List of top Mathematics Questions

If \( \omega \) is the complex cube root of unity and \[ \left( \frac{a + b\omega + c\omega^2}{c + a\omega + b\omega^2} \right)^k + \left( \frac{a + b\omega + c\omega^2}{b + a\omega^2 + c\omega} \right)^2 = 2, \] then \( 2k + 1 \) is always:
If \( z = x + iy \) satisfies the equation \[ z^2 + az + a^2 = 0, \quad a \in \mathbb{R}, \] then:
If \( A = \int_0^{\infty} \frac{1 + x^2}{1 + x^4} dx \) and \( B = \int_0^1 \frac{1 + x^2}{1 + x^4} dx \), then:
The solution of the differential equation \( e^x y dx + e^x dy + xdx = 0 \) is:
The differential equation for which \( ax + by = 1 \) is the general solution is:
If the length of the sub-tangent at any point P on a curve is proportional to the abscissa of the point P, then the equation of that curve is (C is an arbitrary constant):
If \[ y = (x - 1)(x + 2)(x^2 + 5)(x^4 + 8), \] then \[ \lim\limits_{x \to -1} \left( \frac{dy}{dx} \right) = ? \]
Let \( f(x) = \begin{cases 1 + \frac{2x}{a}, & 0 \le x \le 1
ax, & 1<x \le 2 \end{cases} \). If \( \lim_{x \to 1} f(x) \) exists, then the sum of the cubes of the possible values of \( a \) is: }
If the angle \( \theta \) between the line \( \frac{x + 1}{1} = \frac{y - 1}{2} = \frac{z - 2}{2} \) and the plane \( 2x - y + \sqrt{\lambda}z + 4 = 0 \) is such that \( \sin \theta = \frac{1}{3} \), then the value of \( \lambda \) is:
If \( y = \sinh^{-1} \left(\frac{1 - x}{1 + x} \right) \), then \( \frac{dy}{dx} \) is given by:
The area of the quadrilateral formed with the foci of the hyperbola \[ \frac{x^2}{16} - \frac{y^2}{9} = 1 \] and its conjugate hyperbola is (in square units):
If \( P \) is a point which divides the line segment joining the focus of the parabola \( y^2 = 12x \) and a point on the parabola in the ratio 1:2, then the locus of \( P \) is:
If \( y = x + \sqrt{2} \) is a tangent to the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), then equations of its directrices are:
If the direction cosines of two lines are given by \[ l + m + n = 0 \quad \text{and} \quad mn - 2lm - 2nl = 0, \] then the acute angle between those lines is:
Let \( T_1 \) be the tangent drawn at a point \( P(\sqrt{2}, \sqrt{3}) \) on the ellipse \( \frac{x^2}{4} + \frac{y^2}{6} = 1 \). If \( (a, \beta) \) is the point where \( T_1 \) intersects another tangent \( T_2 \) to the ellipse perpendicularly, then \( a^2 + \beta^2 = \):
The length of the internal bisector of angle A in \( \triangle ABC \) with vertices \( A(4,7,8) \), \( B(2,3,4) \), and \( C(2,5,7) \) is:
If \( (1,3) \) is the midpoint of a chord of the circle \( x^2 + y^2 - 4x - 8y + 16 = 0 \), then the area of the triangle formed by that chord with the coordinate axes is:
The combined equation of the bisectors of the angles between the lines joining the origin to the points of intersection of the curve \( x^2 + y^2 + xy + x + 3y + 1 = 0 \) and the line \( x + y + 2 = 0 \) is:
The locus of the midpoint of the portion of the line \( x \cos \alpha + y \sin \alpha = p \) intercepted by the coordinate axes, where \( p \) is a constant, is:
If the straight line passing through \( P(3,4) \) makes an angle \( \frac{\pi}{6} \) with the positive x-axis in the anticlockwise direction and meets the line \( 12x + 5y + 10 = 0 \) at \( Q \), then the length of the segment \( PQ \) is:
A pair of lines drawn through the origin forms a right-angled isosceles triangle with right angle at the origin with the line \( 2x + 3y = 6 \). The area (in square units) of the triangle thus formed is:
Bag A contains 2 white and 3 red balls, and Bag B contains 4 white and 5 red balls. If one ball is drawn at random from one of the bags and is found to be red, then the probability that it was drawn from Bag B is:
In a Binomial distribution \( B(n,p) \), the sum and product of the mean and the variance are 5 and 6 respectively, then \( 6(n + p - q) = \):
Which of the following are true for the trials of a random experiment to be a Bernoulli's trial?
(A) There should be finite number of trials.  
(B) The trials should be dependent.  
(C) Each trial should have at least two outcomes.  
(D) The probability of success remains the same in each trial.  
Choose the correct answer from the options given below:
If \( f(x) \) is given as: \[ f(x) = \begin{cases} \frac{8}{x^3} - 6x, & \text{if } 0 < x \leq 1 \\ \frac{x - 1}{\sqrt{x}-1}, & \text{if } x > 1 \end{cases} \] is a real-valued function, then at \( x = 1 \), \( f \) is: