LFM Pure and Mechanics

jee-main 2023 Q74 Geometric or applied optimisation problem View

A wire of length 20 m is to be cut into two pieces. A piece of length $\ell_1$ is bent to make a square of area $A_1$ and the other piece of length $\ell_2$ is made into a circle of area $A_2$. If $2A_1 + 3A_2$ is minimum then $\pi\ell_1 : \ell_2$ is equal to:
(1) 6:1
(2) $3:1$
(3) $1:6$
(4) $4:1$

jee-main 2023 Q77 Determine intervals of increase/decrease or monotonicity conditions View

Let $f : ( 0,1 ) \rightarrow \mathbb { R }$ be a function defined by $f ( x ) = \frac { 1 } { 1 - e ^ { - x } }$, and $g ( x ) = ( f ( - x ) - f ( x ) )$. Consider two statements (I) $g$ is an increasing function in $( 0,1 )$ (II) $g$ is one-one in $( 0,1 )$ Then,
(1) Only (I) is true
(2) Only (II) is true
(3) Neither (I) nor (II) is true
(4) Both (I) and (II) are true

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

Let the function $f ( x ) = 2 x ^ { 3 } + ( 2 p - 7 ) x ^ { 2 } + 3 ( 2 p - 9 ) x - 6$ have a maxima for some value of $x < 0$ and a minima for some value of $x > 0$. Then, the set of all values of $p$ is
(1) $\left( \frac { 9 } { 2 } , \infty \right)$
(2) $\left( 0 , \frac { 9 } { 2 } \right)$
(3) $\left( - \infty , \frac { 9 } { 2 } \right)$
(4) $\left( - \frac { 9 } { 2 } , \frac { 9 } { 2 } \right)$

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 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 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 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 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 $f(x) = (x+3)^2(x-2)^3$, $x \in [-4, 4]$. If $M$ and $m$ are the maximum and minimum values of $f$, respectively in $[-4, 4]$, then the value of $M - m$ is:
(1) 600
(2) 392
(3) 608
(4) 108

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 Q86 Count or characterize roots using extremum values View

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a thrice differentiable function such that $f ( 0 ) = 0 , f ( 1 ) = 1 , f ( 2 ) = - 1 , f ( 3 ) = 2$ and $f ( 4 ) = - 2$. Then, the minimum number of zeros of $\left( 3 f ^ { \prime } f ^ { \prime \prime } + f f ^ { \prime \prime \prime } \right) ( x )$ is $\_\_\_\_$

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

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LFM Pure and Mechanics

Indices and Surds 107

Exponential Functions 100

Laws of Logarithms 200

Exponential Equations & Modelling 58

Arithmetic Sequences and Series 271

Geometric Sequences and Series 203

Differentiation from First Principles 37

Tangents, normals and gradients 93

Stationary points and optimisation 384

Applied differentiation 164

Indefinite & Definite Integrals 533

Areas Between Curves 83

Areas by integration 109

Volumes of Revolution 61

Numerical integration 32

Vectors Introduction & 2D 204

Vectors 3D & Lines 233

Travel graphs 11

SUVAT in 2D & Gravity 6

Constant acceleration (SUVAT) 32

Variable acceleration (1D) 40

Variable acceleration (vectors) 26

Forces, equilibrium and resultants 9

Newton's laws and connected particles 18

Pulley systems 5

Motion on a slope 4

Friction 10

Momentum and Collisions 15

Moments 13

Parametric curves and Cartesian conversion 11

Parametric differentiation 15

Parametric integration 4

Projectiles 40