List of top Mathematics Questions

If the function \( f(x) \) defined below is continuous at \( x = 2 \), find the value of the constant \( k \): \[ f(x) = \begin{cases} kx^2 & \text{if } x \le 2 3x - 2 & \text{if } x > 2 \end{cases} \]

Find the particular solution of the differential equation \( \frac{dy}{dx} = \frac{y}{x} \) given the initial boundary condition that \( y = 2 \) when \( x = 1 \).

Find the integrating factor (I.F.) for the linear differential equation: \[ \frac{dy}{dx} - y\tan x = e^x \]

In a Linear Programming Problem (LPP), if the objective function to maximize is \( Z = 3x + 4y \) and the corner points of the feasible bounded region are \( (0,0), (4,0), (2,3), \) and \( (0,4) \), find the maximum value of \( Z \).

Let \( A \) be a square matrix of order 3 such that \( |A| = 5 \). Find the value of the determinant of its adjoint matrix, \( |\text{adj}(A)| \).

Find the general solution of the separable differential equation: \[ \frac{dy}{dx} = e^{x-y} + x^2e^{-y} \]

If \( A \) is a square matrix of order 3 such that its determinant value is \( |A| = -2 \), find the value of the scalar-scaled determinant \( |4A| \).

Evaluate the indefinite integral using pattern-based substitution: \[ \int \frac{\ln x - 1}{(\ln x)^2}\,dx \]

Find the area of the region enclosed by the ellipse given parametrically by the coordinates \( x = 2\sin\theta \) and \( y = 3\cos\theta \) where \( 0 \le \theta \le 2\pi \).

Determine the sum of the order and the degree of the following differential equation: \[ \left[1 + \left(\frac{dy}{dx}\right)^2\right]^{frac{3}{2}} = k \frac{d^2y}{dx^2} \]

Find the integrating factor (I.F.) for the linear differential equation: \[ \frac{dy}{dx} + \frac{2x}{1+x^2}y = \frac{1}{(1+x^2)^2} \]

Probability of getting a number \(>4\) on a fair die is:

The natural number 2 is :

If \( 0 < r < s \le n \) and \( ^nP_r = ^nP_s \), then the value of \( (r - s) \) is :

Five candidates are contesting an election, and three members are to be elected. A voter can vote for any number of candidates, but not more than the number of members to be elected. The number of ways a person can cast their vote is :

The product of first \( n \) odd natural numbers is :

The integral of the function \( \frac{1}{9 - 4x^2} \) is :

The interval, in which the function \( f(x) = \frac{3}{x} + \frac{x}{3} \) is strictly decreasing, is :

The exponent of 7 in \( ^{100}C_{50} \) is :

Let \( f(x) = x^3 - 6x^2 + 12x - 3 \), then at \( x = 2 \), \( f(x) \) has :

If \( x = at^4 \) and \( y = 2at^2 \), then \( \frac{d^2y}{dx^2} \) is equal to :

For \( I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \), if \( A = \begin{bmatrix} a & b \\ c & -a \end{bmatrix} \) be such that \( A^2 = I \), then :

If 2 and 3 are the two roots of the equation \[ 2x^3 + mx^2 - 13x + n = 0, \] then the values of \(m, n\) are respectively