Explore topic-wise MCQs in Joint Entrance Exam - Main (JEE Main).

This section includes 8666 Mcqs, each offering curated multiple-choice questions to sharpen your Joint Entrance Exam - Main (JEE Main) knowledge and support exam preparation. Choose a topic below to get started.

8401.

If \[f(x)\] is a continuous periodic function with period \[T,\] then the integral \[I=\int_{a}^{a+T}{f(x)\,dx}\] is

A. Equal to \[2a\]
B. Equal to \[3a\]
C. Independent of \[a\]
D. None of these
Answer» D. None of these
8402.

\[\int_{0}^{a}{f(x)\,dx}=\] [BIT Ranchi 1992]

A. \[\int_{0}^{a}{f(a+x)\,dx}\]
B. \[\int_{0}^{a}{f(2a+x)\,dx}\]
C. \[\int_{0}^{a}{f(x-a)\,dx}\]
D. \[\int_{0}^{a}{f(a-x)\,dx}\]
Answer» E.
8403.

For any integer \[n,\] the integral \[\int_{0}^{\pi }{{{e}^{{{\sin }^{2}}x}}{{\cos }^{3}}(2n+1)x\,dx=}\] [MNR 1982]

A. \[-1\]
B. 0
C. 1
D. \[\pi \]
Answer» C. 1
8404.

If \[f(x)=\left\{ \begin{matrix} 4x+3\,, & \text{if} & 1\le x\le 2 \\ 3x+5\,, & \text{if} & 2

A. 80
B. 20
C. \[-20\]
D. 37
Answer» E.
8405.

\[\int_{\pi /6}^{\pi /3}{\frac{dx}{1+\sqrt{\tan x}}=}\] [Kerala (Engg.) 2005]

A. \[\pi /12\]
B. \[\pi /2\]
C. \[\pi /6\]
D. \[\pi /4\]
Answer» B. \[\pi /2\]
8406.

If \[g(x)=\int_{0}^{x}{{{\cos }^{4}}t\,dt,}\] then \[g(x+\pi )\] equals [IIT 1997 Re-Exam; DCE 2001; UPSEAT 2001; Pb. CET 2002]

A. \[g(x)+g(\pi )\]
B. \[g(x)-g(\pi )\]
C. \[g(x)g(\pi )\]
D. \[g(x)/g(\pi )\]
Answer» B. \[g(x)-g(\pi )\]
8407.

\[\int_{-1}^{1}{\log \frac{2-x}{2+x}\,dx}=\] [Roorkee 1986; Kurukshetra CEE 1998]

A. 2
B. 1
C. \[-1\]
D. 0
Answer» E.
8408.

The area of the loop of the curve \[a{{y}^{2}}={{x}^{2}}(a-x)\] is

A. \[4{{a}^{2}}\]sq. units
B. \[\frac{8{{a}^{2}}}{15}\]sq. units
C. \[\frac{16{{a}^{2}}}{9}\]sq. units
D. None of these
Answer» C. \[\frac{16{{a}^{2}}}{9}\]sq. units
8409.

The area enclosed between the curves \[y=a{{x}^{2}}\] and \[x=a{{y}^{2}}\] (where a > 0) is 1 sq. unit, then the value of a is

A. \[1/\sqrt{3}\]
B. ½
C. 1
D. 44256
Answer» B. ½
8410.

Consider two curves \[{{C}_{1}}:{{y}^{2}}=4[\sqrt{y}]x\] and\[{{C}_{2}}:{{x}^{2}}=4[\sqrt{x}]y\], where [.] denotes the greatest integer function. Then the area of region enclosed by these two curves within the square formed by the lines x =1, y =1, x = 4, y = 4 is

A. 8/3 sq. units
B. 10/3 sq. units
C. 11/3 sq. units
D. 11/4 sq. units
Answer» D. 11/4 sq. units
8411.

\[\int_{0}^{\pi /2}{{{e}^{x}}\sin x\,dx=}\] [Roorkee 1978]

A. \[\frac{1}{2}({{e}^{\pi /2}}-1)\]
B. \[\frac{1}{2}({{e}^{\pi /2}}+1)\]
C. \[\frac{1}{2}(1-{{e}^{\pi /2}})\]
D. \[2({{e}^{\pi /2}}+1)\]
Answer» C. \[\frac{1}{2}(1-{{e}^{\pi /2}})\]
8412.

The area between the parabola \[{{y}^{2}}=4ax\]and \[{{x}^{2}}=8ay\] is [RPET 1997]

A. \[\frac{8}{3}{{a}^{2}}\]
B. \[\frac{4}{3}{{a}^{2}}\]
C. \[\frac{32}{3}{{a}^{2}}\]
D. \[\frac{16}{3}{{a}^{2}}\]
Answer» D. \[\frac{16}{3}{{a}^{2}}\]
8413.

The value of \[\int_{3}^{5}{\frac{{{x}^{2}}}{{{x}^{2}}-4}\,dx}\] is [Roorkee 1992]

A. \[2-{{\log }_{e}}\left( \frac{15}{7} \right)\]
B. \[2+{{\log }_{e}}\left( \frac{15}{7} \right)\]
C. \[2+4{{\log }_{e}}3-4{{\log }_{e}}7+4{{\log }_{e}}5\]
D. \[2-{{\tan }^{-1}}\left( \frac{15}{7} \right)\]
Answer» C. \[2+4{{\log }_{e}}3-4{{\log }_{e}}7+4{{\log }_{e}}5\]
8414.

\[\int_{0}^{\pi /8}{{{\cos }^{3}}4\theta d\theta }=\] [Karnataka CET 2004]

A. \[\frac{2}{3}\]
B. \[\frac{1}{4}\]
C. \[\frac{1}{3}\]
D. \[\frac{1}{6}\]
Answer» E.
8415.

The area bounded by the \[x-\]axis and the curve \[y=\sin x\] and \[x=0,\] \[x=\pi \] is [Kerala (Engg.) 2002]

A. 1
B. 2
C. 3
D. 4
Answer» C. 3
8416.

\[\int_{\,-\,1}^{\,0}{\frac{dx}{{{x}^{2}}+2x+2}=}\] [MP PET 2000]

A. 0
B. \[\pi /4\]
C. \[\pi /2\]
D. \[-\pi /4\]
Answer» C. \[\pi /2\]
8417.

What are the points of intersection of the curve\[4{{x}^{2}}-9{{y}^{2}}=1\] with its conjugate axis?

A. \[(1/2,0)\] and \[(-1/2,0)\]
B. \[(0,2)\] and \[(0,-2)\]
C. \[(0,3)\] and \[(0,-3)\]
D. No such point exists
Answer» E.
8418.

Consider the parabolas \[{{S}_{1}}\equiv {{y}^{2}}-4ax=0\] and \[{{S}_{2}}\equiv {{y}^{2}}-4bx=0.\] \[{{S}_{2}}\] will contain \[{{S}_{1}},\] if

A. \[a>b>0\]
B. \[b>a>0\]
C. \[a>0,\,\,b<0\,\,but\,\,\left| \,b\, \right|>a\]
D. \[a<0,\,\,b>0\,\,but\,\,b>\left| \,a\, \right|\]
Answer» C. \[a>0,\,\,b<0\,\,but\,\,\left| \,b\, \right|>a\]
8419.

A point on the ellipse \[\frac{{{x}^{2}}}{16}+\frac{{{y}^{2}}}{9}=1\] at a distance equal to the mean of the lengths of the semi-major axis and semi-minor axis from the centre is

A. \[\left( \frac{2\sqrt{91}}{7},\frac{3\sqrt{105}}{14} \right)\]
B. \[\left( \frac{2\sqrt{91}}{7}-\frac{3\sqrt{105}}{14} \right)\]
C. \[\left( \frac{2\sqrt{105}}{7},\frac{3\sqrt{91}}{14} \right)\]
D. \[\left( -\frac{2\sqrt{105}}{7}-\frac{3\sqrt{91}}{14} \right)\]
Answer» B. \[\left( \frac{2\sqrt{91}}{7}-\frac{3\sqrt{105}}{14} \right)\]
8420.

Equation of the hyperbola whose directirx is \[2x+y=1\], focus (1, 2) and eccentricity \[\sqrt{3}\] is

A. \[7{{x}^{2}}-2{{y}^{2}}+12xy-2x+14y-22=0\]
B. \[5{{x}^{2}}-2{{y}^{2}}+10xy+2x+5y-20=0\]
C. \[4{{x}^{2}}+8{{y}^{2}}+8xy+2x-2y+10=0\]
D. None of these
Answer» B. \[5{{x}^{2}}-2{{y}^{2}}+10xy+2x+5y-20=0\]
8421.

The sum of the focal distances of a point on the ellipse \[\frac{{{x}^{2}}}{4}+\frac{{{y}^{2}}}{9}=1\] is:

A. 4 units
B. 6 units
C. 8 units
D. 10 units
Answer» B. 6 units
8422.

If the latus rectum of an ellipse is equal to one half its minor axis, what is the eccentricity of the ellipse?

A. \[\frac{1}{2}\]
B. \[\frac{\sqrt{3}}{2}\]
C. \[\frac{3}{4}\]
D. \[\frac{\sqrt{15}}{4}\]
Answer» C. \[\frac{3}{4}\]
8423.

The curve represented by\[x=2(cost+sint),,y=5(cost-sint)\] is

A. A circle
B. A parabola
C. An ellipse
D. A hyperbola
Answer» D. A hyperbola
8424.

The equation of the circle which touches the axes at a distance 5 from the origin is \[{{y}^{2}}+{{x}^{2}}-2ax-2ay+{{a}^{2}}=0.\] what is the value of\[\alpha \]?

A. 4
B. 5
C. 6
D. 7
Answer» C. 6
8425.

If two circles A, B of equal radii pass through the centres of each other, then what is the ratio of the length of the smaller are to the circumference of the circle A cut off by the circle B?

A. \[\frac{1}{2}\]
B. \[\frac{1}{4}\]
C. \[\frac{1}{3}\]
D. \[\frac{2}{3}\]
Answer» D. \[\frac{2}{3}\]
8426.

If the coordinates of four concyclie points on the rectangular hyperbola \[xy={{c}^{2}}\] are \[(c{{t}_{i}},\,\,c/{{t}_{i}}),i=1,\,\,2,\,\,3,\,\,4\] then

A. \[{{t}_{1}}{{t}_{2}}{{t}_{3}}{{t}_{4}}=-1\]
B. \[{{t}_{1}}{{t}_{2}}{{t}_{3}}{{t}_{4}}=1\]
C. \[{{t}_{1}}{{t}_{3}}={{t}_{2}}{{t}_{4}}\]
D. \[{{t}_{1}}+{{t}_{2}}+{{t}_{3}}+{{t}_{4}}={{c}^{2}}\]
Answer» C. \[{{t}_{1}}{{t}_{3}}={{t}_{2}}{{t}_{4}}\]
8427.

What is the equation to circle which touches both the axes and has centre on the line \[x+y=4?\]

A. \[{{x}^{2}}+{{y}^{2}}-4x+4y+4=0\]
B. \[{{x}^{2}}+{{y}^{2}}-4x-4y+4=0\]
C. \[{{x}^{2}}+{{y}^{2}}+4x-4y-4=0\]
D. \[{{x}^{2}}+{{y}^{2}}+4x+4y-4=0\]
Answer» C. \[{{x}^{2}}+{{y}^{2}}+4x-4y-4=0\]
8428.

An equilateral triangle is inscribed in the circle \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\]with one of the vertices at (a, 0). What is the equation of the side opposite to this vertex?

A. \[2x-a=0\]
B. \[x+a=0\]
C. \[2x+a=0\]
D. \[3x-2a=0\]
Answer» D. \[3x-2a=0\]
8429.

If \[P\equiv (x,y),{{F}_{1}}\equiv (3,0),{{F}_{2}}\equiv (-3,0)\] and \[16{{x}^{2}}+25{{y}^{2}}=400,\] then \[P{{F}_{1}}+P{{F}_{2}}\] equals

A. 8
B. 6
C. 10
D. 12
Answer» D. 12
8430.

Equation of the parabola whose vertex is \[(-3,-2),\] axis is horizontal and which passes through the point \[(1,2)\] is

A. \[{{y}^{2}}+4y+4x-8=0\]
B. \[{{y}^{2}}+4y-4x+8=0\]
C. \[{{y}^{2}}+4y-4x-8=0\]
D. None of these
Answer» D. None of these
8431.

If \[P(\theta )\] and \[Q\left( \frac{\pi }{2}+\theta \right)\] are two points on the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\], then locus of the mid-point of PQ is

A. \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=\frac{1}{2},\]
B. \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=4\]
C. \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=2\]
D. None of these
Answer» B. \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=4\]
8432.

Let \[{{S}_{1}},{{S}_{2}}\] be the foci of the ellipse, \[\frac{{{x}^{2}}}{16}+\frac{{{y}^{2}}}{8}=1\]. If \[A(x+y)\] is any point on the ellipse, then the maximum area of the triangle \[A{{S}_{1}}{{S}_{2}}\] (in square units) is

A. \[2\sqrt{2}\]
B. \[2\sqrt{3}\]
C. 8
D. 4
Answer» D. 4
8433.

The normal at the point \[(b{{t}^{2}}_{1},2b{{t}_{1}})\] on a parabola meets the parabola again in the point \[(b{{t}^{2}}_{2},2b{{t}_{2}})\] Then

A. \[{{t}_{2}}={{t}_{1}}+\frac{2}{{{t}_{1}}}\]
B. \[{{t}_{2}}=-{{t}_{1}}-\frac{2}{{{t}_{1}}}\]
C. \[{{t}_{2}}=-{{t}_{1}}+\frac{2}{{{t}_{1}}}\]
D. \[{{t}_{2}}={{t}_{1}}-\frac{2}{{{t}_{1}}}\]
Answer» C. \[{{t}_{2}}=-{{t}_{1}}+\frac{2}{{{t}_{1}}}\]
8434.

An ellipse has OB as semi minor axis, F and F? its foci and the angle FBF? is a right angle. Then the eccentricity of the ellipse is

A. \[\frac{1}{\sqrt{2}}\]
B. \[\frac{1}{2}\]
C. \[\frac{1}{4}\]
D. \[\frac{1}{\sqrt{3}}\]
Answer» B. \[\frac{1}{2}\]
8435.

A point moves such that the square of its distance from a straight line is equal to the difference between the square of it distance from the centre of a circle and the square of the radius of the circle. The locus of the point is

A. A straight line at right angle to the given line
B. A circle concentric with the given circle
C. A parabola with its axis parallel to the given line
D. A parabola with its axis perpendicular to the given line
Answer» E.
8436.

The equation of the ellipse with its centre at \[(1,2),\] focus at (6,2) and passing through the point \[(4,6)\]is \[\frac{{{(x-1)}^{2}}}{{{a}^{2}}}+\frac{{{(y-2)}^{2}}}{{{b}^{2}}}=1\], then

A. \[{{a}^{2}}=1,{{b}^{2}}=25\]
B. \[{{a}^{2}}=25,{{b}^{2}}=20\]
C. \[{{a}^{2}}=20,{{b}^{2}}=25\]
D. None of these
Answer» E.
8437.

A line PQ meets the parabola \[{{y}^{2}}-4ax\] in R such that PQ is bisected at R. if the coordinates of P are \[({{x}_{1}},{{y}_{1}})\] then the locus of Q is the parabola

A. \[{{(y+{{y}_{1}})}^{2}}=8a(x+{{x}_{1}})\]
B. \[{{(y-{{y}_{1}})}^{2}}=8a(x+{{x}_{1}})\]
C. \[{{(y+{{y}_{1}})}^{2}}=8a(x-{{x}_{1}})\]
D. None of these
Answer» B. \[{{(y-{{y}_{1}})}^{2}}=8a(x+{{x}_{1}})\]
8438.

The limiting points of the coaxial system determined by the circles \[{{x}^{2}}+{{y}^{2}}-2x-6y+9=0\] and \[{{x}^{2}}+{{y}^{2}}+6x-2y+1=0\]

A. \[(-1,2),\left( \frac{3}{5},\frac{-14}{5} \right)\]
B. \[(-1,2),\left( \frac{3}{5},\frac{14}{5} \right)\]
C. \[(-1,2),\left( \frac{-3}{5},\frac{14}{5} \right)\]
D. None of these
Answer» C. \[(-1,2),\left( \frac{-3}{5},\frac{14}{5} \right)\]
8439.

Consider a circle of radius R. what is the length of a chord which subtends an angle \[\theta \] at the centre?

A. \[2R\sin \left( \frac{\theta }{2} \right)\]
B. \[2R\sin \theta \]
C. \[2R\tan \left( \frac{\theta }{2} \right)\]
D. \[2R\tan \theta \]
Answer» B. \[2R\sin \theta \]
8440.

If the eccentricity of the hyperbola \[{{x}^{2}}-{{y}^{2}}{{\sec }^{2}}\theta =4\] is \[\sqrt{3}\] times the eccentricity of the ellipse \[{{x}^{2}}{{\sec }^{2}}\theta +{{y}^{2}}=16.\] then the value of \[\theta \] equals.

A. \[\frac{\pi }{6}\]
B. \[\frac{3\pi }{4}\]
C. \[\frac{\pi }{3}\]
D. \[\frac{\pi }{2}\]
Answer» C. \[\frac{\pi }{3}\]
8441.

If tangents are drawn to the parabola \[{{y}^{2}}=4ax\]at points whose abscissae are in the ratio \[{{m}^{2}}:1,\] then the locus of their point of intersection is the curve \[\left( m>0 \right)\]

A. \[{{y}^{2}}={{({{m}^{1/2}}-{{m}^{-1/2}})}^{2}}ax\]
B. \[{{y}^{2}}={{({{m}^{1/2}}+{{m}^{-1/2}})}^{2}}ax\]
C. \[{{y}^{2}}={{({{m}^{1/2}}+{{m}^{-1/2}})}^{2}}x\]
D. None of these
Answer» C. \[{{y}^{2}}={{({{m}^{1/2}}+{{m}^{-1/2}})}^{2}}x\]
8442.

The eccentricity of the hyperbola whose latus rectum is 8 and conjugate axis is equal to half the distance between the foci is:

A. \[\frac{4}{3}\]
B. \[\frac{4}{\sqrt{3}}\]
C. \[\frac{2}{\sqrt{3}}\]
D. None of these
Answer» D. None of these
8443.

If the equation of the common tangent at the point \[(1,-1)\] to the two circles, each of radius 13, is \[12x+5y-7=0\] then the centres of the two circles are

A. \[(13,4),(-11,6)\]
B. \[(13,4),(-11,-6)\]
C. \[(13,-4),(-11,-6)\]
D. \[(-13,4),(-11,-6)\]
Answer» C. \[(13,-4),(-11,-6)\]
8444.

Let d be the perpendicular distance from the centre of the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] to the tangent drawn at a point P on the ellipse. If \[{{F}_{1}}\] and \[{{F}_{2}}\] be the foci of the ellipse, then \[{{(P{{F}_{1}}-P{{F}_{2}})}^{2}}=\]

A. \[4{{a}^{2}}\left( 1-\frac{{{b}^{2}}}{{{d}^{2}}} \right)\]
B. \[{{a}^{2}}\left( 1-\frac{{{b}^{2}}}{{{d}^{2}}} \right)\]
C. \[4{{a}^{2}}\left( 1-\frac{{{a}^{2}}}{{{d}^{2}}} \right)\]
D. \[{{b}^{2}}\left( 1-\frac{{{a}^{2}}}{{{d}^{2}}} \right)\]
Answer» B. \[{{a}^{2}}\left( 1-\frac{{{b}^{2}}}{{{d}^{2}}} \right)\]
8445.

The number of points (a, b) where a and b are positive integers lying on the hyperbola \[{{x}^{2}}-{{y}^{2}}=512\] is

A. 3
B. 4
C. 5
D. 6
Answer» C. 5
8446.

If from any point P, tangents PT, PT? are drawn to two given circles with centres A and B respectively; and if PN is the perpendicular form P on their radical axis, then \[P{{T}^{2}}-PT{{'}^{2}}=\]

A. PN.AB
B. \[2PN.AB\]
C. \[4PN.AB\]
D. None of these
Answer» C. \[4PN.AB\]
8447.

The line passing through the extremity A of the major axis and the extremity B of the minor axis of the ellipse \[{{x}^{2}}+9{{y}^{2}}=9\] meets its auxiliary circle at the point M. Then the area of the triangle with vertices A, M. and the origin O is

A. \[\frac{31}{10}\]
B. \[\frac{29}{10}\]
C. \[\frac{21}{10}\]
D. \[\frac{27}{10}\]
Answer» E.
8448.

If polar of a circle \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\] with respect to \[(x',y')\] is \[Ax+By+C=0,\] then its pole will be

A. \[\left( \frac{{{a}^{2}}A}{-C},\frac{{{a}^{2}}B}{-C} \right)\]
B. \[\left( \frac{{{a}^{2}}A}{C},\frac{{{a}^{2}}B}{C} \right)\]
C. \[\left( \frac{{{a}^{2}}C}{A},\frac{{{a}^{2}}C}{B} \right)\]
D. \[\left( \frac{{{a}^{2}}C}{-A},\frac{{{a}^{2}}C}{-B} \right)\]
Answer» B. \[\left( \frac{{{a}^{2}}A}{C},\frac{{{a}^{2}}B}{C} \right)\]
8449.

Equation of the latus rectum of the hyperbola\[{{(10x-5)}^{2}}+{{(10y-2)}^{2}}=9{{(3x+4y-7)}^{2}}\] is

A. \[y-\frac{1}{5}=-\frac{3}{4}\left( x-\frac{1}{2} \right)\]
B. \[x-\frac{1}{5}=-\frac{3}{4}\left( y-\frac{1}{2} \right)\]
C. \[y+\frac{1}{5}=-\frac{3}{4}\left( x+\frac{1}{2} \right)\]
D. \[x+\frac{1}{5}=-\frac{3}{4}\left( y+\frac{1}{2} \right)\]
Answer» B. \[x-\frac{1}{5}=-\frac{3}{4}\left( y-\frac{1}{2} \right)\]
8450.

The length of the chord \[x+y=3\] intercepted by the circle \[{{x}^{2}}+{{y}^{2}}-2x-2y-2=0\] is

A. \[\frac{7}{2}\]
B. \[\frac{3\sqrt{3}}{2}\]
C. \[\sqrt{14}\]
D. \[\frac{\sqrt{7}}{2}\]
Answer» D. \[\frac{\sqrt{7}}{2}\]