List of top Mathematics Questions

If a function \( f: \mathbb{R} \to \mathbb{R} \) is defined by \( f(x) = x^3 - x \), then \( f \) is:
If the roots of the quadratic equation \( x^2 - 35x + c = 0 \) are in the ratio 2:3 and \( c = 6K \), then \( K \) is:
Find the differential equation representing the family of circles having their centers on the Y-axis. Given that \( y_1 = \frac{dy}{dx} \) and \( y_2 = \frac{d^2y}{dx^2} \).
Find the general solution of the differential equation \( ( \sin y \cos^2 y - x \sec^2 y ) dy = (\tan y) dx \).
Find the general solution of the differential equation \( (x - y -1) dy = (x + y + 1) dx \).
The probability that a person goes to college by car,\(\frac{1}{5}\) bus,\(\frac{2}{5}\) or train \(\frac{3}{5}\) is given. If he reaches college on time, find the probability he traveled by car.
Find the locus of points satisfying the equation \( axy + byz = cy \).
If \( (a, b) \) and \( (c, d) \) are the internal and external centres of similitude of the circles \[ x^2 + y^2 + 4x - 5 = 0 \] and \[ x^2 + y^2 - 6y + 8 = 0 \] respectively, then \( (a + d)(b + c) \) is:
The number of ways a committee of 8 members can be formed from a group of 10 men and 8 women such that the committee contains at most 5 men and at least 5 women is:
P, Q, and R try to hit the same target one after another. Their probabilities of hitting are \( \frac{2}{3}, \frac{3}{5}, \frac{5}{7} \) respectively. Find the probability that the target is hit by P or Q but not by R.
The set of all real values of \( c \) for which the equation \[ zz' + (4 - 3i)z + (4+3i)z + c = 0 \] represents a circle is:
For all positive integers \( n \), if \( 3(5^{2n+1}) + 2^{3n+1 \) is divisible by \( k \), then the number of prime numbers less than or equal to \( k \) is:}
The length of the tangent drawn from the point \( \left(\frac{k}{4}, \frac{k}{3}\right) \) to the circle \( x^2 + y^2 + 8x - 6y - 24 = 0 \) is:
If 5 letters are to be placed in 5-addressed envelopes, then the probability that at least one letter is placed in the wrongly addressed envelope is:
If \( (\alpha, \beta, \gamma) \) are the direction cosines of an angular bisector of two lines whose direction ratios are (2,2,1) and (2,-1,-2), then \( (\alpha + \beta + \gamma)^2 \) is:
If \( f(x + h) = 0 \) represents the transformed equation of $$ f(x) = x^4 + 2x^3 - 19x^2 - 8x + 60 = 0 $$ and this transformation removes the term containing \( x^3 \), then \( h \) is: 
The equation \( 2x^2 - 3xy - 2y^2 = 0 \) represents two lines \( L_1 \) and \( L_2 \). The equation \( 2x^2 - 3xy - 2y^2 - x + 7y - 3 = 0 \) represents another two lines \( L_3 \) and \( L_4 \). Let \( A \) be the point of intersection of lines \( L_1 \) and \( L_3 \), and \( B \) be the point of intersection of lines \( L_2 \) and \( L_4 \). The area of the triangle formed by the lines \( AB \), \( L_3 \), and \( L_4 \) is: .
If the lines \( 3x+y-4=0 \), \( x - \alpha y + 10 = 0 \), \( \beta x + 2y + 4 = 0 \) and \( 3x + y + k = 0 \) represent the sides of a square, then find \( \alpha \beta (k+4)^2 \).
If the coordinate axes are rotated by \( 45^\circ \) about the origin in the counterclockwise direction, then the transformed equation of \( y^2 = 4ax \) is:
Evaluate the integral: \[ I = \int (\tan^7 x + \tan x) dx. \]
If \( f(x) = \sqrt{x - 1 \) and \( g(f(x)) = x + 2\sqrt{x} + 1 \), then \( g(x) \) is:}
For a dataset, if the coefficient of variation is 25 and the mean is 44, find the variance.
If \[ y = \tan^{-1} \left( \frac{2 - 3\sin x}{3 - 2\sin x} \right), \] then find \( \frac{dy}{dx} \).
The line \( x - 2y - 3 = 0 \) cuts the parabola \( y^2 = 4ax \) at points P and Q. If the focus of this parabola is \( \left(\frac{1}{4}, k\right) \), then PQ is:
If the line \( 5x - 2y - 6 = 0 \) is a tangent to the hyperbola \( 5x^2 - ky^2 = 12 \), then the equation of the normal to this hyperbola at \( (\sqrt{6}, p) \) is: