LFM Pure

jee-main 2023 Q89 Piecewise Function Differentiability Analysis View

Let $f: (-2, 2) \rightarrow \mathbb{R}$ be defined by $f(x) = \begin{cases} x\lfloor x\rfloor, & 0 \leq x < 2 \\ (x-1)\lfloor x\rfloor, & -2 < x < 0 \end{cases}$ where $\lfloor x \rfloor$ denotes the greatest integer function. If $m$ and $n$ respectively are the number of points in $(-2, 2)$ at which $y = f(x)$ is not continuous and not differentiable, then $m + n$ is equal to $\_\_\_\_$.

jee-main 2024 Q67 Limit Evaluation Involving Composition or Substitution View

$\lim _ { x \rightarrow 0 } \frac { e ^ { 2 \sin x } - 2 \sin x - 1 } { x ^ { 2 } }$
(1) is equal to $-1$
(2) does not exist
(3) is equal to 1
(4) is equal to 2

jee-main 2024 Q68 Limit Involving Derivative Definition of Composed Functions View

Let $f : ( - \infty , \infty ) - \{ 0 \} \rightarrow \mathbb { R }$ be a differentiable function such that $f ^ { \prime } ( 1 ) = \lim _ { a \rightarrow \infty } a ^ { 2 } f \left( \frac { 1 } { a } \right)$. Then $\lim _ { a \rightarrow \infty } \frac { a ( a + 1 ) } { 2 } \tan ^ { - 1 } \left( \frac { 1 } { a } \right) + a ^ { 2 } - 2 \log _ { e } a$ is equal to
(1) $\frac { 3 } { 2 } + \frac { \pi } { 4 }$
(2) $\frac { 3 } { 4 } + \frac { \pi } { 8 }$
(3) $\frac { 3 } { 8 } + \frac { \pi } { 4 }$
(4) $\frac { 5 } { 2 } + \frac { \pi } { 8 }$

jee-main 2024 Q69 Limit Evaluation Involving Composition or Substitution View

If $a = \lim _ { x \rightarrow 0 } \frac { \sqrt { 1 + \sqrt { 1 + x ^ { 4 } } } - \sqrt { 2 } } { x ^ { 4 } }$ and $b = \lim _ { x \rightarrow 0 } \frac { \sin ^ { 2 } x } { \sqrt { 2 } - \sqrt { 1 + \cos x } }$, then the value of $a b ^ { 3 }$ is:
(1) 36
(2) 32
(3) 25
(4) 30

jee-main 2024 Q72 Higher-order or nth derivative computation View

Let $y = \log _ { e } \left( \frac { 1 - x ^ { 2 } } { 1 + x ^ { 2 } } \right) , - 1 < x < 1$. Then at $x = \frac { 1 } { 2 }$, the value of $225 \left( y ^ { \prime } - y ^ { \prime \prime } \right)$ is equal to
(1) 732
(2) 746
(3) 742
(4) 736

jee-main 2024 Q72 Chain Rule with Composition of Explicit Functions View

Suppose for a differentiable function $h , h ( 0 ) = 0 , h ( 1 ) = 1$ and $h ^ { \prime } ( 0 ) = h ^ { \prime } ( 1 ) = 2$. If $\mathrm { g } ( x ) = h \left( \mathrm { e } ^ { x } \right) \mathrm { e } ^ { h ( x ) }$, then $g ^ { \prime } ( 0 )$ is equal to:
(1) 5
(2) 4
(3) 8
(4) 3

jee-main 2024 Q72 Finding Parameters for Continuity View

For $\mathrm { a } , \mathrm { b } > 0$, let $f ( x ) = \left\{ \begin{array} { c l } \frac { \tan ( ( \mathrm { a } + 1 ) x ) + \mathrm { b } \tan x } { x } , & x < 0 \\ 3 , & x = 0 \\ \frac { \sqrt { \mathrm { a } x + \mathrm { b } ^ { 2 } x ^ { 2 } } - \sqrt { \mathrm { a } x } } { \mathrm {~b} \sqrt { \mathrm { a } } \sqrt { x } } , & x > 0 \end{array} \right.$ be a continous function at $x = 0$. Then $\frac { \mathrm { b } } { \mathrm { a } }$ is equal to : (1) 6 (2) 4 (3) 5 (4) 8

jee-main 2024 Q73 Continuity Conditions via Composition View

Let $g(x)$ be a linear function and $f(x) = \begin{cases} g(x), & x \leq 0 \\ \frac { 1 + x } { 2 + x } , & x > 0 \end{cases}$, is continuous at $x = 0$. If $f'(1) = f(-1)$, then the value of $g(3)$ is
(1) $\frac { 1 } { 3 } \log _ { e } \frac { 4 } { e^{1/3} }$
(2) $\frac { 1 } { 3 } \log _ { e } \frac { 4 } { 9 } + 1$
(3) $\log _ { e } \frac { 4 } { 9 } - 1$
(4) $\log _ { e } \frac { 4 } { 9 e ^ { 1/3 } }$

jee-main 2024 Q73 Higher-order or nth derivative computation View

If $f ( x ) = \left\{ \begin{array} { l } x ^ { 3 } \sin \left( \frac { 1 } { x } \right) , x \neq 0 \\ 0 \quad , x = 0 \end{array} \right.$ then
(1) $f ^ { \prime \prime } \left( \frac { 2 } { \pi } \right) = \frac { 24 - \pi ^ { 2 } } { 2 \pi }$
(2) $f ^ { \prime \prime } \left( \frac { 2 } { \pi } \right) = \frac { 12 - \pi ^ { 2 } } { 2 \pi }$
(3) $f ^ { \prime \prime } ( 0 ) = 1$
(4) $f ^ { \prime \prime } ( 0 ) = 0$

jee-main 2024 Q73 Chain Rule with Composition of Explicit Functions View

Suppose $f ( x ) = \frac { \left( 2 ^ { x } + 2 ^ { - x } \right) \tan x \sqrt { \tan ^ { - 1 } \left( x ^ { 2 } - x + 1 \right) } } { \left( 7 x ^ { 2 } + 3 x + 1 \right) ^ { 3 } }$. Then the value of $f ^ { \prime } ( 0 )$ is equal to
(1) $\pi$
(2) 0
(3) $\sqrt { \pi }$
(4) $\frac { \pi } { 2 }$

jee-main 2024 Q85 Higher-Order Derivatives of Products/Compositions View

Let $f ( x ) = x ^ { 3 } + x ^ { 2 } f ^ { \prime } ( 1 ) + x f ^ { \prime \prime } ( 2 ) + f ^ { \prime \prime \prime } ( 3 ) , x \in R$. Then $f ^ { \prime } ( 10 )$ is equal to

jee-main 2024 Q85 Limit Evaluation Involving Composition or Substitution View

Let $\mathrm { a } > 0$ be a root of the equation $2 x ^ { 2 } + x - 2 = 0$. If $\lim _ { x \rightarrow \frac { 1 } { \mathrm { a } } } \frac { 16 \left( 1 - \cos \left( 2 + x - 2 x ^ { 2 } \right) \right) } { ( 1 - \mathrm { a } x ) ^ { 2 } } = \alpha + \beta \sqrt { 17 }$, where $\alpha , \beta \in Z$, then $\alpha + \beta$ is equal to $\_\_\_\_$

jee-main 2024 Q85 Limit Evaluation Involving Composition or Substitution View

The value of $\lim _ { x \rightarrow 0 } 2 \left( \frac { 1 - \cos x \sqrt { \cos 2 x } \sqrt [ 3 ] { \cos 3 x } \ldots \ldots \sqrt [ 10 ] { \cos 10 x } } { x ^ { 2 } } \right)$ is $\_\_\_\_$

jee-main 2024 Q86 Straightforward Polynomial or Basic Differentiation View

Let for a differentiable function $f : ( 0 , \infty ) \rightarrow R , f ( x ) - f ( y ) \geq \log _ { e } \left( \frac { x } { y } \right) + x - y , \forall x , y \in ( 0 , \infty )$. Then $\sum _ { n = 1 } ^ { 20 } f ^ { \prime } \left( \frac { 1 } { n ^ { 2 } } \right)$ is equal to

jee-main 2025 Q7 Continuity Conditions via Composition View

If the function $f ( x ) = \left\{ \begin{array} { l } \frac { 2 } { x } \left\{ \sin \left( k _ { 1 } + 1 \right) x + \sin \left( k _ { 2 } - 1 \right) x \right\} , \quad x < 0 \\ 4 , \quad x = 0 \end{array} \quad \right.$ is continuous at $\mathrm { x } = 0$, then $\mathrm { k } _ { 1 } ^ { 2 } + \mathrm { k } _ { 2 } ^ { 2 }$ is equal to
(1) 20
(2) 5
(3) 8
(4) 10

jee-main 2025 Q17 Solving Separable DEs with Initial Conditions View

Let $y = y(x)$ be the solution of the differential equation $\left(xy - 5x^2\sqrt{1+x^2}\right)dx + \left(1+x^2\right)dy = 0$, $y(0) = 0$. Then $y(\sqrt{3})$ is equal to
(1) $\sqrt{\frac{15}{2}}$
(2) $\frac{5\sqrt{3}}{2}$
(3) $2\sqrt{2}$
(4) $\sqrt{\frac{14}{3}}$

jee-main 2025 Q25 Simplify or Evaluate a Logarithmic Expression View

Let $f ( x ) = \lim _ { \mathrm { n } \rightarrow \infty } \sum _ { \mathrm { r } = 0 } ^ { \mathrm { n } } \left( \frac { \tan \left( x / 2 ^ { r + 1 } \right) + \tan ^ { 3 } \left( x / 2 ^ { r + 1 } \right) } { 1 - \tan ^ { 2 } \left( x / 2 ^ { r + 1 } \right) } \right)$. Then $\lim _ { x \rightarrow 0 } \frac { \mathrm { e } ^ { x } - \mathrm { e } ^ { f ( x ) } } { ( x - f ( x ) ) }$ is equal to

jee-main 2025 Q68 Limit Involving Derivative Definition of Composed Functions View

Q68. Let $f : ( - \infty , \infty ) - \{ 0 \} \rightarrow \mathbb { R }$ be a differentiable function such that $f ^ { \prime } ( 1 ) = \lim _ { a \rightarrow \infty } a ^ { 2 } f \left( \frac { 1 } { a } \right)$. Then $\lim _ { a \rightarrow \infty } \frac { a ( a + 1 ) } { 2 } \tan ^ { - 1 } \left( \frac { 1 } { a } \right) + a ^ { 2 } - 2 \log _ { e } a$ is equal to
(1) $\frac { 3 } { 2 } + \frac { \pi } { 4 }$
(2) $\frac { 3 } { 4 } + \frac { \pi } { 8 }$
(3) $\frac { 3 } { 8 } + \frac { \pi } { 4 }$
(4) $\frac { 5 } { 2 } + \frac { \pi } { 8 }$

jee-main 2025 Q71 Straightforward Polynomial or Basic Differentiation View

Q71. Let $f ( x ) = a x ^ { 3 } + b x ^ { 2 } + c x + 41$ be such that $f ( 1 ) = 40 , f ^ { \prime } ( 1 ) = 2$ and $f ^ { \prime } ( 1 ) = 4$. Then $\mathrm { a } ^ { 2 } + \mathrm { b } ^ { 2 } + \mathrm { c } ^ { 2 }$ is equal to:
(1) 73
(2) 62
(3) 51
(4) 54

jee-main 2025 Q72 Continuity Conditions via Composition View

Q72. If the function $f ( x ) = \frac { \sin 3 x + \alpha \sin x - \beta \cos 3 x } { x ^ { 3 } } , x \in \mathbf { R }$, is continuous at $x = 0$, then $f ( 0 )$ is equal to :
(1) 2
(2) - 2
(3) 4
(4) - 4

jee-main 2025 Q72 Chain Rule with Composition of Explicit Functions View

Q72. Suppose for a differentiable function $h , h ( 0 ) = 0 , h ( 1 ) = 1$ and $h ^ { \prime } ( 0 ) = h ^ { \prime } ( 1 ) = 2$. If $\mathrm { g } ( x ) = h \left( \mathrm { e } ^ { x } \right) \mathrm { e } ^ { h ( x ) }$, then $g ^ { \prime } ( 0 )$ is equal to:
(1) 5
(2) 4
(3) 8
(4) 3

jee-main 2025 Q72 Continuity Conditions via Composition View

Q72. $\quad$ For $\mathrm { a } , \mathrm { b } > 0$, let $f ( x ) = \left\{ \begin{array} { c l } \frac { \tan ( ( \mathrm { a } + 1 ) x ) + \mathrm { b } \tan x } { x } , & x < 0 \\ 3 , & x = 0 \\ \frac { \sqrt { \mathrm { a } x + \mathrm { b } ^ { 2 } x ^ { 2 } } - \sqrt { \mathrm { a } x } } { \mathrm {~b} \sqrt { \mathrm { a } } \sqrt { x } } , & x > 0 \end{array} \right.$ be a continous function at $x = 0$. Then $\frac { \mathrm { b } } { \mathrm { a } }$ is equal to :
(1) 6
(2) 4
(3) 5
(4) 8

jee-main 2025 Q73 Piecewise Function Differentiability Analysis View

Q73. If $f ( x ) = \left\{ \begin{array} { l } x ^ { 3 } \sin \left( \frac { 1 } { x } \right) , x \neq 0 \\ 0 \quad , x = 0 \end{array} \right.$ then
(1) $f ^ { \prime \prime } \left( \frac { 2 } { \pi } \right) = \frac { 24 - \pi ^ { 2 } } { 2 \pi }$
(2) $f ^ { \prime \prime } \left( \frac { 2 } { \pi } \right) = \frac { 12 - \pi ^ { 2 } } { 2 \pi }$
(3) $f ^ { \prime \prime } ( 0 ) = 1$
(4) $f ^ { \prime \prime } ( 0 ) = 0$

jee-main 2025 Q74 Derivative of Inverse Functions View

Q74. Let $f ( x ) = x ^ { 5 } + 2 \mathrm { e } ^ { x / 4 }$ for all $x \in \mathbf { R }$. Consider a function $g ( x )$ such that $( g \circ f ) ( x ) = x$ for all $x \in \mathbf { R }$. Then the value of $8 g ^ { \prime } ( 2 )$ is :
(1) 2
(2) 8
(3) 4
(4) 16

jee-main 2025 Q85 Limit Involving Derivative Definition of Composed Functions View

Q85. Let $f$ be a differentiable function in the interval $( 0 , \infty )$ such that $f ( 1 ) = 1$ and $\lim _ { t \rightarrow x } \frac { t ^ { 2 } f ( x ) - x ^ { 2 } f ( t ) } { t - x } = 1$ for each $x > 0$. Then $2 f ( 2 ) + 3 f ( 3 )$ is equal to $\_\_\_\_$

1
2
3
4
5
6
7
8
9
10
11
12

Load questions...

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