4
x)
iii. y sin x = tan (
1
4
−
1
4
cos4
x)
iv. y sin x = tan (−
1
4
cos4
x)
26. The particular integral yp(x) of the differential equation
x2
y" +xy' – y =
1
𝑥+1
, x>0 is given by yp(x)=xv1(x) +
1
𝑥
v2(x) where v1(x) & v2(x) are given by
i. xv1’(x) -
1
𝑥2v2'(x) = 0, v1'(x) -
1
𝑥2 v2'(x) =
1
𝑥+1
ii. xv1’(x)+-
1
𝑥2v2'(x) = 0, v1'(x) -
1
𝑥2 v2'(x) =
1
𝑥+1
iii. xv1’(x) -
1
𝑥
v2'(x) = 0, v1'(x) +
1
𝑥2 v2'(x) =
1
𝑥+1
iv. xv1’(x)+
1
𝑥
v2'(x) = 0, v1'(x) +
1
𝑥2 v2'(x) =
1
𝑥+1
27. Consider the group S9 of all the permutations With 9 elements. What is largest order of a permutation in S9?
i. 21 ii. 20 iii. 30 iv. 14
28. Suppose A is an num real symmetric matrix with eigenvalues λ1, λ2, ………………, λn then
i. ∏ λ𝑛
𝑖=1 i < dot A
ii. ∏ λ𝑛
𝑖=1 i > dot A
iii. ∏ λ𝑛
𝑖=1 i = dot A
iv. If dot A =1, then λj=1, for j=1, 2, ……….x.
29. Let f : (0 ∞) → 𝑅 & g : (0 ∞) → 𝑅 be contn function satisfying ∫ 𝑡2
𝑑𝑡
𝑓(𝑥)
0
= x3
(1+x)2
& ∫ 𝑔(𝑡)𝑑𝑡 = 𝑥
𝑥2(1+𝑥)
0
x∈ (0 ∞) then f(2) + g(2) is equal to
i. 0 ii. 5 iii. 6 iv. None of these

4. IIT JAM MATHEMATICS Practice Test Paper
JD INSTITUTE IIT JEE and IIT JAM
Instructor: Prem Chandra Patel(IITB)
30. Let fn (x) ={
1 − 𝑛𝑥
0
,
𝑓𝑜𝑟 𝑥 ∈ [0,
1
𝑛
]
𝑓𝑜𝑟 𝑥 ∈ [
1
𝑛
, 1]
then
i. lim
𝑛→∞
𝑓𝑛(𝑥) defines a continuous function are (0, 1)
ii. {fn} converges uniformly on (0,1)
iii. lim
𝑥→∞
𝑓𝑛(𝑥) = 0 ∀ x ∈ (0, 1)
iv. lim
𝑥→∞
𝑓𝑛(𝑥) exists ∀ x ∈ (0, 1)
Section – B
1. Which of the following are true for the function
f(x) = sin x cos (1/x), ∀x∈ (0, 1)
i. lim
𝑥→0
𝑓(𝑥) = − lim
𝑥→0
𝑓(𝑥)
ii. lim
𝑥→0
𝑓(𝑥) < lim
𝑥→0
𝑓(𝑥)
iii. lim
𝑥→0
𝑓(𝑥) > lim
𝑥→0
𝑓(𝑥)
iv. lim
𝑥→0
𝑓(𝑥) = 0
2. 𝛿 = (
1 2 3 4 5 6 7 8
8 4 3 2 7 6 1 5
), 𝜏 = (
1 2 3 4 5 6 7 8
3 5 2 7 8 1 6 4
)
i. Order of 𝛿 is 4
ii. Order of 𝛿 is 8
iii. Order of 𝛿𝜏 is 8
iv. Order of 𝛿𝜏 is 12.
3. Let 𝛿 be a function with domain x & range y. Let A, B≤ Y & C, D≤ X which of following is not true
i. f-1
(A∩B) = f-1
(A)∩ f-1
(B) ii. f-1
(A∪ 𝐵) =f-1
(A) ∪ f-1
(B)
iii. f (C∩D)=f(C) ∪ f(D) iv. f(C ∩ 𝐷)=f(C) ∩ f(D)
4. Let C1 = [0, 1] be the set of real number x with 0≤ x ≤ 1. Define on operation ∗ on C1 by
x * y = {
𝑥 + 𝑦 𝑖𝑓 𝑥 + 𝑦 < 1
𝑥 + 𝑦 − 1 𝑖𝑓 𝑥 + 𝑦 ≥ 1
then (C1, *)
i. closure property satisfy
ii. ‘o’ is identify
iii. Associative satisfy
iv. (C1, *) is group.
5. (wrong) Let Mn denote the vector space of all nxn real matrices among the following subsets of Mn. decide
which are linear subspace.
i. V1
ii. V
iii. V
iv. V
6. Let ∑ 𝑎𝑛 series converges absolutely to A ∈ R then any management ∑ 𝑏𝑛∞
𝑛=1 of ∑ 𝑎𝑛∞
𝑛+1
i. converges ii. absolutely to A
iii. converges to A iv. may or may not converges

5. IIT JAM MATHEMATICS Practice Test Paper
JD INSTITUTE IIT JEE and IIT JAM
Instructor: Prem Chandra Patel(IITB)
7. For any rational number r in [0, 1], write r=p/q where p & q are integer with no common factor & q>0. Then
define f(x) = 1/q. Define f(x)=0 for all irrational number x in [0, 1], then 𝑓: [0, 1]→[0, 1]
i. F is continuous at any rational
ii. f is not continuous at any rational
iii. f is not continuous at any irrational
iv. f is continuous ∀ 𝑥 ∈ [0, 1]
8. If A x B is open subset of R2
then
i. A & B both are open set
ii. A is open but B is not open
iii. A & B both are not open set
iv. None of these
9. If f'(x) = g' (x) ∀𝑥 in the closed interval [a b] Then
i. (f – g)2
is continuous function
ii. (f – g) is differentiable
iii. (f – g) is decreasing function
iv. (f – g) is increasing function
10. Let D be the region bounded by the lines x =0 , y =4 & the parabola y= x2
. Let f :D → R be function given by
(f(x, y)= x2
+ y2
, than ∬ 𝑓(𝑥, 𝑦)𝑑𝑥𝑑𝑦𝐷
i. 25
(
1
3.5
+
22
7
) ii. 26
(
1
3.5
+
22
7
)
iii.
26
3.5
+
29
7
iv. None of these