LFM Pure and Mechanics

jee-main 2018 Q82 Find critical points and classify extrema of a given function View

Let $f ( x ) = x ^ { 2 } + \frac { 1 } { x ^ { 2 } }$ and $g ( x ) = x - \frac { 1 } { x } , x \in R - \{ - 1,0,1 \}$. If $h ( x ) = \frac { f ( x ) } { g ( x ) }$, then the local minimum value of $h ( x )$ is:
(1) $2 \sqrt { 2 }$
(2) 3
(3) - 3
(4) $- 2 \sqrt { 2 }$

jee-main 2018 Q82 Geometric or applied optimisation problem View

If a right circular cone having maximum volume, is inscribed in a sphere of radius 3 cm , then the curved surface area ( $\mathrm { in } \mathrm { cm } ^ { 2 }$ ) of this cone is
(1) $8 \sqrt { 3 } \pi$
(2) $6 \sqrt { 2 } \pi$
(3) $6 \sqrt { 3 } \pi$
(4) $8 \sqrt { 2 } \pi$

jee-main 2019 Q80 Geometric or applied optimisation problem View

The shortest distance between the line $y = x$ and the curve $y^2 = x - 2$ is
(1) $\frac{7}{4\sqrt{2}}$
(2) $\frac{7}{8}$
(3) $\frac{11}{4\sqrt{2}}$
(4) 2

jee-main 2019 Q80 Find critical points and classify extrema of a given function View

If $f ( x )$ is a non-zero polynomial of degree four, having local extreme points at $x = - 1,0,1$; then the set $S = \{ x \in R : f ( x ) = f ( 0 ) \}$ contains exactly
(1) Two irrational and two rational numbers
(2) Four rational numbers
(3) Two irrational and one rational number
(4) Four irrational numbers

jee-main 2019 Q81 Determine parameters from given extremum conditions View

If the function $f$ given by $f ( x ) = x ^ { 3 } - 3 ( a - 2 ) x ^ { 2 } + 3 a x + 7$, for some $a \in R$ is increasing in $( 0,1 ]$ and decreasing in $[ 1,5 )$, then a root of the equation, $\frac { f ( x ) - 14 } { ( x - 1 ) ^ { 2 } } = 0 , ( x \neq 1 )$ is :
(1) 7
(2) - 7
(3) 6
(4) 5

jee-main 2019 Q81 Find critical points and classify extrema of a given function View

If $S_1$ and $S_2$ are respectively the sets of local minimum and local maximum points of the function, $f(x) = 9x^4 + 12x^3 - 36x^2 + 25$, $x \in R$, then
(1) $S_1 = \{-2\}$; $S_2 = \{0, 1\}$
(2) $S_1 = \{-1\}$; $S_2 = \{0, 2\}$
(3) $S_1 = \{-2, 0\}$; $S_2 = \{1\}$
(4) $S_1 = \{-2, 1\}$; $S_2 = \{0\}$

jee-main 2019 Q82 Geometric or applied optimisation problem View

The shortest distance between the point $\left( \frac { 3 } { 2 } , 0 \right)$ and the curve $y = \sqrt { x } , ( x > 0 )$, is
(1) $\frac { \sqrt { 3 } } { 2 }$
(2) $\frac { 5 } { 4 }$
(3) $\frac { 3 } { 2 }$
(4) $\frac { \sqrt { 5 } } { 2 }$

jee-main 2019 Q82 Geometric or applied optimisation problem View

The height of a right circular cylinder of maximum volume inscribed in a sphere of radius 3 is:
(1) $\sqrt { 3 }$
(2) $\frac { 2 } { 3 } \sqrt { 3 }$
(3) $\sqrt { 6 }$
(4) $2 \sqrt { 3 }$

jee-main 2019 Q84 Find absolute extrema on a closed interval or domain View

Let $I = \int _ { a } ^ { b } \left( x ^ { 4 } - 2 x ^ { 2 } \right) d x$. If $I$ is minimum then the ordered pair $( a , b )$ is
(1) $( 0 , \sqrt { 2 } )$
(2) $( \sqrt { 2 } , - \sqrt { 2 } )$
(3) $( - \sqrt { 2 } , 0 )$
(4) $( - \sqrt { 2 } , \sqrt { 2 } )$

jee-main 2020 Q63 Find absolute extrema on a closed interval or domain View

The minimum value of $2 ^ { \sin x } + 2 ^ { \cos x }$ is:
(1) $2 ^ { - 1 + \frac { 1 } { \sqrt { 2 } } }$
(2) $2 ^ { - 1 + \sqrt { 2 } }$
(3) $2 ^ { 1 - \sqrt { 2 } }$
(4) $2 ^ { 1 - \frac { 1 } { \sqrt { 2 } } }$

jee-main 2020 Q64 Find critical points and classify extrema of a given function View

Let $f ( x )$ be a polynomial of degree 5 such that $x = \pm 1$ are its critical points. If $\lim _ { x \rightarrow 0 } \left( 2 + \frac { f ( x ) } { x ^ { 3 } } \right) = 4$, then which one of the following is not true?
(1) $f$ is an odd function
(2) $f ( 1 ) - 4 f ( - 1 ) = 4$
(3) $x = 1$ is a point of local minimum and $x = - 1$ is a point of local maximum
(4) $x = 1$ is a point of local maxima of $f$

jee-main 2020 Q64 Determine intervals of increase/decrease or monotonicity conditions View

The function, $f ( x ) = ( 3 x - 7 ) x ^ { \frac { 2 } { 3 } } , x \in \mathrm { R }$, is increasing for all $x$ lying in
(1) $( - \infty , 0 ) \cup \left( \frac { 14 } { 15 } , \infty \right)$
(2) $( - \infty , 0 ) \cup \left( \frac { 3 } { 7 } , \infty \right)$
(3) $\left( - \infty , \frac { 14 } { 15 } \right)$
(4) $\left( - \infty , - \frac { 14 } { 15 } \right) \cup ( 0 , \infty )$

jee-main 2020 Q65 Find critical points and classify extrema of a given function View

Let a function $f : [ 0,5 ] \rightarrow R$ be continuous, $f ( 1 ) = 3$ and $F$ be defined as: $F ( x ) = \int _ { 1 } ^ { x } t ^ { 2 } g ( t ) d t$, where $g ( t ) = \int _ { 1 } ^ { t } f ( u ) d u$. Then for the function $F ( x )$, the point $x = 1$ is:
(1) a point of local minima
(2) not a critical point
(3) a point of local maxima
(4) a point of inflection

jee-main 2020 Q65 Determine parameters from given extremum conditions View

If $p(x)$ be a polynomial of degree three that has a local maximum value 8 at $x = 1$ and a local minimum value 4 at $x = 2$ then $p(0)$ is equal to
(1) 6
(2) $-12$
(3) 24
(4) 12

jee-main 2020 Q65 Find critical points and classify extrema of a given function View

If $x = 1$ is a critical point of the function $f(x) = (3x^2 + ax - 2 - a)e^x$, then
(1) $x = 1$ and $x = -\frac{2}{3}$ are local minima of $f$
(2) $x = 1$ and $x = -\frac{2}{3}$ is a local maxima of $f$
(3) $x = 1$ is a local maxima and $x = -\frac{2}{3}$ is a local minima of $f$
(4) $x = 1$ is a local minima and $x = -\frac{2}{3}$ are local maxima of $f$

jee-main 2020 Q66 Geometric or applied optimisation problem View

The area (in sq. units) of the largest rectangle $ABCD$ whose vertices $A$ and $B$ lie on the $x$-axis and vertices $C$ and $D$ lie on the parabola, $y = x ^ { 2 } - 1$ below the $x$-axis, is:
(1) $\frac { 2 } { 3 \sqrt { 3 } }$
(2) $\frac { 1 } { 3 \sqrt { 3 } }$
(3) $\frac { 4 } { 3 }$
(4) $\frac { 4 } { 3 \sqrt { 3 } }$

jee-main 2021 Q63 Geometric or applied optimisation problem View

If $P$ is a point on the parabola $y = x ^ { 2 } + 4$ which is closest to the straight line $y = 4 x - 1$, then the coordinates of $P$ are:
(1) $( - 2,8 )$
(2) $( 1,5 )$
(3) $( 2,8 )$
(4) $( 3,13 )$

jee-main 2021 Q72 Geometric or applied optimisation problem View

The triangle of maximum area that can be inscribed in a given circle of radius ' $r$ ' is :
(1) An equilateral triangle having each of its side of length $\sqrt { 3 } r$.
(2) An isosceles triangle with base equal to $2 r$.
(3) An equilateral triangle of height $\frac { 2 r } { 3 }$.
(4) A right angle triangle having two of its sides of length $2 r$ and $r$.

jee-main 2021 Q72 Determine intervals of increase/decrease or monotonicity conditions View

Let $f$ be a real valued function, defined on $R - \{ - 1,1 \}$ and given by $f ( x ) = 3 \log _ { \mathrm { e } } \left| \frac { x - 1 } { x + 1 } \right| - \frac { 2 } { x - 1 }$. Then in which of the following intervals, function $f ( x )$ is increasing?
(1) $( - \infty , - 1 ) \cup \left( \left[ \frac { 1 } { 2 } , \infty \right) - \{ 1 \} \right)$
(2) $( - \infty , \infty ) - \{ - 1,1 \}$
(3) $\left( - 1 , \frac { 1 } { 2 } \right]$
(4) $\left( - \infty , \frac { 1 } { 2 } \right] - \{ - 1 \}$

jee-main 2021 Q72 Find critical points and classify extrema of a given function View

The sum of all the local minimum values of the twice differentiable function $f : R \rightarrow R$ defined by $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - \frac { 3 f ^ { \prime \prime } ( 2 ) } { 2 } x + f ^ { \prime \prime } ( 1 )$ is:
(1) - 22
(2) 5
(3) - 27
(4) 0

jee-main 2021 Q72 Geometric or applied optimisation problem View

A box open from top is made from a rectangular sheet of dimension $a \times b$ by cutting squares each of side $x$ from each of the four corners and folding up the flaps. If the volume of the box is maximum, then $x$ is equal to: (1) $\frac { a + b + \sqrt { a ^ { 2 } + b ^ { 2 } - a b } } { 6 }$ (2) $\frac { a + b - \sqrt { a ^ { 2 } + b ^ { 2 } - a b } } { 12 }$ (3) $\frac { a + b - \sqrt { a ^ { 2 } + b ^ { 2 } - a b } } { 6 }$ (4) $\frac { a + b - \sqrt { a ^ { 2 } + b ^ { 2 } + a b } } { 6 }$

jee-main 2021 Q73 Determine parameters from given extremum conditions View

If Rolle's theorem holds for the function $f ( x ) = x ^ { 3 } - a x ^ { 2 } + b x - 4 , x \in [ 1,2 ]$ with $f ^ { \prime } \left( \frac { 4 } { 3 } \right) = 0$, then ordered pair $( a , b )$ is equal to :
(1) $( - 5 , - 8 )$
(2) $( - 5,8 )$
(3) $( 5,8 )$
(4) $( 5 , - 8 )$

jee-main 2021 Q73 Find absolute extrema on a closed interval or domain View

Let $M$ and $m$ respectively be the maximum and minimum values of the function $f ( x ) = \tan ^ { - 1 } ( \sin x + \cos x )$ in $\left[ 0 , \frac { \pi } { 2 } \right]$. Then the value of $\tan ( M - m )$ is equal to: (1) $2 - \sqrt { 3 }$ (2) $3 - 2 \sqrt { 2 }$ (3) $3 + 2 \sqrt { 2 }$ (4) $2 + \sqrt { 3 }$

jee-main 2021 Q74 Find critical points and classify extrema of a given function View

Let $a$ be a real number such that the function $f ( x ) = a x ^ { 2 } + 6 x - 15 , x \in R$ is increasing in $( - \infty , \frac { 3 } { 4 } )$ and decreasing in $\left( \frac { 3 } { 4 } , \infty \right)$. Then the function $g ( x ) = a x ^ { 2 } - 6 x + 15 , x \in R$ has a
(1) local maximum at $x = - \frac { 3 } { 4 }$
(2) local minimum at $x = - \frac { 3 } { 4 }$
(3) local maximum at $x = \frac { 3 } { 4 }$
(4) local minimum at $x = \frac { 3 } { 4 }$

jee-main 2021 Q74 Determine parameters from given extremum conditions View

The range of $a \in R$ for which the function $f ( x ) = ( 4 a - 3 ) \left( x + \log _ { e } 5 \right) + 2 ( a - 7 ) \cot \left( \frac { x } { 2 } \right) \sin ^ { 2 } \left( \frac { x } { 2 } \right) , x \neq 2 n \pi , n \in N$, has critical points, is :
(1) $( - 3,1 )$
(2) $\left[ - \frac { 4 } { 3 } , 2 \right]$
(3) $[ 1 , \infty )$
(4) $( - \infty , - 1 ]$

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16

Load questions...

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