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MCQOPTIONS
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.
| 3601. |
If line \[\frac{x-{{x}_{1}}}{l}=\frac{y-{{y}_{1}}}{m}=\frac{z-{{z}_{1}}}{n}\] is parallel to the plane \[ax+by+cz+d=0\], then [MNR 1995: MP PET 1995] |
| A. | \[\frac{a}{l}=\frac{b}{m}=\frac{c}{n}\] |
| B. | \[al+bm+cn=0\] |
| C. | \[\frac{a}{l}+\frac{b}{m}+\frac{c}{n}=0\] |
| D. | None of these |
| Answer» C. \[\frac{a}{l}+\frac{b}{m}+\frac{c}{n}=0\] | |
| 3602. |
The line \[\frac{x+3}{3}=\frac{y-2}{-2}=\frac{z+1}{1}\] and the plane \[4x+5y+3z-5=0\] intersect at a point |
| A. | (3, 1, ?2) |
| B. | (3, ? 2, 1) |
| C. | (2, ?1, 3) |
| D. | (?1, ?2, ?3) |
| Answer» C. (2, ?1, 3) | |
| 3603. |
The ratio in which the line joining the points (a, b, c) and (?a, ?c, ?b) is divided by the xy-plane is [MP PET 1994] |
| A. | \[a:b\] |
| B. | \[b:c\] |
| C. | \[c:a\] |
| D. | \[c:b\] |
| Answer» E. | |
| 3604. |
The line drawn from (4, ?1, 2) to the point (?3, 2, 3) meets a plane at right angles at the point (?10, 5, 4), then the equation of plane is [DSSE 1985] |
| A. | \[7x-3y-z+89=0\] |
| B. | \[7x+3y+z+89=0\] |
| C. | \[7x-3y+z+89=0\] |
| D. | None of these |
| Answer» B. \[7x+3y+z+89=0\] | |
| 3605. |
The distance of the point (?1, ?5, ?10) from the point of intersection of the line \[\frac{x-2}{3}=\frac{y+1}{4}=\frac{z-2}{12}\] and the plane \[x-y+z=5\], is [AISSE 1985; DSSE 1984; MP PET 2002] |
| A. | 10 |
| B. | 11 |
| C. | 12 |
| D. | 13 |
| Answer» E. | |
| 3606. |
The point where the line \[\frac{x-1}{2}=\frac{y-2}{-3}=\frac{z+3}{4}\] meets the plane \[2x+4y-z=1\], is [DSSE 1981] |
| A. | (3, ?1, 1) |
| B. | (3, 1, 1) |
| C. | (1, 1, 3) |
| D. | (1, 3, 1) |
| Answer» B. (3, 1, 1) | |
| 3607. |
The line\[\frac{x-2}{3}=\frac{y-3}{4}=\frac{z-4}{5}\]is parallel to the plane [BIT Ranchi 1991; Pb. CET 1991] |
| A. | \[3x+4y+5z=7\] |
| B. | \[2x+y-2z=0\] |
| C. | \[x+y-z=2\] |
| D. | \[2x+3y+4z=0\] |
| Answer» C. \[x+y-z=2\] | |
| 3608. |
The point at which the line joining the points (2, ?3, 1) and (3, ?4, ?5) intersects the plane \[2x+y+z=7\]is [DSSE 1987; MP PET 1991] |
| A. | (1, 2, 7) |
| B. | (1, ?2, 7) |
| C. | (?1, 2, 7) |
| D. | (1, ?2, ?7) |
| Answer» C. (?1, 2, 7) | |
| 3609. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{3\sin x-\sin 3x}{{{x}^{3}}}=\] [AISSE 1985] |
| A. | 4 |
| B. | ?4 |
| C. | \[\frac{1}{4}\] |
| D. | None of these |
| Answer» B. ?4 | |
| 3610. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin mx}{\tan nx}=\] [DSSE 1987] |
| A. | \[\frac{n}{m}\] |
| B. | \[\frac{m}{n}\] |
| C. | \[mn\] |
| D. | None of these |
| Answer» C. \[mn\] | |
| 3611. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{1-\cos 6x}{x}=\] [DSSE 1982] |
| A. | 0 |
| B. | 6 |
| C. | \[\frac{1}{3}\] |
| D. | None of these |
| Answer» B. 6 | |
| 3612. |
\[\underset{x\to \pi /2}{\mathop{\lim }}\,\frac{1+\cos 2x}{{{(\pi -2x)}^{2}}}=\] [DSSE 1986; AI CBSE 1986] |
| A. | \[1\] |
| B. | \[2\] |
| C. | 3 |
| D. | \[\frac{1}{2}\] |
| Answer» E. | |
| 3613. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin 3x+\sin x}{x}\] = [AISSE 1986] |
| A. | \[\frac{1}{3}\] |
| B. | 3 |
| C. | 4 |
| D. | \[\frac{1}{4}\] |
| Answer» D. \[\frac{1}{4}\] | |
| 3614. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{1-\cos x}{{{\sin }^{2}}x}=\] [DSSE 1987] |
| A. | \[\frac{1}{2}\] |
| B. | \[-\frac{1}{2}\] |
| C. | 2 |
| D. | None of these |
| Answer» B. \[-\frac{1}{2}\] | |
| 3615. |
\[\underset{x\to \infty }{\mathop{\lim }}\,\frac{{{(2x+1)}^{40}}{{(4x-1)}^{5}}}{{{(2x+3)}^{45}}}=\] [IIT 1990] |
| A. | 16 |
| B. | 24 |
| C. | 32 |
| D. | 8 |
| Answer» D. 8 | |
| 3616. |
\[\underset{x\to 0}{\mathop{\lim }}\,\left[ \frac{x}{{{\tan }^{-1}}2x} \right]=\] [IIT 1992; RPET 2001] |
| A. | 0 |
| B. | \[\frac{1}{2}\] |
| C. | 1 |
| D. | \[\infty \] |
| Answer» C. 1 | |
| 3617. |
\[\underset{x\to 0}{\mathop{\lim }}\,{{x}^{x}}=\] [Roorkee 1990] |
| A. | 0 |
| B. | 1 |
| C. | e |
| D. | None of these |
| Answer» C. e | |
| 3618. |
\[\underset{x\to 1}{\mathop{\lim }}\,\frac{\log x}{x-1}=\] [RPET 1996; MP PET 1996; Pb. CET 2002] |
| A. | 1 |
| B. | ?1 |
| C. | 0 |
| D. | \[\infty \] |
| Answer» B. ?1 | |
| 3619. |
\[\underset{h\to 0}{\mathop{\lim }}\,\frac{2\left[ \sqrt{3}\sin \left( \frac{\pi }{6}+h \right)-\cos \left( \frac{\pi }{6}+h \right) \right]}{\sqrt{3}h(\sqrt{3}\cos h-\sin h)}=\] [BIT Ranchi 1987] |
| A. | \[-\frac{2}{3}\] |
| B. | \[-\frac{3}{4}\] |
| C. | \[-2\sqrt{3}\] |
| D. | \[\frac{4}{3}\] |
| Answer» E. | |
| 3620. |
\[\underset{x\to a}{\mathop{\lim }}\,\frac{\cos x-\cos a}{\cos x-\cot a}=\] [BIT Ranchi 1987] |
| A. | \[\frac{1}{2}{{\sin }^{3}}a\] |
| B. | \[\frac{1}{2}\text{cose}{{\text{c}}^{2}}a\] |
| C. | \[{{\sin }^{3}}a\] |
| D. | \[\text{cose}{{\text{c}}^{3}}a\] |
| Answer» D. \[\text{cose}{{\text{c}}^{3}}a\] | |
| 3621. |
\[\underset{x\to \pi /4}{\mathop{\lim }}\,\frac{\sqrt{2}\cos x-1}{\cot x-1}=\] [BIT Ranchi 1989; IIT 1990] |
| A. | \[\frac{1}{\sqrt{2}}\] |
| B. | \[\frac{1}{2}\] |
| C. | \[\frac{1}{2\sqrt{2}}\] |
| D. | 1 |
| Answer» C. \[\frac{1}{2\sqrt{2}}\] | |
| 3622. |
\[\underset{x\to \infty }{\mathop{\lim }}\,\frac{3{{x}^{2}}+2x-1}{2{{x}^{2}}-3x-3}=\] |
| A. | 1 |
| B. | 3 |
| C. | \[\frac{3}{2}\] |
| D. | \[-\frac{3}{2}\] |
| Answer» D. \[-\frac{3}{2}\] | |
| 3623. |
\[\underset{x\to \infty }{\mathop{\lim }}\,\frac{2{{x}^{2}}-3x+1}{{{x}^{2}}-1}=\] |
| A. | 1 |
| B. | 2 |
| C. | ?2 |
| D. | None of these |
| Answer» C. ?2 | |
| 3624. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin (2+x)-\sin (2-x)}{x}=\] [AI CBSE 1983; AISSE 1982, 83] |
| A. | \[\sin 2\] |
| B. | \[2\sin 2\] |
| C. | \[2\cos 2\] |
| D. | 2 |
| Answer» D. 2 | |
| 3625. |
\[\underset{\theta \to 0}{\mathop{\lim }}\,\frac{5\theta \cos \theta -2\sin \theta }{3\theta +\tan \theta }=\] [AI CBSE 1988] |
| A. | \[\frac{3}{4}\] |
| B. | \[-\frac{3}{4}\] |
| C. | 0 |
| D. | None of these |
| Answer» B. \[-\frac{3}{4}\] | |
| 3626. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{x}^{2}}-\tan 2x}{\tan x}=\] [AI CBSE 1990] |
| A. | 2 |
| B. | ?2 |
| C. | 0 |
| D. | None of these |
| Answer» C. 0 | |
| 3627. |
\[\underset{x\to 1}{\mathop{\lim }}\,\frac{1-\sqrt{x}}{{{({{\cos }^{-1}}x)}^{2}}}=\] [AI CBSE 1990] |
| A. | 1 |
| B. | \[\frac{1}{2}\] |
| C. | \[\frac{1}{4}\] |
| D. | Put \[{{\cos }^{-1}}x=y\] and \[x\to 1\,\Rightarrow \,\,y\to 0.\] \[\underset{x\to 1}{\mathop{\lim }}\,\,\frac{1-\sqrt{x}}{{{({{\cos }^{-1}}x)}^{2}}}=\underset{y\to 0}{\mathop{\lim }}\,\,\frac{1-\sqrt{\cos y}}{{{y}^{2}}}\] Now rationalizing it, we get \[\underset{y\to 0}{\mathop{\lim }}\,\,\frac{(1-\cos y)}{{{y}^{2}}(1+\sqrt{\cos y})}\] \[=\underset{y\to 0}{\mathop{\lim }}\,\,\frac{1-\cos y}{{{y}^{2}}}\,.\,\underset{y\to 0}{\mathop{\lim }}\,\,\frac{1}{1+\sqrt{\cos y}}=\frac{1}{2}\times \frac{1}{2}=\frac{1}{4}.\] |
| Answer» E. | |
| 3628. |
If \[f(x)=\left\{ \begin{align} & \,\,\,\,\,\,\,x,\ \text{when }0\le x\le 1 \\ & 2-x,\ \text{when }1 |
| A. | 1 |
| B. | 2 |
| C. | 0 |
| D. | Does not exist |
| Answer» B. 2 | |
| 3629. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{1-\cos x}{x}=\] [AI CBSE 1987; AISSE 1987] |
| A. | 0 |
| B. | \[\frac{1}{2}\] |
| C. | \[\frac{1}{3}\] |
| D. | None of these |
| Answer» B. \[\frac{1}{2}\] | |
| 3630. |
\[\underset{\theta \to 0}{\mathop{\lim }}\,\frac{\sin 3\theta -\sin \theta }{\sin \theta }=\] [AI CBSE 1984; DSSE 1984] |
| A. | 1 |
| B. | 2 |
| C. | 1/3 |
| D. | 3/2 |
| Answer» C. 1/3 | |
| 3631. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{\tan x-\sin x}{{{x}^{3}}}=\] [IIT 1974; AI CBSE 1986, 90; AISSE 1983, 86, 90; RPET 2000] |
| A. | \[\frac{1}{2}\] |
| B. | \[-\frac{1}{2}\] |
| C. | \[\frac{2}{3}\] |
| D. | None of these |
| Answer» B. \[-\frac{1}{2}\] | |
| 3632. |
\[\underset{\theta \to 0}{\mathop{\lim }}\,\frac{1-\cos \theta }{{{\theta }^{2}}}=\] [AI CBSE 1981, 91; DSSE 1981, 83] |
| A. | 1 |
| B. | 2 |
| C. | \[\frac{1}{2}\] |
| D. | \[\frac{1}{4}\] |
| Answer» D. \[\frac{1}{4}\] | |
| 3633. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{x{{.2}^{x}}-x}{1-\cos x}=\] [IIT 1980; BIT Ranchi 1983; RPET 2001] |
| A. | 0 |
| B. | \[\log 4\] |
| C. | \[\log 2\] |
| D. | None of these |
| Answer» C. \[\log 2\] | |
| 3634. |
\[\underset{y\to 0}{\mathop{\lim }}\,\frac{(x+y)\sec (x+y)-x\sec x}{y}=\] [AI CBSE 1990] |
| A. | \[\sec x(x\tan x+1)\] |
| B. | \[x\tan x+\sec x\] |
| C. | \[x\sec x+\tan x\] |
| D. | None of these |
| Answer» B. \[x\tan x+\sec x\] | |
| 3635. |
\[\underset{x\to 0}{\mathop{\lim }}\,\left( \frac{\tan 3x}{x}+\cos x \right)=\] |
| A. | 3 |
| B. | 1 |
| C. | 4 |
| D. | 2 |
| Answer» D. 2 | |
| 3636. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{(1+x)}^{n}}-1}{x}=\] [Kurukshetra CEE 2002] |
| A. | n |
| B. | 1 |
| C. | ?1 |
| D. | None of these |
| Answer» B. 1 | |
| 3637. |
\[\underset{x\to 1}{\mathop{\lim }}\,\frac{1-{{x}^{-1/3}}}{1-{{x}^{-2/3}}}=\] [AI CBSE 1991] |
| A. | \[\frac{1}{3}\] |
| B. | \[\frac{1}{2}\] |
| C. | \[\frac{2}{3}\] |
| D. | \[-\frac{2}{3}\] |
| Answer» C. \[\frac{2}{3}\] | |
| 3638. |
\[\underset{x\to a}{\mathop{\lim }}\,\frac{\sqrt{3x-a}-\sqrt{x+a}}{x-a}=\] |
| A. | \[\sqrt{2}a\] |
| B. | \[1/\sqrt{2a}\] |
| C. | 2a |
| D. | \[1/2a\] |
| Answer» C. 2a | |
| 3639. |
\[\underset{x\to 1}{\mathop{\lim }}\,\frac{{{x}^{3}}-1}{{{x}^{2}}+5x-6}=\] |
| A. | 0 |
| B. | \[\frac{3}{7}\] |
| C. | \[\frac{1}{2}\] |
| D. | \[-\frac{1}{6}\] |
| Answer» C. \[\frac{1}{2}\] | |
| 3640. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{(1+x)}^{1/2}}-{{(1-x)}^{1/2}}}{x}=\] [Roorkee 1979; RPET 1996] |
| A. | 0 |
| B. | 1/2 |
| C. | 1 |
| D. | ?1 |
| Answer» D. ?1 | |
| 3641. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{y}^{2}}}{x}=........\], where \[{{y}^{2}}=ax+b{{x}^{2}}+c{{x}^{3}}\] |
| A. | 0 |
| B. | 1 |
| C. | a |
| D. | None of these |
| Answer» D. None of these | |
| 3642. |
\[\underset{x\to \infty }{\mathop{\lim }}\,\left[ \frac{{{1}^{3}}+{{2}^{3}}+{{3}^{3}}+.......+{{n}^{3}}}{{{n}^{4}}} \right]=\] |
| A. | \[\frac{1}{2}\] |
| B. | \[\frac{1}{3}\] |
| C. | \[\frac{1}{4}\] |
| D. | None of these |
| Answer» D. None of these | |
| 3643. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{x\cos x-\sin x}{{{x}^{2}}\sin x}=\] [MNR 1984,86] |
| A. | \[\frac{1}{3}\] |
| B. | \[-\frac{1}{3}\] |
| C. | 1 |
| D. | None of these |
| Answer» C. 1 | |
| 3644. |
\[\underset{x\to 3}{\mathop{\lim }}\,\left\{ \frac{x-3}{\sqrt{x-2}-\sqrt{4-x}} \right\}=\] [MNR 1991] |
| A. | 1 |
| B. | 2 |
| C. | ?1 |
| D. | ?2 |
| Answer» B. 2 | |
| 3645. |
\[\underset{h\to 0}{\mathop{\lim }}\,\frac{{{(a+h)}^{2}}\sin (a+h)-{{a}^{2}}\sin a}{h}=\] [IIT 1989] |
| A. | \[a\cos a+{{a}^{2}}\sin a\] |
| B. | \[a\sin a+{{a}^{2}}\cos a\] |
| C. | \[2a\sin a+{{a}^{2}}\cos a\] |
| D. | \[2a\cos a+{{a}^{2}}\sin a\] |
| Answer» D. \[2a\cos a+{{a}^{2}}\sin a\] | |
| 3646. |
\[\underset{n\to \infty }{\mathop{\lim }}\,\frac{\sqrt{n}}{\sqrt{n}+\sqrt{n+1}}=\] |
| A. | 1 |
| B. | 1/2 |
| C. | 0 |
| D. | \[\infty \] |
| Answer» C. 0 | |
| 3647. |
\[\underset{x\to \pi /2}{\mathop{\lim }}\,\frac{2x-\pi }{\cos x}=\] [IIT 1973] |
| A. | 2 |
| B. | 1 |
| C. | ?2 |
| D. | None of these |
| Answer» D. None of these | |
| 3648. |
\[\underset{x\to \infty }{\mathop{\lim }}\,\frac{\sqrt{{{x}^{2}}+{{a}^{2}}}-\sqrt{{{x}^{2}}+{{b}^{2}}}}{\sqrt{{{x}^{2}}+{{c}^{2}}}-\sqrt{{{x}^{2}}+{{d}^{2}}}}=\] |
| A. | \[\frac{{{a}^{2}}-{{b}^{2}}}{{{c}^{2}}-{{d}^{2}}}\] |
| B. | \[\frac{{{a}^{2}}+{{b}^{2}}}{{{c}^{2}}-{{d}^{2}}}\] |
| C. | \[\frac{{{a}^{2}}+{{b}^{2}}}{{{c}^{2}}+{{d}^{2}}}\] |
| D. | None of these |
| Answer» B. \[\frac{{{a}^{2}}+{{b}^{2}}}{{{c}^{2}}-{{d}^{2}}}\] | |
| 3649. |
\[\underset{x\to \alpha }{\mathop{\lim }}\,\frac{\sin x-\sin \alpha }{x-\alpha }=\] |
| A. | 0 |
| B. | 1 |
| C. | \[\sin \alpha \] |
| D. | \[\cos \alpha \] |
| Answer» E. | |
| 3650. |
\[\underset{n\to \infty }{\mathop{\lim }}\,\left[ \frac{\Sigma {{n}^{2}}}{{{n}^{3}}} \right]=\] [AMU 1999; RPET 1999, 2002] |
| A. | \[-\frac{1}{6}\] |
| B. | \[\frac{1}{6}\] |
| C. | \[\frac{1}{3}\] |
| D. | \[-\frac{1}{3}\] |
| Answer» D. \[-\frac{1}{3}\] | |