LFM Pure and Mechanics

jee-main 2018 Q85 View

Let $g ( x ) = \cos x ^ { 2 } , f ( x ) = \sqrt { x }$, and $\alpha , \beta ( \alpha < \beta )$ be the roots of the quadratic equation $18 x ^ { 2 } - 9 \pi x + \pi ^ { 2 } = 0$. Then the area (in sq. units) bounded by the curve $y = ( g o f ) ( x )$ and the lines $x = \alpha , x = \beta$ and $y = 0$, is
(1) $\frac { 1 } { 2 } ( \sqrt { 2 } - 1 )$
(2) $\frac { 1 } { 2 } ( \sqrt { 3 } - 1 )$
(3) $\frac { 1 } { 2 } ( \sqrt { 3 } + 1 )$
(4) $\frac { 1 } { 2 } ( \sqrt { 3 } - \sqrt { 2 } )$

jee-main 2018 Q85 View

The area (in sq. units) of the region $\{ x \in R : x \geq 0 , y \geq 0 , y \geq x - 2$ and $y \leq \sqrt { x } \}$ is
(1) $\frac { 13 } { 3 }$
(2) $\frac { 8 } { 3 }$
(3) $\frac { 10 } { 3 }$
(4) $\frac { 5 } { 3 }$

jee-main 2018 Q85 View

The area (in sq. units) of the region $\{ x \in R : x \geq 0 , y \geq 0 , y \geq x - 2$ and $y \leq \sqrt { x } \}$, is
(1) $\frac { 13 } { 3 }$
(2) $\frac { 10 } { 3 }$
(3) $\frac { 5 } { 3 }$
(4) $\frac { 8 } { 3 }$

jee-main 2019 Q72 View

Let $A(4,-4)$ and $B(9,6)$ be points on the parabola, $y^2 = 4x$. Let $C$ be chosen on the arc $AOB$ of the parabola, where $O$ is the origin, such that the area of $\triangle ACB$ is maximum. Then, the area (in sq. units) of $\triangle ACB$, is:
(1) 32
(2) $31\frac{3}{4}$
(3) $30\frac{1}{2}$
(4) $31\frac{1}{4}$

jee-main 2019 Q84 Integral Equation with Symmetry or Substitution View

If $f(x) = \frac{2 - x\cos x}{2 + x\cos x}$ and $g(x) = \log_e x$, then the value of the integral $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} g(f(x))\, dx$ is
(1) $\log_e e$
(2) $\log_e 2$
(3) $\log_e 1$
(4) $\log_e 3$

jee-main 2019 Q85 View

If the area enclosed between the curves $y = k x ^ { 2 }$ and $x = k y ^ { 2 } , ( k > 0 )$, is 1 sq. unit. Then $k$ is
(1) $\sqrt { 3 }$
(2) $\frac { 1 } { \sqrt { 3 } }$
(3) $\frac { \sqrt { 3 } } { 2 }$
(4) $\frac { 2 } { \sqrt { 3 } }$

jee-main 2019 Q85 View

The area (in sq. units) of the region $A = \left\{ ( x , y ) : x ^ { 2 } \leq y \leq x + 2 \right\}$ is
(1) $\frac { 13 } { 6 }$
(2) $\frac { 31 } { 6 }$
(3) $\frac { 9 } { 2 }$
(4) $\frac { 10 } { 3 }$

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 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 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 2021 Q65 Triangle or Quadrilateral Area and Perimeter with Foci View

Let the tangent to the parabola $S : y ^ { 2 } = 2 x$ at the point $P ( 2,2 )$ meet the $x$-axis at $Q$ and normal at it meet the parabola $S$ at the point $R$. Then the area (in sq. units) of the triangle $P Q R$ is equal to:
(1) $\frac { 25 } { 2 }$
(2) $\frac { 35 } { 2 }$
(3) $\frac { 15 } { 2 }$
(4) 25

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 2022 Q64 Area Computation in Coordinate Geometry View

Let $A ( 1,1 ) , B ( - 4,3 ) , C ( - 2 , - 5 )$ be vertices of a triangle $A B C , P$ be a point on side $B C$, and $\Delta _ { 1 }$ and $\Delta _ { 2 }$ be the areas of triangle $A P B$ and $A B C$ respectively. If $\Delta _ { 1 } : \Delta _ { 2 } = 4 : 7$, then the area enclosed by the lines $A P , A C$ and the $x$-axis is
(1) $\frac { 1 } { 4 }$
(2) $\frac { 3 } { 4 }$
(3) $\frac { 1 } { 2 }$
(4) 1

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 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 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) $\frac { 80 } { 3 }$

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LFM Pure and Mechanics

Indices and Surds 96

Exponential Functions 197

Laws of Logarithms 242

Exponential Equations & Modelling 28

Arithmetic Sequences and Series 363

Geometric Sequences and Series 159

Differentiation from First Principles 45

Tangents, normals and gradients 140

Stationary points and optimisation 505

Applied differentiation 76

Indefinite & Definite Integrals 502

Areas Between Curves 71

Areas by integration 221

Volumes of Revolution 68

Numerical integration 49

Vectors Introduction & 2D 211

Vectors 3D & Lines 524

Travel graphs 14

SUVAT in 2D & Gravity 31

Constant acceleration (SUVAT) 78

Variable acceleration (1D) 55

Variable acceleration (vectors) 32

Forces, equilibrium and resultants 24

Newton's laws and connected particles 21

Pulley systems 6

Motion on a slope 6

Friction 8

Momentum and Collisions 34

Moments 28

Parametric curves and Cartesian conversion 9

Parametric differentiation 13

Parametric integration 6

Projectiles 49