List of top Mathematics Questions

The two curves \( x^3 - 3xy^2 + 2 = 0 \) and \( 3x^2y - y^3 = 2 \) are

The circumcenter of a triangle formed by the lines \( xy + 2x + 2y + 4 = 0 \) and \( x + y + 2 = 0 \) is

Each side of a square subtends an angle of \( 60^\circ \) at the top of a tower \( h \) meters high standing in the center of the square. If \( a \) is the length of each side of the square, then

If \( n \in \mathbb{N} \), then \( |\sin nx| \) is

In \( \triangle ABC \), if \( a = 30, b = 24, c = 18 \), then \( r_3 \) is equal to

The number of solutions of the equation \( 2 \sin^{-1} \sqrt{x^2 - x + 1} + \cos^{-1} \sqrt{x^2 - x + 2} = \frac{3\pi}{2} \) in the interval \( [0, 5\pi] \) is

The value of \( \cos^{-1} \left( \cos \frac{7\pi}{6} \right) \) is

In \( \triangle ABC \), \( \frac{b - c}{r_1} + \frac{c - a}{r_2} + \frac{a - b}{r_3} \) is equal to

ABCD is a rectangular field. A vertical lamp post of height 12 m stands at the corner A. If the angle of elevation of its top from B is 60° and from C is 45°, then the area of the field is

The number of values of \( x \) in the interval \( [0, 5\pi] \) satisfying the equation \( 3\sin^2 x - 7\sin x + 2 = 0 \) is

The expression \( \left( 1 + \cos \frac{\pi}{8} \right) \left( 1 + \cos \frac{3\pi}{8} \right) \left( 1 + \cos \frac{5\pi}{8} \right) \left( 1 + \cos \frac{7\pi}{8} \right) \) is equal to

The inequality \( n!>2^{n-1} \) is true for

If \( 2x + 3b + 6c = 0 \), then at least one root of the equation \( ax^2 + bx + c = 0 \) lies in the interval

The ratio in which the line \( 3x + 4y + 2 = 0 \) divides the distance between \( 3x + 4y + 5 = 0 \) and \( 3x + 4y - 5 = 0 \) is

The series \( 1 + \frac{1 + x}{2!} + \frac{1 + x + x^2}{3!} + \frac{1 + x + x^2 + x^3}{4!} + \dots \) is equal to

If the circle \( x^2 + y^2 + 2gx + 2fy + c = 0 \) is touched by \( y = x \) at \( P \) such that \( OP = 6\sqrt{2} \), then the value of \( c \) is

The coefficient of \( x^4 \) in the expansion of \( (1 + x + x^2 + x^3)^n \) is

If the roots of the equation \( \frac{\alpha}{x-\alpha} + \frac{\beta}{x-\beta} = 1 \) are equal in magnitude but opposite in sign, then \( \alpha + \beta \) is equal to

The line \( x + 2y = 4 \) is translated parallel to itself by 3 units in the sense of increasing \( x \) and then rotated by 30° in the anti-clockwise direction about the point where the shifted line cuts the x-axis. The equation of the line in the new position is:

If the equation \( 12x^2 + 7xy - py^2 - 18x + qy + 6 = 0 \) represents a pair of perpendicular straight lines, then

The value of \[ 2 + \frac{1}{5} + \frac{1}{3} + \frac{1}{5^3} + \frac{1}{5^5} + \dots \] is:

The equation of the parabola whose vertex and focus are \( (0, 4) \) and \( (0, 2) \) respectively, is:

The distance between the foci of a hyperbola is double the distance between its vertices and the length of its conjugate axis is 6. The equation of the hyperbola referred to its axes as axes of coordinates are:

The existence of the unique solution of the system \[ x + y + z = \lambda, \quad 5x - y + \mu z = 10, \quad 2x + 3y - z = 6 \] depends on:

Let \[ A = \begin{pmatrix} 1 & -1 & 1 2 & 1 & -3 1 & 1 & 1 \end{pmatrix}, \quad 10B = \begin{pmatrix} -5 & 0 & \alpha 4 & 2 & 2 1 & -2 & 3 \end{pmatrix} \] If \( B \) is the inverse of matrix \( A \), then \( \alpha \) is: