Mathematics
Limits, Continuity and Differentiability
Previous Years Questions

## MCQ (More than One Correct Answer)

Let $f:(0,1) \rightarrow \mathbb{R}$ be the function defined as $f(x)=[4 x]\left(x-\frac{1}{4}\right)^2\left(x-\frac{1}{2}\right)$, where $[x]$ denote...
Let f : R $$\to$$ R be defined by $$f(x) = {{{x^2} - 3x - 6} \over {{x^2} + 2x + 4}}$$Then which of the following statements is (are) TRUE?
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) for all x, y$$\in$$R. If $$\m... Let the function f : R$$ \to $$R be defined by f(x) = x3$$-$$x2 + (x$$-$$1)sin x and let g : R$$ \to $$R be an arbitrary function. Let fg : R ... For$$a \in R,\,|a|\, > 1$$, let$$\mathop {\lim }\limits_{n \to \infty } \left( {{{1 + \root 3 \of 2 + ...\root 3 \of n } \over {{n^{7/3}}\left( ...
Let f : R be a function. We say that f has PROPERTY 1 if $$\mathop {\lim }\limits_{h \to 0} {{f(h) - f(0)} \over {\sqrt {|h|} }}$$ exists and is finit...
Let f : R $$\to$$ R be given by$$f(x) = \left\{ {\matrix{ {{x^5} + 5{x^4} + 10{x^3} + 10{x^2} + 3x + 1,} & {x < 0;} \cr {{x^2} - x + ... Let f : (0,$$\pi $$)$$ \to $$R be a twice differentiable function such that$$\mathop {\lim }\limits_{t \to x} {{f(x)\sin t - f(t)\sin x} \over {t ...
For every twice differentiable function $$f:R \to [ - 2,2]$$ with $${(f(0))^2} + {(f'(0))^2} = 85$$, which of the following statement(s) is(are) TRUE?
Let f : R $$\to$$ R and g : R $$\to$$ R be two non-constant differentiable functions. If f'(x) = (e(f(x) $$-$$ g(x))) g'(x) for all x $$\in$$ R ...
Let $$f(x) = {{1 - x(1 + |1 - x|)} \over {|1 - x|}}\cos \left( {{1 \over {1 - x}}} \right)$$for x $$\ne$$ 1. Then
Let f : R $$\to$$ (0, 1) be a continuous function. Then, which of the following function(s) has (have) the value zero at some point in the interval ...
Let [x] be the greatest integer less than or equals to x. Then, at which of the following point(s) the function $$f(x) = x\cos (\pi (x + [x]))$$ is di...
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
Let $$f:\left[ { - {1 \over 2},2} \right] \to R$$ and $$g:\left[ { - {1 \over 2},2} \right] \to R$$ be function defined by $$f(x) = [{x^2} - 3]$$ and ...
Let $$g:R \to R$$ be a differentiable function with $$g(0) = 0$$, $$g'(0) = 0$$ and $$g'(1) \ne 0$$. Let $$f(x) = \left\{ {\matrix{ {{x \over {|x|}... Let$$f:(a,b) \to [1,\infty )$$be a continuous function and g : R$$\to$$R be defined as$$g(x) = \left\{ {\matrix{ 0 & , & {x b} \cr } } \r...
$$a \in R$$ (the set of all real numbers), a $$\ne$$ $$-$$1, $$\mathop {\lim }\limits_{n \to \infty } {{({1^a} + {2^a} + ... + {n^a})} \over {{{(n + 1... For every integer n, let an and bn be real numbers. Let function f : R$$\to$$R be given by$$f(x) = \left\{ {\matrix{ {{a_n} + \sin \pi x,} & {fo...
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 x = 0, then
If $$f(x) = \left\{ {\matrix{ { - x - {\pi \over 2},} & {x \le - {\pi \over 2}} \cr { - \cos x} & { - {\pi \over 2} 1} \cr } } \right... Let$$L = \mathop {\lim }\limits_{x \to 0} {{a - \sqrt {{a^2} - {x^2}} - {{{x^2}} \over 4}} \over {{x^4}}},a > 0$$. If L is finite, then Let$$f(x)$$be a non-constant twice differentiable function defined on$$\left( { - \infty ,\infty } \right)$$such that$$f\left( x \right) = f\lef...

## Numerical

If $$\beta=\lim \limits_{x \to 0} \frac{e^{x^{3}}-\left(1-x^{3}\right)^{\frac{1}{3}}+\left(\left(1-x^{2}\right)^{\frac{1}{2}}-1\right) \sin x}{x \sin... Let$$\alpha$$be a positive real number. Let$$f: \mathbb{R} \rightarrow \mathbb{R}$$and$$g:(\alpha, \infty) \rightarrow \mathbb{R}$$be the functi... Let the functions$$f:( - 1,1) \to R$$and$$g:( - 1,1) \to ( - 1,1)$$be defined by$$f(x) = |2x - 1| + |2x + 1|$$and$$g(x) = x - [x]$$, where [x] ... The value of the limit$$\mathop {\lim }\limits_{x \to {\pi \over 2}} {{4\sqrt 2 (\sin 3x + \sin x)} \over {\left( {2\sin 2x\sin {{3x} \over 2} + \cos...
let e denote the base of the natural logarithm. The value of the real number a for which the right hand limit$$\mathop {\lim }\limits_{x \to {0^ + }} ... The value of$${({({\log _2}9)^2})^{{1 \over {{{\log }_2}({{\log }_2}9)}}}} \times {(\sqrt 7 )^{{1 \over {{{\log }_4}7}}}}$$is .................... Let f : R$$ \to $$R be a differentiable function such that f(0) = 0,$$f\left( {{\pi \over 2}} \right) = 3$$and f'(0) = 1.If$$g(x) = \int\limits_...
Let $$\alpha$$, $$\beta$$ $$\in$$ R be such that $$\mathop {\lim }\limits_{x \to 0} {{{x^2}\sin (\beta x)} \over {\alpha x - \sin x}} = 1$$. Then 6($$... Let m and n be two positive integers greater than 1. If$$$\mathop {\lim }\limits_{\alpha \to 0} \left( {{{{e^{\cos \left( {{\alpha ^n}} \right)}} - ... The largest value of the non-negative integer a for which $$\mathop {\lim }\limits_{x \to 1} {\left\{ {{{ - ax + \sin (x - 1) + a} \over {x + \sin (x ... Let f : R$$\to$$R and g : R$$\to$$R be respectively given by f(x) = | x | + 1 and g(x) = x2 + 1. Define h : R$$\to$$R by$$h(x) = \left\{ {\matr... ## MCQ (Single Correct Answer) For positive integer$n$, define $$f(n)=n+\frac{16+5 n-3 n^{2}}{4 n+3 n^{2}}+\frac{32+n-3 n^{2}}{8 n+3 n^{2}}+\frac{48-3 n-3 n^{2}}{12 n+3 n^{2}}+\c... Let$${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 R$$and$${f_4}:R \to R$$be f... If f : R$$ \to $$R is a twice differentiable function such that f"(x) > 0 for all x$$ \in $$R, and$$f\left( {{1 \over 2}} \right) = {1 \over 2}$...
If $$\mathop {\lim }\limits_{x \to \infty } \left( {{{{x^2} + x + 1} \over {x + 1}} - ax - b} \right) = 4$$, then
Let $$f(x) = \left\{ {\matrix{ {{x^2}\left| {\cos {\pi \over x}} \right|,} & {x \ne 0} \cr {0,} & {x = 0} \cr } } \right.$$ x$$\in$$R, th...
If $$\mathop {\lim }\limits_{x \to 0} {[1 + x\ln (1 + {b^2})]^{1/x}} = 2b{\sin ^2}\theta$$, $$b > 0$$ and $$\theta \in ( - \pi ,\pi ]$$, then the va...
Let the function $$g:\left( { - \infty ,\infty } \right) \to \left( { - {\pi \over 2},{\pi \over 2}} \right)$$ be given by $$g\left( u \right) = 2{\... Which of the following is true? Let$$g(x) = {{{{(x - 1)}^n}} \over {\log {{\cos }^m}(x - 1)}};0 0$$, and let$$p$$be the left hand derivative of$$|x - 1|$$at$$x = 1$$. If$$\ma...
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