List of top Mathematics Questions asked in GATE Mathematics

Let \( (\mathbb{R}^2, d_1) \) and \( (\mathbb{R}^2, d_2) \) be two metric spaces with \[ d_1\left( (x_1, x_2), (y_1, y_2) \right) = |x_1 - y_1| + |x_2 - y_2| \] \[ {and} \quad d_2\left( (x_1, x_2), (y_1, y_2) \right) = \frac{d_1\left( (x_1, x_2), (y_1, y_2) \right)}{1 + d_1\left( (x_1, x_2), (y_1, y_2) \right)}. \] If the open ball centered at \( (0,0) \) with radius \( \frac{1}{7} \) in \( (\mathbb{R}^2, d_1) \) is equal to the open ball centered at \( (0,0) \) with radius \( \frac{1}{\alpha} \) in \( (\mathbb{R}^2, d_2) \), then the value of \( \alpha \) is (in integer).
Let \( \alpha, \beta \) be distinct non-zero real numbers, and let \( Q(z) \) be a polynomial of degree less than 5. If the function \[ f(z) = \frac{\alpha^6 \sin \beta z - \beta^6 (e^{2az} - Q(z))}{z^6} \] satisfies Morera's theorem in \( \mathbb{C} \setminus \{0\} \), then the value of \( \frac{\alpha}{4\beta} \) is equal to (in integer).

For \( X = (x_1, x_2, x_3)^T \in \mathbb{R}^3 \), consider the quadratic form:

\[ Q(X) = 2x_1^2 + 2x_2^2 + 3x_3^2 + 4x_1x_2 + 2x_1x_3 + 2x_2x_3. \] Let \( M \) be the symmetric matrix associated with the quadratic form \( Q(X) \) with respect to the standard basis of \( \mathbb{R}^3 \). 
Let \( Y = (y_1, y_2, y_3)^T \in \mathbb{R}^3 \) be a non-zero vector, and let

\[ a_n = \frac{Y^T(M + I_3)^{n+1}Y}{Y^T(M + I_3)^n Y}, \quad n = 1, 2, 3, \dots \] Then, the value of \( \lim_{n \to \infty} a_n \) is equal to (in integer).

Consider the inner product space of all real-valued continuous functions defined on \( [-1, 1] \) with the inner product \[ \langle f, g \rangle = \int_{-1}^{1} f(x) g(x) \, dx. \] If \( p(x) = \alpha + \beta x^2 - 30x^4 \), where \( \alpha, \beta \in \mathbb{R} \), is orthogonal to all the polynomials having degree less than or equal to 3, with respect to this inner product, then \( \alpha + 5\beta \) is equal to (in integer).
Let \( E \subset F \) and \( F \subset K \) be field extensions which are not algebraic. Let \( \alpha \in K \) be algebraic over \( F \) and \( \alpha \notin F \). Let \( L \) be the subfield of \( K \) generated over \( E \) by the coefficients of the monic polynomial of minimal degree over \( F \) which has \( \alpha \) as a zero. Then, which of the following is/are TRUE?
All rings considered below are assumed to be associative and commutative with \( 1 \neq 0 \). Further, all ring homomorphisms map 1 to 1.

Consider the following statements about such a ring \( R \):

P1: \( R \) is isomorphic to the product of two rings \( R_1 \) and \( R_2 \).
P2: \( \exists r_1, r_2 \in R \) such that \( r_1^2 = r_1 \neq 0 \), \( r_2^2 = 0 \), and \( r_1 + r_2 = 1 \).
P3: \( R \) has ideals \( I_1, I_2 \subset R \) with \( R \neq I_1 \), \( (0) \neq I_2 \), and \( R = I_1 + I_2 \) and \( I_1 \cap I_2 = (0) \).
P4: \( \exists a, b \in R \) with \( a \neq 0 \), \( b \neq 0 \) such that \( ab = 0 \).

Then, which of the following is/are TRUE?
Let \( X \) be an uncountable set. Let the topology on \( X \) be defined by declaring a subset \( U \subset X \) to be open if \( X - U \) is either empty or finite or countable, and the empty set to be open. Then, which of the following is/are TRUE?
Let \( \{x_k\}_{k=1}^\infty \) be an orthonormal set of vectors in a real Hilbert space \( X \) with inner product \( \langle \cdot, \cdot \rangle \). Let \( n \in \mathbb{N} \), and let \( Y \) be the linear span of \( \{ x_k \}_{k=1}^n \) over \( \mathbb{R} \). For \( x \in X \), let \[ S_n(x) = \sum_{k=1}^n \langle x, x_k \rangle x_k. \] Then, which of the following is/are TRUE?
Let \( u(x, t) \) be the solution of the initial value problem \[ \frac{\partial u}{\partial t} + 3 \frac{\partial u}{\partial x} = u, \quad x \in \mathbb{R}, \quad t>0, \quad u(x, 0) = \cos x, \] and let \( v(x, t) \) be the solution of the initial value problem \[ \frac{\partial v}{\partial t} + 3 \frac{\partial v}{\partial x} = v^2, \quad x \in \mathbb{R}, \quad t>0, \quad v(x, 0) = \cos x. \] Then, which of the following is/are TRUE?

Let \( 0<\alpha<1 \). Define \[ C^\alpha[0, 1] = \left\{ f : [0, 1] \to \mathbb{R} \ : \ \sup_{s \neq t, \, s,t \in [0, 1]} \frac{|f(t) - f(s)|}{|t - s|^\alpha}<\infty \right\}. \] It is given that \( C^\alpha[0, 1] \) is a Banach space with respect to the norm \( \| \cdot \|_\alpha \) given by \[ \| f \|_\alpha = |f(0)| + \sup_{s \neq t, \, s,t \in [0, 1]} \frac{|f(t) - f(s)|}{|t - s|^\alpha}. \] Let \( C[0, 1] \) be the space of all real-valued continuous functions on \( [0, 1] \) with the norm \( \| f \|_\infty = \sup_{0 \leq t \leq 1} |f(t)| \). 
If \( T: C^\alpha[0, 1] \to C[0, 1] \) is the map \( T f = f \), where \( f \in C^\alpha[0, 1] \), then which one of the following is/are TRUE?

Let \( K \) be an algebraically closed field containing a finite field \( F \). Let \( L \) be the subfield of \( K \) consisting of elements of \( K \) that are algebraic over \( F \). 
Consider the following statements: 
S1: \( L \) is algebraically closed. 
S2: \( L \) is infinite. 
Then, which one of the following is correct?

Consider the metric spaces \( X = (\mathbb{R}, d_1) \) and \( Y = ([0, 1], d_2) \) with the metrics defined by \[ d_1(x, y) = |x - y|, \quad x, y \in \mathbb{R}, \quad {and} \quad d_2(x, y) = |x - y|, \quad x, y \in [0, 1], \] respectively. Then, which one of the following is TRUE?

Consider the following two spaces:
\[ \begin{aligned} X &= (C[-1, 1], \| \cdot \|_\infty), \quad \text{the space of all real-valued continuous functions} \\ &\quad \text{defined on } [-1, 1] \text{ equipped with the norm } \| f \|_\infty = \sup_{t \in [-1, 1]} |f(t)|. \\ Y &= (C[-1, 1], \| \cdot \|_2), \quad \text{the space of all real-valued continuous functions} \\ &\quad \text{defined on } [-1, 1] \text{ equipped with the norm } \| f \|_2 = \left( \int_{-1}^1 |f(t)|^2 \, dt \right)^{1/2}. \end{aligned} \]
Let \( W \) be the linear span over \( \mathbb{R} \) of all the Legendre polynomials. Then, which one of the following is correct?

In the following, all subsets of Euclidean spaces are considered with the respective subspace topologies. Define an equivalence relation \( \sim \) on the sphere \[ S = \left\{ (x_1, x_2, x_3) \in \mathbb{R}^3 : x_1^2 + x_2^2 + x_3^2 = 1 \right\} \] by \( (x_1, x_2, x_3) \sim (y_1, y_2, y_3) \) if \( x_3 = y_3 \), for \( (x_1, x_2, x_3), (y_1, y_2, y_3) \in S \). Let \( [x_1, x_2, x_3] \) denote the equivalence class of \( (x_1, x_2, x_3) \), and let \( X \) denote the set of all such equivalence classes. Let \( L : S \to X \) be given by \[ L\left( (x_1, x_2, x_3) \right) = [x_1, x_2, x_3]. \] If \( X \) is provided with the quotient topology induced by the map \( L \), then which one of the following is TRUE?
Let \( C \) be the curve of intersection of the surfaces \( z^2 = x^2 + y^2 \) and \( 4x + z = 7 \). If \( P \) is a point on \( C \) at a minimum distance from the \( xy \)-plane, then the distance of \( P \) from the origin is:
Let \( y_1(x) \) and \( y_2(x) \) be the two linearly independent solutions of the differential equation

\[ (1 + x^2) y'' - x y' + (\cos^2 x) y = 0, \]
satisfying the initial conditions

\[ y_1(0) = 3, \quad y_1'(0) = -1, \quad y_2(0) = -5, \quad y_2'(0) = 2. \]
Define

\[ W(x) = \left| \begin{array}{cc} y_1(x) & y_2(x) \\ y_1'(x) & y_2'(x) \\ \end{array} \right|. \]

Then, the value of \( W\left( \frac{1}{2} \right) \) is:
Let \( y(x) \) be the solution of the differential equation \[ x^2 y'' + 7xy' + 9y = x^{-3} \log_e x, \quad x>0, \] satisfying \( y(1) = 0 \) and \( y'(1) = 0 \). Then, the value of \( y(e) \) is equal to:
Let \( y(x) \) be the solution of the initial value problem \[ \frac{dy}{dx} = \sin(\pi(x + y)), \quad y(0) = 0. \] Using Euler's method, with the step-size \( h = 0.5 \), the approximate value of \( y(1.5) + 2y(1) \) is equal to (in integer):
Let \( \alpha, \beta, \gamma, \delta \in \mathbb{R} \) be such that the quadrature formula \[ \int_{-1}^{1} f(x) \, dx = \alpha f(-1) + \beta f(1) + \gamma f'(-1) + \delta f'(1) \] is exact for all polynomials of degree less than or equal to 3. Then, \( 9(\alpha^2 + \beta^2 + \gamma^2 + \delta^2) \) is equal to (in integer):

Consider \[ I = \frac{1}{2\pi i} \int_C \frac{\sin z}{1 - \cos(z^3)} \, dz, \] where \( C = \{ z \in \mathbb{C} : z = x + iy, |x| + |y| = 1, x, y \in \mathbb{R} \} \) is oriented positively as a simple closed curve. Then, the value of \( 120I \) is equal to _________ (in integer).

Let \( \vec{F} = (y - z)\hat{i} + (z - x)\hat{j} + (x - y)\hat{k} \) be a vector field, and let \( S \) be the surface \( x^2 + y^2 + (z - 1)^2 = 9, 1 \leq z \leq 4 \).
If \( \hat{n} \) denotes the unit outward normal vector to \( S \), then the value of

\[ \frac{1}{\pi} \left| \iint_S (\vec{v} \times \vec{F}) \cdot \hat{n} \, dS \right| \]

is equal to _________ (in integer).
Let \( W \) be the vector space (over \( \mathbb{R} \)) consisting of all bounded real-valued solutions of the differential equation \[ \frac{d^4y}{dx^4} + 2 \frac{d^2y}{dx^2} + y = 0. \] Then, the dimension of \( W \) is _________ (in integer).
The volume of the region bounded by the cylinders \( x^2 + y^2 = 4 \) and \( x^2 + z^2 = 4 \) is _________ (rounded to TWO decimal places).
Let \( \hat{a} \) be a unit vector parallel to the tangent at the point \( P(1, 1, \sqrt{2}) \) to the curve of intersection of the surfaces \( 2x^2 + 3y^2 - z^2 = 3 \) and \( x^2 + y^2 = z^2 \). Then, the absolute value of the directional derivative of \[ f(x, y, z) = x^2 + 2y^2 - 2\sqrt{11} z \] at P in the direction of \( \hat{a} \) is _________ (in integer).

Let \( M \) be a \( 7 \times 7 \) matrix with entries in \( \mathbb{R} \) and having the characteristic polynomial \[ c_M(x) = (x - 1)^\alpha (x - 2)^\beta (x - 3)^2, \] where \( \alpha>\beta \). Let \( {rank}(M - I_7) = {rank}(M - 2I_7) = {rank}(M - 3I_7) = 5 \), where \( I_7 \) is the \( 7 \times 7 \) identity matrix. 
If \( m_M(x) \) is the minimal polynomial of \( M \), then \( m_M(5) \) is equal to __________ (in integer).