List of top Questions asked in GATE MA

A student was supposed to multiply a positive real number \( p \) with another positive real number \( q \). Instead, the student divided \( p \) by \( q \). If the percentage error in the student’s answer is 80\%, the value of \( q \) is:

If the sum of the first 20 consecutive positive odd numbers is divided by 202, the result is:

The ratio of the number of girls to boys in class VIII is the same as the ratio of the number of boys to girls in class IX. The total number of students (boys and girls) in classes VIII and IX is 450 and 360, respectively. If the number of girls in classes VIII and IX is the same, then the number of girls in each class is:

If ‘=’ denotes increasing order of intensity, then the meaning of the words [drizzle — rain — downpour] is analogous to [ — quarrel — feud]. Which one of the given options is appropriate to fill the blank?

Let \(GL_2(\mathbb{C})\) denote the group of \(2 \times 2\) invertible complex matrices with usual matrix multiplication. For \(S, T \in GL_2(\mathbb{C})\), \(\langle S, T \rangle\) denotes the subgroup generated by \(S\) and \(T\). Let
\[ S = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \in GL_2(\mathbb{C}) \] and \(G_1, G_2, G_3\) be three subgroups of \(GL_2(\mathbb{C})\) given by
\[ G_1 = \langle S, T_1 \rangle, \text{ where } T_1 = \begin{pmatrix} i & 0 \\ 0 & i \end{pmatrix}, \]
\[ G_2 = \langle S, T_2 \rangle, \text{ where } T_2 = \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}, \]
\[ G_3 = \langle S, T_3 \rangle, \text{ where } T_3 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}. \]
Let \(Z(G_i)\) denote the center of \(G_i\) for \(i = 1, 2, 3\).
Which of the following statements is correct?

Let \(\ell^2 = \{(x_1, x_2, x_3, \dots) : x_n \in \mathbb{R} \text{ for all } n \in \mathbb{N} \text{ and } \sum_{n=1}^\infty x_n^2 < \infty\}\).
For a sequence \((x_1, x_2, x_3, \dots) \in \ell^2\), define \[ ||(x_1, x_2, x_3, \dots)||_2 = \left(\sum_{n=1}^\infty x_n^2\right)^{1/2}. \]
Let \(S: (\ell^2, ||.||_2) \to (\ell^2, ||.||_2)\) and \(T: (\ell^2, ||.||_2) \to (\ell^2, ||.||_2)\) be defined by
\[ S(x_1, x_2, x_3, \dots) = (y_1, y_2, y_3, \dots), \text{ where } y_n = \begin{cases} 0, & n=1 \\ x_{n-1}, & n \ge 2 \end{cases} \]
\[ T(x_1, x_2, x_3, \dots) = (y_1, y_2, y_3, \dots), \text{ where } y_n = \begin{cases} 0, & n \text{ is odd} \\ x_n, & n \text{ is even} \end{cases} \]
Then

Let \(c_{00} = \{(x_1, x_2, x_3, \dots) : x_i \in \mathbb{R}, i \in \mathbb{N}, x_i \neq 0 \text{ only for finitely many indices } i\}\).
For \((x_1, x_2, x_3, \dots) \in c_{00}\), let \(||(x_1, x_2, x_3, \dots)||_\infty = \sup\{|x_i| : i \in \mathbb{N}\}\).
Define \(F, G: (c_{00}, ||.||_\infty) \to (c_{00}, ||.||_\infty)\) by
\[ F((x_1, x_2, \dots, x_n, \dots)) = ((1+1)x_1, (2+\frac{1}{2})x_2, \dots, (n+\frac{1}{n})x_n, \dots), \]
\[ G((x_1, x_2, \dots, x_n, \dots)) = \left(\frac{x_1}{1+1}, \frac{x_2}{2+\frac{1}{2}}, \dots, \frac{x_n}{n+\frac{1}{n}}, \dots\right), \]
for all \((x_1, x_2, \dots, x_n, \dots) \in c_{00}\).
Then

Consider the linear system \(Mx = b\), where \(M = \begin{pmatrix} 2 & -1 \\ -4 & 3 \end{pmatrix}\) and \(b = \begin{pmatrix} -2 \\ 5 \end{pmatrix}\).
Suppose \(M = LU\), where L and U are lower triangular and upper triangular square matrices, respectively. Consider the following statements:
P: If each element of the main diagonal of L is 1, then \(\text{trace}(U) = 3\).
Q: For any choice of the initial vector \(x^{(0)}\), the Jacobi iterates \(x^{(k)}, k = 1,2,3,...\) converge to the unique solution of the linear system \(Mx = b\).
Then

Let \(D = \{z \in \mathbb{C} : |z| < 1\}\) and \(f: D \to \mathbb{C}\) be an analytic function given by the power series \(f(z) = \sum_{n=0}^{\infty} a_n z^n\), where \(a_0 = a_1 = 1\) and \(a_n = \frac{1}{2^{2n}}\) for \(n \ge 2\).
Consider the following statements:
P: If \(z_0 \in D\), then \(f\) is one-one in some neighbourhood of \(z_0\).
Q: If \(E = \{z \in \mathbb{C} : |z| \le \frac{1}{2}\}\), then \(f(E)\) is a closed subset of \(\mathbb{C}\).
Which of the following statements is/are correct?

Let \(\Omega\) be an open connected subset of \(\mathbb{C}\) containing \(U = \{z \in \mathbb{C} : |z| \le \frac{1}{2}\}\).
Let \(\mathcal{F} = \{ f : \Omega \to \mathbb{C} : f \text{ is analytic and } \sup_{z,w \in U} |f(z) - f(w)| = 1 \}\).
Consider the following statements:
P: There exists \(f \in \mathcal{F}\) such that \(|f'(0)| \ge 2\).
Q: \(|f^{(3)}(0)| \le 48\) for all \(f \in \mathcal{F}\), where \(f^{(3)}\) denotes the third derivative of \(f\).
Then

Let \(\mathcal{R} = \{p(x) \in \mathbb{Q}[x] : p(0) \in \mathbb{Z}\}\), where \(\mathbb{Q}\) denotes the set of rational numbers and \(\mathbb{Z}\) denotes the set of integers. For \(a \in \mathcal{R}\), let \(\langle a \rangle\) denote the ideal generated by \(a\) in \(\mathcal{R}\).
Which of the following statements is/are correct?

Consider the rings
\[ S_1 = \mathbb{Z}[x]/\langle 2, x^3 \rangle \quad \text{and} \quad S_2 = \mathbb{Z}_2[x]/\langle x^2 \rangle \]
where \(\langle 2, x^3 \rangle\) denotes the ideal generated by \(\{2, x^3\}\) in \(\mathbb{Z}[x]\) and \(\langle x^2 \rangle\) denotes the ideal generated by \(x^2\) in \(\mathbb{Z}_2[x]\).
Which of the following statements is/are correct?

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 ................