LFM Pure and Mechanics

jee-main 2015 Q67 Area Between Curves with Parametric or Implicit Region Definition View

The area (in sq. units) of the region described by $\{(x, y) : y^2 \leq 2x \text{ and } y \geq 4x - 1\}$ is:
(1) $\frac{7}{32}$
(2) $\frac{5}{64}$
(3) $\frac{15}{64}$
(4) $\frac{9}{32}$

jee-main 2016 Q71 Area Involving Conic Sections or Circles View

The area (in sq. units) of the region $\{(x, y) : y^2 \geq 2x$ and $x^2 + y^2 \leq 4x, x \geq 0, y \geq 0\}$ is:
(1) $\pi - \frac{4\sqrt{2}}{3}$
(2) $\pi - \frac{8}{3}$
(3) $\pi - \frac{4}{3}$
(4) $\frac{\pi}{2} - \frac{2\sqrt{2}}{3}$

jee-main 2016 Q72 Area Involving Conic Sections or Circles View

The area (in sq. units) of the region $\{(x,y): y^2 \geq 2x$ and $x^2 + y^2 \leq 4x, x \geq 0, y \geq 0\}$ is: (1) $\pi - \frac{4\sqrt{2}}{3}$ (2) $\pi - \frac{8}{3}$ (3) $\pi - \frac{4}{3}$ (4) $\frac{\pi}{2} - \frac{2\sqrt{2}}{3}$

jee-main 2017 Q66 Area Between Curves with Parametric or Implicit Region Definition View

The area (in sq. units) of the region $\{ ( x , y ) : x \geq 0 , x + y \leq 3 , x ^ { 2 } \leq 4 y$ and $y \leq 1 + \sqrt { x } \}$ is:
(1) $\frac { 59 } { 12 }$
(2) $\frac { 3 } { 2 }$
(3) $\frac { 7 } { 3 }$
(4) $\frac { 5 } { 2 }$

jee-main 2020 Q66 Area Involving Conic Sections or Circles View

The area of the region (in sq. units), enclosed by the circle $x ^ { 2 } + y ^ { 2 } = 2$ which is not common to the region bounded by the parabola $y ^ { 2 } = x$ and the straight line $y = x$, is
(1) $\frac { 1 } { 6 } (24 \pi - 1)$
(2) $\frac { 1 } { 3 } (6 \pi - 1)$
(3) $\frac { 1 } { 3 } (12 \pi - 1)$
(4) $\frac { 1 } { 6 } (12 \pi - 1)$

jee-main 2020 Q66 Area Between Curves with Parametric or Implicit Region Definition View

The area (in sq. units) of the region $\left\{ ( x , y ) : 0 \leq y \leq x ^ { 2 } + 1,0 \leq y \leq x + 1 , \frac { 1 } { 2 } \leq x \leq 2 \right\}$ is
(1) $\frac { 23 } { 16 }$
(2) $\frac { 79 } { 24 }$
(3) $\frac { 79 } { 16 }$
(4) $\frac { 23 } { 6 }$

jee-main 2020 Q67 Compute Area Directly (Numerical Answer) View

Let $g ( x ) = \left( x - \frac { 1 } { 2 } \right) ^ { 2 } , x \in R$. Then, the area (in sq. units) of the region bounded by the curves, $y = f ( x )$ and $y = g ( x )$ between the lines $2 x = 1$ and $2 x = \sqrt { 3 }$, is:
(1) $\frac { 1 } { 3 } + \frac { \sqrt { 3 } } { 4 }$
(2) $\frac { \sqrt { 3 } } { 4 } - \frac { 1 } { 3 }$
(3) $\frac { 1 } { 2 } - \frac { \sqrt { 3 } } { 4 }$
(4) $\frac { 1 } { 2 } + \frac { \sqrt { 3 } } { 4 }$

jee-main 2020 Q67 Area Involving Conic Sections or Circles View

Area (in sq. units) of the region outside $\frac{|x|}{2} + \frac{|y|}{3} = 1$ and inside the ellipse $\frac{x^{2}}{4} + \frac{y^{2}}{9} = 1$ is
(1) $6(\pi - 2)$
(2) $3(\pi - 2)$
(3) $3(4 - \pi)$
(4) $6(4 - \pi)$

jee-main 2020 Q67 Find Parameter Given Area Condition View

Consider a region $R = \left\{ ( x , y ) \in R ^ { 2 } : x ^ { 2 } \leq y \leq 2 x \right\}$. If a line $y = \alpha$ divides the area of region $R$ into two equal parts, then which of the following is true?
(1) $\alpha ^ { 3 } - 6 \alpha ^ { 2 } + 16 = 0$
(2) $3 \alpha ^ { 2 } - 8 \alpha ^ { 3 / 2 } + 8 = 0$
(3) $3 \alpha ^ { 2 } - 8 \alpha + 8 = 0$
(4) $\alpha ^ { 3 } - 6 \alpha ^ { 3/2 } + 16 = 0$

jee-main 2020 Q68 Area Involving Piecewise or Composite Functions View

The area (in sq. units) of the region $A = \{(x,y) : (x-1)[x] \leq y \leq 2\sqrt{x},\, 0 \leq x \leq 2\}$, where $[t]$ denotes the greatest integer function, is:
(1) $\frac{8}{3}\sqrt{2} - \frac{1}{2}$
(2) $\frac{4}{3}\sqrt{2} + 1$
(3) $\frac{8}{3}\sqrt{2} - 1$
(4) $\frac{4}{3}\sqrt{2} - \frac{1}{2}$

jee-main 2022 Q75 Area Involving Conic Sections or Circles View

The area of the region bounded by $y ^ { 2 } = 8 x$ and $y ^ { 2 } = 16 ( 3 - x )$ is equal to
(1) $\frac { 32 } { 3 }$
(2) $\frac { 40 } { 3 }$
(3) 16
(4) 9

jee-main 2023 Q72 Area Involving Piecewise or Composite Functions View

The area of the region $\{(x, y): x^2 \leq y \leq |x^2 - 4|, y \geq 1\}$ is
(1) $\frac{4(\sqrt{5}-1)}{3} + 4$
(2) $\frac{4(\sqrt{5}-1)}{3} + 2$
(3) $\frac{2(\sqrt{5}-1)}{3} + 4$
(4) $\frac{2(\sqrt{5}-1)}{3} + 2$

jee-main 2023 Q74 Area Involving Piecewise or Composite Functions View

The area of the region $\{ ( x , y ) : x ^ { 2 } \leq y \leq | x ^ { 2 } - 4 | , y \geq 1 \}$ is
(1) $\frac { 4 } { 3 } ( 4 \sqrt { 2 } - 1 )$
(2) $\frac { 4 } { 3 } ( 4 \sqrt { 2 } + 1 )$
(3) $\frac { 3 } { 4 } ( 4 \sqrt { 2 } + 1 )$
(4) $\frac { 3 } { 4 } ( 4 \sqrt { 2 } - 1 )$

jee-main 2024 Q74 Area Between Curves with Parametric or Implicit Region Definition View

The area of the region $\left\{(x, y) : y ^ { 2 } \leq 4 x , x < 4 , \frac { x y (x - 1)(x - 2) } { (x - 3)(x - 4) } > 0 , x \neq 3 \right\}$ is
(1) $\frac { 16 } { 3 }$
(2) $\frac { 64 } { 3 }$
(3) $\frac { 8 } { 3 }$
(4) $\frac { 32 } { 3 }$

jee-main 2024 Q75 Area Involving Piecewise or Composite Functions View

The area enclosed between the curves $y = x | x |$ and $y = x - | x |$ is :
(1) $\frac { 4 } { 3 }$
(2) 1
(3) $\frac { 2 } { 3 }$
(4) $\frac { 8 } { 3 }$

jee-main 2024 Q76 Compute Area Directly (Numerical Answer) View

Let the area of the region enclosed by the curves $y = 3 x , 2 y = 27 - 3 x$ and $y = 3 x - x \sqrt { x }$ be $A$. Then $10 A$ is equal to
(1) 172
(2) 162
(3) 154
(4) 184

jee-main 2024 Q76 Area Between Curves with Parametric or Implicit Region Definition View

The area (in sq. units) of the region described by $\left\{ ( x , y ) : y ^ { 2 } \leq 2 x \right.$, and $\left. y \geq 4 x - 1 \right\}$ is
(1) $\frac { 11 } { 32 }$
(2) $\frac { 8 } { 9 }$
(3) $\frac { 11 } { 12 }$
(4) $\frac { 9 } { 32 }$

jee-main 2025 Q7 Compute Area Directly (Numerical Answer) View

The area of the region enclosed by the curves $y = x ^ { 2 } - 4 x + 4$ and $y ^ { 2 } = 16 - 8 x$ is :
(1) $\frac { 8 } { 3 }$
(2) $\frac { 4 } { 3 }$
(3) 8
(4) 5

jee-main 2025 Q11 Area Involving Piecewise or Composite Functions View

The area of the region $\left\{(x, y) : x^2 + 4x + 2 \leq y \leq |x+2|\right\}$ is equal to
(1) 7
(2) 5
(3) $24/5$
(4) $20/3$

jee-main 2025 Q11 Area Involving Conic Sections or Circles View

Let the area enclosed between the curves $| y | = 1 - x ^ { 2 }$ and $x ^ { 2 } + y ^ { 2 } = 1$ be $\alpha$. If $9 \alpha = \beta \pi + \gamma ; \beta , \gamma$ are integers, then the value of $| \beta - \gamma |$ equals.
(1) 27
(2) 33
(3) 15
(4) 18

jee-main 2025 Q12 Area Involving Piecewise or Composite Functions View

The area (in sq. units) of the region $\left\{ ( x , y ) : 0 \leq y \leq 2 | x | + 1,0 \leq y \leq x ^ { 2 } + 1 , | x | \leq 3 \right\}$ is
(1) $\frac { 80 } { 3 }$
(2) $\frac { 64 } { 3 }$
(3) $\frac { 32 } { 3 }$
(4) $\frac { 17 } { 3 }$

jee-main 2025 Q13 Area Involving Conic Sections or Circles View

The area of the region, inside the circle $( x - 2 \sqrt { 3 } ) ^ { 2 } + y ^ { 2 } = 12$ and outside the parabola $y ^ { 2 } = 2 \sqrt { 3 } x$ is:
(1) $3 \pi + 8$
(2) $6 \pi - 16$
(3) $3 \pi - 8$
(4) $6 \pi - 8$

jee-main 2025 Q16 Compute Area Directly (Numerical Answer) View

The area of the region bounded by the curves $x \left( 1 + y ^ { 2 } \right) = 1$ and $y ^ { 2 } = 2 x$ is:
(1) $2 \left( \frac { \pi } { 2 } - \frac { 1 } { 3 } \right)$
(2) $\frac { \pi } { 2 } - \frac { 1 } { 3 }$
(3) $\frac { \pi } { 4 } - \frac { 1 } { 3 }$
(4) $\frac { 1 } { 2 } \left( \frac { \pi } { 2 } - \frac { 1 } { 3 } \right)$

jee-main 2025 Q20 Find Parameter Given Area Condition View

If the area of the region $\left\{ ( x , y ) : - 1 \leq x \leq 1 , 0 \leq y \leq a + \mathrm { e } ^ { | x | } - \mathrm { e } ^ { - x } , \mathrm { a } > 0 \right\}$ is $\frac { \mathrm { e } ^ { 2 } + 8 \mathrm { e } + 1 } { \mathrm { e } }$, then the value of $a$ is :
(1) 8
(2) 7
(3) 5
(4) 6

jee-main 2025 Q20 Area Between Curves with Parametric or Implicit Region Definition View

Let the area of the region $\{(x, y): 2y \leq x^2 + 3,\ y + |x| \leq 3,\ y \geq |x-1|\}$ be A. Then $6A$ is equal to:
(1) 16
(2) 12
(3) 14
(4) 18

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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