List of top Mathematics Questions on Functions asked in NEST

The number of solutions of the equation $|\sin(\pi x)| = \frac{1}{50}(x^2 + 1)$ in $\mathbb{R}$ is

Let $f : \mathbb{R} \to \mathbb{R}$ be a function defined by $f(x) = x^5 + x^3$ and let $g(x) = f^{-1}(x)$ be the inverse of $f$. If $g''(-2) = \frac{a}{b}$ where $a$ and $b$ are positive coprime integers, then the value of $a$ is

Let $S_1$ and $S_2$ be two circles drawn inside a unit square ABCD, touching each other externally. Further, the circle $S_1$ touches the sides AD and DC; and the circle $S_2$ touches the sides AB and BC. If the area of $S_2$ is twice the area of $S_1$, then the radius of $S_1$ is

Let $f : \mathbb{R} \to \mathbb{R}$ be a function defined by \[ f(x) = \begin{cases} x \left( \frac{e^{1/x} - e^{-1/x}}{e^{1/x} + e^{-1/x}} \right) & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases} \] Then

Let a variable line $L$ meet $x$-axis and $y$-axis at points A and B, respectively. Suppose the distance of the line $L$ from the origin is 3 units. Then the equation of the locus of the point C that divides the line segment AB internally in the ratio $2 : 1$ is

If $x \in (0, 1)$ and $\sin^{-1} x - \sin^{-1} \frac{1}{4} = \frac{\pi}{3}$, then $x$ equals

For a complex number $z = x + iy$, where $x, y \in \mathbb{R}$, denote $\hat{z} = y + ix$. The locus of $z$ satisfying $|z + \hat{z}| = |z - \hat{z}|$ in the complex plane is

The parabola $y = -x^2 + 16$ intersects the $x$-axis at points A and B. Further, the parabola intersects the line $y = 7$ at points M and N. Then the area of the quadrilateral with vertices A, B, M and N is

The domain of the real function $f(x) = \sin^{-1}\left(\log_2\left(\frac{x^2}{2}\right)\right)$ is

Let $f : \mathbb{R} \to \mathbb{R}$ be a twice differentiable function such that $f''(x) = -f(x)$. Let $h : \mathbb{R} \to \mathbb{R}$ be a differentiable function such that $h'(x) = f(x)^2 + (f'(x))^2$ for all $x \in \mathbb{R}$. If $h(0) = 1$ and $h\left(\frac{1}{3}\right) = \frac{5}{3}$, then the value of $h(100)$ is

Suppose $\lim_{x \to -2} \frac{bx^2 + 15x + 15 + b}{x^2 + x - 2} = L$, where $b$ and $L$ are real numbers. Then

For \(x \in \mathbb{R}\), define \[ \langle x \rangle = x-[x] \] where \([x]\) denotes the greatest integer less than or equal to \(x\). Then for any arbitrary pair of real numbers \(x,y \in \mathbb{R}\), which of the following is always true?