List of top Mathematics Questions

Evaluate: \[ \cosh^{-1} 2 \]
In \( \triangle ABC \), evaluate: \[ \cos A + \cos B + \cos C \]
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 \( \alpha \beta (k+4)^2 \).
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:
If \( Z = x + iy \) is a complex number, then the number of distinct solutions of the equation \[ z^3 + \bar{z} = 0 \] is:
The set of all real values of \( a \) such that the function \( f(x) = x^3 + 2ax^2 + 3(a+1)x + 5 \) is strictly increasing in its entire domain is:
If \( f(x) = \sqrt{x - 1 \) and \( g(f(x)) = x + 2\sqrt{x} + 1 \), then \( g(x) \) 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 \( (x - y -1) dy = (x + y + 1) dx \).
Find the locus of points satisfying the equation \( axy + byz = cy \).
If the sum of two roots \( \alpha, \beta \) of the equation \[ x^4 - x^3 - 8x^2 + 2x + 12 = 0 \] is zero and \( \gamma, \delta \) (\( \gamma\delta \)) are its other roots, then \( 3\gamma + 2\delta \) is:
A random variable \( X \) follows a Poisson distribution with mean 5. Find the probability that \( X3 \).

For real values of $ x $ and $ a $, if the expression $ \frac{x+a}{2x^2 - 3x + 1} $ assumes all real values, then:

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:
Let \( P(a, 4, 7) \) and \( Q(3, \beta, 8) \) be two points. If the YZ-plane divides the join of the points P and Q in the ratio 2:3 and the ZX-plane divides the join of P and Q in the ratio 4:5, then the length of line segment PQ is:
If \[ y = \tan^{-1} \left( \frac{2 - 3\sin x}{3 - 2\sin x} \right), \] then find \( \frac{dy}{dx} \).

Given the function:

\[ f(x) = \begin{cases} \frac{2x e^{1/2x} - 3x e^{-1/2x}}{e^{1/2x} + 4e^{-1/2x}}, & \text{if } x \neq 0 \\ 0, & \text{if } x = 0 \end{cases} \]

Determine the differentiability of \( f(x) \) at \( x = 0 \).

The number of different ways of preparing a garland using 6 distinct white roses and 6 distinct red roses such that no two red roses come together is:
Evaluate \( \int_{1}^{2} \frac{x^4 - 1}{x^6 - 1} dx \):
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:
Evaluate the limit: $$ \lim_{n \to \infty} n^4 \left[ \sum_{k=0}^{\infty} \frac{1}{(n^2 + k)^{5/2}} \right]. $$
Evaluate the integral: \[ I = \int (\tan^7 x + \tan x) dx. \]
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: