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.

2801.

If \[u=\sqrt{{{a}^{2}}{{\cos }^{2}}\theta +{{b}^{2}}{{\sin }^{2}}\theta }+\sqrt{{{a}^{2}}{{\sin }^{2}}\theta +{{b}^{2}}{{\cos }^{2}}\theta }\]then the difference between the maximum and minimum values of \[{{u}^{2}}\]is given by

A. \[2({{a}^{2}}+{{b}^{2}})\]  
B. \[2\sqrt{{{a}^{2}}+{{b}^{2}}}\]
C. \[{{(a+b)}^{2}}\]
D. \[{{(a-b)}^{2}}\]
Answer» E.
2802.

The least positive solution of \[\cot \left( \frac{\pi }{3\sqrt{3}}\sin 2x \right)=\sqrt{3}\]lies in

A. \[\left( 0,\frac{\pi }{6} \right]\]     
B. \[\left( \frac{\pi }{9},\frac{\pi }{6} \right)\]
C. \[\left( \frac{\pi }{12},\frac{\pi }{9} \right]\]         
D. \[\left( \frac{\pi }{3},\frac{\pi }{2} \right]\]
Answer» B. \[\left( \frac{\pi }{9},\frac{\pi }{6} \right)\]
2803.

The number of solution of equation \[6\cos 2\theta +2{{\cos }^{2}}(\theta /2)+2si{{n}^{2}}\theta =0,\]\[-\pi

A. 3                     
B. 4
C. 5                     
D. 6
Answer» B. 4
2804.

Let \[0

A. 6\[\pi \]             
B. 7\[\pi \]
C. 8\[\pi \]                         
D. 4\[\pi \]
Answer» B. 7\[\pi \]
2805.

The most general value for which tan\[\theta \]=-1\[\cos \theta =\frac{1}{\sqrt{2}}\]is (n\[\in \]z)

A. \[n\pi =\frac{7\pi }{4}\] 
B. \[n\pi +{{(-1)}^{n}}\frac{7\pi }{4}\]
C. \[2n\pi {{+}^{}}\frac{7\pi }{4}\]        
D. none of these
Answer» D. none of these
2806.

The number of solutions of \[\sin x+\sin 2x+\sin 3x=\cos x+\cos 2x+\cos 3x,\]\[0\le x\le 2\pi \], is

A. 7                     
B. 5
C. 4                     
D. 6
Answer» E.
2807.

If \[{{\tan }^{2}}\theta =2{{\tan }^{2}}\phi +1\],then \[\cos 2\theta +{{\sin }^{2}}\phi \]equals

A. -1                    
B. 0
C. 1                     
D. none of these
Answer» C. 1                     
2808.

If \[\theta \]is eliminated from the equations \[x=a\,\cos (\theta -\alpha )\]and \[y=b\,\cos (\theta -\beta )\], then \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}-\frac{2xy}{ab}\cos (\alpha -\beta )\]is equal to

A. \[{{\sec }^{2}}(\alpha -\beta )\] 
B. \[\cos e{{c}^{2}}(\alpha -\beta )\]
C. \[{{\cos }^{2}}(-\beta )\]          
D. \[si{{n}^{2}}(\alpha -\beta )\]
Answer» E.
2809.

\[\frac{{{\sin }^{2}}A-{{\sin }^{2}}B}{\sin A\cos A-\sin B\cos B}\]is equal to

A. \[\tan (A-B)\]     
B. \[\tan (A+B)\]
C. \[cot(A-B)\]       
D. \[cot(A+B)\]
Answer» C. \[cot(A-B)\]       
2810.

If A and B are acute postitive angles satisfying the equations 3 \[{{\sin }^{2}}A+2{{\sin }^{2}}B=1\]and 3 \[\sin 2A-2\sin 2B=0\]then A+2B is equal to

A. \[\pi \]
B. \[\frac{\pi }{2}\]
C. \[\frac{\pi }{4}\]
D. \[\frac{\pi }{6}\]
Answer» C. \[\frac{\pi }{4}\]
2811.

If \[2\left| \sin 2\alpha  \right|=\left| \tan \beta +\cot \beta  \right|,\alpha ,\beta \in \left( \frac{\pi }{2},\pi  \right)\],then the value of \[\alpha +\beta \]is

A. \[\frac{3\pi }{4}\]                      
B. \[\pi \]
C. \[\frac{3\pi }{2}\]                                  
D. \[\frac{5\pi }{4}\]
Answer» D. \[\frac{5\pi }{4}\]
2812.

The reflection of the point \[\vec{a}\] in the plane \[\vec{r}\].\[\vec{n}\] =q  is

A. \[\vec{a}+\frac{(\vec{q}-\vec{a}\,\cdot \vec{n})}{\left| {\vec{n}} \right|}\]          
B. \[\vec{a}+2\left( \frac{(\vec{q}-\vec{a}\,\cdot \vec{n})}{{{\left| {\vec{n}} \right|}^{2}}} \right)\vec{n}\]
C. \[\vec{a}+\frac{2(\vec{q}-\vec{a}\,\cdot \vec{n})}{\left| {\vec{n}} \right|}\vec{n}\]
D. none of these
Answer» C. \[\vec{a}+\frac{2(\vec{q}-\vec{a}\,\cdot \vec{n})}{\left| {\vec{n}} \right|}\vec{n}\]
2813.

The length of projection of the line segment joining the points \[\left( 1,\text{ }0,\,\,-1 \right)\] and \[\left( -1,\,\,2,\,\,2 \right)\] on the plane \[x+3y-5z=6\] is equal to

A. 2                     
B. \[\sqrt{\frac{271}{53}}\]
C. \[\sqrt{\frac{472}{31}}\]
D. \[\sqrt{\frac{474}{35}}\]
Answer» E.
2814.

Let A (1, 1, 1), B (2, 3, 5) and C (\[-\]1,0, 2) be three points, then equation of a plane parallel to the plane ABC which is at distance 2 is

A. \[2x-3y+z+2\sqrt{14}=0\]
B. \[2x-3y+z-\sqrt{14}=0\]
C. \[2x-3y+z+2=0\]
D. \[2x-3y+z-2=0\]
Answer» B. \[2x-3y+z-\sqrt{14}=0\]
2815.

The intercept made by the plane \[\vec{r}\cdot \vec{n}=q\]on the x-axis is

A. \[\frac{q}{\hat{i}\cdot \vec{n}}\]           
B. \[\frac{\hat{i}\cdot \vec{n}}{q}\]
C. \[\frac{\hat{i}\cdot \vec{n}}{q}\]           
D. \[\frac{q}{\left| {\vec{n}} \right|}\]
Answer» B. \[\frac{\hat{i}\cdot \vec{n}}{q}\]
2816.

Line \[\vec{r}=\vec{a}+\lambda \vec{b}\] will not meet the plane \[\vec{r}\cdot \vec{n}=q\], if

A. \[\vec{b}\cdot \vec{n}=0,\,\,\vec{a}\cdot \vec{n}=q\]
B. \[\vec{b}\cdot \vec{n}\ne 0,\,\,\vec{a}\cdot \vec{n}\ne q\]
C. \[\vec{b}\cdot \vec{n}=0,\,\,\vec{a}\cdot \vec{n}\ne q\]   
D. \[\vec{b}\cdot \vec{n}\ne 0,\,\,\vec{a}\cdot \vec{n}=q\]
Answer» D. \[\vec{b}\cdot \vec{n}\ne 0,\,\,\vec{a}\cdot \vec{n}=q\]
2817.

The direction ratios (d,r,'s) of the normal to the plane throuth (1, 0, 0) and (0, 1, 0) which makes an angle \[\pi /4\]with the plane \[x+y=3\]are

A. \[1,\,\sqrt{2,}\,1\]          
B. \[1,\,\,1,\,\,\sqrt{2}\]
C. \[1,\,\,1\,,\,\,2\]  
D. \[\sqrt{2,}\,1,\,1\]
Answer» C. \[1,\,\,1\,,\,\,2\]  
2818.

The coordinates of the foot of the perpendicular drawn from the origin to the line joining the points (\[-\]9, 4, 5) and (10, 0, \[-\]1) will be

A. (-3, 2, 1)           
B. (1, 2, 2)
C. (4, 5, 3)            
D. none of these
Answer» E.
2819.

What is the equation of the plane which passes through the z-axis and is perpendicular to the line\[\frac{x-a}{\cos \theta }=\frac{y+2}{\sin \theta }=\frac{z-3}{0}\]?

A. \[x+y\,\text{tan}\,\theta =0\]
B. \[y+x\,\text{tan}\,\theta =0\]
C. \[x\,\cos \theta -y\,\sin \theta =0\]
D. \[x\,\sin \theta -y\,\cos \theta =0\]
Answer» B. \[y+x\,\text{tan}\,\theta =0\]
2820.

The pair of lines whose direction cosines are given by the equations \[3l+m+5n=0\] and \[6mn-2nl+5lm=0\] are

A. parallel 
B. perpendicular
C. inclined at \[{{\cot }^{-1}}\left( \frac{1}{6} \right)\]
D. none of these
Answer» D. none of these
2821.

The coordinates of the point p on the line\[\vec{r}=(\hat{i}+\hat{j}+\hat{k})+\lambda (-\hat{i}+\hat{j}-\hat{k})\] which is nearest to the origin is

A. \[\left( \frac{2}{3},\frac{4}{3},\frac{2}{3} \right)\]
B. \[\left( -\frac{2}{3},-\frac{4}{3},\frac{2}{3} \right)\]
C. \[\left( \frac{2}{3},\frac{4}{3},-\frac{2}{3} \right)\]          
D. None of these
Answer» B. \[\left( -\frac{2}{3},-\frac{4}{3},\frac{2}{3} \right)\]
2822.

The projection of the line \[\frac{x+1}{1}=\frac{y}{2}=\frac{z-1}{3}\]on the plane \[x-2y+z=6\] is the line of intersection of the plane with the plane

A. \[2x+y+2=0\]
B. \[3x+y-z=2\]
C. \[2x-3y+8z=3\]
D. none of these
Answer» B. \[3x+y-z=2\]
2823.

The plane which passes through the point (3, 2, 0) and the line\[\frac{x-3}{1}=\frac{y-6}{5}=\frac{z-4}{4}\]is

A. \[x-y+z=1\]      
B. \[x+y+z=5\]
C. \[x+2y-z=1\]
D. \[2x-y+z=5\]
Answer» B. \[x+y+z=5\]
2824.

The plane \[\vec{r}\,\cdot \vec{n}=q\] will contain the line \[\vec{r}=\vec{a}+\lambda \vec{b}\], If

A. \[\vec{b}\cdot \vec{n}\ne 0\], \[\vec{a}\,\cdot \vec{n}\ne q\]
B. \[\vec{b}\,\cdot \vec{n}=0\],\[\vec{a}\cdot \vec{n}\ne q\]
C. \[\vec{b}\cdot \vec{n}=0\],\[\vec{a}\,\cdot \vec{n}=q\]
D. \[\vec{b}\,\cdot \vec{n}\ne 0\],\[\vec{a}\,\cdot \vec{n}=q\]
Answer» D. \[\vec{b}\,\cdot \vec{n}\ne 0\],\[\vec{a}\,\cdot \vec{n}=q\]
2825.

The radius of the circle in which the sphere \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}+2z-2y-4z-19=0\] is cut by the plane \[x+2y+2z+7=0\]is

A. 2                     
B. 3
C. 4         
D. 1
Answer» C. 4         
2826.

The centre of the circle given by\[\vec{r}\cdot (\hat{i}+2\hat{j}+2\hat{k})=15\] and \[\left| \vec{r}-(\hat{j}+2\hat{k}) \right|=4\]is

A. (0, 1, 2)
B. (1, 3, 4)
C. (-1, 3, 4)           
D. none of these
Answer» C. (-1, 3, 4)           
2827.

The length of the perpendicular drawn from (1, 2, 3) to the line\[\frac{x-6}{3}=\frac{y-7}{2}=\frac{z-7}{-2}\]is

A. 4                     
B. 5
C. 6         
D. 7
Answer» E.
2828.

For the line\[\frac{x-1}{1}=\frac{y-2}{2}=\frac{z-3}{3}\], which of the following is incorrect?

A. It lines in the plane \[x-2y+z=0\]
B. It is same as line\[\frac{x}{1}=\frac{y}{2}=\frac{z}{3}\].
C. It passes through (2, 3, 5).
D. It is parallel of the plane \[x-2y+z-6=0\]
Answer» D. It is parallel of the plane \[x-2y+z-6=0\]
2829.

If the straight lines \[2x+3y-1=0,\text{ }x+2y-1=0,~\text{ }x+2y-1=0,\text{ }and\text{ }ax+by-1=0\] form a triangle with the origin as orthocenter, then (a, b) is given by

A. (6, 4)    
B. (-3, 3)
C. (-8, 8)  
D. (0, 7)
Answer» D. (0, 7)
2830.

The line\[\frac{x}{a}+\frac{y}{b}=1\]meets the x-axis at A, the y-axis at B, and the line y=x at C such that the area of \[\Delta AOC\]is twice the area of \[\Delta BOC\]. Then the coordinates of C are

A. \[\left( \frac{b}{3},\frac{b}{3} \right)\]   
B. \[\left( \frac{2a}{3},\frac{2a}{3} \right)\]
C. \[\left( \frac{2b}{3},\frac{2b}{3} \right)\]
D. none of these
Answer» D. none of these
2831.

A line of fixed length 2 units moves so that its ends are on the positive x-axis and that part of the line x+y=0 which lies in the second quadrant. Then the locus of the midpoint of the line has equation

A.  \[{{x}^{2}}+5{{y}^{2}}+4xy-1=0\]
B.  \[{{x}^{2}}+5{{y}^{2}}+4xy+1=0\]
C.  \[{{x}^{2}}+5{{y}^{2}}-4xy-1=0\]
D.  \[4{{x}^{2}}+5{{y}^{2}}+4xy+1=0\]
Answer» B.  \[{{x}^{2}}+5{{y}^{2}}+4xy+1=0\]
2832.

If \[u={{a}_{1}}x+{{b}_{1}}y={{c}_{1}}=0\], \[v={{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0\]and \[{{a}_{1}}/{{a}_{2}}={{b}_{1}}/{{b}_{2}}={{c}_{1}}/{{c}_{2}}\], then the curve \[u+kv=0\]is

A. the same straight line u
B.  different straight line
C.  not a straight line
D.  none of these
Answer» B.  different straight line
2833.

If each of the points, \[({{x}_{1}},4)\], \[(-2,{{y}_{1}})\]lies on the line joining the points (2, -1) and (5, -3), then the point \[p({{x}_{1}},{{y}_{1}})\]lies on the line

A. \[6(x+y)-25=0\]
B. \[2x+6y+1=0\]
C. \[2x+3y-6=0\]
D. \[6(x+y)+25=0\]
Answer» C. \[2x+3y-6=0\]
2834.

If \[(a,{{a}^{2}})\]falls inside the angle made by the lines y=\[\frac{x}{2}\], x>0, and \[y=3x,\text{ }x>0,\]then a belongs to

A. \[\left( 0,\frac{1}{2} \right)\]      
B. \[\left( 3,\,\infty  \right)\]
C. \[\left( \frac{1}{2},3 \right)\]      
D. \[\left( -3,-\frac{1}{2} \right)\]
Answer» D. \[\left( -3,-\frac{1}{2} \right)\]
2835.

The lines y\[y={{m}_{1}}x,\] \[y={{m}_{2}}x,\]and \[y={{m}_{3}}x,\]make equal intercepts on the line\[x+y=1\]. Then

A.  \[2(1+{{m}_{1}})(1+{{m}_{3}})=(1+{{m}_{2}})(2+{{m}_{1}}+{{m}_{3}})\]
B.  \[(1+{{m}_{1}})(1+{{m}_{3}})=(1+{{m}_{2}})(1+{{m}_{1}}+{{m}_{3}})\]
C.  \[(1+{{m}_{1}})(1+{{m}_{2}})=(1+{{m}_{3}})(2+{{m}_{1}}+{{m}_{3}})\]
D.  \[2(1+{{m}_{1}})(1+{{m}_{3}})=(1+{{m}_{2}})(1+{{m}_{1}}+{{m}_{3}})\]
Answer» B.  \[(1+{{m}_{1}})(1+{{m}_{3}})=(1+{{m}_{2}})(1+{{m}_{1}}+{{m}_{3}})\]
2836.

The line parallel to the x-axis and passing through the intersection of the lines \[ax+2by+3b=0\]and \[bx-2ay-3a=0,\]where (a, b) \[\ne \](0, 0) is

A. below the x-axis at a distance of 3/2 from it.
B.  below the x-axis at a distance of 2/3 from it.
C.  above the x-axis at a distance of 2/3 form it.
D.  above the x-axis at a distance of 2/3 from it.
Answer» B.  below the x-axis at a distance of 2/3 from it.
2837.

Let A, (2,-3) and B (-2, 1) be the vertices of a triangle ABC. If the centroid of this triangle moves on the line \[2x+3y=1,\] then the locus of the vertex C is the line

A. \[2x+3y=9\]    
B. \[2x-3y=7\]
C. \[3x+2y=5\]    
D. \[3x-2y=3\]
Answer» B. \[2x-3y=7\]
2838.

The equation of the straight line passing through the point (4, 3) and making intercepts on the coordinate axes, whose sum is-1, is

A.  \[\frac{x}{2}+\frac{y}{3}=-1\] and \[\frac{x}{-2}+\frac{y}{1}=-1\]
B.  \[\frac{x}{2}-\frac{y}{3}=-1\] and \[\frac{x}{-2}+\frac{y}{1}=-1\]
C.  \[\frac{x}{2}+\frac{y}{3}=1\] and \[\frac{x}{2}+\frac{y}{3}=1\]
D.  \[\frac{x}{2}-\frac{y}{3}=1\] and \[\frac{x}{-2}+\frac{y}{1}=1\]
Answer» E.
2839.

If the equation of the locus of a point equidistant from the points (\[{{a}_{1,}}{{b}_{1}}\]) and (\[{{a}_{2,}}{{b}_{2}}\]) is (\[{{a}_{1,-}}{{a}_{2}}\])x+(\[{{b}_{1,-}}{{b}_{2}}\])\[y+c=0\] then the value of c is

A. \[\frac{1}{2}({{a}_{2}}^{2}+{{b}_{2}}^{2}-{{a}_{1}}^{2}-{{b}_{1}}^{2})\]
B. \[{{a}_{1}}^{2}+{{a}_{2}}^{2}+{{b}_{1}}^{2}-{{b}_{2}}^{2}\]
C. \[\frac{1}{2}({{a}_{1}}^{2}+{{a}_{2}}^{2}-{{b}_{1}}^{2}-{{b}_{2}}^{2})\]
D. \[\sqrt{{{a}_{1}}^{2}+{{b}_{2}}^{2}-{{a}_{2}}^{2}-{{b}_{2}}^{2}}\]
Answer» B. \[{{a}_{1}}^{2}+{{a}_{2}}^{2}+{{b}_{1}}^{2}-{{b}_{2}}^{2}\]
2840.

Line \[ax+by+p=0\]makes angle \[\pi /4\] with \[xcos\,\alpha +ysin\,\alpha =p,\text{ }p\in {{R}^{+}}\]. if these lines and the line \[x\text{ }sin\,\alpha -y\text{ }cos\,\alpha =0\] are concurrent, then

A. \[{{a}^{2}}+{{b}^{2}}=1\]
B. \[{{a}^{2}}+{{b}^{2}}=2\]
C. \[2({{a}^{2}}+{{b}^{2}})=1\]
D. none of these
Answer» C. \[2({{a}^{2}}+{{b}^{2}})=1\]
2841.

If the quadrilateral formed by the lines \[ax+by+c=0,\text{ }a'x+b'y+c'=0,\text{ }ax+by+c'=0,\text{ }a'x+b'y+c'=0\]has perpendicular diagonals, then

A.  \[{{b}^{2}}+{{c}^{2}}=b{{'}^{2}}+c{{'}^{2}}\]
B.  \[{{c}^{2}}+{{a}^{2}}=c{{'}^{2}}+a{{'}^{2}}\]
C.  \[{{a}^{2}}+{{b}^{2}}=a{{'}^{2}}+b{{'}^{2}}\]
D.  none of these
Answer» D.  none of these
2842.

A light ray coming along the line \[3x+4y=5\] gets reflected from the line \[ax+by=1\]and goes along the line\[5x-12y=10\]. Then

A.  \[a=64/115,\text{ }b=112/15\]
B.  \[a=14/15,\text{ }b=-8/115\]
C.  \[a=64/115,\text{ }b=-8/115\]
D.  \[a=64/15,\text{ }b=14/15\]
Answer» D.  \[a=64/15,\text{ }b=14/15\]
2843.

The foot of the perpendicular on the line 3x+y=\[\lambda \]drawn from the origin is C. if the line cuts the x-and the y-axis at A and B, respectively, then BC:CA is

A. 1 : 3                 
B. 0.125694444444444
C. 1 : 3                
D. 0.375694444444444
Answer» E.
2844.

The straight line \[ax+by+c=0\], where \[abc\ne 0\], will pass through the first quadrant if

A.  \[ac>0,bc>0\]
B.  \[c>0\,and\,bc<0\]
C.  \[bc>0\,and/or\,ac>0\]
D.  \[ac<0\,and/or\,bc<0\]
Answer» E.
2845.

If each observation of a raw data whose variance is \[\sigma \]of the new set is

A. \[{{\sigma }^{2}}\]                  
B. \[{{h}^{2}}{{\sigma }^{2}}\]
C. \[h\,{{\sigma }^{2}}\]              
D. \[h+{{\sigma }^{2}}\]
Answer» C. \[h\,{{\sigma }^{2}}\]              
2846.

In a moderately skewed distribution, the values of mean and median are 5 and 6, respectively. The value of mode in such a situation is approximately equal to

A. 8                     
B. 11
C. 6                     
D. None of these
Answer» B. 11
2847.

Following are the marks obtained by 9 students in a mathematics test: 50, 69, 20, 33, 39, 40, 65, 59 the mean deviation from the median is

A. 9                     
B. 10.5
C. 12.67
D. 14.76
Answer» D. 14.76
2848.

Runs scored by a batsman in 10 innings are: 38, 70, 48, 34, 42, 55, 63, 46, 54, 44 the mean deviation is

A. 8.6                   
B. 6.4
C. 10.6     
D. 9.6
Answer» B. 6.4
2849.

The harmonic mean of 4, 8, 16 is

A. 6.4                   
B. 6.7
C. 6.85                 
D. 7.8
Answer» D. 7.8
2850.

Computer the medium form the following table    Marks obtained    No. of students 0-10 2 10-20 18 20-30 30 30-40 45 40-50 35 50-60 20 60-70 6 70-80 3

A. 36.55   
B. 35.55
C. 6.85                 
D. None of these
Answer» B. 35.55