List of top Mathematics Questions

If \( y = \tan^{-1} \left( \frac{\sin 2x}{1 + \cos 2x} \right) \), then \[ \frac{dy}{dx} = \]

If \[ f(x) = \begin{cases} 6\beta - 3x, & \text{if } -4 \leq x<-2,
4x + 1, & \text{if } -2 \leq x \leq 2, \end{cases} \] is continuous on \( [-4, 2] \), then \( \alpha + \beta = \)

Evaluate the integral: \[ \int \left[ \frac{1 - \log x}{1 + (\log x)^2} \right]^2 dx \]

The general solution of \( \tan 3x = 1 \) is

The equation of normal to the curve \( y = \sin \left( \frac{\pi x}{4} \right) \) at the point (2, 5) is

If \[ \sec x + \tan x = 3, \quad x \in \left( 0, \frac{\pi}{2} \right), \] then \( \sin x = \)

The coordinates of the point where the line $\dfrac{x-1}{2}=\dfrac{y-2}{-3}=\dfrac{z+3}{4}$ meets the plane $2x+4y-z=1$ are

If the radius of a circular blot of oil is increasing at the rate of $2$ cm/min, then the rate of change of its area when its radius is $3$ cm is

Evaluate \( \int_0^{\frac{\pi}{2}} \left( e^{\sin x} - e^{\cos x} \right) \, dx \)

If \( f(x) = 2x^2 + bx + c \), \( f(0) = 3 \) and \( f(2) = 1 \), then \( (f \circ f)(1) = \)

If the equation \[ kxy + 5x + 3y + 2 = 0 \text{ represents a pair of lines, then } k = \]

Evaluate the integral: \[ \int_{-5}^{5} \frac{e^x + e^{-x}}{e^x - e^{-x}} \, dx. \]

If \[ \int \sqrt{x - 1} \left( \frac{x^2 + 1}{x^2} \right) dx = \frac{2}{3} (x - 1)^k + c, \quad \text{then the value of } k \text{ is} \]

If \[ A = \begin{bmatrix} 0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0 \end{bmatrix} \] then

If \[ A = \begin{pmatrix} 2 & 5 \\ 0 & 1 \\ 3 & 0 \end{pmatrix}, \quad A^{-1} = \begin{pmatrix} 3 & -1 \\ -6 & 6 \\ -5 & 2 \end{pmatrix} \] then the values of \( \alpha \) and \( \beta \) are, respectively

Evaluate \( \int_{-2}^{1} \left[ x + 1 \right] \, dx \), where \( [x] \) is the greatest integer function not greater than \( x \)

If \[ A = \begin{bmatrix} 2 & -1 \\ -1 & 2 \end{bmatrix}, \] such that \[ A^2 - 4A + 3I = 0, \] then \( A^{-1} \) is

For a sequence if \( S_n = \dfrac{5^n - 2^n}{2^n} \), then its fourth term is

The perimeter of a triangle is \(10\) cm. If one of its sides is \(4\) cm, then the remaining sides of the triangle, when the area of the triangle is maximum, are

The particular solution of the differential equation \[ y \frac{dx}{dy} = x \log x \quad \text{at} \quad x = e \text{ and } y = 1 \] is

If the points \( (1, 1, \lambda) \) and \( (-3, 0, 1) \) are equidistant from the plane \( 3x + 4y - 12z + 13 = 0 \), then the integer value of \( \lambda \) is

If \[ O = (0, 0, 0), \quad P = (1, \sqrt{2}, 1), \] then the acute angles made by the line OP with the \( XOY, \, YOZ, \, ZOX \text{ planes are, respectively,} \)

The logical expression \( [p \wedge (q \vee r)] \vee [\neg r \wedge \neg q \wedge p] \) is equivalent to

The cumulative distribution function of a continuous random variable \( X \) is given by \( F(X = x) = \dfrac{\sqrt{x}}{2} \). Then \( P(X>1) \) is

The particular solution of the differential equation \[ \left( y + x \frac{dy}{dx} \right) \sin y = \cos x \quad \text{at} \quad x = 0 \, \text{is:} \]