List of top Mathematics Questions asked in GATE Mathematics

Let \( \{(a, b) : a, b \in {R, a<b \} }\) be a basis for a topology \( \tau \) on \( {R} \). Which of the following is/are correct?

Let \( T, S : {R}^4 \to {R}^4 \) be two non-zero, non-identity \( {R} \)-linear transformations. Assume \( T^2 = T \). Which of the following is/are true?

Let \( p_1<p_2 \) be the two fixed points of the function \( g(x) = e^x - 2 \), where \( x \in {R} \). For \( x_0 \in {R} \), let the sequence \( (x_n)_{n \geq 1} \) be generated by the fixed-point iteration \[ x_n = g(x_{n-1}), \quad n \geq 1. \] Which one of the following is/are correct?

Which of the following is/are eigenvalue(s) of the Sturm–Liouville problem \[ y'' + \lambda y = 0, \quad 0 \leq x \leq \pi, \] with the boundary conditions \[ y(0) = y'(0), \quad y(\pi) = y'(\pi)? \]
The problem: Let \( f : \mathbb{R}^2 \to \mathbb{R} \) be a function such that \[ f(x, y) = \begin{cases} \left( 1 - \cos\left(\frac{x^2}{y^2}\right) \right) \sqrt{x^2 + y^2}, & \text{if } y \neq 0, x \in \mathbb{R}, \\ 0, & \text{otherwise.} \end{cases} \] Which of the following is/are correct?

For an integer \( n \), let \( f_n(x) = xe^{-nx }\), where \( x \in [0, 1] \). Let \( S := \{f_n : n \geq 1\} \). Consider the metric space \( (C([0, 1]), d) \), where \[ d(f, g) = \sup_{x \in [0, 1]} |f(x) - g(x)|, \quad f, g \in C([0, 1]). \] Which of the following statement(s) is/are true?}

Let \( T : {R}^4 \to {R}^4 \) be an \( {R} \)-linear transformation such that 1 and 2 are the only eigenvalues of \( T \). Suppose the dimensions of \({Kernel}(T - I_4)\) and \({Range}(T - 2I_4)\) are 1 and 2, respectively. Which of the following is/are possible (upper triangular) Jordan canonical form(s) of \( T \)?
Let \( L^2([-1, 1]) \) denote the space of all real-valued Lebesgue square-integrable functions on \( [-1, 1] \), with the usual norm \( \|\cdot\| \). Let \( P_1 \) be the subspace of \( L^2([-1, 1]) \) consisting of all the polynomials of degree at most 1. Let \( f \in L^2([-1, 1]) \) be such that \[ \|f\|^2 = \frac{18}{5}, \quad \int_{-1}^1 f(x) dx = 2, \quad {and} \quad \int_{-1}^1 xf(x) dx = 0. \] Then \[ \inf_{g \in P_1} \|f - g\|^2 = ........... \]

The maximum value of \( f(x, y, z) = 10x + 6y - 8z \) subject to the constraints \[ 5x - 2y + 6z \leq 20, \quad 10x + 4y - 6z \leq 30, \quad x, y, z \geq 0, \] is equal to …………. (round off to TWO decimal places).

Let \( K \subseteq \mathbb{C} \) be the field extension of \( \mathbb{Q} \) obtained by adjoining all the roots of the polynomial equation \( (x^2 - 2)(x^2 - 3) = 0 \). The number of distinct fields \( F \) such that \( \mathbb{Q} \subseteq F \subseteq K \) is equal to ……….. (answer in integer).

Let \( H \) be the subset of \( S_3 \) consisting of all \( \sigma \in S_3 \) such that \[ {Trace}(A_1 A_2 A_3) = {Trace}((A_1 \sigma(A_2) A_3)), \] for all \( A_1, A_2, A_3 \in M_2(\mathbb{C}) \). The number of elements in \( H \) is equal to ……… (answer in integer).

Let \( r : [0,1] \to \mathbb{R}^2 \) be a continuously differentiable path from \( (0,2) \) to \( (3,0) \) and let \[ \mathbf{F} : \mathbb{R}^2 \to \mathbb{R}^2 be  \mathbf{F}(x, y) = (1 - 2y, 1 - 2x). \] The line integral of \( \mathbf{F} \) along \( r \) is equal to \(\_\_\_\_\_\_\) (round off to TWO decimal places).

The problem: Let \( u(x,t) \) be the solution of the initial value problem \[ \frac{\partial^2 u}{\partial t^2} - \frac{\partial^2 u}{\partial x^2} = 0, \quad x \in \mathbb{R}, \, t > 0, \] \[ u(x,0) = 0, \quad x \in \mathbb{R}, \quad \frac{\partial u}{\partial t}(x,0) = \begin{cases} x^4 (1 - x)^4, & 0 < x < 1, \\ 0, & \text{otherwise}. \end{cases} \] If \( \alpha = \inf \{ t > 0 : u(2,t) > 0 \} \), then \( \alpha \) is equal to \(\_\_\_\_\_\_\) (round off to TWO decimal places).

The global maximum of \( f(x, y) = (x^2 + y^2)e^{-x-y} \) on \( {(x, y)}  in \mathbb{R}^2 : x \geq 0, y \geq 0\ \) is equal to \(\_\_\_\_\_\_\) (round off to TWO decimal places).

Let \( k \in \mathbb{R} \) and \( D = \{(r, \theta) : 0<r<2, 0<\theta<\pi\ \). Let \( u(r, \theta) \) be the solution of the following boundary value problem \[ \frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{1}{r^2} \frac{\partial^2 u}{\partial \theta^2} = 0, \quad (r, \theta) \in D, \] \[ u(r, 0) = u(r, \pi) = 0, \quad 0 \leq r \leq 2, \] \[ u(2, \theta) = k \sin(2\theta), \quad 0<\theta<\pi. \] If \( u\left(\frac{1}{4}, \frac{\pi}{4}\right) = 2 \), then the value of \( k \) is equal to ………. (round off to TWO decimal places).

Let \( X \) be the normed space \( ({R}^2, \|\cdot\|) \), where \[ \| (x, y) \| = |x| + |y|, \quad (x, y) \in {R}^2. \] Let \( S = \{ (x, 0) : x \in {R \} \) and \( f : S \to {R} \) be given by \( f((x, 0)) = 2x \) for all \( x \in {R} \). Recall that a Hahn–Banach extension of \( f \) to \( X \) is a continuous linear functional \( F \) on \( X \) such that \( F|_S = f \) and \( \|F\| = \|f\| \), where \( \|F\| \) and \( \|f\| \) are the norms of \( F \) and \( f \) on \( X \) and \( S \), respectively. Which of the following is/are true?}
Let \( \ell^2_{{Z}} = \{ (x_j)_{j \in {Z} : x_j \in {R} { and } \sum_{j = -\infty}^\infty x_j^2<\infty \} \) endowed with the inner product \[ \langle x, y \rangle = \sum_{j = -\infty}^\infty x_j y_j, \quad x = (x_j)_{j \in {Z}}, \, y = (y_j)_{j \in {Z}}. \] Let \( T : \ell^2_{{Z}} \to \ell^2_{{Z}} \) be given by \( T((x_j)_{j \in {Z}}) = (y_j)_{j \in {Z}} \), where \[ y_j = \frac{x_j + x_{-j}}{2}, \quad j \in {Z}. \] Which of the following is/are correct?}
Let \( D = \{(x, y) \in {R^2 : x>0 { and } y>0\} \). If the following second-order linear partial differential equation \[ y^2 \frac{\partial^2 u}{\partial x^2} - x^2 \frac{\partial^2 u}{\partial y^2} + y \frac{\partial u}{\partial y} = 0 \quad {on } D \] is transformed to \[ \left( \frac{\partial^2 u}{\partial \eta^2} - \frac{\partial^2 u}{\partial \xi^2} \right) + \left( \frac{\partial u}{\partial \eta} + \frac{\partial u}{\partial \xi} \right) \frac{1}{2\eta} + \left( \frac{\partial u}{\partial \eta} - \frac{\partial u}{\partial \xi} \right) \frac{1}{2\xi} = 0 \quad {on } D, \] for some \( a, b \in {R} \), via the coordinate transform \( \eta = \frac{x^2}{2} \) and \( \xi = \frac{y^2}{2} \), then which one of the following is correct?}
Let \( A \) be a \( 2 \times 2 \) non-diagonalizable real matrix with a real eigenvalue \( \lambda \) and \( v \) be an eigenvector of \( A \) corresponding to \( \lambda \). Which one of the following is the general solution of the system \( y' = Ay \) of first-order linear differential equations?
Let \( \ell^p = \left\{ x = (x_n)_{n \geq 1 : x_n \in {R}, \|x\|_p = \left( \sum_{n=1}^\infty |x_n|^p \right)^{1/p}<\infty \right\} \) for \( p = 1, 2 \). Let \[ c_0 = \{ (x_n)_{n \geq 1} : x_n = 0 { for all but finitely many } n \geq 1 \}. \] For \( x = (x_n)_{n \geq 1} \in c_0 \), define \( f(x) = \sum_{n=1}^\infty \frac{x_n}{\sqrt{n}} \). Consider the following statements:} 1. There exists a continuous linear functional \( F \) on \( (\ell^1, \|\cdot\|_1) \) such that \( F = f { on } c_0 \).
2. There exists a continuous linear functional \( G \) on \( (\ell^2, \|\cdot\|_2) \) such that \( G = f { on } c_0 \).
Which one of the following is correct?
Consider the initial value problem (IVP): \[ \frac{dy}{dx} = e^{-y}, \quad y(0) = 0. \] 1. The IVP has a unique solution on \( {R} \).
2. Every solution of the IVP is bounded on its maximal interval of existence.
Which one of the following is correct?
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 \( X \) be the space \( {R}/{Z} \) with the quotient topology induced from the usual topology on \( {R} \). Consider the following statements:
1. \( X \) is compact.
2. \( X \setminus \{z\} \) is connected for any \( z \in X \).
Which one of the following is correct?
Let \( f(x) = |x| + |x - 1| + |x - 2|, \, x \in [-1, 2] \). Which one of the following numerical integration rules gives the exact value of the integral \[ \int_{-1}^2 f(x) \, dx? \]
Consider the following statements:
1. There exists a proper subgroup \( G \) of \( ({Q}, +) \) such that \( {Q}/G \) is a finite group.
2. There exists a subgroup \( G \) of \( ({Q}, +) \) such that \( {Q}/G \) is isomorphic to \( ({Z}, +) \).
Which one of the following is correct?