LFM Pure and Mechanics

jee-main 2023 Q79 Determine parameters from given extremum conditions View

If the functions $f(x) = \frac{x^{3}}{3} + 2bx + \frac{ax^{2}}{2}$ and $g(x) = \frac{x^{3}}{3} + ax + bx^{2}$, $a \neq 2b$ have a common extreme point, then $a + 2b + 7$ is equal to
(1) 4
(2) $\frac{3}{2}$
(3) 3
(4) 6

jee-main 2023 Q80 Find critical points and classify extrema of a given function View

Let $x = 2$ be a local minima of the function $f ( x ) = 2 x ^ { 4 } - 18 x ^ { 2 } + 8 x + 12 , x \in ( - 4,4 )$. If $M$ is local maximum value of the function $f$ in $( - 4,4 )$, then $M =$
(1) $12 \sqrt { 6 } - \frac { 33 } { 2 }$
(2) $12 \sqrt { 6 } - \frac { 31 } { 2 }$
(3) $18 \sqrt { 6 } - \frac { 33 } { 2 }$
(4) $18 \sqrt { 6 } - \frac { 31 } { 2 }$

jee-main 2023 Q80 Monotonicity and boundedness analysis View

If $a _ { \alpha }$ is the greatest term in the sequence $a _ { n } = \frac { n ^ { 3 } } { n ^ { 4 } + 147 } , n = 1 , 2 , 3 \ldots$, then $\alpha$ is equal to $\_\_\_\_$

jee-main 2024 Q66 Geometric or applied optimisation problem View

The maximum area of a triangle whose one vertex is at $( 0,0 )$ and the other two vertices lie on the curve $y = - 2 x ^ { 2 } + 54$ at points $( x , y )$ and $( - x , y )$ where $\mathrm { y } > 0$ is :
(1) 88
(2) 122
(3) 92
(4) 108

jee-main 2024 Q70 Determine intervals of increase/decrease or monotonicity conditions View

For the function $f ( x ) = ( \cos x ) - x + 1 , x \in \mathbb { R }$, between the following two statements (S1) $f ( x ) = 0$ for only one value of $x$ in $[ 0 , \pi ]$. (S2) $f ( x )$ is decreasing in $\left[ 0 , \frac { \pi } { 2 } \right]$ and increasing in $\left[ \frac { \pi } { 2 } , \pi \right]$.
(1) Both (S1) and (S2) are correct.
(2) Both (S1) and (S2) are incorrect.
(3) Only (S2) is correct.
(4) Only (S1) is correct.

jee-main 2024 Q71 Find critical points and classify extrema of a given function View

Let $f ( x ) = 4 \cos ^ { 3 } x + 3 \sqrt { 3 } \cos ^ { 2 } x - 10$. The number of points of local maxima of $f$ in interval $( 0,2 \pi )$ is
(1) 3
(2) 4
(3) 1
(4) 2

jee-main 2024 Q72 Continuity and Discontinuity Analysis of Piecewise Functions View

Consider the function $\mathrm { f } : ( 0,2 ) \rightarrow \mathrm { R }$ defined by $\mathrm { f } ( \mathrm { x } ) = \frac { \mathrm { x } } { 2 } + \frac { 2 } { \mathrm { x } }$ and the function $\mathrm { g } ( \mathrm { x } )$ defined by $\mathrm { gx } = \begin{array} { c c } \min \{ \mathrm { f } ( \mathrm { t } ) \} , & 0 < \mathrm { t } \leq \mathrm { x } \text { and } 0 < \mathrm { x } \leq 1 \\ \frac { 3 } { 2 } + \mathrm { x } , & 1 < \mathrm { x } < 2 \end{array}$. Then
(1) g is continuous but not differentiable at $\mathrm { x } = 1$
(2) g is not continuous for all $\mathrm { x } \in ( 0,2 )$
(3) g is neither continuous nor differentiable at $\mathrm { x } = 1$
(4) $g$ is continuous and differentiable for all $x \in ( 0,2 )$

jee-main 2024 Q72 Find critical points and classify extrema of a given function View

The number of critical points of the function $f ( x ) = ( x - 2 ) ^ { 2 / 3 } ( 2 x + 1 )$ is
(1) 1
(2) 2
(3) 0
(4) 3

jee-main 2024 Q73 Find critical points and classify extrema of a given function View

The function $f ( x ) = 2 x + 3 x ^ { \frac { 2 } { 3 } } , x \in R$, has
(1) exactly one point of local minima and no point of local maxima
(2) exactly one point of local maxima and no point of local minima
(3) exactly one point of local maxima and exactly one point of local minima
(4) exactly two points of local maxima and exactly one point of local minima

jee-main 2024 Q73 Convexity and inflection point analysis View

Let $g : R \rightarrow R$ be a non constant twice differentiable such that $g ^ { \prime } \left( \frac { 1 } { 2 } \right) = g ^ { \prime } \left( \frac { 3 } { 2 } \right)$. If a real valued function $f$ is defined as $f ( x ) = \frac { 1 } { 2 } [ g ( x ) + g ( 2 - x ) ]$, then
(1) $f ^ { \prime \prime } ( x ) = 0$ for atleast two $x$ in $( 0,2 )$
(2) $f ^ { \prime \prime } ( x ) = 0$ for exactly one $x$ in $( 0,1 )$
(3) $f ^ { \prime \prime } ( x ) = 0$ for no $x$ in $( 0,1 )$
(4) $f ^ { \prime } \left( \frac { 3 } { 2 } \right) + f ^ { \prime } \left( \frac { 1 } { 2 } \right) = 1$

jee-main 2024 Q73 Find absolute extrema on a closed interval or domain View

If the function $f : (-\infty, -1] \rightarrow [a, b]$ defined by $f(x) = e^{x^3 - 3x + 1}$ is one-one and onto, then the distance of the point $P(2b+4, a+2)$ from the line $x + e^{-3}y = 4$ is:

jee-main 2024 Q73 Determine intervals of increase/decrease or monotonicity conditions View

Let $\mathrm { g } ( \mathrm { x } ) = 3 \mathrm { f } ^ { \mathrm { x } } + \mathrm { f } ( 3 - \mathrm { x } )$ and $\mathrm { f } ^ { \prime \prime } ( \mathrm { x } ) > 0$ for all $\mathrm { x } \in ( 0,3 )$. If g is decreasing in ( $0 , \alpha$ ) and increasing in $( \alpha , 3 )$, then $8 \alpha$ is
(1) 24
(2) 0
(3) 18
(4) 20

jee-main 2024 Q73 Recover a Function from a Composition or Functional Equation View

Let $f: R - \{0\} \rightarrow R$ be a function satisfying $f\left(\frac{x}{y}\right) = \frac{f(x)}{f(y)}$ for all $x, y$, $f(y) \neq 0$. If $f'(1) = 2024$, then
(1) $xf'(x) - 2024f(x) = 0$
(2) $xf'(x) + 2024f(x) = 0$
(3) $xf'(x) + f(x) = 2024$
(4) $xf'(x) - 2023f(x) = 0$

jee-main 2024 Q73 Find absolute extrema on a closed interval or domain View

If the function $f ( x ) = \left( \frac { 1 } { x } \right) ^ { 2 x } ; x > 0$ attains the maximum value at $x = \frac { 1 } { \mathrm { e } }$ then:
(1) $\mathrm { e } ^ { \pi } < \pi ^ { \mathrm { e } }$
(2) $\mathrm { e } ^ { \pi } > \pi ^ { \mathrm { e } }$
(3) $( 2 e ) ^ { \pi } > \pi ^ { ( 2 e ) }$
(4) $\mathrm { e } ^ { 2 \pi } < ( 2 \pi ) ^ { \mathrm { e } }$

jee-main 2024 Q73 Determine parameters from given extremum conditions View

If the function $f ( x ) = 2 x ^ { 3 } - 9 x ^ { 2 } + 12 \mathrm { a } ^ { 2 } x + 1 , \mathrm { a } > 0$ has a local maximum at $x = \alpha$ and a local minimum at $x = \alpha ^ { 2 }$, then $\alpha$ and $\alpha ^ { 2 }$ are the roots of the equation : (1) $x ^ { 2 } - 6 x + 8 = 0$ (2) $x ^ { 2 } + 6 x + 8 = 0$ (3) $8 x ^ { 2 } + 6 x - 1 = 0$ (4) $8 x ^ { 2 } - 6 x + 1 = 0$

jee-main 2024 Q74 Determine intervals of increase/decrease or monotonicity conditions View

The function $f ( x ) = \frac { x } { x ^ { 2 } - 6 x - 16 } , x \in \mathbb { R } - \{ - 2,8 \}$
(1) decreases in $( - 2,8 )$ and increases in $( - \infty , - 2 ) \cup ( 8 , \infty )$
(2) decreases in $( - \infty , - 2 ) \cup ( - 2,8 ) \cup ( 8 , \infty )$
(3) decreases in $( - \infty , - 2 )$ and increases in $( 8 , \infty )$
(4) increases in $( - \infty , - 2 ) \cup ( - 2,8 ) \cup ( 8 , \infty )$

jee-main 2024 Q74 Determine intervals of increase/decrease or monotonicity conditions View

The interval in which the function $f ( x ) = x ^ { x } , x > 0$, is strictly increasing is
(1) $\left( 0 , \frac { 1 } { e } \right]$
(2) $( 0 , \infty )$
(3) $\left[ \frac { 1 } { e } , \infty \right)$
(4) $\left[ \frac { 1 } { e ^ { 2 } } , 1 \right)$

jee-main 2024 Q74 Find absolute extrema on a closed interval or domain View

Let $f ( x ) = 3 \sqrt { x - 2 } + \sqrt { 4 - x }$ be a real valued function. If $\alpha$ and $\beta$ are respectively the minimum and the maximum values of $f$, then $\alpha ^ { 2 } + 2 \beta ^ { 2 }$ is equal to
(1) 42
(2) 38
(3) 24
(4) 44

jee-main 2024 Q87 Find absolute extrema on a closed interval or domain View

Let the maximum and minimum values of $\left( \sqrt { 8 x - x ^ { 2 } - 12 } - 4 \right) ^ { 2 } + ( x - 7 ) ^ { 2 } , x \in \mathbf { R }$ be M and m , respectively. Then $\mathrm { M } ^ { 2 } - \mathrm { m } ^ { 2 }$ is equal to $\_\_\_\_$

jee-main 2024 Q87 Geometric or applied optimisation problem View

Let A be the region enclosed by the parabola $y ^ { 2 } = 2 x$ and the line $x = 24$. Then the maximum area of the rectangle inscribed in the region A is $\_\_\_\_$

jee-main 2024 Q89 Determine intervals of increase/decrease or monotonicity conditions View

Let the set of all values of $p$, for which $f ( x ) = \left( p ^ { 2 } - 6 p + 8 \right) \left( \sin ^ { 2 } 2 x - \cos ^ { 2 } 2 x \right) + 2 ( 2 - p ) x + 7$ does not have any critical point, be the interval $( a , b )$. Then $16 a b$ is equal to $\_\_\_\_$

jee-main 2025 Q4 Composite or piecewise function extremum analysis View

The sum of all local minimum values of the function
$$f ( x ) = \left\{ \begin{array} { l r } 1 - 2 x , & x < - 1 \\ \frac { 1 } { 3 } ( 7 + 2 | x | ) , & - 1 \leq x \leq 2 \\ \frac { 11 } { 18 } ( x - 4 ) ( x - 5 ) , & x > 2 \end{array} \right.$$
is
(1) $\frac { 157 } { 72 }$
(2) $\frac { 131 } { 72 }$
(3) $\frac { 171 } { 72 }$
(4) $\frac { 167 } { 72 }$

jee-main 2025 Q8 Find critical points and classify extrema of a given function View

Let $f ( x ) = \int _ { 0 } ^ { x ^ { 2 } } \frac { \mathrm { t } ^ { 2 } - 8 \mathrm { t } + 15 } { \mathrm { e } ^ { t } } \mathrm { dt } , x \in \mathbf { R }$. Then the numbers of local maximum and local minimum points of $f$, respectively, are :
(1) 2 and 3
(2) 2 and 2
(3) 3 and 2
(4) 1 and 3

jee-main 2025 Q17 Determine intervals of increase/decrease or monotonicity conditions View

Let $(2, 3)$ be the largest open interval in which the function $f(x) = 2\log_{\mathrm{e}}(x - 2) - x^{2} + ax + 1$ is strictly increasing and $(\mathrm{b}, \mathrm{c})$ be the largest open interval, in which the function $\mathrm{g}(x) = (x - 1)^{3}(x + 2 - \mathrm{a})^{2}$ is strictly decreasing. Then $100(a + b - c)$ is equal to:
(1) 420
(2) 360
(3) 160
(4) 280

jee-main 2025 Q18 Find critical points and classify extrema of a given function View

$\lim_{x \rightarrow 0} \operatorname{cosec} x \left(\sqrt{2\cos^2 x + 3\cos x} - \sqrt{\cos^2 x + \sin x + 4}\right)$ is:
(1) 0
(2) $\frac{1}{\sqrt{15}}$
(3) $\frac{1}{2\sqrt{5}}$
(4) $-\frac{1}{2\sqrt{5}}$

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21

Load questions...

LFM Pure and Mechanics

Indices and Surds 96

Exponential Functions 197

Laws of Logarithms 242

Exponential Equations & Modelling 28

Arithmetic Sequences and Series 363

Geometric Sequences and Series 159

Differentiation from First Principles 45

Tangents, normals and gradients 140

Stationary points and optimisation 505

Applied differentiation 76

Indefinite & Definite Integrals 502

Areas Between Curves 71

Areas by integration 221

Volumes of Revolution 68

Numerical integration 49

Vectors Introduction & 2D 211

Vectors 3D & Lines 524

Travel graphs 14

SUVAT in 2D & Gravity 31

Constant acceleration (SUVAT) 78

Variable acceleration (1D) 55

Variable acceleration (vectors) 32

Forces, equilibrium and resultants 24

Newton's laws and connected particles 21

Pulley systems 6

Motion on a slope 6

Friction 8

Momentum and Collisions 34

Moments 28

Parametric curves and Cartesian conversion 9

Parametric differentiation 13

Parametric integration 6

Projectiles 49