LFM Pure and Mechanics

jee-main 2019 Q85 View

The area (in sq. units) of the region $A = \left\{ ( x , y ) : \frac { y ^ { 2 } } { 2 } \leq x \leq y + 4 \right\}$ is:
(1) 30
(2) 18
(3) $\frac { 53 } { 3 }$
(4) 16

jee-main 2019 Q86 View

The area of the region $A = \{(x,y): 0 \leq y \leq x|x| + 1$ and $-1 \leq x \leq 1\}$ in sq. units, is
(1) $\frac{4}{3}$
(2) 2
(3) $\frac{1}{3}$
(4) $\frac{2}{3}$

jee-main 2019 Q86 View

Let $S ( \alpha ) = \{ ( x , y ) : y ^ { 2 } \leq x , 0 \leq x \leq \alpha \}$ and $A ( \alpha )$ is area of the region $S ( \alpha )$. If for a $\lambda$, $0 < \lambda < 4$, $A ( \lambda ) : A ( 4 ) = 2 : 5$, then $\lambda$ equals:
(1) $4 \left( \frac { 2 } { 5 } \right) ^ { \frac { 1 } { 3 } }$
(2) $2 \left( \frac { 4 } { 25 } \right) ^ { \frac { 1 } { 3 } }$
(3) $4 \left( \frac { 4 } { 25 } \right) ^ { \frac { 1 } { 3 } }$
(4) $2 \left( \frac { 2 } { 5 } \right)$

jee-main 2020 Q58 View

The area (in sq. units) of an equilateral triangle inscribed in the parabola $y ^ { 2 } = 8 x$, with one of its vertices on the vertex of this parabola is
(1) $64 \sqrt { 3 }$
(2) $256 \sqrt { 3 }$
(3) $192 \sqrt { 3 }$
(4) $128 \sqrt { 3 }$

jee-main 2021 Q75 View

The area of the region bounded by the parabola $( y - 2 ) ^ { 2 } = ( x - 1 )$, the tangent to it at the point whose ordinate is 3 and the $x$-axis, is: (1) 4 (2) 6 (3) 9 (4) 10

jee-main 2021 Q88 View

Let $f : [ - 3,1 ] \rightarrow R$ be given as $f ( x ) = \left\{ \begin{array} { l l } \min \left\{ ( x + 6 ) , x ^ { 2 } \right\} , & - 3 \leq x \leq 0 \\ \max \left\{ \sqrt { x } , x ^ { 2 } \right\} , & 0 \leq x \leq 1 \end{array} \right.$. If the area bounded by $y = f ( x )$ and $x$-axis is $A$ sq units, then the value of $6 A$ is equal to

jee-main 2021 Q89 View

The graphs of sine and cosine functions, intersect each other at a number of points and between two consecutive points of intersection, the two graphs enclose the same area $A$. Then $A ^ { 4 }$ is equal to

jee-main 2022 Q74 View

The area of the region given by $A = \{(x,y) : x ^ { 2 } \leq y \leq \min(x + 2, 4 - 3x)\}$ is
(1) $\frac { 31 } { 8 }$
(2) $\frac { 17 } { 6 }$
(3) $\frac { 19 } { 6 }$
(4) $\frac { 27 } { 8 }$

jee-main 2022 Q74 View

The area enclosed by the curves $y = \log _ { e } \left( x + e ^ { 2 } \right) , x = \log _ { e } \left( \frac { 2 } { y } \right)$ and $x = \log _ { e } 2$, above the line $y = 1$ is
(1) $2 + e - \log _ { e } 2$
(2) $1 + e - \log _ { e } 2$
(3) $e - \log _ { e } 2$
(4) $1 + \log _ { e } 2$

jee-main 2022 Q75 View

The odd natural number $a$, such that the area of the region bounded by $y = 1 , y = 3 , x = 0 , x = y ^ { a }$ is $\frac { 364 } { 3 }$, equal to:
(1) 3
(2) 5
(3) 7
(4) 9

jee-main 2022 Q75 View

The area of the smaller region enclosed by the curves $y ^ { 2 } = 8 x + 4$ and $x ^ { 2 } + y ^ { 2 } + 4 \sqrt { 3 } x - 4 = 0$ is equal to
(1) $\frac { 1 } { 3 } ( 2 - 12 \sqrt { 3 } + 8 \pi )$
(2) $\frac { 1 } { 3 } ( 2 - 12 \sqrt { 3 } + 6 \pi )$
(3) $\frac { 1 } { 3 } ( 4 - 12 \sqrt { 3 } + 8 \pi )$
(4) $\frac { 1 } { 3 } ( 4 - 12 \sqrt { 3 } + 6 \pi )$

jee-main 2022 Q75 View

The area of the region enclosed by $y \leq 4 x ^ { 2 } , x ^ { 2 } \leq 9 y$ and $y \leq 4$, is equal to
(1) $\frac { 40 } { 3 }$
(2) $\frac { 56 } { 3 }$
(3) $\frac { 112 } { 3 }$
(4) $\frac { 80 } { 3 }$

jee-main 2022 Q76 View

The area bounded by the curve $y = \left| x ^ { 2 } - 9 \right|$ and the line $y = 3$ is
(1) $8 \sqrt { 6 } - 16 \sqrt { 12 } - 72$
(2) $8 \sqrt { 6 } + 8 \sqrt { 12 } - 72$
(3) $16 \sqrt { 6 } + 16 \sqrt { 12 } - 72$
(4) $16 \sqrt { 6 } - 16 \sqrt { 12 } - 64$

jee-main 2022 Q77 View

The area bounded by the curves $y = | x ^ { 2 } - 1 |$ and $y = 1$ is
(1) $\frac { 2 } { 3 } ( \sqrt { 2 } + 1 )$
(2) $\frac { 4 } { 3 } ( \sqrt { 2 } - 1 )$
(3) $2 ( \sqrt { 2 } - 1 )$
(4) $\frac { 8 } { 3 } ( \sqrt { 2 } - 1 )$

jee-main 2022 Q87 View

Let $f ( x ) = \max \{ | x + 1 | , | x + 2 | , \ldots , | x + 5 | \}$. Then $\int _ { - 6 } ^ { 0 } f ( x ) \, dx$ is equal to $\_\_\_\_$.

jee-main 2022 Q88 View

If the area of the region $\left\{ ( x , y ) : x ^ { \frac { 2 } { 3 } } + y ^ { \frac { 2 } { 3 } } \leq 1 , x + y \geq 0 , y \geq 0 \right\}$ is $A$, then $\frac { 256 A } { \pi }$ is

jee-main 2023 Q82 View

Let T and C respectively, be the transverse and conjugate axes of the hyperbola $16 x ^ { 2 } - y ^ { 2 } + 64 x + 4 y + 44 = 0$. Then the area of the region above the parabola $x ^ { 2 } = y + 4$, below the transverse axis T and on the right of the conjugate axis C is:
(1) $4 \sqrt { 6 } + \frac { 44 } { 3 }$
(2) $4 \sqrt { 6 } + \frac { 28 } { 3 }$
(3) $4 \sqrt { 6 } - \frac { 44 } { 3 }$
(4) $4 \sqrt { 6 } - \frac { 28 } { 3 }$

jee-main 2023 Q82 View

Let $A = \left\{ ( x , y ) \in \mathbb { R } ^ { 2 } : y \geq 0,2 x \leq y \leq \sqrt { 4 - ( x - 1 ) ^ { 2 } } \right\}$ and $B = \left\{ ( x , y ) \in \mathbb { R } \times \mathbb { R } : 0 \leq y \leq \min \left\{ 2 x , \sqrt { 4 - ( x - 1 ) ^ { 2 } } \right\} \right\}$. Then the ratio of the area of $A$ to the area of $B$ is
(1) $\frac { \pi - 1 } { \pi + 1 }$
(2) $\frac { \pi } { \pi - 1 }$
(3) $\frac { \pi } { \pi + 1 }$
(4) $\frac { \pi + 1 } { \pi - 1 }$

jee-main 2023 Q82 View

The area of the region enclosed by the curve $y = x ^ { 3 }$ and its tangent at the point $( - 1 , - 1 )$ is
(1) $\frac { 19 } { 4 }$
(2) $\frac { 23 } { 4 }$
(3) $\frac { 31 } { 4 }$
(4) $\frac { 27 } { 4 }$

jee-main 2023 Q83 View

If the area enclosed by the parabolas $P _ { 1 } : 2 y = 5 x ^ { 2 }$ and $P _ { 2 } : x ^ { 2 } - y + 6 = 0$ is equal to the area enclosed by $P _ { 1 }$ and $y = \alpha x , \alpha > 0$, then $\alpha ^ { 3 }$ is equal to $\_\_\_\_$.

jee-main 2023 Q83 View

Let $\Delta$ be the area of the region $\left\{ ( x , y ) \in \mathbb { R } ^ { 2 } : x ^ { 2 } + y ^ { 2 } \leq 21 , y ^ { 2 } \leq 4 x , x \geq 1 \right\}$. Then $\frac { 1 } { 2 } \left( \Delta - 21 \sin ^ { - 1 } \frac { 2 } { \sqrt { 7 } } \right)$ is equal to
(1) $2 \sqrt { 3 } - \frac { 1 } { 3 }$
(2) $\sqrt { 3 } - \frac { 2 } { 3 }$
(3) $2 \sqrt { 3 } - \frac { 2 } { 3 }$
(4) $\sqrt { 3 } - \frac { 4 } { 3 }$

jee-main 2023 Q83 View

The area of the region $A = \left\{ ( x , y ) : | \cos x - \sin x | \leq y \leq \sin x , 0 \leq x \leq \frac { \pi } { 2 } \right\}$ is: (1) $1 - \frac { 3 } { \sqrt { 2 } } + \frac { 4 } { \sqrt { 5 } }$ (2) $\sqrt { 5 } + 2 \sqrt { 2 } - 4.5$ (3) $\frac { 3 } { \sqrt { 5 } } - \frac { 3 } { \sqrt { 2 } } + 1$ (4) $\sqrt { 5 } - 2 \sqrt { 2 } + 1$

jee-main 2023 Q83 View

Let $A$ be the area of the region $\left\{(x, y): y \geq x^{2},\, y \geq (1-x)^{2},\, y \leq 2x(1-x)\right\}$. Then $540A$ is equal to

jee-main 2023 Q83 View

The area of the region $\left\{ ( x , y ) : x ^ { 2 } \leq y \leq 8 - x ^ { 2 } , y \leq 7 \right\}$ is
(1) 27
(2) 18
(3) 20
(4) 21

jee-main 2023 Q87 View

Let $A$ be the area bounded by the curve $y = x(x - 3)$, the $x$-axis and the ordinates $x = -1$ and $x = 2$. Then $12A$ is equal to $\_\_\_\_$.

1
2
3
4
5

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