List of top Mathematics Questions

If \( x = \int_0^y \frac{1}{\sqrt{1+9t^2}} \, dt \) and \( \frac{d^2y}{dx^2} = ay \), then the value of \( a \) is:
If \( \cos^{-1} \alpha + \cos^{-1} \beta + \cos^{-1} \gamma = 3\pi \), then the value of \( \alpha(\beta+\gamma) + \beta(\gamma+\alpha) + \gamma(\alpha+\beta) \) is:
The expression \( 2^{4n} - 15n - 1 \), where \( n \in \mathbb{N} \) (the set of natural numbers), is divisible by:
The sum of the first four terms of an arithmetic progression is 56. The sum of the last four terms is 112. If its first term is 11, then the number of terms is:
A function \( f : \mathbb{R} \to \mathbb{R} \), satisfies \[ \frac{f(x+y)}{3} = \frac{f(x) + f(y) + f(0)}{3} \quad \text{for all} \, x, y \in \mathbb{R}. \] If the function \( f \) is differentiable at \( x = 0 \), then \( f \) is:
If \( f \) is the inverse function of \( g \) and \( g'(x) = \frac{1}{1+x^n} \), then the value of \( f'(x) \) is:
The straight line \[ \frac{x-3}{3} = \frac{y-2}{1} = \frac{z-1}{0} \] is:
Let \( p(x) \) be a real polynomial of least degree which has a local maximum at \( x = 1 \) and a local minimum at \( x = 3 \). If \( p(1) = 6 \) and \( p(3) = 2 \), then \( p'(0) \) is equal to:
The value of the integral \( \int_{0}^{\pi/2} \log\left(\frac{4 + 3\sin x}{4 + 3\cos x}\right) dx \) is:
If \( (1 + x - 2x^2)^6 = 1 + a_1x + a_2x^2 + \ldots + a_{12}x^{12} \), then the value of \( a_2 + a_4 + a_6 + \ldots + a_{12} \) is:
If \( x = -1 \) and \( x = 2 \) are extreme points of \( f(x) = \alpha \log|x| + \beta x^2 + x \), \( x \neq 0 \), then:
A function \( f \) is defined by \( f(x) = 2 + (x - 1)^{2/3} \) on \( [0, 2] \). Which of the following statements is incorrect?
The line \( y - \sqrt{3}x + 3 = 0 \) cuts the parabola \( y^2 = x + 2 \) at the points \( P \) and \( Q \). If the co-ordinates of the point \( X \) are \( (\sqrt{3}, 0) \), then the value of \( XP \cdot XQ \) is:
For what value of \( 'a' \), the sum of the squares of the roots of the equation \( x^2 - (a - 2)x - a + 1 = 0 \) will have the least value?
If \( \vec{\alpha} = 3\hat{i} - \hat{j} + \hat{k} \), \( |\vec{\beta}| = \sqrt{5} \) and \( \vec{\alpha} \cdot \vec{\beta} = 3 \), then the area of the parallelogram for which \( \vec{\alpha} \) and \( \vec{\beta} \) are adjacent sides is:
The line parallel to the x-axis passing through the intersection of the lines \( ax + 2by + 3b = 0 \) and \( bx - 2ay - 3a = 0 \) where \( (a, b) \neq (0, 0) \) is:
Let \( f(x) = |1 - 2x| \), then:
Let \( \omega (\neq 1) \) be a cubic root of unity. Then the minimum value of the set \( \{ |a + b\omega + c\omega^2|^2 : a, b, c \) are distinct non-zero integers \( \} \) equals:
The number of reflexive relations on a set \( A \) of \( n \) elements is equal to:
Let \( \vec{a}, \vec{b}, \vec{c} \) be unit vectors. Suppose \( \vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c} = 0 \) and the angle between \( \vec{b} \) and \( \vec{c} \) is \( \frac{\pi}{6} \). Then \( \vec{a} \) is:
Consider three points \( P(\cos \alpha, \sin \beta) \), \( Q(\sin \alpha, \cos \beta) \) and \( R(0, 0) \), where \( 0<\alpha, \beta<\frac{\pi}{4} \). Then:
Let \( f \) be a function which is differentiable for all real \( x \). If \( f(2) = -4 \) and \( f'(x) \geq 6 \) for all \( x \in [2, 4] \), then:
Evaluate the integral \( \int_{-1}^{1} \frac{x^2 + |x| + 1}{x^2 + 2|x| + 1} \, dx \):
Let $f(x)$ be a second degree polynomial. If $f(1) = f(-1)$ and $p, q, r$ are in A.P., then $f'(p), f'(q), f'(r)$ are
Let \( \phi(x) = f(x) + f(2a - x) \), \( x \in [0, 2a] \) and \( f'(x)>0 \) for all \( x \in [0, a] \). Then \( \phi(x) \) is: