## Numerical

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 eq...

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 t...

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

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 -...

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 $$[x]$$
den...

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 t...

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 ____________...

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)^...

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

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 :...

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...

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 $$...

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 __________

Let the area enclosed by the x-axis, and the tangent and normal drawn to the curve $$4{x^3} - 3x{y^2} + 6{x^2} - 5xy - 8{y^2} + 9x + 14 = 0$$ at the p...

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 r...

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} \r...

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 ...

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 (A...

The area (in sq. units) of the region enclosed between the parabola y2 = 2x and the line x + y = 4 is __________.

Let S be the region bounded by the curves y = x3 and y2 = x. The curve y = 2|x| divides S into two regions of areas R1, R2. If max {R1, R2} = R2, then...

If the line y = mx bisects the area enclosed by the lines x = 0, y = 0, x = $${3 \over 2}$$ and the curve y = 1 + 4x $$-$$ x2, then 12 m is equal to _...

Let a and b respectively be the points of local maximum and local minimum of the function f(x) = 2x3 $$-$$ 3x2 $$-$$ 12x. If A is the total area of th...

The area of the region $$S = \{ (x,y):3{x^2} \le 4y \le 6x + 24\} $$ is ____________.

The area (in sq. units) of the region bounded by the curves x2 + 2y $$-$$ 1 = 0, y2 + 4x $$-$$ 4 = 0 and y2 $$-$$ 4x $$-$$ 4 = 0, in the upper half pl...

Let T be the tangent to the ellipse E : x2 + 4y2 = 5 at the point P(1, 1). If the area of the region bounded by the tangent T, ellipse E, lines x = 1 ...

Let f : [$$-$$3, 1] $$ \to $$ R be given as $$f(x) = \left\{ \matrix{
\min \,\{ (x + 6),{x^2}\}, - 3 \le x \le 0 \hfill \cr
\max \,\{ \sqrt x ,{x...

The area bounded by the lines y = || x $$-$$ 1 | $$-$$ 2 | is ___________.

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...

## MCQ (Single Correct Answer)

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

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

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

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

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

The area bounded by the curves $$y=|x-1|+|x-2|$$ and $$y=3$$ is equal to

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

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 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

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( {\...

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 integ...

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...

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

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

The area enclosed by the curves $$y=\log _{e}\left(x+\mathrm{e}^{2}\right), x=\log _{e}\left(\frac{2}{y}\right)$$ and $$x=\log _{\mathrm{e}} 2$$, abov...

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

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 t...

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

The area bounded by the curves $$y=\left|x^{2}-1\right|$$ and $$y=1$$ is

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 :...

The area of the region given by
$$A=\left\{(x, y): x^{2} \leq y \leq \min \{x+2,4-3 x\}\right\}$$ is :

Let the locus of the centre $$(\alpha, \beta), \beta>0$$, of the circle which touches the circle $$x^{2}+(y-1)^{2}=1$$ externally and also touches the...

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:...

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 :

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

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

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

The area of the region enclosed between the parabolas y2 = 2x $$-$$ 1 and y2 = 4x $$-$$ 3 is

The area, enclosed by the curves $$y = \sin x + \cos x$$ and $$y = \left| {\cos x - \sin x} \right|$$ and the lines $$x = 0,x = {\pi \over 2}$$, is :

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 :...

The area of the region bounded by y $$-$$ x = 2 and x2 = y is equal to :

If the area of the bounded region $$R = \left\{ {(x,y):\max \{ 0,{{\log }_e}x\} \le y \le {2^x},{1 \over 2} \le x \le 2} \right\}$$ is , $$\alpha {({...

The area (in sq. units) of the region, given by the set $$\{ (x,y) \in R \times R|x \ge 0,2{x^2} \le y \le 4 - 2x\} $$ is :

The area bounded by the curve 4y2 = x2(4 $$-$$ x)(x $$-$$ 2) is equal to :

Let A1 be the area of the region bounded by the curves y = sinx, y = cosx and y-axis in the first quadrant. Also, let A2 be the area of the region bou...

The area of the region : $$R = \{ (x,y):5{x^2} \le y \le 2{x^2} + 9\} $$ is :

The area (in sq. units) of the part of the circle x2 + y2 = 36, which is outside the parabola
y2 = 9x, is :

The area (in sq. units) of the region enclosed
by the curves y = x2 – 1 and y = 1 – x2 is equal to :

The area (in sq. units) of the region A = {(x, y) : |x| + |y| $$ \le $$ 1, 2y2 $$ \ge $$ |x|}

The area (in sq. units) of the region
A = {(x, y) : (x – 1)[x] $$ \le $$ y $$ \le $$ 2$$\sqrt x $$, 0 $$ \le $$ x $$ \le $$ 2}, where [t]
denotes the...

The area (in sq. units) of the region
{ (x, y) : 0 $$ \le $$ y $$ \le $$ x2 + 1, 0 $$ \le $$ y $$ \le $$ x + 1,
$${1 \over 2}$$ $$ \le $$ x $$ \le $$...

Consider a region R = {(x, y) $$ \in $$ R : x2 $$ \le $$ y $$ \le $$ 2x}.
if a line y = $$\alpha $$ divides the area of region R into
two equal parts,...

Area (in sq. units) of the region outside
$${{\left| x \right|} \over 2} + {{\left| y \right|} \over 3} = 1$$ and inside the ellipse $${{{x^2}} \over ...

Given : $$f(x) = \left\{ {\matrix{
{x\,\,\,\,\,,} & {0 \le x < {1 \over 2}} \cr
{{1 \over 2}\,\,\,\,,} & {x = {1 \over 2}} \cr
...

The area (in sq. units) of the region
{(x,y) $$ \in $$ R2 : x2 $$ \le $$ y $$ \le $$ 3 – 2x}, is :

For a > 0, let the curves C1 : y2 = ax and
C2 : x2 = ay intersect at origin O and a point P.
Let the line x = b (0 < b < a) intersect the cho...

The area (in sq. units) of the region
{(x, y) $$ \in $$ R2 | 4x2 $$ \le $$ y $$ \le $$ 8x + 12} is :

The area of the region, enclosed by the circle x2 + y2 = 2 which is not common to the region bounded by the parabola y2 = x and the straight line y = ...

If the area (in sq. units) bounded by the parabola y2
= 4$$\lambda $$x and the line y = $$\lambda $$x, $$\lambda $$ > 0, is $${1 \over 9}$$
, then...

If the area (in sq. units) of the region {(x, y) : y2
$$ \le $$ 4x, x + y $$ \le $$ 1, x $$ \ge $$ 0, y $$ \ge $$ 0} is a $$\sqrt 2 $$ + b, then a – b...

The area (in sq.units) of the region bounded by the curves y = 2x
and y = |x + 1|, in the first quadrant is :

The area (in sq. units) of the region
A = {(x, y) : $${{y{}^2} \over 2}$$ $$ \le $$ x $$ \le $$ y + 4} is :-

The area (in sq. units) of the region
A = {(x, y) : x2 $$ \le $$ y $$ \le $$ x + 2} is

Let S($$\alpha $$) = {(x, y) : y2
$$ \le $$ x, 0 $$ \le $$ x $$ \le $$ $$\alpha $$} and A($$\alpha $$)
is area of the region S($$\alpha $$). If for a...

The area (in sq. units) of the region
A = { (x, y) $$ \in $$ R × R| 0 $$ \le $$ x $$ \le $$ 3, 0 $$ \le $$ y $$ \le $$ 4,
y $$ \le $$ x2 ...

The area (in sq. units) of the region bounded by the parabola, y = x2 + 2 and the lines, y = x + 1, x = 0 and x = 3, is

The area (in sq. units) in the first quadrant bounded by the parabola, y = x2 + 1, the tangent to it at the point (2, 5) and the coordinate axes is :...

The area (in sq. units) of the region bounded by the curve x2 = 4y and the straight line x = 4y – 2 is :

If the area enclosed between the curves y = kx2 and x = ky2, (k > 0), is 1 square unit. Then k is -

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

The area (in sq. units) bounded by the parabolae y = x2 – 1, the tangent at the point (2, 3) to it and the y-axis is :

If the area of the region bounded by the curves, $$y = {x^2},y = {1 \over x}$$ and the lines y = 0 and x= t (t >1) is 1 sq. unit, then t is equal...

Let g(x) = cosx2, f(x) = $$\sqrt x $$ and $$\alpha ,\beta \left( {\alpha < \beta } \right)$$ be the roots of the quadratic equation 18x2 - 9$$\pi ...

The area (in sq. units) of the region
{x $$ \in $$ R : x $$ \ge $$ 0, y $$ \ge $$ 0, y $$ \ge $$ x $$-$$ 2 and y $$ \le $$ $$\sqrt x $$}, is :...

The area (in sq. units) of the smaller portion enclosed between the curves, x2 + y2 = 4 and y2 = 3x, is :

The area (in sq. units) of the region
$$\left\{ {\left( {x,y} \right):x \ge 0,x + y \le 3,{x^2} \le 4y\,and\,y \le 1 + \sqrt x } \right\}$$ is

The area (in sq. units) of the region described by
A= {(x, y) $$\left| {} \right.$$y$$ \ge $$ x2 $$-$$ 5x + 4, x + y $$ \ge $$ 1, y $$ \le $$ 0} is : ...

The area (in sq. units) of the region $$\left\{ {\left( {x,y} \right):{y^2} \ge 2x\,\,\,and\,\,\,{x^2} + {y^2} \le 4x,x \ge 0,y \ge 0} \right\}$$ is :

The area (in sq. units) of the region described by
$$\left\{ {\left( {x,y} \right):{y^2} \le 2x} \right.$$ and $$\left. {y \ge 4x - 1} \right\}$$ is ...

The area of the region described by
$$A = \left\{ {\left( {x,y} \right):{x^2} + {y^2} \le 1} \right.$$ and $$\left. {{y^2} \le 1 - x} \right\}$$ is :...

The area (in square units) bounded by the curves $$y = \sqrt {x,} $$ $$2y - x + 3 = 0,$$ $$x$$-axis, and lying in the first quadrant is :

The area between the parabolas $${x^2} = {y \over 4}$$ and $${x^2} = 9y$$ and the straight line $$y=2$$ is :

The area of the region enclosed by the curves $$y = x,x = e,y = {1 \over x}$$ and the positive $$x$$-axis is :

The area bounded by the curves $$y = \cos x$$ and $$y = \sin x$$ between the ordinates $$x=0$$ and $$x = {{3\pi } \over 2}$$ is

The area of the region bounded by the parabola $${\left( {y - 2} \right)^2} = x - 1,$$ the tangent of the parabola at the point $$(2, 3)$$ and the $$x...

The area of the plane region bounded by the curves $$x + 2{y^2} = 0$$ and $$\,x + 3{y^2} = 1$$ is equal to :

The area enclosed between the curves $${y^2} = x$$ and $$y = \left| x \right|$$ is :

The area enclosed between the curve $$y = {\log _e}\left( {x + e} \right)$$ and the coordinate axes is :

The parabolas $${y^2} = 4x$$ and $${x^2} = 4y$$ divide the square region bounded by the lines $$x=4,$$ $$y=4$$ and the coordinate axes. If $${S_1},{S_...

Let $$f(x)$$ be a non - negative continuous function such that the area bounded by the curve $$y=f(x),$$ $$x$$-axis and the ordinates $$x = {\pi \ov...

The area of the region bounded by the curves
$$y = \left| {x - 2} \right|,x = 1,x = 3$$ and the $$x$$-axis is :

The area of the region bounded by the curves $$y = \left| {x - 1} \right|$$ and $$y = 3 - \left| x \right|$$ is :

The area bounded by the curves $$y = \ln x,y = \ln \left| x \right|,y = \left| {\ln {\mkern 1mu} x} \right|$$ and $$y = \left| {\ln \left| x \right|} ...