List of top Mathematics Questions

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:
If \( z = x + iy \) satisfies the equation \[ z^2 + az + a^2 = 0, \quad a \in \mathbb{R}, \] then:
If \( \omega \) is the complex cube root of unity and \[ \left( \frac{a + b\omega + c\omega^2}{c + a\omega + b\omega^2} \right)^k + \left( \frac{a + b\omega + c\omega^2}{b + a\omega^2 + c\omega} \right)^2 = 2, \] then \( 2k + 1 \) is always:
The differential equation for which \( ax + by = 1 \) is the general solution is:
The solution of the differential equation \( e^x y dx + e^x dy + xdx = 0 \) is:
If \( A = \int_0^{\infty} \frac{1 + x^2}{1 + x^4} dx \) and \( B = \int_0^1 \frac{1 + x^2}{1 + x^4} dx \), then:
If the length of the sub-tangent at any point P on a curve is proportional to the abscissa of the point P, then the equation of that curve is (C is an arbitrary constant):
Let \( f(x) = \begin{cases 1 + \frac{2x}{a}, & 0 \le x \le 1
ax, & 1<x \le 2 \end{cases} \). If \( \lim_{x \to 1} f(x) \) exists, then the sum of the cubes of the possible values of \( a \) is: }
If \[ y = (x - 1)(x + 2)(x^2 + 5)(x^4 + 8), \] then \[ \lim\limits_{x \to -1} \left( \frac{dy}{dx} \right) = ? \]
If \( y = \sinh^{-1} \left(\frac{1 - x}{1 + x} \right) \), then \( \frac{dy}{dx} \) is given by:
If the angle \( \theta \) between the line \( \frac{x + 1}{1} = \frac{y - 1}{2} = \frac{z - 2}{2} \) and the plane \( 2x - y + \sqrt{\lambda}z + 4 = 0 \) is such that \( \sin \theta = \frac{1}{3} \), then the value of \( \lambda \) is: