List of top Mathematics Questions on Mathematics asked in GATE MA

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

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?

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

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

Let \(\sigma \in S_8\), where \(S_8\) is the permutation group on 8 elements. Suppose \(\sigma\) is the product of \(\sigma_1\) and \(\sigma_2\), where \(\sigma_1\) is a 4-cycle and \(\sigma_2\) is a 3-cycle in \(S_8\). If \(\sigma_1\) and \(\sigma_2\) are disjoint cycles, then the number of elements in \(S_8\) which are conjugate to \(\sigma\) is ..................

Consider the transportation problem between five sources and four destinations as given in the cost table below. The supply and demand at each of the source and destination are also provided:
\begin{tabular}{|c|c|c|c|c|c|c|} \cline{3-7} \multicolumn{2}{c|}{} & \multicolumn{4}{c|}{DESTINATIONS} & Supply
\cline{3-7} \multicolumn{2}{c|}{} & P & Q & R & S &
\hline \multirow{5}{*}{\rotatebox[origin=c]{90}{SOURCES}} & 1 & 13 & 8 & 12 & 9 & 20
\cline{2-7} & 2 & 10 & 7 & 5 & 20 & 10
\cline{2-7} & 3 & 3 & 19 & 5 & 12 & 50
\cline{2-7} & 4 & 4 & 9 & 7 & 15 & 30
\cline{2-7} & 5 & 14 & 0 & 1 & 7 & 40
\hline \multicolumn{2}{|c|}{Demand} & 60 & 10 & 20 & 60 &
\hline \end{tabular}
Let \(C_N\) and \(C_L\) be the total cost of the initial basic feasible solution obtained from the North-West corner method and the Least-Cost method, respectively. Then \(C_N - C_L\) equals ..................

Consider the following Linear Programming Problem P:
Minimize \(3x_1 + 4x_2\)
subject to \[ x_1 - x_2 \le 1, \] \[ x_1 + x_2 \ge 3, \] \[ x_1 \ge 0, x_2 \ge 0. \] The optimal value of the problem P is ..................

Let \[ \frac{z}{1 - z - z^2} = \sum_{n=0}^{\infty} a_n z^n, \quad a_n \in \mathbb{R} \] for all z in some neighbourhood of 0 in \(\mathbb{C}\).
Then the value of \(a_6 + a_5\) is equal to ................

Let \(D = \{z \in \mathbb{C} : |z|<1\}\) and \(f: D \to \mathbb{C}\) be defined by \[ f(z) = z - 25z^3 + \frac{z^5}{5!} - \frac{z^7}{7!} + \frac{z^9}{9!} - \frac{z^{11}}{11!} \] Consider the following statements:
P: \(f\) has three zeros (counting multiplicity) in D.
Q: \(f\) has one zero in \(U = \{z \in \mathbb{C} : \frac{1}{2}<|z|<1\}\).
Then

Let \(N \subseteq \mathbb{R}\) be a non-measurable set with respect to the Lebesgue measure on \(\mathbb{R}\).
Consider the following statements:
P: If \(M = \{x \in N : x \text{ is irrational}\}\), then \(M\) is Lebesgue measurable.
Q: The boundary of \(N\) has positive Lebesgue outer measure.
Then

For \(k \in \mathbb{N}\), let \(E_k\) be a measurable subset of \([0,1]\) with Lebesgue measure \(\frac{1}{k^2}\).
Define \[ E = \bigcap_{n=1}^{\infty} \bigcup_{k=n}^{\infty} E_k \quad \text{and} \quad F = \bigcup_{n=1}^{\infty} \bigcap_{k=n}^{\infty} E_k \] Consider the following statements:
P: Lebesgue measure of \(E\) is equal to zero.
Q: Lebesgue measure of \(F\) is equal to zero.
Then