List of top Mathematics Questions

If \( \theta \) is the angle between two vectors \( \vec{a} \) and \( \vec{b} \) such that \( |\vec{a}| = 7 \), \( |\vec{b}| = 1 \) and \( |\vec{a} \times \vec{b}|^2 = k^2 - (\vec{a} \cdot \vec{b})^2 \), then the values of \( k \) and \( \theta \) are:
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 value of \( \displaystyle \int_0^{1.5} \left\lfloor x^2 \right\rfloor \, dx \) is:
The expression \( 2^{4n} - 15n - 1 \), where \( n \in \mathbb{N} \) (the set of natural numbers), is divisible by:
The straight line \[ \frac{x-3}{3} = \frac{y-2}{1} = \frac{z-1}{0} \] is:
If \( f \) is the inverse function of \( g \) and \( g'(x) = \frac{1}{1+x^n} \), then the value of \( f'(x) \) is:
If the matrix \[ \begin{pmatrix} 0 & a & a\\ 2b & b & -b\\ c & -c & c \end{pmatrix} \] is orthogonal, then the values of \( a, b, c \) are:
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:
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 \( x = -1 \) 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?
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:
If \( {}^9P_3 + 5 \cdot {}^9P_4 = {}^{10}P_r \), then the value of \( 'r' \) 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?
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:
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:
If for a matrix \( A \), \( |A| = 6 \) and \( \text{adj } A = \begin{bmatrix} 1 & -2 & 4 \\ 4 & 1 & 1 \\ -1 & k & 0 \end{bmatrix} \), then \( k \) is equal to:
If \( g(f(x)) = |\sin x| \) and \( f(g(x)) = (\sin \sqrt{x})^2 \), then:
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 \( \vec{a}, \vec{b}, \vec{c} \) 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:
The number of reflexive relations on a set \( A \) of \( n \) elements is equal to:
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: