List of top Mathematics Questions

Let the functions \( f: \mathbb{R} \to \mathbb{R} \) and \( g: \mathbb{R}^2 \to \mathbb{R} \) be given by \[ f(x_1, x_2) = x_1^2 + x_2^2 - 2x_1x_2, \quad g(x_1, x_2) = 2x_1^2 + 2x_2^2 - x_1x_2. \] Consider the following statements: 
S1: For every compact subset \( K \) of \( \mathbb{R} \), \( f^{-1}(K) \) is compact. 
S2: For every compact subset \( K \) of \( \mathbb{R} \), \( g^{-1}(K) \) is compact. Then, which one of the following is correct?

Consider the function \( F: \mathbb{R}^2 \to \mathbb{R}^2 \) given by \[ F(x, y) = (x^3 - 3xy^2 - 3x, 3x^2y - y^3 - 3y). \] Then, for the function \( F \), the inverse function theorem is:
The partial differential equation \[ (1 + x^2) \frac{\partial^2 u}{\partial x^2} + 2x(1 - y^2) \frac{\partial^2 u}{\partial x \partial y} + (1 - y^2) \frac{\partial^2 u}{\partial y^2} + x \frac{\partial u}{\partial x} + (1 - y^2) \frac{\partial u}{\partial y} = 0 \] is:

Let \( g(x, y) = f(x, y)e^{2x + 3y} \) be defined in \( \mathbb{R}^2 \), where \( f(x, y) \) is a continuously differentiable non-zero homogeneous function of degree 4. Then,

\[ x \frac{\partial g}{\partial x} + y \frac{\partial g}{\partial y} = 0 \text{ holds for} \]

Let \( X = \{ f \in C[0,1] : f(0) = 0 = f(1) \} \) with the norm \( \|f\|_\infty = \sup_{0 \leq t \leq 1} |f(t)| \), where \( C[0,1] \) is the space of all real-valued continuous functions on \( [0,1] \).

Let \( Y = C[0,1] \) with the norm \( \|f\|_2 = \left( \int_0^1 |f(t)|^2 \, dt \right)^{\frac{1}{2}} \). Let \( U_X \) and \( U_Y \) be the closed unit balls in \( X \) and \( Y \) centered at the origin, respectively. Consider \( T: X \to \mathbb{R} \) and \( S: Y \to \mathbb{R} \) given by

\[ T(f) = \int_0^1 f(t) \, dt \quad \text{and} \quad S(f) = \int_0^1 f(t) \, dt. \]

Consider the following statements:
S1: \( \sup |T(f)| \) is attained at a point of \( U_X \).
S2: \( \sup |S(f)| \) is attained at a point of \( U_Y \).

Then, which one of the following is correct?

Consider the system of ordinary differential equations \[ \frac{dX}{dt} = MX, \] where \( M \) is a \( 6 \times 6 \) skew-symmetric matrix with entries in \( \mathbb{R} \). Then, for this system, the origin is a stable critical point for

Consider the following subsets of the Euclidean space \( \mathbb{R}^4 \): 
\( S = \{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1^2 + x_2^2 + x_3^2 - x_4^2 = 0 \} \), 
\( T = \{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1^2 + x_2^2 + x_3^2 - x_4^2 = 1 \} \), 
\( U = \{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1^2 + x_2^2 + x_3^2 - x_4^2 = -1 \} \). 
Then, which one of the following is TRUE?

Given that the Laplace transforms of \( J_0(x), J_0'(x), \) and \( J_0''(x) \) exist, where \( J_0(x) \) is the Bessel function. Let \( Y = Y(s) \) be the Laplace transform of the Bessel function \( J_0(x) \). Then, which one of the following is TRUE?


Then, which one of the following is TRUE?

Let \( y = P_n(x) \) be the unique polynomial of degree \( n \) satisfying the Legendre differential equation \[ (1 - x^2)y'' - 2xy' + n(n + 1)y = 0 \quad {and} \quad y(1) = 1. \] Then, the value of \( P_{11}'(1) \) is equal to _________ (in integer).
The integral \[ 80 \int_0^{\frac{\pi}{4}} \frac{(\sin \theta + \cos \theta)}{(9 + 16 \sin 2\theta)} \, d\theta \] is equal to:
The system of equations \[ x + y + z = 6, \] \[ x + 2y + 5z = 9, \] \[ x + 5y + \lambda z = \mu, \] has no solution if:

The following empirical relationship describes how the number of trees \( N(t) \) in a patch changes over time \( t \): \[ N(t) = -2t^2 + 12t + 24 \] where \( t = 0 \) is when the number of trees were first counted. Given this relationship, the maximum number of trees that occur in the patch is

In a lake with a large population of fish, there are 4 blue fish for every red fish. Every fish has an equal probability of being caught. If you dip a net into this lake and pick up 4 individuals at random, the probability that you will get 2 fish of each colour is:
There are 9 different rock lizard species, each with a unique colour. Lizards of 2 different species sit on a rock at any given time. The number of possible colour combinations of rock lizards seen together on a rock is:

If \( \text{Re} \left( \frac{2z + i}{z + i} \right) + \text{Re} \left( \frac{2z - i}{z - i} \right) = 2 \) is a circle of radius \( r \) and centre \( (a, b) \), then \( \frac{15ab}{r^2} \) is equal to:

If two vectors \( \mathbf{a} \) and \( \mathbf{b} \) satisfy the equation:

\[ \frac{|\mathbf{a} + \mathbf{b}| + |\mathbf{a} - \mathbf{b}|}{|\mathbf{a} + \mathbf{b}| - |\mathbf{a} - \mathbf{b}|} = \sqrt{2} + 1, \]

then the value of

\[ \frac{|\mathbf{a} + \mathbf{b}|}{|\mathbf{a} - \mathbf{b}|} \]

is equal to:

If a box contains 19 unbiased coins and 1 biased coin with both faces heads, and a coin is randomly chosen from this box and tossed. If the head appears, then the probability that the unbiased coin was selected is:
If \( f(\theta) = \frac{\tan(\tan(\theta)) - \tan(\sin(\theta))}{\tan(\theta) - \sin(\theta)} \) is continuous at \( \theta = 0 \), then the value of \( f(\theta) \) at \( \theta = 0 \) is equal to.
A circle with center at \( (x, y) = (0.5, 0) \) and radius = 0.5 intersects with another circle with center at \( (x, y) = (1, 1) \) and radius = 1 at two points. One of the points of intersection \( (x, y) \) is:

A quadratic polynomial \( (x - \alpha)(x - \beta) \) over complex numbers is said to be square invariant if \[ (x - \alpha)(x - \beta) = (x - \alpha^2)(x - \beta^2). \] Suppose from the set of all square invariant quadratic polynomials we choose one at random. The probability that the roots of the chosen polynomial are equal is ___________. (rounded off to one decimal place) 

\[ \int_9^{10} \frac{\sqrt{1 + x^2 + x}}{\sqrt{1 + x^2 - x}} \, dx = \frac{1}{m} \left[ \left( \sqrt{1 + x^2 + x} + x \right)^n \left( n \sqrt{1 + x^2 - x} - x \right) \right] + c \] where \( c \) is the constant of integration and \( m, n \in \mathbb{N} \), then \( m + n \) is ______.
Let \( L_1 : \frac{x - 1}{3} = \frac{y}{4} = \frac{z}{5} \) and \( L_2 : \frac{x - p}{2} = \frac{y}{3} = \frac{z}{4} \). If the shortest distance between \( L_1 \) and \( L_2 \) is \( \frac{1}{\sqrt{6}} \), then the possible value of \( p \) is:
\[ \cot^{-1}\left(\frac{7}{4}\right) + \cot^{-1}\left(\frac{19}{4}\right) + \cot^{-1}\left(\frac{39}{4}\right) + \dots \, \infty \]
\[ \sum_{k=1}^{n} \left( \alpha^k + \frac{1}{\alpha^k} \right)^2 = 20, \quad \alpha \text{ is one of the roots of } x^2 + x + 1 = 0, \text{ then } n = ? \]