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

For σ ∈ S₈, let o(σ) denote the order of σ. Then max{o(σ) : σ ∈ S₈} is equal to __________.

Let G be a finite group. Then G is necessarily a cyclic group if the order of G is

Suppose f : (−1, 1) → \(\R\) is an infinitely differentiable function such that the series \(\sum\limits_{j=0}^{\infin}a_j\frac{x^j}{j^!}\) converges to f(x) for each x ∈ (−1, 1), where,
\(a_j=\int\limits_{0}^{\pi/2}\theta^j\cos^j(\tan\theta)d\theta+\int\limits^{\pi}_{\pi/2}(\theta-\pi)^2\cos^j(\tan\theta)d\theta\)
for j ≥ 0. Then

Let y : \(\R → \R\) be a twice differentiable function such that y" is continuous on [0, 1] and y(0) = y(1) = 0. Suppose y"(x) + x² < 0 for all x ∈ [0, 1]. Then

Let p(x) = x⁵⁷ + 3x¹⁰ - 21x³ + x² + 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

Let R₁ and R₂ be the radii of convergence of the power series \(\sum\limits_{n=1}^{\infin}(-1)^nx^{n-1}\) and \(\sum\limits_{n=1}^{\infin}(-1)^n\frac{x^{n+1}}{n(n+1)}\), respectively. Then

Let v₁, . . . , v₉ be the column vectors of a non-zero 9 × 9 real matrix A. Let a₁, . . . , a₉ ∈ \(\R\), 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

Let T : \(P_2(\R) → P_4(\R)\) be the linear transformation given by T(p(x)) = p(x²). Then the rank of T is equal to __________.

Let S be the set of all real numbers α such that the solution y of the initial value problem
\(\frac{dy}{dx}=y(2-y),\\y(0)=\alpha,\)
exists on [0, ∞). Then the minimum of the set S is equal to __________. (rounded off to two decimal places)

Let y : (1, ∞) → \(\R\) be the solution of the differential equation
\(y"-\frac{2y}{(1-x)^2}=0\)
satisfying y(2) = 1 and\(\lim\limits_{x→∞}y(x) = 0\). Then y(3) is equal to __________. (rounded off to two decimal places)

Let V be the volume of the region S ⊆ \(\R^3\) defined by
S = {(x, y, z) ∈ \(\R^3\) : xy ≤ z ≤ 4, 0 ≤ x² + y² ≤ 1}.
Then \(\frac{V}{π}\) is equal to ________ . (rounded off to two decimal places)

Let V be a nonzero subspace of the complex vector space 𝑀₇(ℂ) such that every nonzero matrix in 𝑉 is invertible. Then, the dimension of V over ℂ is

Which one of the following groups has elements of order 1, 2, 3, 4, 5 but does not have an element of order greater than or equal to 6 ?

Let 𝑦(𝑥) be the solution of the differential equation
\(\frac{dy}{dx}=1+y\sec x\ \ \text{for}\ x \in(-\frac{\pi}{2},\frac{\pi}{2})\)
that satisfies 𝑦(0) = 0. Then, the value of \(y(\frac{\pi}{6})\) equals

For a positive integer n, let U(n) = {\(\bar{r}\) ∈ ℤ_n ∶ gcd(r, n) = 1} be the group under multiplication modulo n. Then, which one of the following statements is TRUE ?

Let 𝑔: ℝ → ℝ be a continuous function. Which one of the following is the solution of the differential equation
\(\frac{d^2y}{dx^2}+y=g(x)\ \ \ \ \text{for}\ x \in \R\),
satisfying the conditions y(0) = 0, y'(0) = 1 ?

Let G be a finite group containing a non-identity element which is conjugate to its inverse. Then, which one of the following is TRUE ?

Consider the following statements.
P : If a system of linear equations Ax = b has a unique solution, where A is an m × n matrix and b is an m × 1 matrix, then m = n.
Q : For a subspace W of a nonzero vector space V, whenever u ∈ V ∖ W and v ∈ V ∖ W, then u + v ∈ V ∖ W.
Which one of the following holds ?

Let G be a group of order 39 such that it has exactly one subgroup of order 3 and exactly one subgroup of order 13. Then, which one of the following statements is TRUE ?

Let F be the family of curves given by
x² + 2hxy + y² = 1, -1 < h < 1.
Then, the differential equation for the family of orthogonal trajectories to F is

Let \( V \) be the real vector space consisting of all polynomials in one variable with real coefficients and having degree at most 6, together with the zero polynomial. Then which one of the following is true?

On the open interval \((-c, c)\), where \( c \) is a positive real number, \( y(x) \) is an infinitely differentiable solution of the differential equation\[ \frac{dy}{dx} = y^2 - 1 + \cos x, \] with the initial condition \( y(0) = 0 \). Then which one of the following is correct?

Let P be a fixed 3×3 matrix with entries in \(\R\). Which of the following maps from \(M_3(\R)\) to \(M_3(\R)\) is/are linear?

Define f : \(\R^2 → \R\) by
f(x, y) = x² + 2y² - x for (x, y) ∈ \(\R^2\).
Let D = {(x, y) ∈ \(\R^2\) ∶ x² + y² ≤ 1} and \(E=\left\{(x,y)\in \R^2:\frac{x^2}{4}+\frac{y^2}{9} \le 1\right\}.\)
Consider the sets
D_max = {(a, b) ∈ D ∶ f has absolute maximum on D at (a, b)}, 
D_min = {(a, b) ∈ D ∶ f has absolute minimum on D at (a, b)}, 
E_max = {(c, d) ∈ E ∶ f has absolute maximum on E at (c, d)}, 
E_min = {(c, d) ∈ E ∶ f has absolute minimum on E at (c, d)}.
Then the total number of elements in the set
D_max ∪ D_min ∪ E_max ∪ E_min
is equal to _________.

Let \(\theta \in \left( \frac{\pi}{4}, \frac{\pi}{2} \right)\). Consider the functions
\[ u : \mathbb{R}^2 - \{ (0, 0) \} \to \mathbb{R} \quad \text{and} \quad v : \mathbb{R}^2 - \{ (0, 0) \} \to \mathbb{R} \]
given by
\[ u(x,y) = x - \frac{x}{x^2 + y^2} \quad \text{and} \quad v(x,y) = y + \frac{y}{x^2 + y^2}. \]
The value of the determinant \(\begin{vmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \end{vmatrix}\) at the point \((\cos \theta, \sin \theta)\) is equal to