List of top Mathematics Questions asked in GATE Mathematics

Let \( (\cdot, \cdot) \) denote the standard inner product on \( {R}^n \). Let \( V = \{v_1, v_2, v_3, v_4, v_5\} \subset {R}^n \) be a set of unit vectors such that \( (v_i, v_j) \) is a non-positive integer for all \( 1 \leq i \neq j \leq 5 \). Define \( N(V) \) to be the number of pairs \( (r, s) \), \( 1 \leq r, s \leq 5 \), such that \( (v_r, v_s) \neq 0 \). The maximum possible value of \( N(V) \) is equal to:

Let \( a \in {R} \) and \( h \) be a positive real number. For any twice-differentiable function \( f : {R} \to {R} \), let \( P_f(x) \) be the interpolating polynomial of degree at most two that interpolates \( f \) at the points \( a - h, a, a + h \). Define \( d \) to be the largest integer such that any polynomial \( g \) of degree \( d \) satisfies \[ g''(a) = P_f''(a). \] The value of \( d \) is equal to (answer in integer):

Let \( P_f(x) \) be the interpolating polynomial of degree at most two that interpolates the function \( f(x) = x^2|x| \) at the points \( x = -1, 0, 1 \). Then \[ \sup_{x \in [-1, 1] |f(x) - P_f(x)| = \, {(round off to TWO decimal places)}. \] }

The maximum of the function \( f(x, y, z) = xyz \) subject to the constraints \[ xy + yz + zx = 12, \quad x>0, y>0, z>0, \] is equal to (round off to TWO decimal places):

Let \( f : [0, 1] \to {R} \) be a function. Which one of the following is a sufficient condition for \( f \) to be Lebesgue measurable?

Let \( {H} = \{ z \in {C} : \operatorname{Im}(z)>0 \ \) and \( {D} = \{ z \in {C} : |z|<1 \} \). Then \[ \sup \{ |f'(0)| : f { is an analytic function from } {D} { to } {H} { and } f(0) = \frac{i}{2} \} \] is equal to:}

If the outward flux of \( F(x, y, z) = (x^3, y^3, z^3) \) through the unit sphere \( x^2 + y^2 + z^2 = 1 \) is \( \alpha \pi \), then \( \alpha \) is equal to (round off to TWO decimal places):

Let \( S^1 = \{ z \in \mathbb{C} : |z| = 1 \} \). For which one of the following functions \( f \) does there exist a sequence of polynomials in \( z \) that uniformly converges to \( f \) on \( S^1 \)?

Let \( f(x, y) = (x^2 - y^2, 2xy) \), where \( x>0, y>0 \). Let \( g \) be the inverse of \( f \) in a neighborhood of \( f(2, 1) \). Then the determinant of the Jacobian matrix of \( g \) at \( f(2, 1) \) is equal to (round off to TWO decimal places):

Let \( p = (1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}) \in {R^4 \) and \( f : {R}^4 \to {R} \) be a differentiable function such that \( f(p) = 6 \) and \( f(Ax) = A^3 f(x) \), for every \( A \in (0, \infty) \) and \( x \in {R}^4 \). The value of \[ 12 \frac{\partial f}{\partial x_1}(p) + 6 \frac{\partial f}{\partial x_2}(p) + 4 \frac{\partial f}{\partial x_3}(p) + 3 \frac{\partial f}{\partial x_4}(p) \] is equal to (answer in integer):}

The number of non-isomorphic finite groups with exactly 3 conjugacy classes is equal to (answer in integer):

Let \( {F}_3 \) be the field with exactly 3 elements. The number of elements in \( GL_2({F}_3) \) is equal to (answer in integer):

Given a real subspace \( W \) of \( {R}^4 \), let \( W^\perp \) denote its orthogonal complement with respect to the standard inner product on \( {R}^4 \). Let \( W_1 = {Span}\{(1, 0, 0, -1)\ \) and \( W_2 = {Span}\{(2, 1, 0, -1)\} \). The dimension of \( W_1^\perp \cap W_2^\perp \) over \( {R} \) is equal to (answer in integer):}

The number of group homomorphisms from \( {Z}/47 \) to \( S_4 \) is equal to (answer in integer):

Let \( A \in M_n({C}) \) be a normal matrix. Consider the following statements:
1. If all the eigenvalues of \( A \) are real, then \( A \) is Hermitian.
2. If all the eigenvalues of \( A \) have absolute value 1, then \( A \) is unitary.
Which one of the following is correct?

For the initial value problem \[ \frac{dy}{dx} = f(x, y), \quad y(x_0) = y_0, \] generate approximations \( y_n \) to \( y(x_n) \) using the recursion formula \[ y_n = y_{n-1} + a k_1 + b k_2, \] where \[ k_1 = h f(x_{n-1}, y_{n-1}), \quad k_2 = h f(x_{n-1} + \beta h, y_{n-1} + \beta k_1). \] Which one of the following choices of \( a, b, \alpha, \beta \) gives the Runge-Kutta method of order 2?

Let \( u = u(x, t) \) be the solution of \[ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}, \quad 0<x<1, \, t>0, \] with boundary conditions \( u(0, t) = u(1, t) = 0 \) and initial condition \( u(x, 0) = \sin(\pi x) \). Define \[ g(t) = \int_0^1 u^2(x, t) \, dx. \] Which one of the following is correct?

If \( y_1 \) and \( y_2 \) are two different solutions of the ordinary differential equation \[ y'' + \sin(e^x)y = \cos(e^x), \quad 0<x<1, \] then which one of the following is its general solution on \( [0, 1] \)?

Let \( A \) be a \( 3 \times 3 \) real matrix and \( b \) be a \( 3 \times 1 \) real column vector. Consider the statements:
1. The Jacobi iteration method for the system \( (A + \epsilon I_3)x = b \) converges for any initial approximation and \( \epsilon>0 \).
2. The Gauss-Seidel iteration method for the system \( (A + \epsilon I_3)x = b \) converges for any initial approximation and \( \epsilon>0 \).
Which one of the following is correct?

Consider the following Linear Programming Problem \( P \): Minimize \( x_1 + 2x_2 \), subject to \[ 2x_1 + x_2 \leq 2, \quad x_1 + x_2 = 1, \quad x_1, x_2 \geq 0. \] The optimal value of the problem \( P \) is equal to:

Consider the following statements:
1. Every compact Hausdorff space is normal.
2. Every metric space is normal.
Which one of the following is correct?

Let \( {R}[X^2, X^3] \) be the subring of \( {R}[X] \) generated by \( X^2 \) and \( X^3 \). Consider the following statements: 1. The ring \( {R}[X^2, X^3] \) is a unique factorization domain.
2. The ring \( {R}[X^2, X^3] \) is a principal ideal domain.
Which one of the following is correct?

Given a prime number \( p \), let \( n_p(G) \) denote the number of \( p \)-Sylow subgroups of a finite group \( G \). Which one of the following is TRUE for every group \( G \) of order \( 2024 \)?

Consider the following limit: \[ \lim_{\epsilon \to 0} \frac{1}{\epsilon} \int_{0}^{\infty} e^{-x / \epsilon} \left( \cos(3x) + x^2 + \sqrt{x + 4} \right) dx. \] Which one of the following is correct?

Let \( A \in M_2({C}) \) be given by \[ A = \begin{bmatrix} 0 & 2
2 & 0 \end{bmatrix}. \] Let \( T: M_2({C}) \to M_2({C}) \) be the linear transformation given by \( T(B) = AB \). The characteristic polynomial of \( T \) is: