LFM Pure and Mechanics

jee-main 2021 Q85 Geometric or applied optimisation problem View

If the point on the curve $y ^ { 2 } = 6 x$, nearest to the point $\left( 3 , \frac { 3 } { 2 } \right)$ is $( \alpha , \beta )$, then $2 ( \alpha + \beta )$ is equal to $\underline{\hspace{1cm}}$.

jee-main 2021 Q86 Find absolute extrema on a closed interval or domain View

Let $f : [ - 1,1 ] \rightarrow R$ be defined as $f ( x ) = a x ^ { 2 } + b x + c$ for all $x \in [ - 1,1 ]$, where $a , b , c \in R$ such that $f ( - 1 ) = 2 , f ^ { \prime } ( - 1 ) = 1$ and for $x \in ( - 1,1 )$ the maximum value of $f ^ { \prime \prime } ( x )$ is $\frac { 1 } { 2 }$. If $f ( x ) \leq \alpha , x \in [ - 1,1 ]$, then the least value of $\alpha$ is equal to

jee-main 2021 Q86 Determine intervals of increase/decrease or monotonicity conditions View

If $R$ is the least value of $a$ such that the function $f ( x ) = x ^ { 2 } + \mathrm { a } x + 1$ is increasing on $[ 1,2 ]$ and $S$ is the greatest value of $a$ such that the function $f ( x ) = x ^ { 2 } + a x + 1$ is decreasing on $[ 1,2 ]$, then the value of $| R - S |$ is

jee-main 2021 Q88 Find absolute extrema on a closed interval or domain View

Let $f : [ - 1,1 ] \rightarrow R$ be defined as $f ( x ) = a x ^ { 2 } + b x + c$ for all $x \in [ - 1,1 ]$, where $a , b , c \in R$ such that $f ( - 1 ) = 2 , f ^ { \prime } ( - 1 ) = 1$ and for $x \in ( - 1,1 )$ the maximum value of $f ^ { \prime \prime } ( x )$ is $\frac { 1 } { 2 }$. If $f ( x ) \leq \alpha , x \in [ - 1,1 ]$, then the least value of $\alpha$ is equal to

jee-main 2021 Q88 Determine parameters from given extremum conditions View

Let $f ( x )$ be a polynomial of degree 6 in $x$, in which the coefficient of $x ^ { 6 }$ is unity and it has extrema at $x = - 1$ and $x = 1$. If $\lim _ { x \rightarrow 0 } \frac { f ( x ) } { x ^ { 3 } } = 1$, then $5 \cdot f ( 2 )$ is equal to

jee-main 2022 Q62 Geometric or applied optimisation problem View

Let $x , y > 0$. If $x ^ { 3 } y ^ { 2 } = 2 ^ { 15 }$, then the least value of $3 x + 2 y$ is
(1) 30
(2) 32
(3) 36
(4) 40

jee-main 2022 Q62 Find absolute extrema on a closed interval or domain View

If the minimum value of $f(x) = \frac{5x^2}{2} + \frac{\alpha}{x^5}$, $x > 0$, is 14, then the value of $\alpha$ is equal to
(1) 32
(2) 64
(3) 128
(4) 256

jee-main 2022 Q71 Find absolute extrema on a closed interval or domain View

If the absolute maximum value of the function $f(x) = (x ^ { 2 } - 2x + 7) e ^ { (4x ^ { 3 } - 12x ^ { 2 } - 180x + 31)}$ in the interval $[-3,0]$ is $f(\alpha)$, then
(1) $\alpha = 0$
(2) $\alpha = - 3$
(3) $\alpha \in (-1,0)$
(4) $\alpha \in (-3,-1)$

jee-main 2022 Q71 Determine intervals of increase/decrease or monotonicity conditions View

The function $f ( x ) = x e ^ { x ( 1 - x ) } , x \in R$, is
(1) increasing in $\left( - \frac { 1 } { 2 } , 1 \right)$
(2) decreasing in $\left( \frac { 1 } { 2 } , 2 \right)$
(3) increasing in $\left( - 1 , - \frac { 1 } { 2 } \right)$
(4) decreasing in $\left( - \frac { 1 } { 2 } , \frac { 1 } { 2 } \right)$

jee-main 2022 Q72 Find concavity, inflection points, or second derivative properties View

Let $f ( x ) = 3 ^ { \left( x ^ { 2 } - 2 \right) ^ { 3 } + 4 } , \mathrm { x } \in R$. Then which of the following statements are true? $P : x = 0$ is a point of local minima of $f$ $Q : x = \sqrt { 2 }$ is a point of inflection of $f$ $R : f ^ { \prime }$ is increasing for $x > \sqrt { 2 }$
(1) Only $P$ and $Q$
(2) Only $P$ and $R$
(3) Only $Q$ and $R$
(4) All $P , Q$ and $R$

jee-main 2022 Q72 Find critical points and classify extrema of a given function View

The curve $y(x) = ax ^ { 3 } + bx ^ { 2 } + cx + 5$ touches the $x$-axis at the point $P(-2,0)$ and cuts the $y$-axis at the point $Q$, where $y'$ is equal to 3. Then the local maximum value of $y(x)$ is
(1) $\frac { 27 } { 4 }$
(2) $\frac { 29 } { 4 }$
(3) $\frac { 37 } { 4 }$
(4) $\frac { 9 } { 2 }$

jee-main 2022 Q72 Find absolute extrema on a closed interval or domain View

The sum of the absolute maximum and absolute minimum values of the function $f ( x ) = \tan ^ { - 1 } ( \sin x - \cos x )$ in the interval $[ 0 , \pi ]$ is
(1) $0$
(2) $\tan ^ { - 1 } \left( \frac { 1 } { \sqrt { 2 } } \right) - \frac { \pi } { 4 }$
(3) $\cos ^ { - 1 } \left( \frac { 1 } { \sqrt { 3 } } \right) - \frac { \pi } { 4 }$
(4) $\frac { - \pi } { 12 }$

jee-main 2022 Q73 Determine intervals of increase/decrease or monotonicity conditions View

Let $\lambda ^ { * }$ be the largest value of $\lambda$ for which the function $f _ { \lambda } ( x ) = 4 \lambda x ^ { 3 } - 36 \lambda x ^ { 2 } + 36 x + 48$ is increasing for all $x \in \mathbb { R }$. Then $f _ { \lambda ^ { * } } ( 1 ) + f _ { \lambda ^ { * } } ( - 1 )$ is equal to:
(1) 36
(2) 48
(3) 64
(4) 72

jee-main 2022 Q73 Determine intervals of increase/decrease or monotonicity conditions View

For the function $f ( x ) = 4 \log _ { e } ( x - 1 ) - 2 x ^ { 2 } + 4 x + 5 , x > 1$, which one of the following is NOT correct?
(1) $f ( x )$ is increasing in $( 1,2 )$ and decreasing in $( 2 , \infty )$
(2) $f ( x ) = - 1$ has exactly two solutions
(3) $f ^ { \prime } ( \mathrm { e } ) - f ^ { \prime \prime } ( 2 ) < 0$
(4) $f ( x ) = 0$ has a root in the interval $( e , e + 1 )$

jee-main 2022 Q73 Geometric or applied optimisation problem View

Consider a cuboid of sides $2 x , 4 x$ and $5 x$ and a closed hemisphere of radius $r$. If the sum of their surface areas is constant $k$, then the ratio $x : r$, for which the sum of their volumes is maximum, is
(1) $2 : 5$
(2) $19 : 45$
(3) $3 : 8$
(4) $19 : 15$

jee-main 2022 Q73 Find absolute extrema on a closed interval or domain View

The sum of the absolute minimum and the absolute maximum values of the function $f ( x ) = \left| 3 x - x ^ { 2 } + 2 \right| - x$ in the interval $[ - 1 , 2 ]$ is
(1) $\frac { \sqrt { 17 } + 3 } { 2 }$
(2) $\frac { \sqrt { 17 } + 5 } { 2 }$
(3) 5
(4) $\frac { 9 - \sqrt { 17 } } { 2 }$

jee-main 2022 Q73 Find critical points and classify extrema of a given function View

If $m$ and $n$ respectively are the number of local maximum and local minimum points of the function $f ( x ) = \int _ { 0 } ^ { x ^ { 2 } } \frac { t ^ { 2 } - 5 t + 4 } { 2 + e ^ { t } } d t$, then the ordered pair $( m , n )$ is equal to
(1) $( 2,3 )$
(2) $( 3,2 )$
(3) $( 2,2 )$
(4) $( 3,4 )$

jee-main 2022 Q73 Composite or piecewise function extremum analysis View

Let $f ( x ) = \left\{ \begin{array} { c c } x ^ { 3 } - x ^ { 2 } + 10 x - 7 , & x \leq 1 \\ - 2 x + \log _ { 2 } \left( b ^ { 2 } - 4 \right) , & x > 1 \end{array} \right.$ Then the set of all values of $b$, for which $f ( x )$ has maximum value at $x = 1$, is:
(1) $( - 6 , - 2 )$
(2) $( 2,6 )$
(3) $[ - 6 , - 2 ) \cup ( 2,6 ]$
(4) $[ - \sqrt { 6 } , - 2 ) \cup ( 2 , \sqrt { 6 } ]$

jee-main 2022 Q73 Composite or piecewise function extremum analysis View

Let a function $f : \mathbb { R } \rightarrow \mathbb { R }$ be defined as: $f ( x ) = \begin{cases} \int _ { 0 } ^ { x } ( 5 - | t - 3 | ) d t , & x > 4 \\ x ^ { 2 } + b x , & x \leq 4 \end{cases}$ where $b \in \mathbb { R }$. If $f$ is continuous at $x = 4$, then which of the following statements is NOT true?
(1) $f$ is not differentiable at $x = 4$
(2) $f ^ { \prime } ( 3 ) + f ^ { \prime } ( 5 ) = \frac { 35 } { 4 }$
(3) $f$ is increasing in $\left( - \infty , \frac { 1 } { 8 } \right) \cup ( 8 , \infty )$
(4) $f$ has a local minima at $x = \frac { 1 } { 8 }$

jee-main 2022 Q74 Determine intervals of increase/decrease or monotonicity conditions View

If the maximum value of $a$, for which the function $f _ { a } ( x ) = \tan ^ { - 1 } 2 x - 3 a x + 7$ is non-decreasing in $\left[ - \frac { \pi } { 6 } , \frac { \pi } { 6 } \right]$, is $\bar { a }$, then $f _ { \bar { a } } \left( \frac { \pi } { 8 } \right)$ is equal to
(1) $8 - \frac { 9 \pi } { 49 + \pi ^ { 2 } }$
(2) $8 - \frac { 4 \pi } { 94 + \pi ^ { 2 } }$
(3) $8 \frac { 1 + \pi ^ { 2 } } { 9 + \pi ^ { 2 } }$
(4) $8 - \frac { \pi } { 4 }$

jee-main 2022 Q75 Geometric or applied optimisation problem View

A wire of length 22 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of the square and the equilateral triangle is minimum, is equal to (if the full question text was truncated, this is the standard formulation of this problem).

jee-main 2022 Q75 Find absolute extrema on a closed interval or domain View

Let $f ( x ) = 2 \cos ^ { - 1 } x + 4 \cot ^ { - 1 } x - 3 x ^ { 2 } - 2 x + 10 , x \in [ - 1 , 1 ]$. If $[ a , b ]$ is the range of the function, then $4a - b$ is equal to
(1) 11
(2) $11 - \pi$
(3) $11 + \pi$
(4) $15 - \pi$

jee-main 2023 Q69 Geometric or applied optimisation problem View

Let $S$ be the set of all $a \in N$ such that the area of the triangle formed by the tangent at the point $P(b, c)$, $b, c \in N$, on the parabola $y^2 = 2ax$ and the lines $x = b$, $y = 0$ is 16 unit$^2$, then $\sum_{a \in S} a$ is equal to $\_\_\_\_$.

jee-main 2023 Q71 Geometric or applied optimisation problem View

A square piece of tin of side 30 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. If the volume of the box is maximum, then its surface area (in $\mathrm{cm}^2$) is equal to
(1) 800
(2) 675
(3) 1025
(4) 900

jee-main 2023 Q74 Find critical points and classify extrema of a given function View

Let $f(x) = \int_0^x t(t-1)(t-2)\,dt$, $x > 0$. Then the number of points in the interval $(0, 3)$ at which $f(x)$ has a local maximum is $\_\_\_\_$.

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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