List of top Mathematics Questions

If \( a,b,c \) are in A.P. and the equations \[ (b-c)x^2 + (c-a)x + (a-b) = 0 \] \[ 2(c+a)x^2 + (b+c)x = 0 \] have a common root, then:

Let \( a_n \) denote the term independent of \( x \) in the expansion of \[ \left[x + \frac{\sin(1/n)}{x^2}\right]^{3n}, \] then \[ \lim_{n\to\infty} \frac{(a_n)n!}{\,{}^{3n}P_n} \] equals:

The number of solutions of \[ \sin^{-1} x + \sin^{-1}(1-x) = \cos^{-1} x \] is:

If \( x=-1 \) and \( x=2 \) are extreme points of \[ f(x) = \alpha \log|x| + \beta x^2 + x \quad (x \ne 0), \] then:

If \( f \) is the inverse function of \( g \) and \( g'(x) = \dfrac{1{1 + x^n} \), then the value of \( f'(x) \) is:}

Evaluate \[ \lim_{x \to 0} \frac{\tan\!\left(\lfloor -\pi^2 \rfloor x^2\right) - x^2 \tan\!\left(\lfloor -\pi^2 \rfloor\right)}{\sin^2 x} \]

Let \[ f_n(x) = \tan\frac{x}{2}(1+\sec x)(1+\sec 2x)\cdots(1+\sec 2^{n-1}x), \] then:

If \( \theta \) is the angle between two vectors \( \vec{a} \) and \( \vec{b} \) such that \( |\vec{a}| = 7, |\vec{b}| = 1 \) and \[ |\vec{a} \times \vec{b}|^2 = k^2 - (\vec{a} - \vec{b})^2, \] then the values of \( k \) and \( \theta \) are:

Let \( \omega (\ne 1) \) be a cube root of unity. Then the minimum value of the set \[ \left\{ |a + b\omega + c\omega^2|^2 : a,b,c \text{ are distinct non-zero integers} \right\} \] equals:

If \( (1 + x - 2x^2)^6 = 1 + a_1 x + a_2 x^2 + \cdots + a_{12x^{12} \), then the value of \( a_2 + a_4 + a_6 + \cdots + a_{12} \) 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:

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:

If \( \operatorname{adj} B = A, \ |P|=|Q|=1 \), then \[ \operatorname{adj}(Q^{-1} B P^{-1}) = \ ? \]

The value of the integral \[ \int_0^{\pi/2} \log\!\left(\frac{4+3\sin x}{4+3\cos x}\right) dx is:

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: