List of top Questions asked in Indian Institute Of Technology Joint Admission Test for MSc

Let \( u: \mathbb{R} \to \mathbb{R} \) be a twice continuously differentiable function such that \( u(0) > 0 \) and \( u'(0) > 0 \). Suppose \( u \) satisfies \[ u''(x) = \frac{u(x)}{1 + x^2} \]
for all \( x \in \mathbb{R} \).
Consider the following two statements:
I. The function \( uu' \) is monotonically increasing on \([0, \infty)\).
II. The function \( u \) is monotonically increasing on \([0, \infty)\).
Then which one of the following is correct?

For a 4 x 4 matrix \(M\isin M_4(\mathbb{C})\), let M denote the matrix obtained from M by replacing each entry of M by its complex conjugate. Consider the real vector space
H = \(\{M\isin M_4(\mathbb{C}):M^T=\overline M\)
where M^T denotes the transpose of M. The dimension of H as a vector space over \(\R\) is equal to

Consider the group \( (\mathbb{Q}, +) \) and its subgroup \( (\mathbb{Z}, +) \).
For the quotient group \( \mathbb{Q}/\mathbb{Z} \), which one of the following is FALSE?

Let \( (x_n) \) and \( (y_n) \) be sequences of real numbers defined by
\[ x_1 = 1, \quad y_1 = \frac{1}{2}, \quad x_{n+1} = \frac{x_n + y_n}{2}, \quad \text{and} \quad y_{n+1} = \sqrt{x_n y_n} \quad \text{for all} \ n \in \mathbb{N}. \]
Then which one of the following is true?

Consider the family F1 of curve lying in the region
{(x,y) εR²: y>0 and 0<x<π }
and given by 
\(y=\frac{c(1-cosx)}{sinx},\) Where c is a positive real number.
Let F₂ be the family of orthogonal trajectories to f₁. consider the curve C belonging to the family F2 passing through the point (\(\frac{π}{3}\),1). if a is a real number such that (\(\frac{π}{4}\), a) lies on C, then the value of a⁴ is equal _____to (Rounded off to two decimal places).

Let \(A=\begin{pmatrix} 1 & 1 \\ 0 & 1 \\ -1 & 1 \end{pmatrix}\)and let A^T denote the transpose of A. Let \(u=\begin{pmatrix} u_1 \\ u_2 \end{pmatrix}\) and \(v=\begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}\)be column vectors with entries in \(\R\) such that \(u^2_1+u_2^2=1\) and \(v_1^2+v_2^2+v^2_3=1\). Suppose
\(Au=\sqrt2v\) and \(A^Tv=\sqrt2u.\)
Then |\(𝑢1 + 2 \sqrt2 𝑣_1\)| is equal to _________. (Rounded off to two decimal places)

Let y(x) be the solution of the deferential equation \(\frac{dy}{dx}+3x^2y=x^2\), for xεR, 
satisfying the initial condition y(0)=4. 
Then \(lim \,\,y_{n→∞}\) y(x) is equal to ____ (Rounded off to decimal places)

Let y(x) be the solution of the differential equation
\(x,y^2y'+y^3=\frac{sin x}{x} for x>0\)
satisfying (\(\frac{π}{2}\))=0 
The the value of y (\(\frac{5π}{2}\)) is equal to ____. (Rounded off to two decimal places)

The number of elements of order 12 in the symmetric group S₇ is equal to____

The number of distinct subgroup of Z₉₉₉ is_____.

Suppose \[ a_n = \frac{3^n + 3}{5^n - 5} \quad \text{and} \quad b_n = \frac{1}{(1 + n^2)^{1/4}} \quad \text{for} \ n = 2, 3, 4, \ldots \] Then which one of the following is true?

Consider the \(2 \times 2\) matrix \( M = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix} \in M_2(\mathbb{R}) \). If the eighth power of \( M \) satisfies \( M^8 \begin{pmatrix} 1 \\ 1 \end{pmatrix} = \begin{pmatrix} x \\ y \end{pmatrix} \), then the value of \( x \) is

Consider the open rectangle \( G = \{ (s,t) \in \mathbb{R}^2 : 0 < s < 1 \ \text{and} \ 0 < t < 1 \} \) and the map \( T : G \to \mathbb{R}^2 \) given by \[ T(s,t) = \left( \frac{\pi s (1-t)}{2}, \ \frac{\pi (1-s)t}{2} \right) \ \text{for} \ (s,t) \in G. \] Then the area of the image \( T(G) \) of the map \( T \) is equal to

Let σ be the permutation in the symmetric group S₅ given by
σ(1) = 2, σ(2) = 3, σ(3) = 1, σ(4) = 5, σ(5) = 4.
Define
N(σ) = {𝜏 ∈ S₅ ∶ σ ∘ 𝜏 = 𝜏 ∘ σ}.
Then the number of elements in N(σ) is equal to _________.

For \(g\isin\Z\), let \(\bar{g}\isin\Z_{37}\) denote the residue class of g modulo 37. Consider the group U₃₇ = {\(\bar{g}\isin\Z_{37}:1\le g\le37\) with gcd(g, 37) = 1} with respect to multiplication modulo 37. Then which one of the following is FALSE?

Let \( c \) be a positive real number and let \( u: \mathbb{R}^2 \to \mathbb{R} \) be defined by \[ u(x,t) = \frac{1}{2c} \int_{x-ct}^{x+ct} e^{s^2} \, ds \quad \text{for} \ (x,t) \in \mathbb{R}^2. \] Then which one of the following is true?

Consider the region G={(x,y,z) εR³: 0<z<x²-y², x²+y²<1}.
Then the volume of G is equal to____. (Rounded off to two decimal place).

Let f : [0, π] → \(\R\) be the function defined by
\(f(x)=\begin{cases} (x-\pi)e^{\sin x} & \text{if } 0 \le x \le \frac{\pi}{2},\\ xe^{\sin x}+\frac{4}{\pi} & \text{if }\frac{\pi}{2}\lt x \le \pi. \end{cases}\)
Then the value of
\(\int\limits_0^{\pi}f(x)dx\)
is equal to _________. (Rounded off to two decimal places)

Consider the series \[\sum_{n=1}^{\infty} n^m \left(\frac{1}{1 + \frac{1}{n^p}}\right)\] where \( m \) and \( p \) are real numbers. Under which of the following conditions does the above series converge?

Let f : (−1, 1) → \(\R\) and g : (−1, 1) → \(\R\) be thrice continuously differentiable functions such that f(x) ≠ g(x) for every nonzero x ∈ (−1, 1). Suppose
f(0) = ln 2, f'(0) = π, f"(0) = π² , and f"'(0) = π⁹
and
g(0) = ln 2, g'(0) = π, g′′(0) = π², and g'"(0) = π³.
Then the value of the limit
\(\lim\limits_{x\rightarrow0}\frac{e^{f(x)}-e^{g(x)}}{f(x)-g(x)}\)
is equal to _________. (Rounded off to two decimal places)

Consider the system of linear equations\[ \begin{cases}x + y + t = 4, \\2x - 4t = 7, \\x + y + z = 5, \\x - 3y - z - 10t = \lambda,\end{cases}\]where \( x, y, z, t \) are variables and \( \lambda \) is a constant. Then which one of the following is true?

For \(n\isin\N\) and \(x\isin[1,\infin)\), let
\(f_n(x)=\int\limits_{0}^{\pi}(x^2+(cos\theta)\sqrt{x^2-1})^nd\theta\)
Then which one of the following is true?

For \( P \in M_5(\mathbb{R}) \) and \( i,j \in \{1,2, \ldots, 5\} \), let \( p_{ij} \) denote the \((i,j)\)th entry of \( P \). Let\[ S = \{ P \in M_5(\mathbb{R}) : p_{ij} = p_{sr} \text{ for } i,j,s,r \in \{1,2, \ldots, 5\} \text{ with } i + r = j + s \}.\]Then which one of the following is FALSE?

For some real number c with 0 < c < 1, let φ:(1-c, 1 + c) → (0,∞) be a differentiable function such that φ(1) = 1 and y = φ(x) is a solution of the differential equation (x² + y²)dx-4xy dy = 0. Then which one of the following is true?

Let \( G \) be a group of order 2022. Let \( H \) and \( K \) be the subgroups of \( G \) of order 337 and 674, respectively. If \( H \cup K \) is also a subgroup of \( G \), then which one of the following is FALSE?