LFM Pure

jee-main 2012 Q73 Limit Evaluation Involving Composition or Substitution View

$\lim _ { x \rightarrow 0 } \frac { \sin \left( \pi \cos ^ { 2 } x \right) } { x ^ { 2 } }$ equals
(1) $- \pi$
(2) $1$
(3) $-1$
(4) $\pi$

jee-main 2012 Q78 Chain Rule with Composition of Explicit Functions View

If $f^{\prime}(x) = \sin(\log x)$ and $y = f\left(\frac{2x+3}{3-2x}\right)$, then $\frac{dy}{dx}$ at $x = 1$ is equal to (the question continues with answer options as given in the paper).

jee-main 2013 Q74 Limit Evaluation Involving Composition or Substitution View

The value of $\lim_{x \rightarrow 0} \frac{(1 - \cos 2x)(3 + \cos x)}{x \tan 4x}$ is equal to
(1) 1
(2) 2
(3) $-\frac{1}{4}$
(4) $\frac{1}{2}$

jee-main 2013 Q81 Chain Rule with Composition of Explicit Functions View

If $y = \sec\left(\tan^{-1}x\right)$, then $\frac{dy}{dx}$ at $x = 1$ is equal to
(1) 1
(2) $\sqrt{2}$
(3) $\frac{1}{\sqrt{2}}$
(4) $\frac{1}{2}$

jee-main 2014 Q74 Evaluate Composition from Algebraic Definitions View

If $f ( x )$ is continuous and $f \left( \frac { 9 } { 2 } \right) = \frac { 2 } { 9 }$, then $\lim _ { x \rightarrow 0 } f \left( \frac { 1 - \cos 3 x } { x ^ { 2 } } \right)$ equals to
(1) $\frac { 8 } { 9 }$
(2) 0
(3) $\frac { 2 } { 9 }$
(4) $\frac { 9 } { 2 }$

jee-main 2014 Q80 Higher-Order Derivatives of Products/Compositions View

If $y = e ^ { n x }$, then $\frac { d ^ { 2 } y } { d x ^ { 2 } } \cdot \frac { d ^ { 2 } x } { d y ^ { 2 } }$ is equal to:
(1) $n e ^ { - n x }$
(2) $- n e ^ { - n x }$
(3) $n e ^ { n x }$
(4) 1

jee-main 2015 Q73 Limit Evaluation Involving Composition or Substitution View

$\lim _ { x \rightarrow 0 } \frac { ( 1 - \cos 2 x ) ( 3 + \cos x ) } { x \tan 4 x } =$
(1) $\frac { 1 } { 2 }$
(2) 4
(3) 3
(4) 2

jee-main 2015 Q80 Continuity Conditions via Composition View

If the function $g ( x ) = \left\{ \begin{array} { c c } k \sqrt { x + 1 } , & 0 \leq x \leq 3 \\ m x + 2 , & 3 < x \leq 5 \end{array} \right.$ is differentiable, then the value of $k + m$ is
(1) 4
(2) 2
(3) $\frac { 16 } { 5 }$
(4) $\frac { 10 } { 3 }$

jee-main 2016 Q81 Limit Evaluation Involving Composition or Substitution View

$\lim_{x \to \pi/2} \frac{\cot x - \cos x}{(\pi - 2x)^3}$ equals:
(1) $\frac{1}{24}$
(2) $\frac{1}{16}$
(3) $\frac{1}{8}$
(4) $\frac{1}{4}$

jee-main 2016 Q82 Compute derivative of transcendental function View

For $x \in \mathbb{R}$, $f(x) = |\log 2 - \sin x|$ and $g(x) = f(f(x))$, then:
(1) $g$ is not differentiable at $x = 0$
(2) $g'(0) = \cos(\log 2)$
(3) $g'(0) = -\cos(\log 2)$
(4) $g$ is differentiable at $x = 0$ and $g'(0) = -\sin(\log 2)$

jee-main 2016 Q84 Compute derivative of transcendental function View

For $x \in \mathbb{R}$, $f(x) = |\log 2 - \sin x|$ and $g(x) = f(f(x))$, then: (1) $g'(0) = \cos(\log 2)$ (2) $g'(0) = -\cos(\log 2)$ (3) $g$ is differentiable at $x=0$ and $g'(0) = -\sin(\log 2)$ (4) $g$ is not differentiable at $x=0$

jee-main 2016 Q85 Limit Evaluation Involving Composition or Substitution View

$\lim_{x \to \pi/2} \frac{\cot x - \cos x}{(\pi - 2x)^3}$ equals: (1) $\frac{1}{24}$ (2) $\frac{1}{16}$ (3) $\frac{1}{8}$ (4) $\frac{1}{4}$

jee-main 2017 Q69 Evaluate Composition from Algebraic Definitions View

For $x \in \mathbb { R }$, $f ( x ) = | \log 2 - \sin x |$ and $g ( x ) = f ( f ( x ) )$, then:
(1) $g$ is not differentiable at $x = 0$
(2) $g ^ { \prime } ( 0 ) = \cos ( \log 2 )$
(3) $g ^ { \prime } ( 0 ) = - \cos ( \log 2 )$
(4) $g$ is differentiable at $x = 0$ and $g ^ { \prime } ( 0 ) = - \sin ( \log 2 )$

jee-main 2017 Q71 Limit Evaluation Involving Composition or Substitution View

$\lim_{x \to \frac{\pi}{2}} \dfrac{\cot x - \cos x}{(\pi - 2x)^3}$ equals
(1) $\dfrac{1}{24}$
(2) $\dfrac{1}{16}$
(3) $\dfrac{1}{8}$
(4) $\dfrac{1}{4}$

jee-main 2017 Q72 Limit Computation from Algebraic Expressions View

$\lim _ { x \rightarrow 3 } \frac { \sqrt { 3 x } - 3 } { \sqrt { 2 x - 4 } - \sqrt { 2 } }$ is equal to
(1) $\frac { 1 } { \sqrt { 2 } }$
(2) $\frac { 1 } { 2 \sqrt { 2 } }$
(3) $\frac { \sqrt { 3 } } { 2 }$
(4) $\sqrt { 3 }$

jee-main 2018 Q78 Limit Involving Derivative Definition of Composed Functions View

If $f ( x ) = \left| \begin{array} { c c c } \cos x & x & 1 \\ 2 \sin x & x ^ { 2 } & 2 x \\ \tan x & x & 1 \end{array} \right|$, then $\lim _ { x \rightarrow 0 } \frac { f ^ { \prime } ( x ) } { x }$
(1) does not exist
(2) exists and is equal to $-2$
(3) exists and is equal to 0
(4) exists and is equal to 2

jee-main 2018 Q78 Limit Involving Derivative Definition of Composed Functions View

$f ( x ) = \left| \begin{array} { c c c } \cos x & x & 1 \\ 2 \sin x & x ^ { 2 } & 2 x \\ \tan x & x & 1 \end{array} \right|$, then $\lim _ { x \rightarrow 0 } \frac { f ^ { \prime } ( x ) } { x }$
(1) Exists and is equal to - 2
(2) Does not exist
(3) Exist and is equal to 0
(4) Exists and is equal to 2

jee-main 2018 Q80 Piecewise Function Differentiability Analysis View

Let $S = \left\{ t \in R : f ( x ) = | x - \pi | \cdot \left( e ^ { | x | } - 1 \right) \sin | x |\right.$ is not differentiable at $\left. t \right\}$. Then, the set $S$ is equal to:
(1) $\{ 0 , \pi \}$
(2) $\phi$ (an empty set)
(3) $\{ 0 \}$
(4) $\{ \pi \}$

jee-main 2018 Q81 Second derivative via implicit differentiation View

If $x ^ { 2 } + y ^ { 2 } + \sin y = 4$, then the value of $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ at the point $( - 2,0 )$ is
(1) - 34
(2) - 32
(3) - 2
(4) 4

jee-main 2019 Q72 Limit Evaluation Involving Composition or Substitution View

$\lim _ { x \rightarrow 1 ^ { - } } \frac { \sqrt { \pi } - \sqrt { 2 \sin ^ { - 1 } x } } { \sqrt { 1 - x } }$ is equal to
(1) $\sqrt { \pi }$
(2) $\sqrt { \frac { 2 } { \pi } }$
(3) $\frac { 1 } { \sqrt { 2 \pi } }$
(4) $\sqrt { \frac { \pi } { 2 } }$

jee-main 2019 Q72 Limit Involving Derivative Definition of Composed Functions View

Let $f : R \rightarrow R$ be a differentiable function satisfying $f ^ { \prime } ( 3 ) + f ^ { \prime } ( 2 ) = 0$. Then $\lim _ { x \rightarrow 0 } \left( \frac { 1 + f ( 3 + x ) - f ( 3 ) } { 1 + f ( 2 - x ) - f ( 2 ) } \right)$ is equal to
(1) 1
(2) e
(3) $e ^ { 2 }$
(4) $e ^ { - 1 }$

jee-main 2019 Q72 Limit Evaluation Involving Composition or Substitution View

$\lim_{x \rightarrow 0} \frac{\sin^2 x}{\sqrt{2} - \sqrt{1 + \cos x}}$ equals
(1) $4\sqrt{2}$
(2) $2\sqrt{2}$
(3) $\sqrt{2}$
(4) 4

jee-main 2019 Q79 Chain Rule with Composition of Explicit Functions View

Let $f$ be a differentiable function such that $f ( 1 ) = 2$ and $f ^ { \prime } ( x ) = f ( x )$ for all $x \in R$. If $h ( x ) = f ( f ( x ) )$, then $h ^ { \prime } ( 1 )$ is equal to :
(1) $4 e ^ { 2 }$
(2) $2 e$
(3) $4 e$
(4) $2 e ^ { 2 }$

jee-main 2019 Q79 Compute derivative of transcendental function View

If $2y = \cot^{-1}\left(\frac{\sqrt{3}\cos x + \sin x}{\cos x - \sqrt{3}\sin x}\right)$, $\forall x \in \left(0, \frac{\pi}{2}\right)$, then $\frac{dy}{dx}$ is equal to
(1) $\frac{\pi}{6} - x$
(2) $2x - \frac{\pi}{3}$
(3) $x - \frac{\pi}{6}$
(4) None of these

jee-main 2019 Q81 Higher-Order Derivatives of Products/Compositions View

Let, $f : R \rightarrow R$ be a function such that $f ( x ) = x ^ { 3 } + x ^ { 2 } f \prime ( 1 ) + x f \prime \prime ( 2 ) + f \prime \prime \prime ( 3 ) , \forall x \in R$. Then $f ( 2 )$ equals
(1) 30
(2) 8
(3) $- 4$
(4) $- 2$

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LFM Pure

Straight Lines & Coordinate Geometry 242

Circles 702

Simultaneous equations 134

Proof 868

Proof by induction 36

Sine and Cosine Rules 185

Radians, Arc Length and Sector Area 22

Trig Graphs & Exact Values 95

Standard trigonometric equations 142

Trig Proofs 92

Quadratic trigonometric equations 8

Trigonometric equations in context 11

Modulus function 37

Matrices 1085

Linear transformations 72

Invariant lines and eigenvalues and vectors 339

Reciprocal Trig & Identities 63

Addition & Double Angle Formulae 88

Harmonic Form 14

Small angle approximation 14

Chain Rule 281

Product & Quotient Rules 13

Differentiating Transcendental Functions 160

Connected Rates of Change 48

Standard Integrals and Reverse Chain Rule 82

Integration by Substitution 213

Integration by Parts 44

Implicit equations and differentiation 27

Differential equations 308

3x3 Matrices 164