Area Under The Curves PYQs - Last 10 Years

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Practice 153 JEE Main Mathematics questions from Area Under The Curves. Use the year-wise and type-wise breakdown to prioritize recent PYQs, then continue into the question list below.

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Last 10 Years Area Under The Curves Questions

Showing 50 of 153 filtered questions.

1Area Under The Curves

The area of the region enclosed by the parabola \((y-2)^2=x-1\), the line \(x-2 y+4=0\) and the positive coordinate axes is _________.

INTEGER+4 / -12024

2Area Under The Curves

If the points of intersection of two distinct conics \(x^2+y^2=4 b\) and \(\frac{x^2}{16}+\frac{y^2}{b^2}=1\) lie on the curve \(y^2=3 x^2\), then \(3 \sqrt{3}\) times the area of the rectangle formed by the intersection points is _________...

INTEGER+4 / -12024

3Area Under The Curves

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

INTEGER+4 / -12024

4Area Under The Curves

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

INTEGER+4 / -12024

5Area Under The Curves

Let the area of the region $\left\{(x, y): x-2 y+4 \geqslant 0, x+2 y^2 \geqslant 0, x+4 y^2 \leq 8, y \geqslant 0\right\}$ be $\frac{\mathrm{m}}{\mathrm{n}}$, where $\mathrm{m}$ and $\mathrm{n}$ are coprime numbers. Then $\mathrm{m}+\mathr...

INTEGER+4 / -12024

6Area Under The Curves

If the area of the region \(\left\{(x, y): 0 \leq y \leq \min \left\{2 x, 6 x-x^2\right\}\right\}\) is \(\mathrm{A}\), then \(12 \mathrm{~A}\) is equal to ________.

INTEGER+4 / -12024

7Area Under The Curves

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

8Area Under The Curves

The sum of squares of all possible values of $k$, for which area of the region bounded by the parabolas $2 y^2=\mathrm{k} x$ and $\mathrm{ky}^2=2(y-x)$ is maximum, is equal to :

INTEGER+4 / -12024

9Area Under The Curves

Three points $\mathrm{O}(0,0), \mathrm{P}\left(\mathrm{a}, \mathrm{a}^2\right), \mathrm{Q}\left(-\mathrm{b}, \mathrm{b}^2\right), \mathrm{a}>0, \mathrm{~b}>0$, are on the parabola $y=x^2$. Let $\mathrm{S}_1$ be the area of the region bounde...

INTEGER+4 / -12024

10Area Under The Curves

The area of the region \(\left\{(x, y): x^{2} \leq y \leq 8-x^{2}, y \leq 7\right\}\) is :

11Area Under The Curves

Let the area enclosed by the lines \(x+y=2, \mathrm{y}=0, x=0\) and the curve \(f(x)=\min \left\{x^{2}+\frac{3}{4}, 1+[x]\right\}\) where \([x]\)
denotes the greatest integer \(\leq x\), be \(\mathrm{A}\). Then the value of $$12 \mathrm{~A...

INTEGER+4 / -12023

12Area Under The Curves

If the area of the region \(S=\left\{(x, y): 2 y-y^{2} \leq x^{2} \leq 2 y, x \geq y\right\}\) is equal to \(\frac{n+2}{n+1}-\frac{\pi}{n-1}\), then the natural number \(n\) is equal to ___________.

INTEGER+4 / -12023

13Area Under The Curves

The area bounded by the curves \(y=|x-1|+|x-2|\) and \(y=3\) is equal to :

14Area Under The Curves

Let for \(x \in \mathbb{R}\),
$$ f(x)=\frac{x+|x|}{2} \text { and } g(x)=\left\{\begin{array}{cc} x, & x<0 \\ x^{2}, & x \geq 0 \end{array}\right. \text {. } $$
Then area bounded by the curve \(y=(f \circ g)(x)\) and the lines $$y=0,2 y-x=1...

INTEGER+4 / -12023

15Area Under The Curves

Let the area of the region
$\left\{(x, y):|2 x-1| \leq y \leq\left|x^{2}-x\right|, 0 \leq x \leq 1\right\}$ be $\mathrm{A}$.
Then $(6 \mathrm{~A}+11)^{2}$ is equal to

INTEGER+4 / -12023

16Area Under The Curves

Let \(\alpha\) be the area of the larger region bounded by the curve \(y^{2}=8 x\) and the lines \(y=x\) and \(x=2\), which lies in the first quadrant. Then the value of \(3 \alpha\) is equal to ___________.

INTEGER+4 / -12023

17Area Under The Curves

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

INTEGER+4 / -12023

18Area Under The Curves

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

19Area Under The Curves

Let \(A=\left\{(x, y) \in \mathbb{R}^{2}: y \geq 0,2 x \leq y \leq \sqrt{4-(x-1)^{2}}\right\}\) and \(B=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: 0 \leq y \leq \min \left\{2 x, \sqrt{4-(x-1)^{2}}\right\}\right\} \text {. }\).
Then t...

20Area Under The Curves

Let \([x]\) denote the greatest integer \(\le x\). Consider the function \(f(x) = \max \left\{ {{x^2},1 + [x]} \right\}\). Then the value of the integral \(\int\limits_0^2 {f(x)dx}\) is

21Area Under The Curves

Let \(\Delta\) be the area of the region \(\left\{ {(x,y) \in {R^2}:{x^2} + {y^2} \le 21,{y^2} \le 4x,x \ge 1} \right\}\). Then \({1 \over 2}\left( {\Delta - 21{{\sin }^{ - 1}}{2 \over {\sqrt 7 }}} \right)\) is equal to

22Area Under The Curves

The area of the region \(A = \left\{ {(x,y):\left| {\cos x - \sin x} \right| \le y \le \sin x,0 \le x \le {\pi \over 2}} \right\}\) is

23Area Under The Curves

If the area enclosed by the parabolas \(\mathrm{P_1:2y=5x^2}\) and \(\mathrm{P_2:x^2-y+6=0}\) is equal to the area enclosed by \(\mathrm{P_1}\) and \(\mathrm{y=\alpha x,\alpha > 0}\), then \(\alpha^3\) is equal to ____________.

INTEGER+4 / -12023

24Area Under The Curves

The area enclosed by the curves \({y^2} + 4x = 4\) and \(y - 2x = 2\) is :

25Area Under The Curves

If the area of the region bounded by the curves \(y^2-2y=-x,x+y=0\) is A, then 8 A is equal to __________

INTEGER+4 / -12023

26Area Under The Curves

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 \(12 A\) is equal to ____________.

INTEGER+4 / -12023

27Area Under The Curves

The area of the region given by \(\{ (x,y):xy \le 8,1 \le y \le {x^2}\}\) is :

28Area Under The Curves

If the area bounded by the curve $2 y^{2}=3 x$, lines $x+y=3, y=0$ and outside the circle $(x-3)^{2}+y^{2}=2$ is $\mathrm{A}$, then $4(\pi+4 A)$ is equal to ____________.

INTEGER+4 / -12023

29Area Under The Curves

The area of the region enclosed by the curve
\(f(x)=\max \{\sin x, \cos x\},-\pi \leq x \leq \pi\) and the \(x\)-axis is

30Area Under The Curves

The area of the region \(\left\{(x, y): x^{2} \leq y \leq\left|x^{2}-4\right|, y \geq 1\right\}\) is

31Area Under The Curves

The area of the region enclosed by the curve \(y=x^{3}\) and its tangent at the point \((-1,-1)\) is :

32Area Under The Curves

Area of the region \(\left\{(x, y): x^{2}+(y-2)^{2} \leq 4, x^{2} \geq 2 y\right\}\) is

33Area Under The Curves

If A is the area in the first quadrant enclosed by the curve \(\mathrm{C: 2 x^{2}-y+1=0}\), the tangent to \(\mathrm{C}\) at the point \((1,3)\) and the line \(\mathrm{x}+\mathrm{y}=1\), then the value of \(60 \mathrm{~A}\) is _________.

INTEGER+4 / -12023

34Area Under The Curves

Let \(y = p(x)\) be the parabola passing through the points \(( - 1,0),(0,1)\) and \((1,0)\). If the area of the region \(\{ (x,y):{(x + 1)^2} + {(y - 1)^2} \le 1,y \le p(x)\}\) is A, then \(12(\pi - 4A)\) is equal to ___________.

INTEGER+4 / -12023

35Area Under The Curves

If the area of the region \(\left\{(x, \mathrm{y}):\left|x^{2}-2\right| \leq y \leq x\right\}\) is \(\mathrm{A}\), then \(6 \mathrm{A}+16 \sqrt{2}\) is equal to __________.

INTEGER+4 / -12023

36Area Under The Curves

If for some \(\alpha\) > 0, the area of the region \(\{ (x,y):|x + \alpha | \le y \le 2 - |x|\}\) is equal to \({3 \over 2}\), then the area of the region \(\{ (x,y):0 \le y \le x + 2\alpha ,\,|x| \le 1\}\) is equal to ____________.

INTEGER+4 / -12022

37Area Under The Curves

The area enclosed by y2 = 8x and y = \(\sqrt2\) x that lies outside the triangle formed by y = \(\sqrt2\) x, x = 1, y = 2\(\sqrt2\), is equal to:

38Area Under The Curves

For real numbers a, b (a > b > 0), let
Area \(\left\{ {(x,y):{x^2} + {y^2} \le {a^2}\,and\,{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} \ge 1} \right\} = 30\pi\)
and
Area $$\left\{ {(x,y):{x^2} + {y^2} \le {b^2}\,and\,{{{x^2}} \over ...

INTEGER+4 / -12022

39Area Under The Curves

The area of the region
\(\left\{(x, y):|x-1| \leq y \leq \sqrt{5-x^{2}}\right\}\) is equal to :

40Area Under The Curves

The area of the region S = {(x, y) : y2 \(\le\) 8x, y \(\ge\) \(\sqrt2\)x, x \(\ge\) 1} is

41Area Under The Curves

The area of the bounded region enclosed by the curve \(y = 3 - \left| {x - {1 \over 2}} \right| - |x + 1|\) and the x-axis is :

42Area Under The Curves

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

43Area Under The Curves

Let
\({A_1} = \left\{ {(x,y):|x| \le {y^2},|x| + 2y \le 8} \right\}\) and
\({A_2} = \left\{ {(x,y):|x| + |y| \le k} \right\}\). If 27 (Area A1) = 5 (Area A2), then k is equal to :

INTEGER+4 / -12022

44Area Under The Curves

If the area of the region \(\left\{ {(x,y):{x^{{2 \over 3}}} + {y^{{2 \over 3}}} \le 1,\,x + y \ge 0,\,y \ge 0} \right\}\) is A, then \({{256A} \over \pi }\) is equal to __________.

INTEGER+4 / -12022

45Area Under The Curves

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

46Area Under The Curves

Consider a curve \(y=y(x)\) in the first quadrant as shown in the figure. Let the area \(\mathrm{A}_{1}\) is twice the area \(\mathrm{A}_{2}\). Then the normal to the curve perpendicular to the line \(2 x-12 y=15\) does NOT pass through the...

47Area Under The Curves

The area of the region enclosed by \(y \leq 4 x^{2}, x^{2} \leq 9 y\) and \(y \leq 4\), is equal to :

48Area Under The Curves

The area bounded by the curve y = |x2 \(-\) 9| and the line y = 3 is :

49Area Under The Curves

The area of the region bounded by y2 = 8x and y2 = 16(3 \(-\) x) is equal to:

50Area Under The Curves

The odd natural number a, such that the area of the region bounded by y = 1, y = 3, x = 0, x = ya is \({{364} \over 3}\), is equal to :