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s45VVInIfdRU4FBS1N1kls4lvsb | maths | application-of-derivatives | tangent-and-normal | If the curves, $${{{x^2}} \over a} + {{{y^2}} \over b} = 1$$ and $${{{x^2}} \over c} + {{{y^2}} \over d} = 1$$ intersect each other at an angle of 90$$^\circ$$, then which of the following relations is TRUE? | [{"identifier": "A", "content": "a $$-$$ c = b + d"}, {"identifier": "B", "content": "a + b = c + d "}, {"identifier": "C", "content": "$$ab = {{c + d} \\over {a + b}}$$"}, {"identifier": "D", "content": "a $$-$$ b = c $$-$$ d"}] | ["D"] | null | $${{{x^2}} \over a} + {{{y^2}} \over b} = 1$$ ..........(1)<br><br>Differentiating both sides :
<br><br>$${{2x} \over a} + {{2y} \over b}{{dy} \over {dx}} = 0 \Rightarrow {y \over b}{{dy} \over {dx}} = {{ - x} \over a}$$<br><br>$$ \Rightarrow $$ $${{dy} \over {dx}} = {{ - bx} \over {ay}}$$ ............(2)<br><br>$${{{... | mcq | jee-main-2021-online-25th-february-morning-slot | 4,828 |
Qvcb8eUQ8LwGYFPJYw1klta2a04 | maths | application-of-derivatives | tangent-and-normal | If the curves x = y<sup>4</sup> and xy = k cut at right angles, then (4k)<sup>6</sup> is equal to __________. | [] | null | 4 | $$x = {y^4}$$ and $$xy = k$$<br><br>for intersection $${y^5} = k$$ ..... (1)<br><br>Also $$x = {y^4}$$ <br><br>$$ \Rightarrow 1 = 4{y^3}{{dy} \over {dx}} \Rightarrow {{dy} \over {dx}} = {1 \over {4{y^3}}}$$<br><br>for $$xy = k \Rightarrow x = {k \over y}$$<br><br>$$ \Rightarrow 1 = - {k \over {{y^2}}}.{{dy} \over {dx... | integer | jee-main-2021-online-25th-february-evening-slot | 4,829 |
onv7vutPC56pWmNGtD1kluxctnr | maths | application-of-derivatives | tangent-and-normal | Let slope of the tangent line to a curve at any point P(x, y) be given by $${{x{y^2} + y} \over x}$$. If the curve intersects the line x + 2y = 4 at x = $$-$$2, then the value of y, for which the point (3, y) lies on the curve, is : | [{"identifier": "A", "content": "$$ - {{18} \\over {19}}$$"}, {"identifier": "B", "content": "$$ - {{4} \\over {3}}$$"}, {"identifier": "C", "content": "$${{18} \\over {35}}$$"}, {"identifier": "D", "content": "$$ - {{18} \\over {11}}$$"}] | ["A"] | null | $${{dy} \over {dx}} = {{x{y^2} + y} \over x}$$<br><br>$$ \Rightarrow {{xdy - ydx} \over {{y^2}}} = xdx$$<br>$$ \Rightarrow - d\left( {{x \over y}} \right) = d\left( {{{{x^2}} \over 2}} \right)$$<br><br>$$ \Rightarrow {{ - x} \over y} = {{{x^2}} \over 2} + C$$<br><br>Curve intersect the line x + 2y = 4 at x = $$-$$ 2<b... | mcq | jee-main-2021-online-26th-february-evening-slot | 4,830 |
kZ58r1wbH4gdw8wrHl1kluyhkmg | maths | application-of-derivatives | tangent-and-normal | Let the normals at all the points on a given curve pass through a fixed point (a, b). If the curve passes through (3, $$-$$3) and (4, $$-$$2$$\sqrt 2 $$), and given that a $$-$$ 2$$\sqrt 2 $$ b = 3, <br/>then (a<sup>2</sup> + b<sup>2</sup> + ab) is equal to __________. | [] | null | 9 | Let the equation of normal is Y $$-$$ y = $$-$$$${1 \over m}(X - x)$$, where, m = $${{dy} \over {dx}}$$<br><br>As it passes through (a, b)<br><br>$$b - y = - {1 \over m}(a - x) = - {{dx} \over {dy}}(a - x)$$<br><br>$$ \Rightarrow (b - y)dy = (x - a)dx$$<br><br>by $$ - {{{y^2}} \over 2} = {{{x^2}} \over 2} - ax + c$$ ... | integer | jee-main-2021-online-26th-february-evening-slot | 4,831 |
1l567ppaq | maths | application-of-derivatives | tangent-and-normal | <p>Let l be a line which is normal to the curve y = 2x<sup>2</sup> + x + 2 at a point P on the curve. If the point Q(6, 4) lies on the line l and O is origin, then the area of the triangle OPQ is equal to ___________.</p> | [] | null | 13 | <p> <img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l5oc5ao6/7ae3148f-ca07-4559-92f9-0d6735c2710d/0346fe60-0545-11ed-987f-3938cfc0f7f1/file-1l5oc5ao7.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l5oc5ao6/7ae3148f-ca07-4559-92f9-0d6735c2710d/0346fe60-0545-11ed-987f-3938cfc0f7f... | integer | jee-main-2022-online-28th-june-morning-shift | 4,833 |
1l589kpc2 | maths | application-of-derivatives | tangent-and-normal | <p>Let S be the set of all the natural numbers, for which the line $${x \over a} + {y \over b} = 2$$ is a tangent to the curve $${\left( {{x \over a}} \right)^n} + {\left( {{y \over b}} \right)^n} = 2$$ at the point (a, b), ab $$\ne$$ 0. Then :</p> | [{"identifier": "A", "content": "S = $$\\phi$$"}, {"identifier": "B", "content": "n(S) = 1"}, {"identifier": "C", "content": "S = {2k : k $$\\in$$ N}"}, {"identifier": "D", "content": "S = N"}] | ["D"] | null | <p>$${\left( {{x \over a}} \right)^n} + {\left( {{y \over b}} \right)^n} = 2$$</p>
<p>Differentiating both sides with respect to x, we get</p>
<p>$$ \Rightarrow n\,.\,{\left( {{x \over a}} \right)^{n - a}}\,.\,{1 \over a} + n\,.\,{\left( {{y \over b}} \right)^{n - 1}}\,.\,{1 \over b}\,.\,{{dy} \over {dx}} = 0$$</p>
<p>... | mcq | jee-main-2022-online-26th-june-morning-shift | 4,834 |
1l59km5jz | maths | application-of-derivatives | tangent-and-normal | <p>If the angle made by the tangent at the point (x<sub>0</sub>, y<sub>0</sub>) on the curve $$x = 12(t + \sin t\cos t)$$, $$y = 12{(1 + \sin t)^2}$$, $$0 < t < {\pi \over 2}$$, with the positive x-axis is $${\pi \over 3}$$, then y<sub>0</sub> is equal to:</p> | [{"identifier": "A", "content": "$$6\\left( {3 + 2\\sqrt 2 } \\right)$$"}, {"identifier": "B", "content": "$$3\\left( {7 + 4\\sqrt 3 } \\right)$$"}, {"identifier": "C", "content": "27"}, {"identifier": "D", "content": "48"}] | ["C"] | null | <p>$$\because$$ $${{dy} \over {dx}} = {{24(1 + \sin t)\cos t} \over {12(1 + \cos 2t)}} = {{1 + \sin t} \over {\cos t}} = \tan \left( {{\pi \over 4} + {t \over 2}} \right)$$</p>
<p>$$\because$$ $${{dy} \over {d{x_{({x_0},{y_0})}}}} = \sqrt 3 = \tan \left( {{\pi \over 4} + {t \over 2}} \right)$$</p>
<p>$$ \Rightarrow ... | mcq | jee-main-2022-online-25th-june-evening-shift | 4,835 |
1l5c1hf56 | maths | application-of-derivatives | tangent-and-normal | <p>Let $$\lambda x - 2y = \mu $$ be a tangent to the hyperbola $${a^2}{x^2} - {y^2} = {b^2}$$. Then $${\left( {{\lambda \over a}} \right)^2} - {\left( {{\mu \over b}} \right)^2}$$ is equal to :</p> | [{"identifier": "A", "content": "$$-$$2"}, {"identifier": "B", "content": "$$-$$4"}, {"identifier": "C", "content": "2"}, {"identifier": "D", "content": "4"}] | ["D"] | null | $\frac{x^{2}}{\left(\frac{b^{2}}{a^{2}}\right)}-\frac{y^{2}}{b^{2}}=1$
<br/><br/>
Tangent in slope form $\Rightarrow y=m x \pm \sqrt{\frac{b^{2}}{a^{2}} m^{2}-b^{2}}$
<br/><br/>
i.e., same as $y=\frac{\lambda x}{2}-\frac{\mu}{2}$
<br/><br/>
Comparing coefficients,
<br/><br/>
$$
\begin{aligned}
&m=\frac{\lambda}{2}, \fr... | mcq | jee-main-2022-online-24th-june-morning-shift | 4,838 |
1l6jeeqfu | maths | application-of-derivatives | tangent-and-normal | <p>Let $$M$$ and $$N$$ be the number of points on the curve $$y^{5}-9 x y+2 x=0$$, where the tangents to the curve are parallel to $$x$$-axis and $$y$$-axis, respectively. Then the value of $$M+N$$ equals ___________.</p> | [] | null | 2 | <p>Here equation of curve is</p>
<p>$${y^5} - 9xy + 2x = 0$$ ...... (i)</p>
<p>On differentiating : $$5{y^4}{{dy} \over {dx}} - 9y - 9x{{dy} \over {dx}} + 2 = 0$$</p>
<p>$$\therefore$$ $${{dy} \over {dx}} = {{9y - 2} \over {5{y^4} - 9x}}$$</p>
<p>When tangents are parallel to x-axis then $$9y - 2 = 0$$</p>
<p>$$\theref... | integer | jee-main-2022-online-27th-july-morning-shift | 4,839 |
1l6rfwzo1 | maths | application-of-derivatives | tangent-and-normal | <p>If the tangent to the curve $$y=x^{3}-x^{2}+x$$ at the point $$(a, b)$$ is also tangent to the curve $$y = 5{x^2} + 2x - 25$$ at the point (2, $$-$$1), then $$|2a + 9b|$$ is equal to __________.</p> | [] | null | 195 | Slope of tangent to curve $y=5 x^{2}+2 x-25$
<br/><br/>$$
=m=\left(\frac{d y}{d x}\right)_{\mathrm{at}(2,-1)}=22
$$
<br/><br/>$\therefore \quad$ Equation of tangent $: y+1=22(x-2)$
<br/><br/>$\therefore \quad y=22 x-45$.
<br/><br/>Slope of tangent to $y=x^{3}-x^{2}+x$ at point $(a, b)$
<br/><br/>$$
=3 a^{2}-2 a+1
... | integer | jee-main-2022-online-29th-july-evening-shift | 4,840 |
1ldr7heoj | maths | application-of-derivatives | tangent-and-normal | <p>The number of points on the curve $$y=54 x^{5}-135 x^{4}-70 x^{3}+180 x^{2}+210 x$$ at which the normal lines are parallel to $$x+90 y+2=0$$ is :</p> | [{"identifier": "A", "content": "2"}, {"identifier": "B", "content": "3"}, {"identifier": "C", "content": "4"}, {"identifier": "D", "content": "0"}] | ["C"] | null | <p>$$y'=270x^4-540x^3-210x^2+360x+210$$</p>
<p>Slope of normal $$=-\frac{1}{90}$$</p>
<p>$$\therefore$$ Slope of tangent = 90</p>
<p>$$\therefore$$ Number of normal will be number of solutions of</p>
<p>$$270x^4-540x^3-210x^2+360x+210=90$$</p>
<p>$$\Rightarrow 9x^4-18x^3-7x^2+12x+4=0$$</p>
<p>$$\therefore x=1,2,-\frac{... | mcq | jee-main-2023-online-30th-january-morning-shift | 4,841 |
1ldsg4b6k | maths | application-of-derivatives | tangent-and-normal | <p>If the equation of the normal to the curve $$y = {{x - a} \over {(x + b)(x - 2)}}$$ at the point (1, $$-$$3) is $$x - 4y = 13$$, then the value of $$a + b$$ is equal to ___________.</p> | [] | null | 4 | <p>Given curve : $$y = {{x - a} \over {(x + b)(x - 2)}}$$ at $$(1, - 3)$$</p>
<p>$$\therefore$$ $$ - 3 = {{1 - a} \over {(1 + b)( - 1)}} \Rightarrow 3 + 3b = 1 - a$$</p>
<p>$$\beta \Rightarrow a + 3b + 2 = 0$$</p>
<p>$$y = {{x - a} \over {(x + b)(x - 2)}}$$</p>
<p>$${{dy} \over {dx}} = {{(x + b)(x - 2) - (x - a)[(x + ... | integer | jee-main-2023-online-29th-january-evening-shift | 4,842 |
1lgvqx8pc | maths | application-of-derivatives | tangent-and-normal | <p>Let the quadratic curve passing through the point $$(-1,0)$$ and touching the line $$y=x$$ at $$(1,1)$$ be $$y=f(x)$$. Then the $$x$$-intercept of the normal to the curve at the point $$(\alpha, \alpha+1)$$ in the first quadrant is __________.</p> | [] | null | 11 | Let the quadratic curve be $f(x)=a x^2+b x+c$
<br/><br/>The curve passes through $(-1,0)$
<br/><br/>$0=a-b+c \Rightarrow a+c=b$ ..........(i)
<br/><br/>The curve also passes through $(1,1)$
<br/><br/>$$
\begin{gathered}
a+b+c=1 .........(ii)\\\\
2 b=1 \Rightarrow b=\frac{1}{2}
\end{gathered}
$$
<br/><br/>$$
f^{\prime}... | integer | jee-main-2023-online-10th-april-evening-shift | 4,843 |
1lgxsy1zh | maths | application-of-derivatives | tangent-and-normal | <p>The slope of tangent at any point (x, y) on a curve $$y=y(x)$$ is $${{{x^2} + {y^2}} \over {2xy}},x > 0$$. If $$y(2) = 0$$, then a value of $$y(8)$$ is :</p> | [{"identifier": "A", "content": "$$ - 4\\sqrt 2 $$"}, {"identifier": "B", "content": "$$2\\sqrt 3 $$"}, {"identifier": "C", "content": "$$4\\sqrt 3 $$"}, {"identifier": "D", "content": "$$ - 2\\sqrt 3 $$"}] | ["C"] | null | Let the slope of tangent at any point
<br/><br/>$(x, y)$ on a curve $y=y(x)$ is $\frac{d y}{d x}$
<br/><br/>According to the question, $\frac{d y}{d x}=\frac{x^2+y^2}{2 x y}(x>0)$
[Given]
<br/><br/>$$
\begin{aligned}
&\text { Let } y=v x \Rightarrow \frac{d y}{d x}=v+x \frac{d v}{d x} \\\\
&\Rightarrow v+x \frac{d v... | mcq | jee-main-2023-online-10th-april-morning-shift | 4,844 |
v8p9syKOxJlo7k47 | maths | area-under-the-curves | area-bounded-between-the-curves | The area bounded by the curves $$y = \ln x,y = \ln \left| x \right|,y = \left| {\ln {\mkern 1mu} x} \right|$$ and $$y = \left| {\ln \left| x \right|} \right|$$ is : | [{"identifier": "A", "content": "$$4$$sq. units "}, {"identifier": "B", "content": "$$6$$sq. units "}, {"identifier": "C", "content": "$$10$$sq. units"}, {"identifier": "D", "content": "none of these "}] | ["A"] | null | First we draw each curve as separate graph
<br><br><img class="question-image" src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266365/exam_images/eejvpxyhkrguzqnnbwmv.webp" loading="lazy" alt="AIEEE 2002 Mathematics - Area Under The Curves Question 133 English Explanation 1">
<br><br><b>NOTE :</b> Graph of... | mcq | aieee-2002 | 4,846 |
0zJaHFqpedZaI9Xm | maths | area-under-the-curves | area-bounded-between-the-curves | The area of the region bounded by the curves
<br/>$$y = \left| {x - 2} \right|,x = 1,x = 3$$ and the $$x$$-axis is : | [{"identifier": "A", "content": "$$4$$"}, {"identifier": "B", "content": "$$2$$"}, {"identifier": "C", "content": "$$3$$"}, {"identifier": "D", "content": "$$1$$"}] | ["D"] | null | The required area is shown by shaded region
<br><br><img class="question-image" src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266321/exam_images/ixjlsq9plxoilqztxxsw.webp" loading="lazy" alt="AIEEE 2004 Mathematics - Area Under The Curves Question 131 English Explanation">
<br><br>$$A = \int\limits_1^3 ... | mcq | aieee-2004 | 4,848 |
0AwCj1YqAxpB6L96 | maths | area-under-the-curves | area-bounded-between-the-curves | The parabolas $${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_2},{S_3}$$ are respectively the areas of these parts numbered from top to bottom ; then $${S_1},{S_2},{S_3}$$ is : | [{"identifier": "A", "content": "$$1:2:1$$"}, {"identifier": "B", "content": "$$1:2:3$$"}, {"identifier": "C", "content": "$$2:1:2$$"}, {"identifier": "D", "content": "$$1:1:1$$"}] | ["D"] | null | Intersection points of $${x^2} = 4y$$ and $${y^2} = 4x$$ are $$\left( {0,0} \right)$$ and $$\left( {4,4} \right).$$ The graph is as shown in the figure.
<br><br><img class="question-image" src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734267219/exam_images/qhldynplznxc2j2lyapa.webp" loading="lazy" alt="AIEE... | mcq | aieee-2005 | 4,849 |
OJxelJmkjNf7amKn | maths | area-under-the-curves | area-bounded-between-the-curves | The area enclosed between the curves $${y^2} = x$$ and $$y = \left| x \right|$$ is : | [{"identifier": "A", "content": "$$1/6$$"}, {"identifier": "B", "content": "$$1/3$$"}, {"identifier": "C", "content": "$$2/3$$"}, {"identifier": "D", "content": "$$1$$"}] | ["A"] | null | The area enclosed between the curves
<br><br>$${y^2} = x$$ and $$y = \left| x \right|$$
<br><br>From the figure, area lies between
<br><br>$${y^2} = x$$ and $$y = x$$
<br><br><img class="question-image" src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266518/exam_images/peptywbkfivgg2xjbmvn.webp" loading="l... | mcq | aieee-2007 | 4,850 |
pMzAZwRFlpht8r8n | maths | area-under-the-curves | area-bounded-between-the-curves | The area of the plane region bounded by the curves $$x + 2{y^2} = 0$$ and $$\,x + 3{y^2} = 1$$ is equal to : | [{"identifier": "A", "content": "$${5 \\over 3}$$ "}, {"identifier": "B", "content": "$${1 \\over 3}$$"}, {"identifier": "C", "content": "$${2 \\over 3}$$"}, {"identifier": "D", "content": "$${4 \\over 3}$$"}] | ["D"] | null | $$x + 2{y^2} = 0 \Rightarrow {y^2} = - {x \over 2}$$
<br><br>$$\left[ {} \right.$$ Left handed parabola with vertex at $$\left( {0,0} \right)$$ $$\left. {} \right]$$
<br><br>$$x + 3{y^2} = 1 \Rightarrow {y^2} = - {1 \over 3}\left( {x - 1} \right)$$
<br><br>$$\left[ {} \right.$$ Left handed parabola with vertex at $$... | mcq | aieee-2008 | 4,851 |
IFwRcsOuCCngSl5t | maths | area-under-the-curves | area-bounded-between-the-curves | 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 $$x$$-axis is : | [{"identifier": "A", "content": "$$6$$"}, {"identifier": "B", "content": "$$9$$"}, {"identifier": "C", "content": "$$12$$"}, {"identifier": "D", "content": "$$3$$"}] | ["B"] | null | The given parabola is $${\left( {y - 2} \right)^2} = x - 1$$
<br><br>Vertex $$\left( {1,2} \right)$$ and it meets $$x$$-axis at $$\left( {5,0} \right)$$
<br><br>Also it gives $${y^2} - 4y - x + 5 = 0$$
<br><br>So, that equation of tangent to the parabola at $$\left( {2,3} \right)$$ is
<br><br>$$y.3 - 2\left( {y + 3} \... | mcq | aieee-2009 | 4,852 |
AX7qp2uZbAWLtltG | maths | area-under-the-curves | area-bounded-between-the-curves | The area bounded by the curves $$y = \cos x$$ and $$y = \sin x$$ between the ordinates $$x=0$$ and $$x = {{3\pi } \over 2}$$ is | [{"identifier": "A", "content": "$$4\\sqrt 2 + 2$$ "}, {"identifier": "B", "content": "$$4\\sqrt 2 - 1$$"}, {"identifier": "C", "content": "$$4\\sqrt 2 + 1$$"}, {"identifier": "D", "content": "$$4\\sqrt 2 - 2$$"}] | ["D"] | null | <img class="question-image" src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734267132/exam_images/qwadcct8cbggvfesgjha.webp" loading="lazy" alt="AIEEE 2010 Mathematics - Area Under The Curves Question 124 English Explanation">
<br><br>$$\therefore$$ Required area
<br><br>$$ = \left[ {\int\limits_0^{{\pi \ove... | mcq | aieee-2010 | 4,853 |
SVlnixqVTARTxEc5 | maths | area-under-the-curves | area-bounded-between-the-curves | The area between the parabolas $${x^2} = {y \over 4}$$ and $${x^2} = 9y$$ and the straight line $$y=2$$ is : | [{"identifier": "A", "content": "$$20\\sqrt 2 $$ "}, {"identifier": "B", "content": "$${{10\\sqrt 2 } \\over 3}$$ "}, {"identifier": "C", "content": "$${{20\\sqrt 2 } \\over 3}$$"}, {"identifier": "D", "content": "$$10\\sqrt 2 $$"}] | ["C"] | null | Given curves $${x^2} = {y \over 4}$$ and $${x^2} = 9y$$ are the parabolas whose equations can be written as $$y = 4{x^2}$$ and $$y = {1 \over 9}{x^2}.$$
<br><br>Also, given $$y=2.$$
<br><br>Now, shaded portion shows the required area which is symmetric.
<br><br><img class="question-image" src="https://res.cloudinary.... | mcq | aieee-2012 | 4,855 |
2vnQTfUpqrB6aQZg | maths | area-under-the-curves | area-bounded-between-the-curves | 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 : | [{"identifier": "A", "content": "$$9$$ "}, {"identifier": "B", "content": "$$36$$ "}, {"identifier": "C", "content": "$$18$$"}, {"identifier": "D", "content": "$${{27} \\over 4}$$ "}] | ["A"] | null | Given curves are
<br><br>$$y = \sqrt x $$ $$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( 1 \right)$$
<br><br>and $$2y - x + 3 = 0\,\,\,\,\,\,\,\,\,...\left( 2 \right)$$
<br><br>On solving both we get $$y=-1,3$$
<br><br><img class="question-image" src="https://res.cloudinary.com/dck... | mcq | jee-main-2013-offline | 4,856 |
DI501ESEKnVjEWBw | maths | area-under-the-curves | area-bounded-between-the-curves | The area of the region described by
<br/>$$A = \left\{ {\left( {x,y} \right):{x^2} + {y^2} \le 1} \right.$$ and $$\left. {{y^2} \le 1 - x} \right\}$$ is : | [{"identifier": "A", "content": "$${\\pi \\over 2} - {2 \\over 3}$$ "}, {"identifier": "B", "content": "$${\\pi \\over 2} + {2 \\over 3}$$"}, {"identifier": "C", "content": "$${\\pi \\over 2} + {4 \\over 3}$$"}, {"identifier": "D", "content": "$${\\pi \\over 2} - {4 \\over 3}$$"}] | ["C"] | null | Given curves are $${x^2} + {y^2} = 1$$ and $${y^2} = 1 - x.$$
<br><br>Intersection points are $$x = 0,1$$
<br><br>Area of shaded portion is the required area.
<br><br>So, Required Area $$=$$ Area of semi-circle $$+$$ Area bounded by parabola
<br><br>$$ = {{\pi {r^2}} \over 2} + 2\int\limits_0^1 {\sqrt {1 - x} dx} $... | mcq | jee-main-2014-offline | 4,857 |
XTxcDJGWpMyOANNI | maths | area-under-the-curves | area-bounded-between-the-curves | The area (in sq. units) of the region described by
<br/><br>$$\left\{ {\left( {x,y} \right):{y^2} \le 2x} \right.$$ and $$\left. {y \ge 4x - 1} \right\}$$ is :</br> | [{"identifier": "A", "content": "$${{15} \\over {64}}$$ "}, {"identifier": "B", "content": "$${{9} \\over {32}}$$"}, {"identifier": "C", "content": "$${{7} \\over {32}}$$"}, {"identifier": "D", "content": "$${{5} \\over {64}}$$"}] | ["B"] | null | Required area
<br><br><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l91nf3yh/b4e886d1-7f41-4b72-ab7a-245bf94417c9/ef2e8480-47fb-11ed-8757-0f869593f41f/file-1l91nf3yi.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l91nf3yh/b4e886d1-7f41-4b72-ab7a-245bf94417c9/ef2e8480-47fb-11e... | mcq | jee-main-2015-offline | 4,858 |
pgV7X2hCYOLs7nkDCGAqG | maths | area-under-the-curves | area-bounded-between-the-curves | The area (in sq. units) of the region described by
<br/><br/>A= {(x, y) $$\left| {} \right.$$y$$ \ge $$ x<sup>2</sup> $$-$$ 5x + 4, x + y $$ \ge $$ 1, y $$ \le $$ 0} is : | [{"identifier": "A", "content": "$${7 \\over 2}$$ "}, {"identifier": "B", "content": "$${{19} \\over 6}$$ "}, {"identifier": "C", "content": "$${{13} \\over 6}$$"}, {"identifier": "D", "content": "$${{17} \\over 6}$$"}] | ["B"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265211/exam_images/xedy87gs6fsevx9ehium.webp" style="max-width: 100%; height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2016 (Online) 9th April Morning Slot Mathematics - Area Under The Curves Question 112 English Explanation">
<b... | mcq | jee-main-2016-online-9th-april-morning-slot | 4,859 |
UNVOXbPQzfaJPPXv | maths | area-under-the-curves | area-bounded-between-the-curves | The area (in sq. units) of the region $$\left\{ {\left( {x,y} \right):{y^2} \ge 2x\,\,\,and\,\,\,{x^2} + {y^2} \le 4x,x \ge 0,y \ge 0} \right\}$$ is : | [{"identifier": "A", "content": "$$\\pi - {{4\\sqrt 2 } \\over 3}$$ "}, {"identifier": "B", "content": "$${\\pi \\over 2} - {{2\\sqrt 2 } \\over 3}$$ "}, {"identifier": "C", "content": "$$\\pi - {4 \\over 3}$$ "}, {"identifier": "D", "content": "$$\\pi - {8 \\over 3}$$"}] | ["D"] | null | <img class="question-image" src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264764/exam_images/zyqeh2enbhsl6lnxeebb.webp" loading="lazy" alt="JEE Main 2016 (Offline) Mathematics - Area Under The Curves Question 121 English Explanation">
<br><br>Points of intersection of the two curves are $$\left( {0,0} \ri... | mcq | jee-main-2016-offline | 4,860 |
FEfb6CgsB2oAvcfP | maths | area-under-the-curves | area-bounded-between-the-curves | The area (in sq. units) of the region
<br/><br/>$$\left\{ {\left( {x,y} \right):x \ge 0,x + y \le 3,{x^2} \le 4y\,and\,y \le 1 + \sqrt x } \right\}$$ is | [{"identifier": "A", "content": "$${3 \\over 2}$$"}, {"identifier": "B", "content": "$${7 \\over 3}$$"}, {"identifier": "C", "content": "$${5 \\over 2}$$"}, {"identifier": "D", "content": "$${59 \\over 12}$$"}] | ["C"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266336/exam_images/dfewqownwnzl1h0lfyw7.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2017 (Offline) Mathematics - Area Under The Curves Question 116 English Explanation">
<br>Area of shaded region
... | mcq | jee-main-2017-offline | 4,862 |
ThczWi1ecPYlB5lQwPItg | maths | area-under-the-curves | area-bounded-between-the-curves | The area (in sq. units) of the region
<br/><br/>{x $$ \in $$ <b>R</b> : x $$ \ge $$ 0, y $$ \ge $$ 0, y $$ \ge $$ x $$-$$ 2 <i>and</i> y $$ \le $$ $$\sqrt x $$}, is : | [{"identifier": "A", "content": "$${{13} \\over 3}$$"}, {"identifier": "B", "content": "$${{8} \\over 3}$$"}, {"identifier": "C", "content": "$${{10} \\over 3}$$"}, {"identifier": "D", "content": "$${{5} \\over 3}$$"}] | ["C"] | null | y = $$\sqrt x $$
<br><br>y = x $$-$$ 2
<br><br>$$\therefore\,\,\,$$ $$\sqrt x $$ = x $$-$$ 2
<br><br>$$ \Rightarrow $$$$\,\,\,$$ x = x<sup>2</sup> $$-$$ 4x + 4
<br><br>x<sup>2</sup> $$-$$ 5x + 4 = 0
<br><br>x<sup>2</sup> $$-$$ 4x $$-$$ x + 4 = 0
<br><br>$$ \Rightarrow $$$$\,\,\,$$ x(x $$-$$ 4) $$-$$ (x $$-$$ 4) = 0
<br... | mcq | jee-main-2018-online-15th-april-morning-slot | 4,863 |
8L5Ej5Etn8PQ5O4I0IQ6b | maths | area-under-the-curves | area-bounded-between-the-curves | 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 to : | [{"identifier": "A", "content": "$${e^{{3 \\over 2}}}$$"}, {"identifier": "B", "content": "$${4 \\over 3}$$"}, {"identifier": "C", "content": "$${3 \\over 2}$$"}, {"identifier": "D", "content": "$${e^{{2 \\over 3}}}$$"}] | ["D"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265062/exam_images/b58nqrkplxkhfj2zdyvj.webp" style="max-width: 100%; height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2018 (Online) 16th April Morning Slot Mathematics - Area Under The Curves Question 114 English Explanation">
<... | mcq | jee-main-2018-online-16th-april-morning-slot | 4,864 |
olQ2CuKJqbEOapJsuh3rsa0w2w9jxaoline | maths | area-under-the-curves | area-bounded-between-the-curves | If the area (in sq. units) bounded by the parabola y<sup>2</sup>
= 4$$\lambda $$x and the line y = $$\lambda $$x, $$\lambda $$ > 0, is $${1 \over 9}$$
, then $$\lambda $$ is equal to : | [{"identifier": "A", "content": "$$4\\sqrt 3 $$"}, {"identifier": "B", "content": "2$$\\sqrt 6 $$"}, {"identifier": "C", "content": "48"}, {"identifier": "D", "content": "24"}] | ["D"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266672/exam_images/z4km1mkoieljdfzlmsq9.webp" style="max-width: 100%; height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2019 (Online) 12th April Evening Slot Mathematics - Area Under The Curves Question 99 English Explanation">
y<... | mcq | jee-main-2019-online-12th-april-evening-slot | 4,866 |
VgYeNczCUuvuSa2x6S3rsa0w2w9jx61y3q1 | maths | area-under-the-curves | area-bounded-between-the-curves | If the area (in sq. units) of the region {(x, y) : y<sup>2</sup>
$$ \le $$ 4x, x + y $$ \le $$ 1, x $$ \ge $$ 0, y $$ \ge $$ 0} is a $$\sqrt 2 $$ + b, then a – b is equal
to : | [{"identifier": "A", "content": "$${8 \\over 3}$$"}, {"identifier": "B", "content": "$$ - {2 \\over 3}$$"}, {"identifier": "C", "content": "6"}, {"identifier": "D", "content": "$${{10} \\over 3}$$"}] | ["C"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734267762/exam_images/pgiki0cdc9cjw0ntf4lc.webp" style="max-width: 100%; height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2019 (Online) 12th April Morning Slot Mathematics - Area Under The Curves Question 100 English Explanation">
L... | mcq | jee-main-2019-online-12th-april-morning-slot | 4,867 |
I1nmB5bUtKNBoasg383rsa0w2w9jx25emr2 | maths | area-under-the-curves | area-bounded-between-the-curves | The area (in sq.units) of the region bounded by the curves y = 2<sup>x</sup>
and y = |x + 1|, in the first quadrant is :
| [{"identifier": "A", "content": "$${1 \\over 2}$$"}, {"identifier": "B", "content": "$${3 \\over 2}$$"}, {"identifier": "C", "content": "$${3 \\over 2} - {1 \\over {\\log _e^2}}$$"}, {"identifier": "D", "content": "$$\\log _e^2 + {3 \\over 2}$$"}] | ["C"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734263379/exam_images/pwnxyxw58zdpddygd57j.webp" style="max-width: 100%; height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2019 (Online) 10th April Evening Slot Mathematics - Area Under The Curves Question 101 English Explanation"><b... | mcq | jee-main-2019-online-10th-april-evening-slot | 4,868 |
TsupCJrGClDjxoGzUl18hoxe66ijvww5mu1 | maths | area-under-the-curves | area-bounded-between-the-curves | The area (in sq. units) of the region<br/>
A = {(x, y) : $${{y{}^2} \over 2}$$ $$ \le $$ x $$ \le $$ y + 4} is :- | [{"identifier": "A", "content": "30"}, {"identifier": "B", "content": "18"}, {"identifier": "C", "content": "$${{53} \\over 3}$$"}, {"identifier": "D", "content": "16"}] | ["B"] | null | <picture><source media="(max-width: 320px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265986/exam_images/g7vhs2kmokzw9dgrvimz.webp"><source media="(max-width: 500px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264472/exam_images/wbpeghezagympwrgiinu.webp"><img src="https://res.c... | mcq | jee-main-2019-online-9th-april-evening-slot | 4,869 |
mdpcBn9fdbD5jQ5ut30AV | maths | area-under-the-curves | area-bounded-between-the-curves | The area (in sq. units) in the first quadrant bounded by the parabola, y = x<sup>2</sup> + 1, the tangent to it at the point (2, 5) and the coordinate axes is : | [{"identifier": "A", "content": "$${8 \\over 3}$$"}, {"identifier": "B", "content": "$${{14} \\over 3}$$"}, {"identifier": "C", "content": "$${{187} \\over {24}}$$"}, {"identifier": "D", "content": "$${{37} \\over {24}}$$"}] | ["D"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264058/exam_images/uhew9xrggiezdnledix0.webp" style="max-width: 100%; height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2019 (Online) 11th January Evening Slot Mathematics - Area Under The Curves Question 107 English Explanation">... | mcq | jee-main-2019-online-11th-january-evening-slot | 4,871 |
Ln6s5OIuo2KWZAUu98dpB | maths | area-under-the-curves | area-bounded-between-the-curves | If the area enclosed between the curves y = kx<sup>2</sup> and x = ky<sup>2</sup>, (k > 0), is 1 square unit. Then k is - | [{"identifier": "A", "content": "$$\\sqrt 3 $$"}, {"identifier": "B", "content": "$${{\\sqrt 3 } \\over 2}$$"}, {"identifier": "C", "content": "$${2 \\over {\\sqrt 3 }}$$"}, {"identifier": "D", "content": "$${1 \\over {\\sqrt 3 }}$$"}] | ["D"] | null | Area bounded by
<br><br>y<sup>2</sup> = 4ax & x<sup>2</sup> = 4by, a, b $$ \ne $$ 0
<br><br>is $$\left| {{{16ab} \over 3}} \right|$$
<br><br>by using formula :
<br><br>4a $$=$$ $${1 \over k} = 4b,k > 0$$
<br><br>Area $$ = \left| {{{16.{1 \over {4k}}.{1 \over {4k}}} \over 3}} \right| = 1$$
<br><br>$$ \Righ... | mcq | jee-main-2019-online-10th-january-morning-slot | 4,873 |
uQTkAboyY20GxQ1XzsGfr | maths | area-under-the-curves | area-bounded-between-the-curves | The area (in sq. units) bounded by the parabolae y = x<sup>2</sup> – 1, the tangent at the point (2, 3) to it and the y-axis is : | [{"identifier": "A", "content": "$$56\\over3$$"}, {"identifier": "B", "content": "$$32\\over3$$"}, {"identifier": "C", "content": "$$8\\over3$$"}, {"identifier": "D", "content": "$$14\\over3$$"}] | ["C"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265696/exam_images/atq9a0prxhuwrfqaru4w.webp" style="max-width: 100%; height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2019 (Online) 9th January Morning Slot Mathematics - Area Under The Curves Question 111 English Explanation">
... | mcq | jee-main-2019-online-9th-january-morning-slot | 4,874 |
vc73pE7QbP2xsKjFnTjgy2xukg395b9n | maths | area-under-the-curves | area-bounded-between-the-curves | The area (in sq. units) of the region enclosed
<br/>by the curves y = x<sup>2</sup> – 1 and y = 1 – x<sup>2</sup> is equal to : | [{"identifier": "A", "content": "$${8 \\over 3}$$"}, {"identifier": "B", "content": "$${4 \\over 3}$$"}, {"identifier": "C", "content": "$${7 \\over 2}$$"}, {"identifier": "D", "content": "$${{16} \\over 3}$$"}] | ["A"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734267732/exam_images/mtmqm6fur1hjtylyhwhz.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2020 (Online) 6th September Evening Slot Mathematics - Area Under The Curves Question 88 English Explanation">
<... | mcq | jee-main-2020-online-6th-september-evening-slot | 4,875 |
TjLSpk6Qv1h0WchkY2jgy2xukfqfasd9 | maths | area-under-the-curves | area-bounded-between-the-curves | The area (in sq. units) of the region
<br/><br>A = {(x, y) : (x – 1)[x] $$ \le $$ y $$ \le $$ 2$$\sqrt x $$, 0 $$ \le $$ x $$ \le $$ 2}, where [t]
<br/><br>denotes the greatest integer function, is :</br></br> | [{"identifier": "A", "content": "$${8 \\over 3}\\sqrt 2 - 1$$"}, {"identifier": "B", "content": "$${4 \\over 3}\\sqrt 2 + 1$$"}, {"identifier": "C", "content": "$${8 \\over 3}\\sqrt 2 - {1 \\over 2}$$"}, {"identifier": "D", "content": "$${4 \\over 3}\\sqrt 2 - {1 \\over 2}$$"}] | ["C"] | null | y = (x – 1)[x] = $$\left\{ {\matrix{
{0,} & {0 \le x < 1} \cr
{x - 1,} & {1 \le x < 2} \cr
{2,} & {x = 2} \cr
} } \right.$$
<br><img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264666/exam_images/koeynuqfjc3pdhjdcnwf.webp" style="max-width: 100%;height: auto;display: ... | mcq | jee-main-2020-online-5th-september-evening-slot | 4,877 |
oliGjHCWAC079qXRzK7k9k2k5higv7o | maths | area-under-the-curves | area-bounded-between-the-curves | The area (in sq. units) of the region<br/>
<br>{(x,y) $$ \in $$ R<sup>2</sup> : x<sup>2</sup> $$ \le $$ y $$ \le $$ 3 – 2x}, is :</br> | [{"identifier": "A", "content": "$${{34} \\over 3}$$"}, {"identifier": "B", "content": "$${{29} \\over 3}$$"}, {"identifier": "C", "content": "$${{31} \\over 3}$$"}, {"identifier": "D", "content": "$${{32} \\over 3}$$"}] | ["D"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265543/exam_images/g6ehrq5oh7fip48nkm6y.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2020 (Online) 8th January Evening Slot Mathematics - Area Under The Curves Question 95 English Explanation">
<br... | mcq | jee-main-2020-online-8th-january-evening-slot | 4,878 |
gHNdC9wBr4VGCvWafkjgy2xukf0p4kl4 | maths | area-under-the-curves | area-bounded-between-the-curves | The area (in sq. units) of the region
<br/><br/>{ (x, y) : 0 $$ \le $$ y $$ \le $$ x<sup>2</sup> + 1, 0 $$ \le $$ y $$ \le $$ x + 1,
<br/><br> $${1 \over 2}$$ $$ \le $$ x $$ \le $$ 2 } is :</br> | [{"identifier": "A", "content": "$${{79} \\over {16}}$$"}, {"identifier": "B", "content": "$${{79} \\over {24}}$$"}, {"identifier": "C", "content": "$${{23} \\over {6}}$$"}, {"identifier": "D", "content": "$${{23} \\over {16}}$$"}] | ["B"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734267143/exam_images/d1iaq40mqsj3mmajo5mu.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2020 (Online) 3rd September Morning Slot Mathematics - Area Under The Curves Question 91 English Explanation">
$... | mcq | jee-main-2020-online-3rd-september-morning-slot | 4,879 |
vVmxpIPbu31oaPTkNFjgy2xukezery5f | maths | area-under-the-curves | area-bounded-between-the-curves | Consider a region R = {(x, y) $$ \in $$ R : x<sup>2</sup> $$ \le $$ y $$ \le $$ 2x}.
if a line y = $$\alpha $$ divides the area of region R into
two equal parts, then which of the following is
true? | [{"identifier": "A", "content": "3$$\\alpha $$<sup>2</sup> - 8$$\\alpha $$ + 8 = 0"}, {"identifier": "B", "content": "$$\\alpha $$<sup>3</sup> - 6$$\\alpha $$<sup>3/2</sup> - 16 = 0"}, {"identifier": "C", "content": "3$$\\alpha $$<sup>2</sup> - 8$$\\alpha $$<sup>3/2</sup> + 8 = 0"}, {"identifier": "D", "content": "$$\\... | ["C"] | null | <picture><source media="(max-width: 320px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264598/exam_images/wusjy1qensakgeqp8glu.webp"><source media="(max-width: 500px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734267704/exam_images/ngz0ndqarqosp7kzmh1p.webp"><source media="(max-wid... | mcq | jee-main-2020-online-2nd-september-evening-slot | 4,880 |
P3CagPVm1jMSVgFFUg7k9k2k5kgx0xy | maths | area-under-the-curves | area-bounded-between-the-curves | Given : $$f(x) = \left\{ {\matrix{
{x\,\,\,\,\,,} & {0 \le x < {1 \over 2}} \cr
{{1 \over 2}\,\,\,\,,} & {x = {1 \over 2}} \cr
{1 - x\,\,\,,} & {{1 \over 2} < x \le 1} \cr
} } \right.$$<br/><br/>
and $$g(x) = \left( {x - {1 \over 2}} \right)^2,x \in R$$ <br/><br/>Then the area
(in sq.... | [{"identifier": "A", "content": "$${1 \\over 2} + {{\\sqrt 3 } \\over 4}$$"}, {"identifier": "B", "content": "$${1 \\over 2} - {{\\sqrt 3 } \\over 4}$$"}, {"identifier": "C", "content": "$${1 \\over 3} + {{\\sqrt 3 } \\over 4}$$"}, {"identifier": "D", "content": "$${{\\sqrt 3 } \\over 4} - {1 \\over 3}$$"}] | ["D"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264106/exam_images/d94wa1tsnhgimjfnfsfb.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2020 (Online) 9th January Evening Slot Mathematics - Area Under The Curves Question 94 English Explanation">
<br... | mcq | jee-main-2020-online-9th-january-evening-slot | 4,881 |
97vagoGtOJjPTsqqEL7k9k2k5gryhb0 | maths | area-under-the-curves | area-bounded-between-the-curves | For a > 0, let the curves C<sub>1</sub> : y<sup>2</sup> = ax and
C<sub>2</sub> : x<sup>2</sup> = ay intersect at origin O and a point P.
Let the line x = b (0 < b < a) intersect the chord
OP and the x-axis at points Q and R,
respectively. If the line x = b bisects the area
bounded by the curves, C<sub>1</sub> ... | [{"identifier": "A", "content": "x<sup>6</sup> \u2013 12x<sup>3</sup> + 4 = 0"}, {"identifier": "B", "content": "x<sup>6</sup> \u2013 12x<sup>3</sup> \u2013 4 = 0"}, {"identifier": "C", "content": "x<sup>6</sup> + 6x<sup>3</sup> \u2013 4 = 0"}, {"identifier": "D", "content": "x<sup>6</sup> \u2013 6x<sup>3</sup> + 4 = 0... | ["A"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264200/exam_images/yig7dqvbpytiruqt2hnf.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2020 (Online) 8th January Morning Slot Mathematics - Area Under The Curves Question 96 English Explanation">
C<s... | mcq | jee-main-2020-online-8th-january-morning-slot | 4,882 |
ev69psbeho98LVk3LE7k9k2k5fjbh1k | maths | area-under-the-curves | area-bounded-between-the-curves | The area (in sq. units) of the region
<br/>{(x, y) $$ \in $$ R<sup>2</sup> | 4x<sup>2</sup> $$ \le $$ y $$ \le $$ 8x + 12} is : | [{"identifier": "A", "content": "$${{125} \\over 3}$$"}, {"identifier": "B", "content": "$${{128} \\over 3}$$"}, {"identifier": "C", "content": "$${{127} \\over 3}$$"}, {"identifier": "D", "content": "$${{124} \\over 3}$$"}] | ["B"] | null | For point of intersection
4x<sup>2</sup>
= 8x + 12
<br><br>$$ \Rightarrow $$ x<sup>2</sup> - 2x - 3 = 0
<br>$$ \Rightarrow $$ x = –1, 3
<img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734263719/exam_images/casyif6hyghpfwzxghwt.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" lo... | mcq | jee-main-2020-online-7th-january-evening-slot | 4,883 |
mrazloIgO7RLHAf87O7k9k2k5e2xp0r | maths | area-under-the-curves | area-bounded-between-the-curves | The area of the region, enclosed by the circle x<sup>2</sup> + y<sup>2</sup> = 2 which is not common to the region bounded by the parabola y<sup>2</sup> = x and the straight line y = x, is: | [{"identifier": "A", "content": "$${1 \\over 6}\\left( {24\\pi - 1} \\right)$$"}, {"identifier": "B", "content": "$${1 \\over 3}\\left( {12\\pi - 1} \\right)$$"}, {"identifier": "C", "content": "$${1 \\over 3}\\left( {6\\pi - 1} \\right)$$"}, {"identifier": "D", "content": "$${1 \\over 6}\\left( {12\\pi - 1} \\righ... | ["D"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734263874/exam_images/ilutwsifhzc063d2ocbw.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2020 (Online) 7th January Morning Slot Mathematics - Area Under The Curves Question 98 English Explanation">
<br... | mcq | jee-main-2020-online-7th-january-morning-slot | 4,884 |
d6xFyNgEj2CmZ6SHmgjgy2xukewn1vf0 | maths | area-under-the-curves | area-bounded-between-the-curves | Area (in sq. units) of the region outside
<br/><br>$${{\left| x \right|} \over 2} + {{\left| y \right|} \over 3} = 1$$ and inside the ellipse $${{{x^2}} \over 4} + {{{y^2}} \over 9} = 1$$ is :</br> | [{"identifier": "A", "content": "$$6\\left( {4 - \\pi } \\right)$$"}, {"identifier": "B", "content": "$$3\\left( {4 - \\pi } \\right)$$"}, {"identifier": "C", "content": "$$6\\left( {\\pi - 2} \\right)$$"}, {"identifier": "D", "content": "$$3\\left( {\\pi - 2} \\right)$$"}] | ["C"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266337/exam_images/yov949hdcyfww9kn13ai.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2020 (Online) 2nd September Morning Slot Mathematics - Area Under The Curves Question 93 English Explanation">
<... | mcq | jee-main-2020-online-2nd-september-morning-slot | 4,885 |
yPCA6PkvYr7HYrsAwf1klrmdwhc | maths | area-under-the-curves | area-bounded-between-the-curves | The area of the region : $$R = \{ (x,y):5{x^2} \le y \le 2{x^2} + 9\} $$ is : | [{"identifier": "A", "content": "$$6\\sqrt 3 $$ square units"}, {"identifier": "B", "content": "$$12\\sqrt 3 $$ square units"}, {"identifier": "C", "content": "$$11\\sqrt 3 $$ square units"}, {"identifier": "D", "content": "$$9\\sqrt 3 $$ square units"}] | ["B"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264729/exam_images/jm9chvj4tml7xrg2bqui.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2021 (Online) 24th February Evening Shift Mathematics - Area Under The Curves Question 86 English Explanation">
... | mcq | jee-main-2021-online-24th-february-evening-slot | 4,887 |
cffEOVLEI7ehYNp5lZ1kmko9iza | maths | area-under-the-curves | area-bounded-between-the-curves | Let f : [$$-$$3, 1] $$ \to $$ R be given as <br/><br/>$$f(x) = \left\{ \matrix{
\min \,\{ (x + 6),{x^2}\}, - 3 \le x \le 0 \hfill \cr
\max \,\{ \sqrt x ,{x^2}\} ,\,0 \le x \le 1. \hfill \cr} \right.$$<br/><br/>If the area bounded by y = f(x) and x-axis is A, then the value of 6A is equal to ___________. | [] | null | 41 | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265792/exam_images/adjt0f0zaxeqso6stpct.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2021 (Online) 17th March Evening Shift Mathematics - Area Under The Curves Question 82 English Explanation">
<br... | integer | jee-main-2021-online-17th-march-evening-shift | 4,890 |
1krub9ptn | maths | area-under-the-curves | area-bounded-between-the-curves | The area (in sq. units) of the region bounded by the curves x<sup>2</sup> + 2y $$-$$ 1 = 0, y<sup>2</sup> + 4x $$-$$ 4 = 0 and y<sup>2</sup> $$-$$ 4x $$-$$ 4 = 0, in the upper half plane is _______________. | [] | null | 2 | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266774/exam_images/u1ozidkzbjd8lcnvbnev.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2021 (Online) 22th July Evening Shift Mathematics - Area Under The Curves Question 79 English Explanation"><br>R... | integer | jee-main-2021-online-22th-july-evening-shift | 4,892 |
1krvzemnw | maths | area-under-the-curves | area-bounded-between-the-curves | The area (in sq. units) of the region, given by the set $$\{ (x,y) \in R \times R|x \ge 0,2{x^2} \le y \le 4 - 2x\} $$ is : | [{"identifier": "A", "content": "$${8 \\over 3}$$"}, {"identifier": "B", "content": "$${{17} \\over 3}$$"}, {"identifier": "C", "content": "$${{13} \\over 3}$$"}, {"identifier": "D", "content": "$${7 \\over 3}$$"}] | ["D"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264240/exam_images/tijy7o0iderdha3gedzq.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2021 (Online) 25th July Morning Shift Mathematics - Area Under The Curves Question 78 English Explanation"><br>R... | mcq | jee-main-2021-online-25th-july-morning-shift | 4,893 |
1krxjjxgo | maths | area-under-the-curves | area-bounded-between-the-curves | The area of the region bounded by y $$-$$ x = 2 and x<sup>2</sup> = y is equal to : | [{"identifier": "A", "content": "$${{16} \\over 3}$$"}, {"identifier": "B", "content": "$${{2} \\over 3}$$"}, {"identifier": "C", "content": "$${{9} \\over 2}$$"}, {"identifier": "D", "content": "$${{4} \\over 3}$$"}] | ["C"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264963/exam_images/u8k9qqbaojrvwvtkepl5.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2021 (Online) 27th July Evening Shift Mathematics - Area Under The Curves Question 77 English Explanation"><br><... | mcq | jee-main-2021-online-27th-july-evening-shift | 4,894 |
1ktbilndb | maths | area-under-the-curves | area-bounded-between-the-curves | The area of the region $$S = \{ (x,y):3{x^2} \le 4y \le 6x + 24\} $$ is ____________. | [] | null | 27 | <picture><source media="(max-width: 320px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734267515/exam_images/gnxx2fmvzigd5eqlwpbs.webp"><source media="(max-width: 500px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265574/exam_images/sa1cxk6zsgh0gri62bbr.webp"><source media="(max-wid... | integer | jee-main-2021-online-26th-august-morning-shift | 4,896 |
1ktd2yfdk | maths | area-under-the-curves | area-bounded-between-the-curves | Let a and b respectively be the points of local maximum and local minimum of the function f(x) = 2x<sup>3</sup> $$-$$ 3x<sup>2</sup> $$-$$ 12x. If A is the total area of the region bounded by y = f(x), the x-axis and the lines x = a and x = b, then 4A is equal to ______________. | [] | null | 114 | f'(x) = 6x<sup>2</sup> $$-$$ 6x $$-$$ 12 = 6(x $$-$$ 2) (x + 1)<br><br>Point = (2, $$-$$20) & ($$-$$1, 7)<br><br><img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734263659/exam_images/k3ibititsvzikcqrsdja.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE ... | integer | jee-main-2021-online-26th-august-evening-shift | 4,897 |
1ktg2vo7y | maths | area-under-the-curves | area-bounded-between-the-curves | The area of the region bounded by the parabola (y $$-$$ 2)<sup>2</sup> = (x $$-$$ 1), the tangent to it at the point whose ordinate is 3 and the x-axis is : | [{"identifier": "A", "content": "9"}, {"identifier": "B", "content": "10"}, {"identifier": "C", "content": "4"}, {"identifier": "D", "content": "6"}] | ["A"] | null | y = 3 $$\Rightarrow$$ x = 2<br><br>Point is (2, 3)<br><br>Diff. w.r.t x<br><br>2 (y $$-$$ 2) y' = 1<br><br>$$\Rightarrow$$ $$y' = {1 \over {2(y - 2)}}$$<br><br>$$ \Rightarrow y{'_{(2,3)}} = {1 \over 2}$$<br><br>$$ \Rightarrow {{y - 3} \over {x - 2}} = {1 \over 2} \Rightarrow x - 2y + 4 = 0$$<br><br>Area $$ = \int\limit... | mcq | jee-main-2021-online-27th-august-evening-shift | 4,898 |
1kto652vo | maths | area-under-the-curves | area-bounded-between-the-curves | 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 : | [{"identifier": "A", "content": "$$2\\sqrt 2 (\\sqrt 2 - 1)$$"}, {"identifier": "B", "content": "$$2(\\sqrt 2 + 1)$$"}, {"identifier": "C", "content": "$$4(\\sqrt 2 - 1)$$"}, {"identifier": "D", "content": "$$2\\sqrt 2 (\\sqrt 2 + 1)$$"}] | ["A"] | null | $$A = \int_0^{{\pi \over 2}} {\left( {(\sin x + \cos x) - \left| {\cos x - \sin x} \right|} \right)\,dx} $$<br><br>$$A = \int_0^{{\pi \over 2}} {\left( {(\sin x + \cos x) - (\cos x - \sin x)} \right)\,dx} + \int_{{\pi \over 4}}^{{\pi \over 2}} {\left( {(\sin x + \cos x) - (\sin x - \cos x)} \right)\,dx} $$<br><br>... | mcq | jee-main-2021-online-1st-september-evening-shift | 4,899 |
1l544ulfw | maths | area-under-the-curves | area-bounded-between-the-curves | <p>The area enclosed by y<sup>2</sup> = 8x and y = $$\sqrt2$$ x that lies outside the triangle formed by y = $$\sqrt2$$ x, x = 1, y = 2$$\sqrt2$$, is equal to:</p> | [{"identifier": "A", "content": "$${{16\\sqrt 2 } \\over 6}$$"}, {"identifier": "B", "content": "$${{11\\sqrt 2 } \\over 6}$$"}, {"identifier": "C", "content": "$${{13\\sqrt 2 } \\over 6}$$"}, {"identifier": "D", "content": "$${{5\\sqrt 2 } \\over 6}$$"}] | ["C"] | null | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l5niit1g/6da14760-cb2e-4c6b-b5f6-e80807f5e0ce/291c7440-04d1-11ed-93b8-936002ac8631/file-1l5niit1h.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l5niit1g/6da14760-cb2e-4c6b-b5f6-e80807f5e0ce/291c7440-04d1-11ed-93b8-936002ac8631... | mcq | jee-main-2022-online-29th-june-morning-shift | 4,900 |
1l566hjjl | maths | area-under-the-curves | area-bounded-between-the-curves | <p>The area of the region S = {(x, y) : y<sup>2</sup> $$\le$$ 8x, y $$\ge$$ $$\sqrt2$$x, x $$\ge$$ 1} is</p> | [{"identifier": "A", "content": "$${{13\\sqrt 2 } \\over 6}$$"}, {"identifier": "B", "content": "$${{11\\sqrt 2 } \\over 6}$$"}, {"identifier": "C", "content": "$${{5\\sqrt 2 } \\over 6}$$"}, {"identifier": "D", "content": "$${{19\\sqrt 2 } \\over 6}$$"}] | ["B"] | null | <p> <img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l5obth1c/3674f2a1-bb5f-42c5-bcda-0e644fc0ece5/ba7c4f10-0543-11ed-987f-3938cfc0f7f1/file-1l5obth1d.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l5obth1c/3674f2a1-bb5f-42c5-bcda-0e644fc0ece5/ba7c4f10-0543-11ed-987f-3938cfc0f7f... | mcq | jee-main-2022-online-28th-june-morning-shift | 4,902 |
1l56u5mze | maths | area-under-the-curves | area-bounded-between-the-curves | <p>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 __________.</p> | [] | null | 36 | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l5p7lae4/469ec989-bde2-4216-be88-8dbb75d383c8/fb02f1c0-05bf-11ed-8617-d71e6444d1a0/file-1l5p7lae5.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l5p7lae4/469ec989-bde2-4216-be88-8dbb75d383c8/fb02f1c0-05bf-11ed-8617-d71e6444d1a0... | integer | jee-main-2022-online-27th-june-evening-shift | 4,903 |
1l589ntbl | maths | area-under-the-curves | area-bounded-between-the-curves | <p>The area bounded by the curve y = |x<sup>2</sup> $$-$$ 9| and the line y = 3 is :</p> | [{"identifier": "A", "content": "$$4(2\\sqrt 3 + \\sqrt 6 - 4)$$"}, {"identifier": "B", "content": "$$4(4\\sqrt 3 + \\sqrt 6 - 4)$$"}, {"identifier": "C", "content": "$$8(4\\sqrt 3 + 3\\sqrt 6 - 9)$$"}, {"identifier": "D", "content": "$$8(4\\sqrt 3 + \\sqrt 6 - 9)$$"}] | ["D"] | null | <p>$$y = 3$$ and $$y = |{x^2} - 9|$$</p>
<p>Intersect in first quadrant at $$x = \sqrt 6 $$ and $$x = \sqrt {12} $$</p>
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l5rh7hff/4ef2bdc8-3288-4017-9611-a67305104639/27b58cb0-06ff-11ed-b821-f5ba0940c0a2/file-1l5rh7hfg.png?format=png" data-orsrc="https:... | mcq | jee-main-2022-online-26th-june-morning-shift | 4,905 |
1l58f80fj | maths | area-under-the-curves | area-bounded-between-the-curves | <p>The area of the region bounded by y<sup>2</sup> = 8x and y<sup>2</sup> = 16(3 $$-$$ x) is equal to:</p> | [{"identifier": "A", "content": "$${{32} \\over 3}$$"}, {"identifier": "B", "content": "$${{40} \\over 3}$$"}, {"identifier": "C", "content": "16"}, {"identifier": "D", "content": "19"}] | ["C"] | null | <p>$${c_1}:{y^2} = 8x$$</p>
<p>$${c_2}:{y^2} = 16(3 - x)$$</p>
<p> <img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l5qp2am3/c5e50fc7-f237-4ff6-b8a1-b5d6a9b181f3/17c870b0-0691-11ed-93bf-f57702a71509/file-1l5qp2am4.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l5qp2am3/c5e50fc7-... | mcq | jee-main-2022-online-26th-june-evening-shift | 4,906 |
1l5bb4eq9 | maths | area-under-the-curves | area-bounded-between-the-curves | <p>The area (in sq. units) of the region enclosed between the parabola y<sup>2</sup> = 2x and the line x + y = 4 is __________.</p> | [] | null | 18 | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l5v6294y/2f8c41d0-8e8c-4db4-b12e-8da89b735e16/b46f9a20-0906-11ed-a790-b11fa70c8a36/file-1l5v6294z.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l5v6294y/2f8c41d0-8e8c-4db4-b12e-8da89b735e16/b46f9a20-0906-11ed-a790-b11fa70c8a36... | integer | jee-main-2022-online-24th-june-evening-shift | 4,908 |
1l5c2geb9 | maths | area-under-the-curves | area-bounded-between-the-curves | <p>Let S be the region bounded by the curves y = x<sup>3</sup> and y<sup>2</sup> = x. The curve y = 2|x| divides S into two regions of areas R<sub>1</sub>, R<sub>2</sub>. If max {R<sub>1</sub>, R<sub>2</sub>} = R<sub>2</sub>, then $${{{R_2}} \over {{R_1}}}$$ is equal to ______________.</p> | [] | null | 19 | <img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l8lc1nby/e7dc3cf8-3c01-44b7-9f93-5093bb7a6fcf/ea0230d0-3f02-11ed-8d74-051dc2e154aa/file-1l8lc1nbz.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l8lc1nby/e7dc3cf8-3c01-44b7-9f93-5093bb7a6fcf/ea0230d0-3f02-11ed-8d74-051dc2e154aa/fi... | integer | jee-main-2022-online-24th-june-morning-shift | 4,909 |
1l5w1a9en | maths | area-under-the-curves | area-bounded-between-the-curves | <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 region $$\{ (x,y):0 \le y \le x + 2\alpha ,\,|x| \le 1\} $$ is equal to ____________.</p> | [] | null | 4 | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l65usgec/bd118792-9d01-40fb-b8a3-8435f69bbadb/3ead1c40-0ee7-11ed-a7de-eff776fdb55c/file-1l65usged.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l65usgec/bd118792-9d01-40fb-b8a3-8435f69bbadb/3ead1c40-0ee7-11ed-a7de-eff776fdb55c... | integer | jee-main-2022-online-30th-june-morning-shift | 4,910 |
1l6dvcsqp | maths | area-under-the-curves | area-bounded-between-the-curves | <p>The area of the region given by</p>
<p>$$A=\left\{(x, y): x^{2} \leq y \leq \min \{x+2,4-3 x\}\right\}$$ is :</p> | [{"identifier": "A", "content": "$$\\frac{31}{8}$$"}, {"identifier": "B", "content": "$$\\frac{17}{6}$$"}, {"identifier": "C", "content": "$$\\frac{19}{6}$$"}, {"identifier": "D", "content": "$$\\frac{27}{8}$$"}] | ["B"] | null | $A=\left\{(x, y): x^{2} \leq y \leq \min \{x+2,4-3 x\}\right.$<br><br>
<img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l97rmv56/3d3d2137-f038-407b-8cca-4933b5be4cbf/1f426190-4b59-11ed-bfde-e1cb3fafe700/file-1l97rmv57.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l97rmv56/3d3d2... | mcq | jee-main-2022-online-25th-july-morning-shift | 4,911 |
1l6f3sq57 | maths | area-under-the-curves | area-bounded-between-the-curves | <p>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 point ($$-$$2, 3) be A. Then 8A is equal to ______________.</p> | [] | null | 170 | $$
\begin{aligned}
& 4 x^3-3 x y^2+6 x^2-5 x y-8 y^2+9 x+14=0 \text { at } P(-2,3) \\\\
& 12 x^2-3\left(y^2+2 y x y^{\prime}\right)+12 x-5\left(x y^{\prime}+y\right)-16 y y^{\prime} + 9=0 \\\\
& 48-3\left(9-12 y^{\prime}\right)-24-5\left(-2 y^{\prime}+3\right)-48 y^{\prime}+9 =0 \\\\
& y^{\prime}=-9 /... | integer | jee-main-2022-online-25th-july-evening-shift | 4,913 |
1l6ggjssj | maths | area-under-the-curves | area-bounded-between-the-curves | <p>The odd natural number a, such that the area of the region bounded by y = 1, y = 3, x = 0, x = y<sup>a</sup> is $${{364} \over 3}$$, is equal to :</p> | [{"identifier": "A", "content": "3"}, {"identifier": "B", "content": "5"}, {"identifier": "C", "content": "7"}, {"identifier": "D", "content": "9"}] | ["B"] | null | <p>$$\mathrm{a}$$ is a odd natural number and</p>
<p>$$\left| {\int\limits_1^3 {{y^a}dy} } \right| = {{364} \over 3}$$</p>
<p>$$ \Rightarrow \left| {{1 \over {a + 1}}\left( {{y^{a + 1}}} \right)_1^3} \right| = {{364} \over 3}$$</p>
<p>$$ \Rightarrow {{{3^{a + 1}} - 1} \over {a + 1}} = \, \pm \,{{364} \over 3}$$</p>
<p>... | mcq | jee-main-2022-online-26th-july-morning-shift | 4,914 |
1l6hzgs0d | maths | area-under-the-curves | area-bounded-between-the-curves | <p>The area bounded by the curves $$y=\left|x^{2}-1\right|$$ and $$y=1$$ is</p> | [{"identifier": "A", "content": "$$\\frac{2}{3}(\\sqrt{2}+1)$$"}, {"identifier": "B", "content": "$$\\frac{4}{3}(\\sqrt{2}-1)$$"}, {"identifier": "C", "content": "$$2(\\sqrt{2}-1)$$"}, {"identifier": "D", "content": "$$\\frac{8}{3}(\\sqrt{2}-1)$$"}] | ["D"] | null | <p> <img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l7nnzsdz/1978812c-886a-41da-a99e-ba9650919e28/fdc43870-2c7e-11ed-a18d-5933e4fde865/file-1l7nnzse0.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l7nnzsdz/1978812c-886a-41da-a99e-ba9650919e28/fdc43870-2c7e-11ed-a18d-5933e4fde86... | mcq | jee-main-2022-online-26th-july-evening-shift | 4,915 |
1l6jbsxxf | maths | area-under-the-curves | area-bounded-between-the-curves | <p>The area of the smaller region enclosed by the curves $$y^{2}=8 x+4$$ and $$x^{2}+y^{2}+4 \sqrt{3} x-4=0$$ is equal to</p> | [{"identifier": "A", "content": "$$\\frac{1}{3}(2-12 \\sqrt{3}+8 \\pi)$$"}, {"identifier": "B", "content": "$$\\frac{1}{3}(2-12 \\sqrt{3}+6 \\pi)$$"}, {"identifier": "C", "content": "$$\\frac{1}{3}(4-12 \\sqrt{3}+8 \\pi)$$"}, {"identifier": "D", "content": "$$\\frac{1}{3}(4-12 \\sqrt{3}+6 \\pi)$$"}] | ["C"] | null | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l7ps543w/90436956-c0bb-4a4d-b417-ec2e224aeb75/c7ae54c0-2da8-11ed-8542-f96181a425b5/file-1l7ps543x.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l7ps543w/90436956-c0bb-4a4d-b417-ec2e224aeb75/c7ae54c0-2da8-11ed-8542-f96181a425b5... | mcq | jee-main-2022-online-27th-july-morning-shift | 4,916 |
1l6kk0129 | maths | area-under-the-curves | area-bounded-between-the-curves | <p>Consider a curve $$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$$ does NOT pass through the point.</p>
<p><img src="data:image/png;base64,UklGRioRAABXRUJQVlA4IB4RAABQ2g... | [{"identifier": "A", "content": "(6, 21)"}, {"identifier": "B", "content": "(8, 9)"}, {"identifier": "C", "content": "(10, $$-$$4)"}, {"identifier": "D", "content": "(12, $$-$$15)"}] | ["C"] | null | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l7qa2cbj/f16ef20a-d0a5-47f2-a99e-77b83471fd40/deec2e00-2dee-11ed-a744-1fb8f3709cfa/file-1l7qa2cbk.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l7qa2cbj/f16ef20a-d0a5-47f2-a99e-77b83471fd40/deec2e00-2dee-11ed-a744-1fb8f3709cfa... | mcq | jee-main-2022-online-27th-july-evening-shift | 4,918 |
1l6nmba5c | maths | area-under-the-curves | area-bounded-between-the-curves | <p>The area enclosed by the curves $$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$$ is:</p> | [{"identifier": "A", "content": "$$2+\\mathrm{e}-\\log _{\\mathrm{e}} 2$$"}, {"identifier": "B", "content": "$$1+e-\\log _{e} 2$$"}, {"identifier": "C", "content": "$$e-\\log _{e} 2$$"}, {"identifier": "D", "content": "$$1+\\log _{e} 2$$"}] | ["B"] | null | <img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l97u3d0g/088f2998-8bd4-4720-8b55-fa1efe186501/bc448a00-4b62-11ed-80b9-4154b7faa509/file-1l97u3d0h.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l97u3d0g/088f2998-8bd4-4720-8b55-fa1efe186501/bc448a00-4b62-11ed-80b9-4154b7faa509/fi... | mcq | jee-main-2022-online-28th-july-evening-shift | 4,919 |
1l6p2kie8 | maths | area-under-the-curves | area-bounded-between-the-curves | <p>The area of the region</p>
<p>$$\left\{(x, y):|x-1| \leq y \leq \sqrt{5-x^{2}}\right\}$$ is equal to :</p> | [{"identifier": "A", "content": "$$\\frac{5}{2} \\sin ^{-1}\\left(\\frac{3}{5}\\right)-\\frac{1}{2}$$"}, {"identifier": "B", "content": "$$\\frac{5 \\pi}{4}-\\frac{3}{2}$$"}, {"identifier": "C", "content": "$$\\frac{3 \\pi}{4}+\\frac{3}{2}$$"}, {"identifier": "D", "content": "$$\\frac{5 \\pi}{4}-\\frac{1}{2}$$"}] | ["D"] | null | <p>$$A = \int\limits_{ - 1}^1 {\left( {\sqrt {5 - {x^2}} - (1 - x)} \right)dx + \int\limits_1^2 {\left( {\sqrt {5 - {x^2}} - (x - 1)} \right)dx} } $$</p>
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l7ssldo3/57b8d8ee-cd1f-4f78-8605-15da9040ba0e/e5f60c30-2f50-11ed-85dd-19dc023e9ad1/file-1l7ssldo... | mcq | jee-main-2022-online-29th-july-morning-shift | 4,920 |
ldoah9qp | maths | area-under-the-curves | area-bounded-between-the-curves | Let the area of the region
<br/><br/>$\left\{(x, y):|2 x-1| \leq y \leq\left|x^{2}-x\right|, 0 \leq x \leq 1\right\}$ be $\mathrm{A}$.
<br/><br/>Then $(6 \mathrm{~A}+11)^{2}$ is equal to | [] | null | 125 | For $B$,
<br><br>$$
\begin{aligned}
& x-x^{2}=2 x-1 \\\\
& x^{2}+x-1=0 \\\\
& x=\frac{-1+\sqrt{5}}{2}
\end{aligned}
$$
<img src="https://app-content.cdn.examgoal.net/fly/@width/image/1leeiutst/47959be2-54bf-4c50-8327-bce6176bbe32/5bf957e0-b20d-11ed-9c9a-c3bd979bc154/file-1leeiutsu.png?format=png" data-ors... | integer | jee-main-2023-online-31st-january-evening-shift | 4,922 |
1ldoocgjv | maths | area-under-the-curves | area-bounded-between-the-curves | <p>Let $$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$$ is equal to ____________.</p> | [] | null | 62 | <img src="https://app-content.cdn.examgoal.net/fly/@width/image/1le9xvce9/cce9188c-fc73-4ab5-a87d-33da42a11b53/28073f10-af88-11ed-bd02-1d2a2a7b6687/file-1le9xvcea.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1le9xvce9/cce9188c-fc73-4ab5-a87d-33da42a11b53/28073f10-af88-11ed-bd02-1d2a2a7b6687/fi... | integer | jee-main-2023-online-1st-february-morning-shift | 4,923 |
ldqwiizg | maths | area-under-the-curves | area-bounded-between-the-curves | Let $q$ be the maximum integral value of $p$ in $[0,10]$ for which the roots of the equation $x^2-p x+\frac{5}{4} p=0$ are rational. Then the area of the region $\left\{(x, y): 0 \leq y \leq(x-q)^2, 0 \leq x \leq q\right\}$ is : | [{"identifier": "A", "content": "$\\frac{125}{3}$"}, {"identifier": "B", "content": "243"}, {"identifier": "C", "content": "164"}, {"identifier": "D", "content": "25"}] | ["B"] | null | <p>Given equation : $$4{x^2} - 4px + 5p = 0$$</p>
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1leolev6m/9d2f1ed0-3ff3-4daa-b4d0-ae63d5dd71a4/31fa37e0-b797-11ed-b103-ed967fad3dff/file-1leolev6n.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1leolev6m/9d2f1ed0-3ff3-4daa-b4d0... | mcq | jee-main-2023-online-30th-january-evening-shift | 4,925 |
ldr01u60 | maths | area-under-the-curves | area-bounded-between-the-curves | Let $A$ be the area of the region
<br/><br/>$\left\{(x, y): y \geq x^2, y \geq(1-x)^2, y \leq 2 x(1-x)\right\}$.
<br/><br/>Then $540 \mathrm{~A}$ is equal to : | [] | null | 25 | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1leol5gd1/d39e5f85-8fa2-457b-8f2b-aa98ad9bb499/2c3ee450-b796-11ed-b103-ed967fad3dff/file-1leol5gd2.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1leol5gd1/d39e5f85-8fa2-457b-8f2b-aa98ad9bb499/2c3ee450-b796-11ed-b103-ed967fad3dff... | integer | jee-main-2023-online-30th-january-evening-shift | 4,926 |
1ldr7y2pf | maths | area-under-the-curves | area-bounded-between-the-curves | <p>Let $$\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$$ is equal to ___________.</p> | [] | null | 22 | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1leq0fubs/f2abef4a-3124-4350-839e-3e7bfc1aa857/beb62c80-b85e-11ed-9fed-b1659a6c339b/file-1leq0fubt.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1leq0fubs/f2abef4a-3124-4350-839e-3e7bfc1aa857/beb62c80-b85e-11ed-9fed-b1659a6c339b... | integer | jee-main-2023-online-30th-january-morning-shift | 4,927 |
1ldsf7k2m | maths | area-under-the-curves | area-bounded-between-the-curves | <p>The area of the region $$A = \left\{ {(x,y):\left| {\cos x - \sin x} \right| \le y \le \sin x,0 \le x \le {\pi \over 2}} \right\}$$ is</p> | [{"identifier": "A", "content": "$$\\sqrt 5 + 2\\sqrt 2 - 4.5$$"}, {"identifier": "B", "content": "$$1 - {3 \\over {\\sqrt 2 }} + {4 \\over {\\sqrt 5 }}$$"}, {"identifier": "C", "content": "$$\\sqrt 5 - 2\\sqrt 2 + 1$$"}, {"identifier": "D", "content": "$${3 \\over {\\sqrt 5 }} - {3 \\over {\\sqrt 2 }} + 1$$"}] | ["C"] | null | <p>$$
|\cos x-\sin x| \leq y \leq \sin x
$$
<br><br>Intersection point of $\cos x-\sin x=\sin x$
<br><br>$$
\Rightarrow \tan x=\frac{1}{2}
$$
<br><br>Let $\psi=\tan ^{-1} \frac{1}{2}$
<br><br>So, $\tan \psi=\frac{1}{2}, \sin \psi=\frac{1}{\sqrt{5}}, \cos \psi=\frac{2}{\sqrt{5}}$</p>
<p><img src="https://app-content.cdn... | mcq | jee-main-2023-online-29th-january-evening-shift | 4,928 |
1ldsv2h4c | maths | area-under-the-curves | area-bounded-between-the-curves | <p>Let $$\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)$$ is equal to</p> | [{"identifier": "A", "content": "$$2\\sqrt 3 - {1 \\over 3}$$"}, {"identifier": "B", "content": "$$2\\sqrt 3 - {2 \\over 3}$$"}, {"identifier": "C", "content": "$$\\sqrt 3 - {4 \\over 3}$$"}, {"identifier": "D", "content": "$$\\sqrt 3 - {2 \\over 3}$$"}] | ["C"] | null | <img src="https://app-content.cdn.examgoal.net/fly/@width/image/1leksols2/a3491772-b199-41ab-9761-0d39a13abce7/7e1c5220-b580-11ed-b843-fd540edb80bf/file-1leksols3.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1leksols2/a3491772-b199-41ab-9761-0d39a13abce7/7e1c5220-b580-11ed-b843-fd540edb80bf/fi... | mcq | jee-main-2023-online-29th-january-morning-shift | 4,929 |
1ldsvuk9z | maths | area-under-the-curves | area-bounded-between-the-curves | <p>Let $$[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} $$ is</p> | [{"identifier": "A", "content": "$${{5 + 4\\sqrt 2 } \\over 3}$$"}, {"identifier": "B", "content": "$${{4 + 5\\sqrt 2 } \\over 3}$$"}, {"identifier": "C", "content": "$${{8 + 4\\sqrt 2 } \\over 3}$$"}, {"identifier": "D", "content": "$${{1 + 5\\sqrt 2 } \\over 3}$$"}] | ["A"] | null | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1ldt0ucjn/f3408cd8-1181-40fe-aecc-1f2a69b3243c/74ae7e20-a63a-11ed-a341-61f046f8f7d7/file-1ldt0ucjo.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1ldt0ucjn/f3408cd8-1181-40fe-aecc-1f2a69b3243c/74ae7e20-a63a-11ed-a341-61f046f8f7d7... | mcq | jee-main-2023-online-29th-january-morning-shift | 4,930 |
1ldsvyln7 | maths | area-under-the-curves | area-bounded-between-the-curves | <p>Let $$A=\left\{(x, y) \in \mathbb{R}^{2}: y \geq 0,2 x \leq y \leq \sqrt{4-(x-1)^{2}}\right\}$$ and<br/><br/> $$
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 {. }
$$.</p>
<p>Then the ratio of the area of A to the area of B is</p> | [{"identifier": "A", "content": "$$\\frac{\\pi}{\\pi+1}$$"}, {"identifier": "B", "content": "$$\\frac{\\pi-1}{\\pi+1}$$"}, {"identifier": "C", "content": "$$\\frac{\\pi}{\\pi-1}$$"}, {"identifier": "D", "content": "$$\\frac{\\pi+1}{\\pi-1}$$"}] | ["B"] | null | <p>$y^{2}+(x-1)^{2}=4$</p>
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lekro7yb/473e481f-1c3a-4e67-a338-a22cacc1490d/8a4d0930-b57c-11ed-9ad9-3b1cedbe69d8/file-1lekro7yc.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lekro7yb/473e481f-1c3a-4e67-a338-a22cacc1490d/8a4d0930-... | mcq | jee-main-2023-online-29th-january-morning-shift | 4,931 |
1ldv36uu9 | maths | area-under-the-curves | area-bounded-between-the-curves | <p>If the area enclosed by the parabolas $$\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$$ is equal to ____________.</p> | [] | null | 600 | $x^{2}+6=\frac{5}{2} x^{2} \Rightarrow x=\pm 2$
<br/><br/>
$$
\begin{aligned}
& \text { Area between } P_{1} \text { and } P_{2} \quad \text { [Say } \left.A_{1}\right] \\\\
& =\int\limits_{-2}^{2}\left(x^{2}+6\right)-\frac{5}{2} x^{2} d x \\\\
& =2 \int\limits_{0}^{2}\left(6-\frac{3}{2} x^{2}\right) d x=2\left[6 x-\f... | integer | jee-main-2023-online-25th-january-morning-shift | 4,932 |
1ldwxskv5 | maths | area-under-the-curves | area-bounded-between-the-curves | <p>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 __________</p> | [] | null | 36 | Area enclosed by
<br><br>
$$
\begin{aligned}
& y^{2}-2 y=-x \\\\
& x+y=0
\end{aligned}
$$<br><br>
<img src="https://app-content.cdn.examgoal.net/fly/@width/image/1le5i5l5v/1c86c1a0-48dd-4e1e-9be0-6401a21b507d/903a5b30-ad17-11ed-8a8c-4d67f5492755/file-1le5i5l5w.png?format=png" data-orsrc="https://app-content.cdn... | integer | jee-main-2023-online-24th-january-evening-shift | 4,933 |
1lgoxjwx4 | maths | area-under-the-curves | area-bounded-between-the-curves | <p>The area of the region $$\left\{(x, y): x^{2} \leq y \leq\left|x^{2}-4\right|, y \geq 1\right\}$$ is</p> | [{"identifier": "A", "content": "$$\\frac{4}{3}(4 \\sqrt{2}+1)$$"}, {"identifier": "B", "content": "$$\\frac{3}{4}(4 \\sqrt{2}+1)$$"}, {"identifier": "C", "content": "$$\\frac{4}{3}(4 \\sqrt{2}-1)$$"}, {"identifier": "D", "content": "$$\\frac{3}{4}(4 \\sqrt{2}-1)$$"}] | ["C"] | null | <img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lh2jl21d/1eaa8b9f-d025-4054-8bc8-91786b0e9ad1/73b7e010-e6db-11ed-948f-4b963ec65c15/file-1lh2jl21e.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lh2jl21d/1eaa8b9f-d025-4054-8bc8-91786b0e9ad1/73b7e010-e6db-11ed-948f-4b963ec65c15/fi... | mcq | jee-main-2023-online-13th-april-evening-shift | 4,936 |
1lgpxipld | maths | area-under-the-curves | area-bounded-between-the-curves | <p>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</p> | [{"identifier": "A", "content": "$$2 \\sqrt{2}(\\sqrt{2}+1)$$"}, {"identifier": "B", "content": "4"}, {"identifier": "C", "content": "$$2(\\sqrt{2}+1)$$"}, {"identifier": "D", "content": "$$4(\\sqrt{2})$$"}] | ["B"] | null | <img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lh3urauz/df812b2b-41d3-412b-9523-6e99a63e5a07/ee7558b0-e793-11ed-9b1f-65ec4fb7911b/file-1lh3urav0.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lh3urauz/df812b2b-41d3-412b-9523-6e99a63e5a07/ee7558b0-e793-11ed-9b1f-65ec4fb7911b/fi... | mcq | jee-main-2023-online-13th-april-morning-shift | 4,937 |
1lgrgc23k | maths | area-under-the-curves | area-bounded-between-the-curves | <p>The area of the region enclosed by the curve $$y=x^{3}$$ and its tangent at the point $$(-1,-1)$$ is :</p> | [{"identifier": "A", "content": "$$\\frac{23}{4}$$"}, {"identifier": "B", "content": "$$\\frac{19}{4}$$"}, {"identifier": "C", "content": "$$\\frac{27}{4}$$"}, {"identifier": "D", "content": "$$\\frac{31}{4}$$<br/><br/>"}] | ["C"] | null | <img src="https://app-content.cdn.examgoal.net/fly/@width/image/1li62ldz6/e3e1ec7b-7f4a-4711-90a5-23ea1380cf4d/5bb8a710-fc98-11ed-bb91-ad55392bec91/file-1li62ldz7.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1li62ldz6/e3e1ec7b-7f4a-4711-90a5-23ea1380cf4d/5bb8a710-fc98-11ed-bb91-ad55392bec91/fi... | mcq | jee-main-2023-online-12th-april-morning-shift | 4,938 |
1lgswa4aj | maths | area-under-the-curves | area-bounded-between-the-curves | <p>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 _________.</p> | [] | null | 16 | <img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lie8i8gh/9671b86e-ee06-496d-abb6-609a695b7ace/c0004c10-0115-11ee-9f57-5de63d0f488b/file-1lie8i8gi.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lie8i8gh/9671b86e-ee06-496d-abb6-609a695b7ace/c0004c10-0115-11ee-9f57-5de63d0f488b/fi... | integer | jee-main-2023-online-11th-april-evening-shift | 4,939 |
1lgxw9wy2 | maths | area-under-the-curves | area-bounded-between-the-curves | <p>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 ___________.</p> | [] | null | 16 | Let, $y=p(x)$ be the parabola passing through the points $(-1,0)(0,1)(1,0)$.
<br><br>Now, to find the area of the region
<br><br>$$
\left\{(x, y) ;(x+1)^2+(y-1)^2 \leq 1, y \leq p(x)\right\}
$$
<br><br><img src="https://app-content.cdn.examgoal.net/fly/@width/image/6y3zli1lnda9au3/d6943506-1c0a-4dc0-87a7-f279a28202cc/e... | integer | jee-main-2023-online-10th-april-morning-shift | 4,942 |
1lgyom5cr | maths | area-under-the-curves | area-bounded-between-the-curves | <p>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 $$12 \mathrm{~A}$$ is _____________.</p> | [] | null | 17 | <img src="https://app-content.cdn.examgoal.net/fly/@width/image/6y3zli1lmyue4ph/cd17397d-b57f-4fb7-a489-8f746007324f/e1368850-5b9b-11ee-83da-a3f80d422da4/file-6y3zli1lmyue4pi.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/6y3zli1lmyue4ph/cd17397d-b57f-4fb7-a489-8f746007324f/e1368850-5b9b-11ee-83... | integer | jee-main-2023-online-8th-april-evening-shift | 4,943 |
1lh23quiq | maths | area-under-the-curves | area-bounded-between-the-curves | <p>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 ___________.</p> | [] | null | 5 | Given region,
<br><br>$$
S=\left\{(x, y): 2 y-y^2 \leq x^2 \leq 2 y, x \geq y\right\}
$$
<br><br>Here, we have three curves
<br><br>$$
\begin{aligned}
&2 y-y^2 =x^2 ..........(i)\\\\
&x^2 =2 y ..........(2)\\\\
&\text {and}~~ x = y ...........(3)
\end{aligned}
$$
<br><br><img src="https://app-content.cd... | integer | jee-main-2023-online-6th-april-morning-shift | 4,945 |
1lh2xt7m5 | maths | area-under-the-curves | area-bounded-between-the-curves | <p>The area bounded by the curves $$y=|x-1|+|x-2|$$ and $$y=3$$ is equal to :</p> | [{"identifier": "A", "content": "5"}, {"identifier": "B", "content": "4"}, {"identifier": "C", "content": "6"}, {"identifier": "D", "content": "3"}] | ["B"] | null | Given equation of curve $y=|x-1|+|x-2|$ and $y=3$
<br><br><img src="https://app-content.cdn.examgoal.net/fly/@width/image/6y3zli1lo8c5sm6/ed381416-4cb6-4a32-b174-452035f058cf/a2f26cd0-74a0-11ee-8723-4d48ce392782/file-6y3zli1lo8c5sm7.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/6y3zli1lo8c5sm6/... | mcq | jee-main-2023-online-6th-april-evening-shift | 4,946 |
lsan0p30 | maths | area-under-the-curves | area-bounded-between-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 bounded by the line $\mathrm{PQ}$ and the parabola, and $\mathrm{S}_2$ be the ar... | [] | null | 7 | <img src="https://app-content.cdn.examgoal.net/fly/@width/image/6y3zli1lsogvt13/49cb1ddc-aaf9-4143-8cc4-b6d4940befaa/82ba6c70-ccb0-11ee-aa98-13f456b8f7af/file-6y3zli1lsogvt14.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/6y3zli1lsogvt13/49cb1ddc-aaf9-4143-8cc4-b6d4940befaa/82ba6c70-ccb0-11ee-aa... | integer | jee-main-2024-online-1st-february-evening-shift | 4,947 |
lsan1ar0 | maths | area-under-the-curves | area-bounded-between-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 : | [] | null | 8 | Given $k y^2=2(y-x)$ .........(i)
<br/><br/>$$
2 y^2=k x
$$ .........(ii)
<br/><br/>Point of intersection of (i) and (ii)
<br/><br/>$$
\begin{aligned}
& k y^2=2\left(y-\frac{2 y^2}{k}\right) \\\\
& \Rightarrow y=0, k y=2\left(1-\frac{2 y}{k}\right)
\end{aligned}
$$
<br/><br/>$\begin{aligned} & k y+\frac{4 y}{k}=2 \\\\... | integer | jee-main-2024-online-1st-february-evening-shift | 4,948 |
lsaonhys | maths | area-under-the-curves | area-bounded-between-the-curves | The area enclosed by the curves $x y+4 y=16$ and $x+y=6$ is equal to : | [{"identifier": "A", "content": "$28-30 \\log _{\\mathrm{e}} 2$"}, {"identifier": "B", "content": "$30-28 \\log _{\\mathrm{e}} 2$"}, {"identifier": "C", "content": "$30-32 \\log _{\\mathrm{e}} 2$"}, {"identifier": "D", "content": "$32-30 \\log _{\\mathrm{e}} 2$"}] | ["C"] | null | <p>To find the enclosed area between the two curves $ x y+4 y=16 $ and $ x+y=6 $, we need to determine the region of intersection and integrate the difference of the functions over the interval where they intersect.</p>
<p>First, let's solve the equations simultaneously to find the points of intersection.</p>
<p>The ... | mcq | jee-main-2024-online-1st-february-morning-shift | 4,949 |
lsblh7jy | maths | area-under-the-curves | area-bounded-between-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 $\mathrm{m}+\mathrm{n}$ is equal to _____________. | [] | null | 119 | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lt1g6xtj/f303b300-000e-4c93-be80-f4117fb2f938/ffa44560-d3d3-11ee-a874-bd0f61840084/file-1lt1g6xtk.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lt1g6xtj/f303b300-000e-4c93-be80-f4117fb2f938/ffa44560-d3d3-11ee-a874-bd0f61840084... | integer | jee-main-2024-online-27th-january-morning-shift | 4,950 |
jaoe38c1lscojzx5 | maths | area-under-the-curves | area-bounded-between-the-curves | <p>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 ________.</p> | [] | null | 304 | <p>We have</p>
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lt1wj8og/d813b308-3b51-4315-93b6-2504383585b1/e7ce7b00-d413-11ee-b9d5-0585032231f0/file-1lt1wj8oh.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lt1wj8og/d813b308-3b51-4315-93b6-2504383585b1/e7ce7b00-d413-11ee-b9... | integer | jee-main-2024-online-27th-january-evening-shift | 4,951 |
jaoe38c1lsd4h6fi | maths | area-under-the-curves | area-bounded-between-the-curves | <p>The area of the region enclosed by the parabolas $$y=4 x-x^2$$ and $$3 y=(x-4)^2$$ is equal to :</p> | [{"identifier": "A", "content": "$$\\frac{32}{9}$$\n"}, {"identifier": "B", "content": "$$\\frac{14}{3}$$"}, {"identifier": "C", "content": "4"}, {"identifier": "D", "content": "6"}] | ["D"] | null | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/6y3zli1lsjwhl5f/5e904568-fdcb-45b9-b0dc-a84be5bf8837/9e21fa30-ca2d-11ee-8854-3b5a6c9e9092/file-6y3zli1lsjwhl5g.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/6y3zli1lsjwhl5f/5e904568-fdcb-45b9-b0dc-a84be5bf8837/9e21fa30-ca2d-11ee... | mcq | jee-main-2024-online-31st-january-evening-shift | 4,952 |
jaoe38c1lse5f37p | maths | area-under-the-curves | area-bounded-between-the-curves | <p>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</p> | [{"identifier": "A", "content": "$$\\frac{32}{3}$$\n"}, {"identifier": "B", "content": "$$\\frac{16}{3}$$\n"}, {"identifier": "C", "content": "$$\\frac{8}{3}$$\n"}, {"identifier": "D", "content": "$$\\frac{64}{3}$$"}] | ["A"] | null | <p>$$\begin{aligned}
& y^2 \leq 4 x, x<4 \\
& \frac{x y(x-1)(x-2)}{(x-3)(x-4)}>0 \\
& \text { Case - I}: y>0 \\
& \frac{x(x-1)(x-2)}{(x-3)(x-4)}>0 \\
& x \in(0,1) \cup(2,3) \\
& \text { Case }- \text { II : y<0 } \\
& \frac{x(x-1)(x-2)}{(x-3)(x-4)}<0, x \in(1,2) \cup(3,4)
\... | mcq | jee-main-2024-online-31st-january-morning-shift | 4,953 |
jaoe38c1lsf0no1u | maths | area-under-the-curves | area-bounded-between-the-curves | <p>If the points of intersection of two distinct conics $$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}$$ times the area of the rectangle formed by the intersection points is _________.</p> | [] | null | 432 | <p>Putting $$y^2=3 x^2$$ in both the conics</p>
<p>We get $$x^2=b$$ and $$\frac{b}{16}+\frac{3}{b}=1$$</p>
<p>$$\Rightarrow \mathrm{b}=4,12 \quad(\mathrm{b}=4$$ is rejected because curves coincide)</p>
<p>$$\therefore \mathrm{b}=12$$</p>
<p>Hence points of intersection are</p>
<p>$$( \pm \sqrt{12}, \pm 6) \Rightarrow \... | integer | jee-main-2024-online-29th-january-morning-shift | 4,955 |
jaoe38c1lsfl0gg5 | maths | area-under-the-curves | area-bounded-between-the-curves | <p>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 __________.</p> | [] | null | 164 | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/6y3zli1lsr8vkvl/77732020-1aec-484b-950e-aed3c54f53c2/8d586e10-ce37-11ee-9412-cd4f9c6f2c40/file-6y3zli1lsr8vkvm.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/6y3zli1lsr8vkvl/77732020-1aec-484b-950e-aed3c54f53c2/8d586e10-ce37-11ee... | integer | jee-main-2024-online-29th-january-evening-shift | 4,956 |
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