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

Let \( f(x) = 2x - \sin(x) \), for all \( x \in \mathbb{R} \). Let \( k \in \mathbb{N} \) be such that \[ \lim_{x \to 0} \left( \frac{1}{x} \sum_{i=1}^{k} i^2 f \left( \frac{x}{i} \right) \right) = 45. \] Then, the value of \( k \) is equal to ..............

Let \( S \) be the surface area of the portion of the plane \( z = x + y + 3 \), which lies inside the cylinder \( x^2 + y^2 = 1 \). Then, the value of \( \left( \frac{S}{\pi} \right)^2 \) is equal to ............. (rounded off to two decimal places).

Let \( T \) denote the triangle in the \( xy \)-plane bounded by the \( x \)-axis and the lines \( y = x \) and \( x = 1 \). The value of the double integral (over \( T \)) \[ \iint_T (5 - y) \, dx \, dy \] is equal to ............. (rounded off to two decimal places).

Consider the following subspaces of \( \mathbb{R}^4 \): \[ V_1 = \left\{ (x, y, z, w) \in \mathbb{R}^4 : x + y + 2w = 0 \right\}, \quad V_2 = \left\{ (x, y, z, w) \in \mathbb{R}^4 : 2y + z + w = 0 \right\}, \quad V_3 = \left\{ (x, y, z, w) \in \mathbb{R}^4 : x + 3y + z + 3w = 0 \right\}. \] Then, the dimension of the subspace \( V_1 \cap V_2 \cap V_3 \) is equal to ............... (rounded off to two decimal places).

Let \( T, S : P_4(\mathbb{R}) \to P_4(\mathbb{R}) \) be the linear transformations defined by \[ T(p(x)) = xp'(x), \quad S(p(x)) = (x + 1)p'(x) \] for all \( p(x) \in P_4(\mathbb{R}) \). Then, the nullity of the composition \( S \circ T \) is ................

Consider the real vector space \( \mathbb{R}^3 \). Let \( T : \mathbb{R}^3 \to \mathbb{R} \) be a linear transformation such that \[ T(1, 1, 1) = 0, \quad T(1, -1, 1) = 0, \quad T(0, 0, 1) = 16. \] Then, the value of \( T \left( \frac{1}{2}, \frac{2}{3}, \frac{3}{4} \right) \) is equal to ............... (rounded off to two decimal places).

Let \( \alpha \) be the real number such that \[ \lim_{x \to 0} \frac{(1 - \cos x)(22x^2 + x - 4)}{x^3} = \alpha \ln 2. \] Then, the value of \( \alpha \) is equal to ............ (rounded off to two decimal places).

The radius of convergence of the power series \[ \sum_{n=1}^{\infty} \frac{(x + \frac{1}{4})^n}{(-2)^n n^2} \] about \( x = -\frac{1}{4} \) is equal to ............ (rounded off to two decimal places).

Let \( \varphi : \mathbb{R} \to \mathbb{R} \) be the solution of the differential equation \[ 4 \frac{d^2 y}{dx^2} + 16 \frac{dy}{dx} + 25y = 0 \] satisfying \( \varphi(0) = 1 \) and \( \varphi'(0) = -\frac{1}{2} \). Then, the value of \( \lim_{x \to \infty} e^{2x} \varphi(x) \) is equal to ............ (rounded off to two decimal places).

The value of \[ \lim_{n \to \infty} 8n \left( \left( e^{\frac{1}{2n}} - 1 \right) \left( \sin \frac{1}{2n} + \cos \frac{1}{2n} \right) \right) \] is equal to ............... (rounded off to two decimal places).

Let \( f : \mathbb{R} \to \mathbb{R} \) be a twice differentiable function such that \[ f(0) = 4, \, f(1) = -2, \, f(2) = 8, \, f(3) = 2. \] Then, which of the following is/are TRUE?

For \( n \in \mathbb{N} \), consider the set \( U(n) = \{ x \in \mathbb{Z}_n : \gcd(x, n) = 1 \} \) as a group under multiplication modulo \( n \). Then, which of the following is/are TRUE?

Let \( u_1 = (1, 0, 0, -1) \), \( u_2 = (2, 0, 0, -1) \), \( u_3 = (0, 0, 1, -1) \), \( u_4 = (0, 0, 0, 1) \) be elements in the real vector space \( \mathbb{R}^4 \). Then, which of the following is/are TRUE?

Let \( f : \mathbb{R} \to \mathbb{R} \) be a twice differentiable function such that \[ f(0) = 0, \, f'(0) = 2, \, f(1) = -3. \] Then, which of the following is/are TRUE?

For \( n \in \mathbb{N} \), let \[ x_n = \sum_{k=1}^{n} \frac{k}{n^2 + k}. \] Then, which of the following is/are TRUE?

If \( M, N, \mu, w : \mathbb{R}^2 \to \mathbb{R} \) are differentiable functions with continuous partial derivatives, satisfying \[ \mu(x, y) M(x, y) \, dx + \mu(x, y) N(x, y) \, dy = dw, \] then which one of the following is TRUE?

Let \( \varphi : (-1, \infty) \to (0, \infty) \) be the solution of the differential equation \[ \frac{dy}{dx} = 2 y e^x = 2 e^x \sqrt{y}, \] satisfying \( \varphi(0) = 1 \). Then, which of the following is/are TRUE?

Let \( \Omega \) be the bounded region in \( \mathbb{R}^3 \) lying in the first octant \( (x \geq 0, y \geq 0, z \geq 0) \), and bounded by the surfaces \( z = x^2 + y^2 \), \( z = 4 \), \( x = 0 \) and \( y = 0 \). Then, the volume of \( \Omega \) is equal to:

The number of elements in the set \[ \{ x \in \mathbb{R} : 8x^2 + x^4 + x^8 = \cos x \} \] is equal to:

Let \( x_1 = 1 \). For \( n \in \mathbb{N} \), define \[ x_{n+1} = \left( \frac{1}{2} + \frac{\sin^2 n}{n} \right) x_n. \] Then, which one of the following is TRUE?

Let \( f : \mathbb{R}^2 \to \mathbb{R} \) be defined by \[ f(x, y) = e^{y}(x^2 + y^2) \quad \text{for all } (x, y) \in \mathbb{R}^2. \] Then, which one of the following is TRUE?

Let \( x_1 = 1 \). For \( n \in \mathbb{N} \), define \[ x_{n+1} = \left( \frac{1}{2} + \frac{\sin^2 n}{n} \right) x_n. \] Then, which one of the following is TRUE?

Let \( x_1>0 \). For \( n \in \mathbb{N} \), define \[ x_{n+1} = x_n + 4. \] If \[ \lim_{n \to \infty} \left( \frac{1}{x_1 x_2 x_3} + \frac{1}{x_2 x_3 x_4} + \cdots + \frac{1}{x_{n+1} x_{n+2} x_{n+3}} \right) = \frac{1}{24}, \] then the value of \( x_1 \) is equal to:

Let \( \mathbb{R}/\mathbb{Z} \) denote the quotient group, where \( \mathbb{Z} \) is considered as a subgroup of the additive group of real numbers \( \mathbb{R} \). Let \( m \) denote the number of injective (one-one) group homomorphisms from \( \mathbb{Z}_3 \) to \( \mathbb{R}/\mathbb{Z} \) and \( n \) denote the number of group homomorphisms from \( \mathbb{R}/\mathbb{Z} \) to \( \mathbb{Z}_3 \). Then, which one of the following is TRUE?

Let \( f_1, f_2, f_3 \) be nonzero linear transformations from \( \mathbb{R}^4 \) to \( \mathbb{R} \) and \[ \ker(f_1) \subset \ker(f_2) \cap \ker(f_3). \] Let \( T : \mathbb{R}^4 \to \mathbb{R}^3 \) be the linear transformation defined by \[ T(v) = (f_1(v), f_2(v), f_3(v)) \quad \text{for all } v \in \mathbb{R}^4. \] Then, the nullity of \( T \) is equal to: