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JEE Advanced

Limits, Continuity and Differentiability

Mathematics

Previous Years Questions

MCQ (Single Correct Answer)

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For positive integer $n$, define $$ f(n)=n+\frac{16+5 n-3 n^{2}}{4 n+3 n^{2}}+\frac{32+n-3 n^{2}}{8 n+3 n^{2}}+\frac{48...
JEE Advanced 2022 Paper 2 Online
Let $${f_1}:R \to R,\,{f_2}:\left( { - {\pi \over 2},{\pi \over 2}} \right) \to R,\,{f_3}:( - 1,{e^{\pi /2}} - 2) \to ...
JEE Advanced 2018 Paper 2 Offline
If f : R $$ \to $$ R is a twice differentiable function such that f"(x) > 0 for all x$$ \in $$R, and $$f\left( {{1 \o...
JEE Advanced 2017 Paper 2 Offline
Let $$f(x) = \left\{ {\matrix{ {{x^2}\left| {\cos {\pi \over x}} \right|,} & {x \ne 0} \cr {0,} & {x = 0} \cr ...
IIT-JEE 2012 Paper 1 Offline
If $$\mathop {\lim }\limits_{x \to \infty } \left( {{{{x^2} + x + 1} \over {x + 1}} - ax - b} \right) = 4$$, then
IIT-JEE 2012 Paper 1 Offline
If $$\mathop {\lim }\limits_{x \to 0} {[1 + x\ln (1 + {b^2})]^{1/x}} = 2b{\sin ^2}\theta $$, $$b > 0$$ and $$\theta \in...
IIT-JEE 2011 Paper 2 Offline

Numerical

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If $$ \beta=\lim \limits_{x \to 0} \frac{e^{x^{3}}-\left(1-x^{3}\right)^{\frac{1}{3}}+\left(\left(1-x^{2}\right)^{\frac{...
JEE Advanced 2022 Paper 2 Online
Let $$\alpha$$ be a positive real number. Let $$f: \mathbb{R} \rightarrow \mathbb{R}$$ and $$g:(\alpha, \infty) \rightar...
JEE Advanced 2022 Paper 1 Online
The value of the limit$$\mathop {\lim }\limits_{x \to {\pi \over 2}} {{4\sqrt 2 (\sin 3x + \sin x)} \over {\left( {2\si...
JEE Advanced 2020 Paper 2 Offline
Let the functions $$f:( - 1,1) \to R$$ and $$g:( - 1,1) \to ( - 1,1)$$ be defined by $$f(x) = |2x - 1| + |2x + 1|$$ and ...
JEE Advanced 2020 Paper 2 Offline
let e denote the base of the natural logarithm. The value of the real number a for which the right hand limit$$\mathop {...
JEE Advanced 2020 Paper 1 Offline
The value of $${({({\log _2}9)^2})^{{1 \over {{{\log }_2}({{\log }_2}9)}}}} \times {(\sqrt 7 )^{{1 \over {{{\log }_4}7}}...
JEE Advanced 2018 Paper 1 Offline
Let f : R $$ \to $$ R be a differentiable function such that f(0) = 0, $$f\left( {{\pi \over 2}} \right) = 3$$ and f'(0...
JEE Advanced 2017 Paper 1 Offline
Let $$\alpha$$, $$\beta$$ $$\in$$ R be such that $$\mathop {\lim }\limits_{x \to 0} {{{x^2}\sin (\beta x)} \over {\alpha...
JEE Advanced 2016 Paper 1 Offline
Let m and n be two positive integers greater than 1. If $$$\mathop {\lim }\limits_{\alpha \to 0} \left( {{{{e^{\cos \le...
JEE Advanced 2015 Paper 2 Offline
Let f : R $$\to$$ R and g : R $$\to$$ R be respectively given by f(x) = | x | + 1 and g(x) = x2 + 1. Define h : R $$\to$...
JEE Advanced 2014 Paper 1 Offline
The largest value of the non-negative integer a for which $$\mathop {\lim }\limits_{x \to 1} {\left\{ {{{ - ax + \sin (x...
JEE Advanced 2014 Paper 1 Offline

MCQ (More than One Correct Answer)

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Let f : R $$\to$$ R be defined by $$f(x) = {{{x^2} - 3x - 6} \over {{x^2} + 2x + 4}}$$Then which of the following statem...
JEE Advanced 2021 Paper 1 Online
Let f : R $$ \to $$ R and g : R $$ \to $$ R be functions satisfying f(x + y) = f(x) + f(y) + f(x)f(y) and f(x) = xg(x) f...
JEE Advanced 2020 Paper 2 Offline
Let the function f : R $$ \to $$ R be defined by f(x) = x3 $$-$$ x2 + (x $$-$$ 1)sin x and let g : R $$ \to $$ R be an a...
JEE Advanced 2020 Paper 1 Offline
Let f : R be a function. We say that f has PROPERTY 1 if $$\mathop {\lim }\limits_{h \to 0} {{f(h) - f(0)} \over {\sqrt ...
JEE Advanced 2019 Paper 2 Offline
For $$a \in R,\,|a|\, > 1$$, let $$\mathop {\lim }\limits_{n \to \infty } \left( {{{1 + \root 3 \of 2 + ...\root 3 \...
JEE Advanced 2019 Paper 2 Offline
Let f : R $$ \to $$ R be given by$$f(x) = \left\{ {\matrix{ {{x^5} + 5{x^4} + 10{x^3} + 10{x^2} + 3x + 1,} & {x &...
JEE Advanced 2019 Paper 1 Offline
Let f : (0, $$\pi $$) $$ \to $$ R be a twice differentiable function such that $$\mathop {\lim }\limits_{t \to x} {{f(x)...
JEE Advanced 2018 Paper 2 Offline
Let f : R $$ \to $$ R and g : R $$ \to $$ R be two non-constant differentiable functions. If f'(x) = (e(f(x) $$-$$ g(x))...
JEE Advanced 2018 Paper 1 Offline
For every twice differentiable function $$f:R \to [ - 2,2]$$ with $${(f(0))^2} + {(f'(0))^2} = 85$$, which of the follow...
JEE Advanced 2018 Paper 1 Offline
Let $$f(x) = {{1 - x(1 + |1 - x|)} \over {|1 - x|}}\cos \left( {{1 \over {1 - x}}} \right)$$for x $$ \ne $$ 1. Then...
JEE Advanced 2017 Paper 2 Offline
Let [x] be the greatest integer less than or equals to x. Then, at which of the following point(s) the function $$f(x) =...
JEE Advanced 2017 Paper 1 Offline
Let f : R $$ \to $$ (0, 1) be a continuous function. Then, which of the following function(s) has (have) the value zero ...
JEE Advanced 2017 Paper 1 Offline
Let $$f:\left[ { - {1 \over 2},2} \right] \to R$$ and $$g:\left[ { - {1 \over 2},2} \right] \to R$$ be function defined ...
JEE Advanced 2016 Paper 2 Offline
Let a, b $$\in$$ R and f : R $$\to$$ R be defined by $$f(x) = a\cos (|{x^3} - x|) + b|x|\sin (|{x^3} + x|)$$. Then f is...
JEE Advanced 2016 Paper 2 Offline
Let $$g:R \to R$$ be a differentiable function with $$g(0) = 0$$, $$g'(0) = 0$$ and $$g'(1) \ne 0$$. Let $$f(x) = \left\...
JEE Advanced 2015 Paper 1 Offline
Let $$f:(a,b) \to [1,\infty )$$ be a continuous function and g : R $$\to$$ R be defined as $$g(x) = \left\{ {\matrix{ ...
JEE Advanced 2014 Paper 1 Offline
$$a \in R$$ (the set of all real numbers), a $$\ne$$ $$-$$1, $$\mathop {\lim }\limits_{n \to \infty } {{({1^a} + {2^a} +...
JEE Advanced 2013 Paper 2 Offline
For every integer n, let an and bn be real numbers. Let function f : R $$\to$$ R be given by $$f(x) = \left\{ {\matrix{ ...
IIT-JEE 2012 Paper 2 Offline
If $$f(x) = \left\{ {\matrix{ { - x - {\pi \over 2},} & {x \le - {\pi \over 2}} \cr { - \cos x} & { - {\pi \...
IIT-JEE 2011 Paper 2 Offline
Let f : R $$\to$$ R be a function such that $$f(x + y) = f(x) + f(y),\,\forall x,y \in R$$. If f(x) is differentiable at...
IIT-JEE 2011 Paper 1 Offline

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