List of top Mathematics Questions

A real differentiable function \(f\) satisfies \(f(x)+f(y)+2xy=f(x+y)\). Given \(f''(0)=0\), then \[ \int_0^{\pi/2} f(\sin x)\,dx = \]

Let \( f:\mathbb{N} \to \mathbb{N} \) be defined as \[ f(n)= \begin{cases} \frac{n+1}{2}, & \text{if } n \text{ is odd} \\ \frac{n}{2}, & \text{if } n \text{ is even} \end{cases} \] Then \( f \) is:

If \(x = \sin(2\tan^{-1}2)\), \(y = \sin\left(\frac{1}{2}\tan^{-1}\frac{4}{3}\right)\), then:

In \(\triangle ABC\), \(\sin A, \sin B, \sin C\) are in A.P. and \(C>90^\circ\). Then \(\cos A\) is:

If in a \(\triangle ABC\), \(\sin^2 A + \sin^2 B + \sin^2 C = 2\), then the triangle is always:

Let \( f(x) = \left[\frac{\sin x}{x}\right] + \left[\frac{2\sin x}{x}\right] + \cdots + \left[\frac{10\sin x}{x}\right] \) (where \([\,]\) is the greatest integer function). Find \( \lim_{x \to 0} f(x)\).

If two tangents from point \((h,k)\) to parabola \(y^2 = 64x\) have slopes such that one is 8 times the other, then value of \( \frac{k^2}{2h} \) is:

The equation of mirror image of the circle \(x^2 + y^2 - 6x - 10y + 33 = 0\) about the line \(y = x\) is:

If the angle between the pair of straight lines formed by joining the points of intersection of \(x^2 + y^2 = 4\) and \(y = 3x + c\) to the origin is a right angle, then \(c^2\) is:

If the coefficient of \(x^m\) in the expansion of \(\left(\sqrt{2x} + \sqrt[3]{\frac{3}{x^2}}\right)^9\) is equal to \(k\), then \(k\) is:

If the expression \(x + \frac{1}{x^2},\; x>0\) attains minimum value at \(x=\alpha\), then \(\alpha^3\) is:

If \(N\) denotes number of 8-digit numbers that contain exactly four nines, then unit digit of \(N\) is:

Total number of even divisors of \(2079000\) which are divisible by \(15\) are:

If \( \frac{a}{b} = \frac{1}{3} \) and \( \frac{b}{c} = \frac{3}{4} \), then the value of \( \frac{a+2b}{b+2c} \) is:

The solution of the equation \( \log\left(\log_4(\sqrt{x+4} + \sqrt{x})\right) = 0 \) is:

The range of \( 2 \left| \sin x + \cos x \right| - \sqrt{2} \) is:

The derivative of  \(\sin^2\)\(( \cot^{-1} {\sqrt {( \frac{1 + x}{1 - x}} })\) with respect to \( x \) is equal to: (a) 0
 

Let \( \mathbb{Z} \) denote the set of integers. Then, the function \( f : \mathbb{Z} \to \mathbb{Z} \) defined as \( f(x) = x^3 - 1 \) is:

The value of \( \tan 15^\circ \) is:

The value of \( \int_0^\infty \frac{dx}{(x^2 + a^2)(x^2 + b^2)} \) is:

The function \[ f(x) = \frac{\cos x}{\left\lfloor \frac{2x}{\pi} \right\rfloor + \frac{1}{2}}, \] where \( x \) is not an integral multiple of \( \pi \) and \( \lfloor \cdot \rfloor \) denotes the greatest integer function, is:

Evaluate the integral: \[ \int \frac{x^2 (x \sec^2 x + \tan x)}{(x \tan x + 1)^2} dx \]