List of top Mathematics Questions asked in GATE Mathematics

Consider the sequence of Lebesgue measurable functions \(f_n: \mathbb{R} \to \mathbb{R}\) given by
\[ f_n(x) = \begin{cases} n^2(x-n), & \text{if } x \in [n, n + \frac{1}{n^2}] \\ 0, & \text{otherwise} \end{cases} \]
For a measurable subset E of \(\mathbb{R}\), denote m(E) to be the Lebesgue measure of E.
Which of the following statements is/are correct?

Define the characteristic function \(\chi_E\) of a subset E in \(\mathbb{R}\) by
\[ \chi_E(x) = \begin{cases} 1, & \text{if } x \in E \\ 0, & \text{if } x \notin E \end{cases} \]
For \(1 \le p < 2\), let \(L^p[0,1] = \{f: [0,1] \to \mathbb{R} : f \text{ is Lebesgue measurable and } \int_0^1 |f(x)|^p dx < \infty\}\).
Let \(f: [0,1] \to \mathbb{R}\) be defined by
\[ f(x) = \sum_{n=1}^\infty \frac{2^n}{n^3} \chi_{[\frac{1}{2^{n+1}}, \frac{1}{2^n}]}(x). \]
Consider the following two statements:
P: \(f \in L^p[0,1]\) for every \(p \in (1, 2)\).
Q: \(f \in L^1[0,1]\).
Then

Let \(x(t), y(t), t \in \mathbb{R}\), be two functions satisfying the following system of differential equations:
\[ x'(t) = y(t), \]
\[ y'(t) = x(t), \]
and \(x(0) = \alpha, y(0) = \beta\), where \(\alpha, \beta\) are real numbers.
Which of the following statements is/are correct?

Let \( f: \mathbb{R}^2 \to \mathbb{R} \) be defined by \[ f(x, y) = \begin{cases} \frac{2x^2y}{x^2 + y^2}, & (x, y) \neq (0,0) \\ 0, & (x, y) = (0,0) \end{cases} \] Then

Let \( C[0,1] = \{ f: [0,1] \to \mathbb{R} : f \text{ is continuous}\} \) and \[ d_\infty(f,g) = \sup\{ |f(x) - g(x)|: x \in [0, 1]\} \] for \( f, g \in C[0,1] \).
For each \( n \in \mathbb{N} \), define \( f_n: [0,1] \to \mathbb{R} \) by \( f_n(x) = x^n \) for all \( x \in [0,1] \).
Let \( P = \{f_n: n \in \mathbb{N}\} \).
Which of the following statements is/are correct?

Let \( G \) be an abelian group and \( \Phi: G \to (\mathbb{Z},+) \) be a surjective group homomorphism. Let \( 1 = \Phi(a) \) for some \( a \in G \).
Consider the following statements:
\( P \): For every \( g \in G \), there exists an \( n \in \mathbb{Z} \) such that \( g a^n \in \ker(\Phi) \).
\( Q \): Let \( e \) be the identity of G and \( \langle a \rangle \) be the subgroup generated by \( a \). Then \( G = \ker(\Phi) \langle a \rangle \) and \( \ker(\Phi) \cap \langle a \rangle = \{e\} \).
Which of the following statements is/are correct?

Let \( l^2 = \{(x_1, x_2, x_3, \dots) : x_n \in \mathbb{R} \text{ for all } n \in \mathbb{N} \text{ and } \sum_{n=1}^\infty x_n^2 < \infty\} \).
For a sequence \( (x_1, x_2, x_3, \dots) \in l^2 \), define \( \|(x_1, x_2, x_3, \dots)\|_2 = \left( \sum_{n=1}^\infty x_n^2 \right)^{1/2} \).
Consider the subspace \( M = \{(x_1, x_2, x_3, \dots) \in l^2 : \sum_{n=1}^\infty \frac{x_n}{4^n} = 0\} \).
Let \( M^\perp \) denote the orthogonal complement of \( M \) in the Hilbert space \( (l^2, \|.\|_2) \).
Consider \( \left(1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots\right) \in l^2 \).
If the orthogonal projection of \( \left(1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots\right) \) onto \( M^\perp \) is given by \( \alpha \left(\frac{1}{4}, \frac{1}{4^2}, \frac{1}{4^3}, \dots\right) \) for some \( \alpha \in \mathbb{R} \), then \( \alpha \) equals ..................

Let \(A = [a_{ij}]\) be a \(3 \times 3\) real matrix such that
\[ A \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix} = 2 \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}, \quad A \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} = 2 \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}, \quad \text{and} \quad A \begin{bmatrix} -1 \\ 1 \\ 0 \end{bmatrix} = 4 \begin{bmatrix} -1 \\ 1 \\ 0 \end{bmatrix}. \]
If \(m\) is the degree of the minimal polynomial of \(A\), then \(a_{11} + a_{21} + a_{31} + m\) equals ..................

Consider the Linear Programming Problem P:
\[ \text{Maximize } 3x_1 + 2x_2 + 5x_3 \]
subject to
\[ \begin{aligned} x_1 + 2x_2 + x_3 &\le 44, \\ x_1 + 2x_3 &\le 48, \\ x_1 + 4x_2 &\le 52, \\ x_1 \ge 0, \ x_2 \ge 0, \ x_3 &\ge 0 \end{aligned} \]
The optimal value of the problem P is equal to ..................

Let \( M = \begin{bmatrix} 3 & -1 & -2 \\ 0 & 2 & 4 \\ 0 & 0 & 1 \end{bmatrix} \) and \( I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \). If \(6M^{-1} = M^2 - 6M + \alpha I\) for some \(\alpha \in \mathbb{R}\), then the value of \(\alpha\) is equal to ................

Consider \(\mathbb{R}^4\) with the inner product \(\langle x, y \rangle = \sum_{i=1}^4 x_i y_i\), for \(x = (x_1, x_2, x_3, x_4)\) and \(y = (y_1, y_2, y_3, y_4)\).
Let \(M = \{(x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1 = x_3\}\) and \(M^\perp\) denote the orthogonal complement of M. The dimension of \(M^\perp\) is equal to ................

Let \(y(t)\) be the solution of the initial value problem
\[ y'' + 4y = \begin{cases} t, & 0 \le t \le 2, \\ 2, & 2 < t < \infty, \end{cases} \quad \text{and} \quad y(0) = y'(0) = 0. \]
If \(\alpha = y(\pi/2)\), then the value of \(\frac{4}{\pi}\alpha\) is ................................ (rounded off to 2 decimal places).

Let \(L^2[-1, 1] = \{f: [-1,1] \to \mathbb{R} : f \text{ is Lebesgue measurable and } \int_{-1}^1 |f(x)|^2dx < \infty\}\) and the norm \(||f||_2 = \left(\int_{-1}^1 |f(x)|^2 dx\right)^{1/2}\) for \(f \in L^2[-1,1]\).
Let \(F: (L^2[-1,1], ||.||_2) \to \mathbb{R}\) be defined by \[ F(f) = \int_{-1}^1 f(x)x^2dx \quad \text{for all } f \in L^2[-1,1]. \] If \(||F||\) denotes the norm of the linear functional F, then \(5||F||^2\) is equal to ................

Let X and Y be two topological spaces. A continuous map \(f: X \to Y\) is said to be proper if \(f^{-1}(K)\) is compact in X for every compact subset K of Y, where \(f^{-1}(K)\) is defined by \(f^{-1}(K) = \{x \in X : f(x) \in K\}\).
Consider \(\mathbb{R}\) with the usual topology. If \(\mathbb{R} \setminus \{0\}\) has the subspace topology induced from \(\mathbb{R}\) and \(\mathbb{R} \times \mathbb{R}\) has the product topology, then which of the following maps is proper?

Let C[0,1] denote the set of all real valued continuous functions defined on [0,1] and \(||f||_\infty = \sup\{|f(x)| : x \in [0,1]\}\) for all \(f \in C[0,1]\). Let
\[ X = \{ f \in C[0,1] : f(0) = f(1) = 0 \}. \]
Define \(F : (C[0,1], ||.||_\infty) \to \mathbb{R}\) by \(F(f) = \int_0^1 f(t)dt\) for all \(f \in C[0,1]\).
Denote \(S_X = \{f \in X : ||f||_\infty = 1\}\).
Then the set \(\{f \in X : F(f) = ||F||\} \cap S_X\) has

Consider the metrics \(\rho_1\) and \(\rho_2\) on \(\mathbb{R}\), defined by
\[ \rho_1(x, y) = |x-y| \quad \text{and} \quad \rho_2(x, y) = \begin{cases} 0, & \text{if } x = y \\ 1, & \text{if } x \neq y \end{cases} \]
Let \(X = \{n \in \mathbb{N} : n \ge 3\}\) and \(Y = \{n + \frac{1}{n} : n \in \mathbb{N}\}\).
Define \(f: X \cup Y \to \mathbb{R}\) by \[ f(x) = \begin{cases} 2, & \text{if } x \in X \\ 3, & \text{if } x \in Y \end{cases} \]
Consider the following statements:
P: The function \(f: (X \cup Y, \rho_1) \to (\mathbb{R}, \rho_1)\) is uniformly continuous.
Q: The function \(f: (X \cup Y, \rho_2) \to (\mathbb{R}, \rho_1)\) is uniformly continuous.
Then

Let \(C[0, 1] = \{ f : [0, 1] \to \mathbb{R} : f \text{ is continuous}\}\).
Consider the metric space \((C[0,1], d_\infty)\), where \[ d_\infty(f, g) = \sup\{ |f(x) - g(x)| : x \in [0, 1] \} \text{ for } f, g \in C[0,1]. \] Let \(f_0(x) = 0\) for all \(x \in [0,1]\) and \[ X = \{f \in (C[0, 1], d_\infty) : d_\infty(f_0, f) \ge \frac{1}{2}\}. \] Let \(f_1, f_2 \in C[0, 1]\) be defined by \(f_1(x) = x\) and \(f_2(x) = 1-x\) for all \(x \in [0,1]\).
Consider the following statements:
P: \(f_1\) is in the interior of X.
Q: \(f_2\) is in the interior of X.
Which of the following statements is correct?

Let \( M = \begin{bmatrix} 4 & -3 \\ 1 & 0 \end{bmatrix} \).
Consider the following statements:
P: \(M^8 + M^{12}\) is diagonalizable.
Q: \(M^7 + M^9\) is diagonalizable.
Which of the following statements is correct?

Consider \(\mathbb{R}^2\) with the usual Euclidean metric. Let
\[ X = \left\{\left(x, x \sin\frac{1}{x}\right) \in \mathbb{R}^2 : x \in (0,1]\right\} \cup \{(0, y) \in \mathbb{R}^2 : -\infty < y < \infty\}, \]
\[ Y = \left\{\left(x, \sin\frac{1}{x}\right) \in \mathbb{R}^2 : x \in (0,1]\right\} \cup \{(0, y) \in \mathbb{R}^2 : -\infty < y < \infty\}. \]

Consider the following statements:
P: \(X\) is a connected subset of \(\mathbb{R}^2\).
Q: \(Y\) is a connected subset of \(\mathbb{R}^2\).
Then

Let \(T: \mathbb{R}^3 \to \mathbb{R}^3\) be a linear transformation satisfying
\(T(1, 0, 0) = (0, 1, 1)\), \(T(1, 1, 0) = (1, 0, 1)\) and \(T(1, 1, 1) = (1, 1, 2)\).
Then

Let \(P(x) = 1 + e^{2\pi i x} + 2 e^{3\pi i x}\), \(x \in \mathbb{R}\), \(i = \sqrt{-1}\). Then
\[ \lim_{N \to \infty} \frac{1}{N} \sum_{k=0}^{N-1} P(k\sqrt{2}) \] is equal to

Let \(\mathbb{R}^3\) be a topological space with the usual topology and \(\mathbb{Q}\) denote the set of rational numbers. Define the subspaces X, Y, Z and W of \(\mathbb{R}^3\) as follows:
\(X = \{(x, y, z) \in \mathbb{R}^3 : |x| + |y| + |z| \in \mathbb{Q}\}\)
\(Y = \{(x, y, z) \in \mathbb{R}^3 : xyz = 1\}\)
\(Z = \{(x, y, z) \in \mathbb{R}^3 : x^2 + y^2 + z^2 = 1\}\)
\(W = \{(x, y, z) \in \mathbb{R}^3 : xyz = 0\}\)
Which of the following statements is correct?

Let \( f, g: \mathbb{R}^2 \to \mathbb{R} \) be defined by
\[ f(x, y) = x^2 - \frac{3}{2}xy^2 \quad \text{and} \quad g(x, y) = 4x^4 - 5x^2y + y^2 \] for all \( (x, y) \in \mathbb{R}^2 \).
Consider the following statements:
P: \( f \) has a saddle point at (0, 0).
Q: \( g \) has a saddle point at (0, 0).
Then

Let \(u(x, t)\) be the solution of the non-homogeneous wave equation \[ \frac{\partial^2 u}{\partial x^2} - \frac{\partial^2 u}{\partial t^2} = \sin x \sin(2t), \quad 0<x<\pi, \ t>0 \] \[ u(x, 0) = 0, \text{ and } \frac{\partial u}{\partial t}(x, 0) = 0, \quad \text{for } 0 \le x \le \pi, \] \[ u(0, t) = 0, \quad u(\pi, t) = 0, \quad \text{for } t \ge 0. \] Then the value of \( u \left( \frac{\pi}{2}, \frac{3\pi}{2} \right) \) is ................. (rounded off to 2 decimal places).