List of top Mathematics Questions

In a triangle ABC, if \( r_1 = 6 \), \( r_2 = 9 \), \( r_3 = 18 \), then \( \cos A \) is:
Let \( r \) and \( \theta \) respectively be the modulus and amplitude of the complex number \( z = 2 - i \left( 2 \tan \frac{5\pi}{8} \right) \), then \( (r, \theta) \) is equal to
If \( A(1, 2, -3) \), \( B(2, 3, -1) \), and \( C(3, 1, 1) \) are the vertices of triangle \( \Delta ABC \), then find \( \left| \frac{\cos A}{\cos B} \right|.\)
If \( (p, q) \) is the center of the circle which cuts the three circles \( x^2 + y^2 - 2x - 4y + 4 = 0 \), \( x^2 + y^2 + 2x - 4y + 1 = 0 \), and \( x^2 + y^2 - 4x - 2y - 11 = 0 \) orthogonally, then find \( p + q \).
If the probability distribution of a random variable \( X \) is as follows, then the variance of \( X \) is: \[ X = x \quad 2 \quad 3 \quad 5 \quad 9 \] \[ P(X = x) = k \quad 2k \quad 3k^2 \quad k^2 \]
There are 6 cards numbered 1 to 6, one number on one card. Two cards are drawn at random without replacement. Let X denote the sum of the numbers on the two cards drawn. Then P(X > 3) is:

In a triangle \(ABC\), if

\[ (a - b)^2 \cos^2 \frac{C}{2} + (a + b)^2 \sin^2 \frac{C}{2} = a^2 + b^2, \]

then \( \cos A \) is:

Evaluate the expression: \[ \cos^{-1} \frac{3}{5} + \sin^{-1} \frac{5}{13} + \tan^{-1} \frac{16}{63} \]
If 4 letters are selected at random from the letters of the word PROBABILITY, compute the probability that at least one letter is repeated in the selection.
The pole of the line \(x - 5y - 7 = 0\) with respect to the circle \(S \equiv x^2 + y^2 - 2x - 2y + 1 = 0\) is \(P(a,b)\). If \(C\) is the centre of the circle \(S = 0\) then \(PC =\):
If \( f(x) = kx^3 - 3x^2 - 12x + 8 \) is strictly decreasing for all \( x \in \mathbb{R} \), then:
The end of a latus rectum of the ellipse \(3x^2 + 4y^2 = 12\) is lying in the third quadrant. If the normal drawn at \(L_1\) to this ellipse intersects the ellipse again at the point \(P(a,b)\), then find the value of \(a\):
The equations of the directrices of the ellipse \(9x^2 + 4y^2 - 18x - 16y - 11 = 0\) are:
If the line \(L = x \cos \alpha + y \sin \alpha - p = 0\) represents a line perpendicular to the line \(x + y + 1 = 0\) and \(p\) is positive, \(a\) lies in the fourth quadrant and perpendicular distance from \(\left(\sqrt{2}, \sqrt{2}\right)\) to the line \(L = 0\) is 5 units, then find \(p\):
The sum of two roots of the equation \(x^4 - x^3 - 16x^2 + 4x + 48 = 0\) is zero. If \(\alpha, \beta, \gamma, \delta\) are the roots of this equation, then \(\alpha^4 + \beta^4 + \gamma^4 + \delta^4\) is:
If \( \alpha, \beta \) (\(\alpha<\beta\)) are the values of \( x \) such that the determinant of the matrix \[ \begin{bmatrix} x - 2 & 0 & 1 \\ 1 & x + 3 & 2 \\ 2 & 0 & 2x - 1 \end{bmatrix} \] is zero (i.e., the matrix is singular), then the value of \( 2\alpha + 3\beta + 4\gamma \) is:
Given the two parabolas: \[ S = y^2 - 4ax = 0, \quad S' = y^2 + ax = 0 \] where \( P(t) \) is a point on the parabola \( S' = 0 \). If \( A \) and \( B \) are the feet of the perpendiculars from \( P \) to the coordinate axes and \( AB \) is a tangent to the parabola \( S = 0 \) at the point \( Q(t_1) \), then the value of \( t_1 \) is:
If \( A = \begin{bmatrix} 2 & 3 \\ 1 & k \end{bmatrix} \) is a singular matrix, then the quadratic equation having the roots \( k \) and \( \frac{1}{k} \) is:
If the direction ratios of two lines are \( (3,0,2) \) and \( (0,2,k) \), and \( \theta \) is the angle between them, and if \( |\cos \theta| = \frac{6}{13} \), then \( k = \)
A(1, -2, 1) and B(2, -1, 2) are the end points of a line segment. If \(D(\alpha, \beta, \gamma)\) is the foot of the perpendicular drawn from \(C(1, 2, 3)\) to AB, then \(\alpha^2 + \beta^2 + \gamma^2 =\)
If the mean deviation about the mean is \(m\) and the variance is \(\sigma^2\) for the following data, then \(m + \sigma^2 =\) % Data table \[ \begin{array}{|c|c|c|c|c|c|} \hline x & 1 & 3 & 5 & 7 & 9 \\ \hline f & 4 & 24 & 28 & 16 & 8 \\ \hline \end{array} \]
P is a variable point such that the distance of P from A(4,0) is twice the distance of P from B(-4,0). If the line \( 3y - 3x - 20 = 0 \) intersects the locus of P at the points C and D, then the distance between C and D is:
In \( \triangle ABC \), if \( AB:BC:CA = 6:4.5 \), then \( R : r = \)
The number of ways in which 15 identical gold coins can be distributed among 3 persons such that each one gets at least 3 gold coins is
$$ x^5 + 4x^4 - 13x^3 - 52x^2 + 36x + 144 = 0 $$