List of top Mathematics Questions

If \( L \) is the line of intersection of two planes \[ x + 2y + 2z = 15 \quad \text{and} \quad x - y + z = 4 \] and the direction ratios of the line \( L \) are \( (a, b, c) \), then the value of \[ \frac{a^2 + b^2 + c^2}{b^2} \] is:

The foot of the perpendicular drawn from \( A(1,2,2) \) onto the plane \[ x + 2y + 2z - 5 = 0 \] is \( B(a, \beta, \gamma) \). If \( \pi(x,y,z) = x + 2y + 2z + 5 = 0 \) is a plane then \(-\pi(A):\pi(B) \) is: 

If \[ y = \log \left( x - \sqrt{x^2 - 1} \right), \] then \[ (x^2 - 1)y'' + xy' + e^y + \sqrt{x^2 - 1} = \] evaluates to:
The sum of the maximum and minimum values of the function \[ f(x) = \frac{x^2 - x + 1}{x^2 + x + 1} \] is:
If the area of the region enclosed by the curve \( ay = x^2 \) and the line \( x + y = 2a \) is \( k a^3 \), then \( k \) is:
The order and degree of the differential equation \[ \frac{dy}{dx} + \left( \frac{d^2y}{dx^2} + 2 \right)^{\frac{1}{2}} + \frac{d^3y}{dx^3} + 5 = 0 \] are respectively:

If \( y = \sin x + A \cos x \) is the general solution of \[ \frac{dy}{dx} + f(x)y = \sec x, \] then an integrating factor of the differential equation is: 

If the mean of the data 7, 8, 9, 7, 8, 7, \(\lambda\), 8 is 8, then the variance of the data:
If \( AB = 2i + 3j - 6k \), \( BC = 6i - 2j + 3k \) are the vectors along two sides of a triangle ABC, then the perimeter of triangle ABC is:
In \( \triangle ABC \), \( (r_2 + r_3) \csc^2 \frac{A}{2} =\)
If \( 0<x<\frac{1}{2} \) and \( \alpha = \sin^{-1} x + \cos^{-1} \left(\frac{x}{2} +\frac{\sqrt{3} - 3x^2}{2} \right) \), then \( \tan \alpha + \cot \alpha = \):
Evaluate the sum: \[ \tan^2 \frac{\pi}{16} + \tan^2 \frac{2\pi}{16} + \tan^2 \frac{3\pi}{16} + \tan^2 \frac{4\pi}{16} + \tan^2 \frac{5\pi}{16} + \tan^2 \frac{6\pi}{16} + \tan^2 \frac{7\pi}{16} \]
If \(f(x) = \begin{cases} ax^2 + bx - \frac{13}{8}, & x \le 1 \\ 3x - 3, & 1<x \le 2 \\ bx^3 + 1, & x>2 \end{cases}\)is differentiable \(\forall x \in \mathbb{R}\), then \( a - b = \)
\[ f(x) = \begin{cases} \frac{x - |x|}{x}, & x \neq 0 \\[8pt] 2, & x = 0 \end{cases} \] Which of the following is true for \( f(x) \)
Evaluate \( \int (\log x)^m x^n dx \).
If \[ \int \frac{\sqrt[4]{x}}{\sqrt{x} + \sqrt[4]{x}} \, dx = \frac{2}{3} \left[ A \sqrt[4]{x^3} + B \sqrt[4]{x^2} + C \sqrt[4]{x} + D \log \left( 1 + \sqrt[4]{x} \right) \right] + K \] then \( \frac{2}{3} (A + B + C + D) = \)}
Evaluate the integral: \[ \int \sin^{-1} \left( \sqrt{\frac{x - a}{x}} \right) dx \]
Find the domain of \( f(x) \) given: \[ \int \frac{\sin x \cos x}{\sqrt{\cos^4 x - \sin^4 x}} dx = -\frac{f(x)}{2} + C. \] then domain of f{x) is
If \( \int_0^{2\pi} (\sin^4 x + \cos^4 x) \, dx = K \int_0^\pi \sin^2 x \, dx + L \int_0^\frac{\pi}{2} \cos^2 x \, dx \) and \( K, L \in \mathbb{N} \), then the number of possible ordered pairs \( (K, L) \) is}
Among the options given below, from which option a differential equation of order two can be formed?
Evaluate the integral: \[ I = \int \frac{1}{x^m \sqrt[m]{x^m + 1}} dx. \]
If \( m \) is the slope and \( P(\beta, \beta) \) is the midpoint of a chord of contact of the circle \[ x^2 + y^2 = 125, \] then the number of values of \( \beta \) such that \( \beta \) and \( m \) are integers is: \
A rhombus is inscribed in the region common to the two circles \[ x^2 + y^2 - 4x - 12 = 0 \] and \[ x^2 + y^2 + 4x - 12 = 0. \] If the line joining the centres of these circles and the common chord of them are the diagonals of this rhombus, then the area (in Sq. units) of the rhombus is:
The equation of the straight line whose slope is \( -\frac{2}{3} \) and which divides the line segment joining \( (1,2) \) and \( (-3,5) \) in the ratio 4:3 externally is:
If \( X \sim B(6, p) \) is a binomial variate and \[ \frac{P(X=4)}{P(X=2)} = \frac{1}{9}, \] then the value of \( p \) is: