Continuity Questions - JEE

If $x = \sec\theta - \cos\theta$ and $y = \sec^n\theta - \cos^n\theta$, then $\left(x^2+4\right)\!\left(\dfrac{dy}{dx}\right)^{\!2}$ equals $n^2(y^2+k)$. Find $k$.

If $f(x) = \frac{2\sin x \cos x + 2x \cot x}{2\sin x}$, find $f'\left(\frac{\pi}{4}\right)$ and verify it equals $\sqrt{2}$

Let $f(x) = \sin^2(\sin x)$. If $g$ is the inverse of $f$, find $g''(3)$.Given: $f'(0) = \sin^2(\sin 1)$, $f''(0) = 2\sin(\sin 1)\cos(\sin 1)\cos 1$\[g''(y) = \frac{-1}{[f'(x)]^3} f''(x)\]

Let $y = \cot^{-1}(1) + \cot^{-1}(2) + \cot^{-1}(3) + \cdots$. If $\dfrac{d}{dn}\!\left[\sum_{r=1}^{n}\cot^{-1}(r)\right]$ is evaluated and the sum $\sum_{r=1}^{10}\cot^{-1}(r^2-r+1)$ equals $\tan^{-1}(k/l)$, find $10k+l$ (answer 101 from key).

If $y^2 + \ln(\cos^2 x) = y$, then $|y''(0)+y'(0)|$ equals: [Integer type]

If $f(x) = \begin{cases} \frac{\tan^2\{x\}}{x}, & x > 0 \\ 1, & x = 0 \\ \{x\}\cot\{x\}, & x where \([x]$ is the integral part of $x$ and $\{x\}$ is the fractional part of $x$, then

Let $f:(−\infty,\infty)−\{0\}\to\mathbb{R}$ be a differentiable function such that $f'(1)=\displaystyle\lim_{a\to\infty}a^2f\!\left(\frac{1}{a}\right)$. Then $\displaystyle\lim_{a\to\infty}\frac{a(a+1)}{2}\tan^{-1}\!\left(\frac{1}{a}\right)+a^2-2\log_e a$ is equal to

For each $ x \in \mathbb{R} $, let $ [x] $ be the greatest integer less than or equal to x. Then $ \lim_{x \to 0} \frac{x[|x|] \sin[x]}{|x|} $ is equal to

For each $ t \in \mathbb{R} $, let $ [t] $ be the greatest integer less than or equal to t. Then $ \lim_{x \to 1^-} \frac{(1-|x|+\sin[1-x])\sin\left(\frac{\pi}{2}[1-x]\right)}{|1-x|[|1-x|]} $

If $\lim_{x \to -1} \frac{x^2 - ax + b}{x - 1} = 5$, then $a + b$ is equal to :-

\[ \lim_{x \to 2} \frac{\sqrt{1 - \cos{2(x - 2)}}}{x - 2} \]

If $ \lim_{x \to 2} \frac{\tan{(x - 2)}{(x^2 + (k - 2)x - 2k)}}{x^2 - 4x + 4} = 5 $ then k is equal to

Given: $x^2 + y^2 + \sin y = 4$. Find $-\dfrac{d^2y}{dx^2}\bigg|_{(-2,0)}$.

If $f(x) = \log_x(\log_e x)$, then $f'(e)$ equals:

If $f(2) = 2$, $f'(2) = 1$ then $\lim_{x \to 2} \frac{2x^2 - 4f(x)}{x - 2} =$ ______

Given $f'(3) + f'(2) = 0$. Let \[y = \lim_{x \to 0} \left[\frac{1 + f(3+x) - f(3)}{1 + f(2-x) - f(2)}\right]^{1/x}\] Find the value of $y$.

If α = lim x→0+ e √ tan x −e √x √ tan x −√x and β = lim x→0(1 + sin x) 1 2 cot x are the roots of the quadratic equation ax2 + bx −√e = 0, then 12 loge(a + b) is equal to

$f(x)=\begin{cases}1-x & 0\le x\le 1\\ x+2 & 1. For \(y=f(f(x))$ on $[0,4]$:

Let $g(x)=\begin{cases}3x^2-4\sqrt{x}+1 & x. If \(g(x)$ is continuous and differentiable at $x=1$, find $a$ and $b$.

Let f : (−1, 1) → ℝ be continuous and \[\int_0^{\sin x} f(t)\,dt = \frac{\sqrt{3}}{2}\,x.\] Find $f\!\left(\dfrac{\sqrt{3}}{2}\right)$.

The set of all values of a for which \lim_{x \to a}\left([x-5] - [2x+2]\right) = 0, where [\alpha] denotes the greatest integer less than or equal to \alpha, is equal to

Let $f$ be a function defined by $y = f(x)$ where $x = 2t - |t|$ and $y = t^2 + t|t|$ for $t \in \mathbb{R}$, then:

If $y = e^{nx}$ then $\left(\dfrac{d^2y}{dx^2}\right)\left(\dfrac{d^2x}{dy^2}\right)$ is equal to

Given: $2y = \left(\cot^{-1}\left(\dfrac{\sqrt{3}\cos x + \sin x}{\cos x - \sqrt{3}\sin x}\right)\right)^2$. Find $\dfrac{dy}{dx}$ (or simplify $2y$).

If $f(x)$ and $g(x)$ are not differentiable finitely at a point then will $f(x) \cdot g(x)$ will also be nondifferentiable finitely at that point?

Given: $x = \sqrt{2^{\cosec^{-1}t}}$ and $y = \sqrt{2^{\sec^{-1}t}}$ where $|t| \geq 1$. Find $\dfrac{dy}{dx}$.

Let $f(x)$ be a continuous and differentiable function such that $\displaystyle\lim_{h \to 0} \frac{f(3+7h) - f(3+4h)}{h} = 4$. Then the value of $f'(3)$ equals:

$\lim_{x \to 2} \dfrac{\sqrt{2\sin^2(x-2)}}{x-2}$

$\dfrac{d}{dx}\left[\tan^{-1}\\!\left(\dfrac{\sqrt{2-x}}{1+x^2}\right)\right]$ equals $(x \ge 0)$:

Find $a$, $b$ and $c$ such that \[\lim_{x \to 0} \frac{ax e^x - b\log(1+x) + cx e^{-x}}{x^2 \sin x} = 2\]

Let $f(x) = x + \sin x$. Suppose $g$ denotes the inverse function of $f$. The value of $g'\\!\left(\frac{\pi}{4} + \frac{1}{\sqrt{2}}\right)$ has the value equal to:

If $y = x + e^x$, then $\dfrac{d^2x}{dy^2}$ is:

If $y = 1 + \dfrac{c_1}{x - c_1} + \dfrac{c_2 x}{(x - c_1)(x - c_2)} + \dfrac{c_3 x^2}{(x - c_1)(x - c_2)(x - c_3)}$, then $\dfrac{dy}{dx}$ is equal to:

Let $f(x) = ax^4 + bx^3 + cx^2 + dx + e$. If $\displaystyle\lim_{x \to 0}\left(\dfrac{f(x)}{x^2} + 1\right) = 3$, $f'(1) = 0$ and $f'(2) = 0$, then the value of $a$ is:

Given that $f(a) = g(a) = k$; $f^n(a) \neq g^n(a)$ for some $n \in \mathbb{N}$ and \[\lim_{x \to a} \frac{f(a)\cdot g(x) - f(a) - g(a)f(x) + g(a)}{g(x) - f(x)} = 4\] Find the value of $k$.

Let $ f: (-1, 1) \to R $ be a differentiable function with $ f(0) = -1 $ and $ f'(0) = 1 $. Let $ g(x) = [f(2f(x)+2)]^2 $. Then $ g'(0) = $

$f(x)=[\sin x]+\sqrt{\sin x-[\sin x]}$, where [.] is GIF. $f(x)$ is:

$f:\mathbb{R}\to\mathbb{R}$, $g(x)=|f(x)|$. Which are NOT always true?

Let f and g be two functions such that g(f(x)) is defined. If f is differentiable at x and g is differentiable at f(x), then find the value of $7g'(2\pi) + 3g''(2\pi)$, given that $f'\!\left(\dfrac{3\pi}{2}\right) = \dfrac{1}{3}$ and $f''\!\left(\dfrac{3\pi}{2}\right) = 0$.

If $f(4x) = 4f(x)$ for all $x$ and $f'(1)=2$, then $\displaystyle\lim_{x\to 1}\frac{\sqrt{f(x)}-\sqrt{f(1)}}{\sqrt{x}-1}$ is equal to:

If $2y = \cot^{-1}\!\left(\sqrt{\dfrac{\sqrt{3}\cos x + \sin x}{\sqrt{3}\cos x - \sin x}}\right)$, then $\dfrac{dy}{dx}$ is equal to:

If $f(x) = \{x + \sin x\} + [x - \sin x] + [x]$ where $[y]$ and $\{y\}$ denote greatest integer function and fractional part function of $y$ respectively, then find the number of points of discontinuity in $[0, \pi]$.

If $f(x) = \log_{\sec x} |\cos 4x| + |\sin x|$, then find $\frac{dy}{dx}$ at $x = -\frac{\pi}{6}$ from the first principle.

tignum function $\text{tgn}(x)=\begin{cases}1 & [x]\text{ even}\\ -1 & [x]\text{ odd}\end{cases}$. $f(x)=\text{tgn}(x)\cdot\sin(x)\cdot|x|$. Points of discontinuity in $[0,10]$:

limt→0(11/ sin2 t + · · · + n1/ sin2 t)sin2 t

If $\lim_{x \to 0} \frac{x(1 + m\cos x) - n\sin x}{x^3} = 1$ then $m =$ ______, $n =$ ______

We have $|f(x) - f(y)| \leq (x-y)^2$ for all $x, y \in \mathbb{R}$ and $f(0) = 0$. Find $f(1)$.

The value of $ p $ and $ q $ for which the function \[ f(x) = \begin{cases} \dfrac{\sin(p+1)x + \sin x}{x}, & x 0 \end{cases} \] is continuous for all $ x $ in $ R $, is

If $f(x) = (2x-3x^2)^4 + \cos x$ and $g$ is the inverse of $f$, then which of the following is/are correct?

Let $f(a) = g(a) = k$ and their $n$th derivatives $f^n(a)$, $g^n(a)$ exist and are not equal for some $n$. Further, if $\lim_{x \to a} \dfrac{f(a)g(x) - f(a) - g(a)f(x) + g(a)}{g(x) - f(x)} = 4$, then the value of $k$ is

Continuity Questions (1086)