List of top Mathematics Questions

The length of major axis and minor axis of an ellipse are, respectively, \( m \) and \( n \). If \( m^2 - n^2 = 45 \) and the eccentricity of the ellipse is \( \frac{\sqrt{5}}{3} \), then the length of the major axis is

A circle touches the \( x \)-axis at \( (9,0) \). If it also touches the straight line \( y = 14 \), then the equation of the circle is

The focus of the parabola \( x^2 - 4x + 8y + 4 = 0 \) is

Let \( ABC \) be an equilateral triangle. If the coordinates of \( A \) are \( (-2,2) \) and the side \( BC \) is along the line \( x + y = 6 \), then the length of the side of the triangle is

The straight line \( ax + by + c = 0 \) passes through the point \( (-10,7) \). If the line is perpendicular to \( 11x - 7y = 13 \), then the value of \( c \) is equal to

If \( \theta = \cot^{-1}\sqrt{\frac{1-x}{1+x}} \), then \( \sec^2 \theta \) is equal to

If \( 5\sin^{-1}\alpha + 3\cos^{-1}\alpha = \pi \), then \( \alpha \) is equal to

If \( f(x) = \tan^{-1}\left(\frac{2x}{1 - x^2}\right) \), then \( f\left(\frac{1}{\sqrt{3}}\right) \) is equal to

If \( \sin \alpha = \frac{12}{13} \), where \( \frac{\pi}{2}<\alpha<\frac{3\pi}{2} \), then the value of \( \tan \alpha \) is equal to

If \( \sin x + \sin y = a \), \( \cos x + \cos y = b \) and \( x + y = \frac{2\pi}{3} \), then the value of \( \frac{a}{b} \) is equal to

If \( x + z = 2y \) and \( y = \frac{\pi}{4} \), then \( \tan x \tan y \tan z = \)

\( \tan 15^\circ + \tan 75^\circ = \)

\( -5<x \le -1 \) implies \( -21<5x + 4 \le b \), the least value of \( b \) is

The set of all \( x \) satisfying the inequalities \( -4 \le 2 - 3x<7 \) is

The following system of equations \[ x + y + z = 1 \] \[ 2x + 3y - mz = 2 \] \[ 3x + 5y + 3z = 3 \] has no unique solution. Then the value of \( m \) is equal to

If the matrix \( \begin{pmatrix} 8-k & 2 -2 & 4-k \end{pmatrix} \) is singular, then the value of \( k \) is equal to

If \( \begin{pmatrix} -1 & 2 3 & -4 -5 & 6 \end{pmatrix} \begin{pmatrix} 7 8 \end{pmatrix} = \begin{pmatrix} \alpha \beta 13 \end{pmatrix} \), then the value of \( \alpha + \beta \) is equal to

Let \( A \) be a \(3 \times 3\) matrix and let \( B = 3A \). If \( |A| = 5 \), then the value of \( \frac{|\text{adj } B|}{|3A|} \) is equal to

Let \( A = \{0,2,4,6,8\} \). The number of 5-digit numbers that can be formed using the digits in \( A \) without replacement, is

In the binomial expansion of \( (2x + \alpha)^8 \), the co-efficients of \( x^2 \) and \( x^3 \) are equal. Then the value of \( \alpha \) is equal to

If \( ^{11}P_r = 7920 \), then the value of \( r \) is equal to

If \( \alpha = {^nC_r} \) and \( \beta = {^nC_{r-1}} \), then \( 1 + \frac{\alpha}{\beta} \) is equal to

Let \( \alpha = \sum_{k=0}^{5} {^{10}C_{2k}} \) and \( \beta = \sum_{k=0}^{4} {^{10}C_{2k+1}} \). Then \( \alpha - \beta \) is equal to

The product of first 5 terms of a G.P., whose terms are increasing, is 32. The third term of the G.P. is

The general term of a sequence is \( t_n = \frac{n(n+6)}{n+4}, \, n = 1,2,3,\ldots \). If \( t_n = 5 \), then the value of \( n \) is