Differentiability Questions - JEE

If $y = f(x)$ is differentiable for all $x \in \mathbb{R}$, then:

If $x^2 + y^2 = r^2$ and $z = \dfrac{1}{r}$ then $z = \sqrt{1 + \left(\dfrac{dy}{dx}\right)^2}$.State whether the statement is true or false.

Let $\text{Lt}_{n \to \infty} f(n) = l$. Given that $\text{Lt}_{n \to \infty} f(n+1) = \dfrac{1}{2} \text{Lt}_{n \to \infty} \left[f(n) + \dfrac{9}{f(n)}\right]$ and $f(n) > 0 \; \forall n \in N$, find $\text{Lt}_{n \to \infty} f(n)$.

It is given that $f(x) = \text{Min}\{x+1, |x|+1\}$. Then which of the following is true?

Let BC be the diameter of a circle centred at O. Point A is variable on circumference. If BC = 1, then limA→B BM (Area of sector OAB)2 is:

$\lim_{x\to\infty} \dfrac{\left(\displaystyle\int_0^x e^{x^2}\,dx\right)^2}{\displaystyle\int_0^x e^{2x^2}\,dx} = $

The value of a for which $\lim_{x \to 0} \dfrac{\left(e^x - 1\right)^4}{\sin\!\left(\dfrac{x^2}{a^2}\right)\ln\!\left(1 + \dfrac{x^2}{2}\right)} = 8$ is

If $y=e^{a\cos^{-1}x}$, then $(1-x^2)y_2-xy_1=$

$\lim_{x \to 0} \dfrac{\tan(\pi \sin^2 x) + (|x| - \sin(x[x]))^2}{x^2}$ is equal to: (where $[\,]$ denotes greatest integer function)

Evaluate: $\lim_{n \to \infty} \dfrac{(3n+3)(3n+2)(3n+1)(3n!)}{(n+1)^3 (3n)!}$

Let $f(x)$ be continuous on $[a,b]$ and differentiable on $(a,b)$. Number of correct statements: I) If $f(x)$ strictly increasing on $(a,b)$ then $f'(x)\ge 0$ for all $x\in(a,b)$. II) If $f(x)$ strictly decreasing on $(a,b)$ then $f'(x)<0$ for all $x\in(a,b)$. III) $f(x)$ and $f'(x)$ have opposite sign for all $x$, then $f^2(x)$ is decreasing. IV) $f(x)$ and $f'(x)$ have opposite sign for all $x$, then $|f(x)|$ is increasing.

Evaluate $\lim_{x \to \infty} \dfrac{x^a \sin\dfrac{1}{x^a} + x^2}{1 + |x|^3}$ when $x \to \infty$.

Evaluate $\lim_{x \to \pi/2} \tan x \cdot \log \sin x$.

Differentiability Questions (1063)