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IIT JAM MA
List of top Questions asked in IIT JAM MA
The value of
\(\lim\limits_{n\rightarrow \infin}\left(n\int\limits^1_0\frac{x^n}{x+1}dx\right)\)
is equal to __________ . (rounded off to two decimal places)
IIT JAM MA - 2023
IIT JAM MA
Multivariable Calculus
Integral Calculus
Let
\(a_n=\left(1+\frac{1}{n}\right)^n\)
and
\(b_n=n\cos(\frac{n!\pi}{2^{10}})\)
for n ∈
\(\N\)
. Then
IIT JAM MA - 2023
IIT JAM MA
Real Analysis
Sequences and Series
The limit
\(\lim\limits_{a\rightarrow0}\left(\frac{\int\limits_{0}^{a}\sin(x^2)dx}{\int\limits_{0}^{a}(\ln(x+1))^2dx}\right)\)
is
IIT JAM MA - 2023
IIT JAM MA
Multivariable Calculus
Integral Calculus
The maximum number of linearly independent eigenvectors of the matrix
\(\begin{pmatrix} 1 & 1 & 0 & 0 \\ 2 & 2 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 1 & 3 \\ \end{pmatrix}\)
is equal to _________.
IIT JAM MA - 2023
IIT JAM MA
Linear Algebra
Matrices
Let (an) be a sequence of real numbers defined by
\(a_n=\begin{cases} 1 & \text{if } n \text{ is prime}\\ -1 & \text{if } n \text{ is not prime} \end{cases}\)
Let
\(b_n=\frac{a_n}{n}\)
for n ∈
\(\N\)
. Then
IIT JAM MA - 2023
IIT JAM MA
Real Analysis
Sequences and Series
Consider the initial value problem
\(\frac{dy}{dx}+αy=0, \\ y(0)=1,\)
where α ∈
\(\R\)
. Then
IIT JAM MA - 2023
IIT JAM MA
Differential Equations
Differential Equations
For each t ∈ (0, 1), the surface P
t
in
\(\R^3\)
is defined by
\(P_t = \left\{(x, y, z) : (x^2 + y^2 )z = 1, t^2 ≤ x^2 + y^2 ≤ 1\right\}.\)
Let a
t
∈ R be the surface area of P
t
. Then
IIT JAM MA - 2023
IIT JAM MA
Multivariable Calculus
Functions of Two or Three Real Variables
A subset S ⊆
\(\R^2\)
is said to be bounded if there is an M > 0 such that |x| ≤ M and |y| ≤ M for all (x, y) ∈ S. Which of the following subsets of
\(\R^2\)
is/are bounded ?
IIT JAM MA - 2023
IIT JAM MA
Multivariable Calculus
Functions of Two or Three Real Variables
Consider the following statements :
I. Every infinite group has infinitely many subgroups.
II. There are only finitely many non-isomorphic groups of a given finite order.
Then
IIT JAM MA - 2023
IIT JAM MA
Linear Algebra
Groups
Let S and T be non-empty subsets of
\(\R^2\)
, and W be a non-zero proper subspace of
\(\R^2\)
. Consider the following statements :
I. If span(S) =
\(\R^2\)
, then span(S ∩ W) = W.
II. span(S ∪ T) = span(S) ∪ span(T).
Then
IIT JAM MA - 2023
IIT JAM MA
Linear Algebra
Finite Dimensional Vector Spaces
Which of the following functions is/are Riemann integrable on [0, 1] ?
IIT JAM MA - 2023
IIT JAM MA
Multivariable Calculus
Integral Calculus
For g ∈
\(\Z\)
, let
\(\bar{g}\)
∈
\(\Z_8\)
denote the residue class of g modulo 8. Consider the group
\(\Z^×_8\)
= {
\(\bar{x}\)
∈
\(\Z_8\)
: 1 ≤ x ≤ 7, gcd(x, 8) = 1} with respect to multiplication modulo 8. The number of group isomorphisms from
\(\Z^×_8\)
onto itself is equal to ________
IIT JAM MA - 2023
IIT JAM MA
Linear Algebra
Groups
Let f :
\(\R^2 → \R^2\)
be defined by f(x, y) = (e
x
cos(y), e
x
sin(y)). Then the number of points in
\(\R^2\)
that do NOT lie in the range of f is
IIT JAM MA - 2023
IIT JAM MA
Multivariable Calculus
Functions of Two or Three Real Variables
Let f :
\(\R^2 → \R\)
be defined as follows :
\(f(x,y)=\begin{cases} \frac{x^4y^3}{x^6+y^6} & \text{if }(x,y) \ne (0,0)\\ 0 & \text{if } (x,y)=(0,0) \end{cases}\)
Then
IIT JAM MA - 2023
IIT JAM MA
Multivariable Calculus
Functions of Two or Three Real Variables
Let f :
\(\R^2 → \R\)
be the function defined as follows :
\(f(x,y)=\begin{cases} (x^2-1)^2\cos^2(\frac{y^2}{(x^2-1)^2}) & \text{if }x \ne ±1 \\ 0 & \text{if } x=±1\end{cases}\)
The number of points of discontinuity of f(x, y) is equal to _________.
IIT JAM MA - 2023
IIT JAM MA
Multivariable Calculus
Functions of Two or Three Real Variables
Let f :
\(\R → \R\)
be an infinitely differentiable function such that f" has exactly two distinct zeroes. Then
IIT JAM MA - 2023
IIT JAM MA
Real Analysis
Functions of One Real Variable
Let f(x) =
\(\sqrt[3]{x}\)
for x ∈ (0, ∞), and θ(h) be a function such that
f(3 + h) − f(3) = hf′ (3 + θ(h)h)
for all h ∈ (−1, 1). Then
\(\lim\limits_{h→0} θ(h)\)
is equal to _________. (rounded off to two decimal places)
IIT JAM MA - 2023
IIT JAM MA
Real Analysis
Functions of One Real Variable
Let
\(f(x,y)=\iint\limits_{(u-x^2)+(v-y)^2 \le 1}e^{-\sqrt{(u-x)^2+(v-y)^2}}du\ dv.\)
Then
\(\lim\limits_{n \rightarrow \infin}f(n,n^2)\)
is
IIT JAM MA - 2023
IIT JAM MA
Multivariable Calculus
Integral Calculus
Let G be a finite group. Then G is necessarily a cyclic group if the order of G is
IIT JAM MA - 2023
IIT JAM MA
Linear Algebra
Groups
For σ ∈ S
8
, let o(σ) denote the order of σ. Then max{o(σ) : σ ∈ S
8
} is equal to __________.
IIT JAM MA - 2023
IIT JAM MA
Linear Algebra
Groups
Let W be the subspace of
\(M_3(\R)\)
consisting of all matrices with the property that the sum of the entries in each row is zero and the sum of the entries in each column is zero. Then the dimension of W is equal to ___________.
IIT JAM MA - 2023
IIT JAM MA
Linear Algebra
Matrices
Let v
1
, . . . , v
9
be the column vectors of a non-zero 9 × 9 real matrix A. Let a
1
, . . . , a
9
∈
\(\R\)
, not all zero, be such that
\(\sum^9_{i=1}a_iv_i=0\)
. Then the system
\(Ax=\sum^9_{i=1}v_i\)
has
IIT JAM MA - 2023
IIT JAM MA
Linear Algebra
Matrices
Let S be the triangular region whose vertices are (0, 0),
\((0,\frac{\pi}{2})\)
and
\((\frac{\pi}{2},0)\)
. The value of
\(\iint\limits_S\sin(x)\cos(y)dx\ dy\)
is equal to ________. (rounded off to two decimal places)
IIT JAM MA - 2023
IIT JAM MA
Multivariable Calculus
Integral Calculus
Let p(x) = x
57
+ 3x
10
- 21x
3
+ x
2
+ 21 and
\(q(x)=p(x)+\sum\limits_{j=1}^{57}p^{(j)}(x) \ \text{for all }x \in \R,\)
where p
(j)
(x) denotes the j
th
derivative of p(x). Then the function q admits
IIT JAM MA - 2023
IIT JAM MA
Real Analysis
Functions of One Real Variable
Which of the following is/are true ?
IIT JAM MA - 2023
IIT JAM MA
Linear Algebra
Finite Dimensional Vector Spaces
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