Question

The area (in sq. units) of the region $A=\left\{\right(x,y):(x-1\left)\right[x]\le y\le 2\sqrt{x},0\le x\le 2\},$ where $\left[t\right]$ denotes the greatest integer function, is :

JEE Main 2020 (05 Sep Shift 2)

Options

• A: $\frac{8}{3}\sqrt{2}-\frac{1}{2}$
• B: $\frac{4}{3}\sqrt{2}+1$
• C: $\frac{8}{3}\sqrt{2}-1$
• D: $\frac{4}{3}\sqrt{2}-\frac{1}{2}$

Question

The area (in sq. units) of the region $A=\left\{(x,y):\left|x\right|+\left|y\right|\le 1,2y^2\ge \left|x\right|\right\}$
 

JEE Main 2020 (06 Sep Shift 1)

Options

• A: $\frac{1}{3}$
• B: $\frac{7}{6}$
• C: $\frac{1}{6}$
• D: $\frac{5}{6}$

Question

The area (in sq. unit) bounded by the curve $4y^2=x^2(4-x)(x-2)$ is equal to

JEE Main 2021 (18 Mar Shift 2)

Options

• A: $\frac{\pi }{8}$
• B: $\frac{3\pi }{8}$
• C: $\frac{3\pi }{2}$
• D: $\frac{\pi }{16}$

Question

If the area of the bounded region $R=\left\{\left(x,y\right):\max \left\{0,{\log }_{e}x\right\}\le y\le 2^x,\frac{1}{2}\le x\le 2\right\}$ is, $\alpha {\left({\log }_{e}2\right)}^{-1}+\beta \left({\log }_{\mathrm{e}}2\right)+\gamma$ then the value of ${\left(\alpha +\beta -2\gamma \right)}^2$ is equal to:

JEE Main 2021 (27 Jul Shift 1)

Options

• A: $8$
• B: $2$
• C: $4$
• D: $1$

Question

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

JEE Main 2021 (17 Mar Shift 2)

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Question

Let $a$ and $b$ respectively be the points of local maximum and local minimum of the function $f\left(x\right)=2x^3-3x^2-12x$. If $A$ is the total area of the region bounded by $y=f\left(x\right),$ the $x$-axis and the lines $x=a$ and $x\mathit{=}b\mathit{,}$ then $4\mathrm{A}$ is equal to ______.

JEE Main 2021 (26 Aug Shift 2)

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Question

The area of the region bounded by $y^2=8x$ and $y^2=16\left(3-x\right)$ is equal to

JEE Main 2022 (26 Jun Shift 2)

Options

• A: $\frac{32}{3}$
• B: $\frac{40}{3}$
• C: $16$
• D: $9$

Question

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

JEE Main 2022 (28 Jul Shift 2)

Options

• A: $2+e-{\log }_{e}2$
• B: $1+e-{\log }_{\mathrm{e}}2$
• C: $e-{\log }_{e}2$
• D: $1+{\log }_{\mathrm{e}}2$

Question

Let $q$ be the maximum integral value of $p$ in $\left[0,10\right]$ for which the roots of the equation $x^2-px+\frac{5}{4}p=0$ are rational. Then the area of the region $\left\{\left(x,y\right):0\le y\le (x-q{)}^2,0\le x\le q\right\}$ is

JEE Main 2023 (30 Jan Shift 2)

Options

• A: $243$
• B: $25$
• C: $\frac{125}{3}$
• D: $164$

Question

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

JEE Main 2023 (25 Jan Shift 2)

Options

• A: $4\sqrt{6}+\frac{44}{3}$
• B: $4\sqrt{6}+\frac{28}{3}$
• C: $4\sqrt{6}-\frac{44}{3}$
• D: $4\sqrt{6}-\frac{28}{3}$

Question

The area of the region enclosed by the parabolas $y=x^2-5 x$ and $y=7 x-x^2$ is

JEE Main 2024 (05 Apr Shift 1)

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Question

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

JEE Main 2024 (06 Apr Shift 1)

Options

• A: 172
• B: 162
• C: 154
• D: 184

Question

Let $f(x)$ be a positive function such that the area bounded by $y=f(x), y=0$ from $x=0$ to $x=a>0$ is $e^{-a}+4 a^2+a-1$. Then the differential equation, whose general solution is $y=c_1 f(x)+c_2$, where $c_1$ and $c_2$ are arbitrary constants, is

JEE Main 2024 (08 Apr Shift 1)

Options

• A: $\left(8 e^x-1\right) \frac{d^2 y}{d x^2}+\frac{d y}{d x}=0$
• B: $\left(8 e^x-1\right) \frac{d^2 y}{d x^2}-\frac{d y}{d x}=0$
• C: $\left(8 e^x+1\right) \frac{d^2 y}{d x^2}-\frac{d y}{d x}=0$
• D: $\left(8 e^x+1\right) \frac{d^2 y}{d x^2}+\frac{d y}{d x}=0$

Question

The area of the region in the first quadrant inside the circle $x^2+y^2=8$ and outside the parabola $y^2=2 x$ is equal to :

JEE Main 2024 (08 Apr Shift 2)

Options

• A: $\frac{\pi}{2}-\frac{1}{3}$
• B: $\pi-\frac{1}{3}$
• C: $\frac{\pi}{2}-\frac{2}{3}$
• D: $\pi-\frac{2}{3}$

Question

Let the area of the region $\left\{\right(x,y):0\le x\le 3,0\le y\le$ $\left.\min \left\{x^2+2,2x+2\right\}\right\}$ be $A$. Then $12A$ is equal to ______.

JEE Main 2024 (29 Jan Shift 2)

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Question

The area (in sq. units) of the part of circle $x^2+y^2=169$ which is below the line $5x-y=13$ is $\frac{\pi \alpha }{2\beta }-\frac{65}{2}+\frac{\alpha }{\beta }{\sin }^{-1}\left(\frac{12}{13}\right)$ where $\alpha ,\beta$ are coprime numbers. Then $\alpha +\beta$ is equal to

JEE Main 2024 (29 Jan Shift 1)

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Question

Three points $O\left(0,0\right), P\left(a,a^2\right), Q\left(-b,b^2\right), a>0, b>0,$ are on the parabola $y=x^2$. Let $S_1$ be the area of the region bounded by the line $PQ$ and the parabola, and $S_2$ be the area of the triangle $OPQ$. If the minimum value of $\frac{S_1}{S_2}$ is $\frac{m}{n}, \gcd \left(m,n\right)=1,$ then $m+n$ is equal to:

JEE Main 2024 (01 Feb Shift 2)

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Question

The area enclosed by the curves $xy+4y=16$ and $x+y=6$ is equal to:

JEE Main 2024 (01 Feb Shift 1)

Options

• A: $28-30{\log }_{e}2$
• B: $30-28{\log }_{e}2$
• C: $30-32{\log }_{e}2$
• D: $32-30{\log }_{e}2$

Question

The area of the region $\left\{\left(x, y\right):y^2\le 4x, x<4, \frac{xy\left(x-1\right)\left(x-2\right)}{\left(x-3\right)\left(x-4\right)}>0, x\ne 3\right\}$ is

JEE Main 2024 (31 Jan Shift 1)

Options

• A: $\frac{16}{3}$
• B: $\frac{64}{3}$
• C: $\frac{8}{3}$
• D: $\frac{32}{3}$