List of top Mathematics Questions

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:
Let \( f(x) = |1 - 2x| \), then
If \( \operatorname{adj} B = A, \ |P|=|Q|=1 \), then \[ \operatorname{adj}(Q^{-1} B P^{-1}) = \ ? \]
Suppose \( \alpha, \beta, \gamma \) are the roots of the equation \( x^3 + qx + r = 0 \) (with \( r \ne 0 \)) and they are in A.P. Then the rank of the matrix \[ \begin{pmatrix} \alpha & \beta & \gamma
\beta & \gamma & \alpha
\gamma & \alpha & \beta \end{pmatrix} \] is:
The line parallel to the x-axis passing through the intersection of the lines \[ ax + 2by + 3b = 0 \quad \text{and} \quad bx - 2ay - 3a = 0 \] where \( (a,b) \neq (0,0) \), 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:
If the sum of the squares of the roots of the equation \[ x^2 - (a-2)x - (a+1) = 0 \] is least for an appropriate real parameter \( a \), then the value of \( a \) will be:
The value of the integral \[ \int_0^{\pi/2} \log\!\left(\frac{4+3\sin x}{4+3\cos x}\right) dx is:
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:
If \( z_1, z_2 \) are complex numbers such that \( \dfrac{2z_1}{3z_2} \) is a purely imaginary number, then the value of \[ \left|\frac{z_1 - z_2}{z_1 + z_2}\right| \] is:
If the sum of \( n \) terms of an A.P. is \( 3n^2 + 5n \) and its \( m \)-th term is 164, then the value of \( m \) is:
If for a matrix \( A \), \( |A| = 6 \) and \[ \operatorname{adj}A = \begin{bmatrix} 1 & -2 & 4
4 & 1 & 1
-1 & k & 0 \end{bmatrix}, \] then \( k \) is equal to:
The straight line \[ \frac{x-3}{3} = \frac{y-2}{1} = \frac{z-1}{0} \] is:
If \( g(f(x)) = |\sin x| \) and \( f(g(x)) = (\sin\sqrt{x})^2 \), then:
If \( E \) and \( F \) are two independent events with \( P(E)=0.3 \) and \( P(E\cup F)=0.5 \), then \( P(E/F) - P(F/E) \) equals:
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:
If \( {}^9P_5 + 5 \cdot {}^9P_4 = {}^{10}P_r \), then the value of \( r \) is:
The value of the expression \[ {}^{47}C_4 + \sum_{j=1}^{5} {}^{52-j}C_3 \] is:
If \[ x = \int_0^y \frac{1}{\sqrt{1+9t^2}} \, dt \quad \text{and} \quad \frac{d^2 y}{dx^2} = ay, \] then \( a \) is equal to:
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:
The value of \[ \int_{-1}^{1} \frac{x^3 + |x| + 1}{x^2 + 2|x| + 1} \, dx is equal to: \]
A function \( f \) is defined by \( f(x) = 2 + (x-1)^{2/3 \) on \( [0,2] \). Which of the following statements is incorrect?}
Let \( \vec{a}, \vec{b}, \vec{c} \) be vectors of equal magnitude such that the angle between \( \vec{a} \) and \( \vec{b} \) is \( \alpha \), between \( \vec{b} \) and \( \vec{c} \) is \( \beta \), and between \( \vec{c} \) and \( \vec{a} \) is \( \gamma \). Then the minimum value of \( \cos\alpha + \cos\beta + \cos\gamma \) is:
If \( \vec{a}, \vec{b}, \vec{c} \) are non-coplanar vectors and \( \lambda \) is a real number, then the vectors \[ \vec{a} + 2\vec{b} + 3\vec{c}, \quad \lambda\vec{b} + 4\vec{c}, \quad (2\lambda - 1)\vec{c} \] are non-coplanar for:
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?