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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.
| 3701. |
The value of \[\underset{x\to \infty }{\mathop{\lim }}\,\frac{(x+1)(3x+4)}{{{x}^{2}}(x-8)}\] is equal to [Pb. CET 2002] |
| A. | 2 |
| B. | 3 |
| C. | 1 |
| D. | 0 |
| Answer» E. | |
| 3702. |
The value of \[\underset{x\to 0}{\mathop{\lim }}\,\frac{2}{x}\log (1+x)\] is equal to [Pb. CET 2000] |
| A. | e |
| B. | \[{{e}^{2}}\] |
| C. | \[\frac{1}{2}\] |
| D. | 2 |
| Answer» E. | |
| 3703. |
The value of \[\underset{x\to -1}{\mathop{\lim }}\,\frac{{{x}^{2}}+3x+2}{{{x}^{2}}+4x+3}\]is equal to [Pb. CET 2000] |
| A. | 0 |
| B. | 1 |
| C. | 2 |
| D. | ½ |
| Answer» E. | |
| 3704. |
\[\underset{x\to 0}{\mathop{\lim }}\,\left[ \frac{{{e}^{x}}-{{e}^{\sin x}}}{x-\sin x} \right]\]is equal to [UPSEAT 2004] |
| A. | ?1 |
| B. | 0 |
| C. | 1 |
| D. | None of these |
| Answer» D. None of these | |
| 3705. |
\[\underset{x\to \infty }{\mathop{\lim }}\,{{\left( 1-\frac{4}{x-1} \right)}^{3x-1}}=\] [Karnataka CET 2004] |
| A. | \[{{e}^{12}}\] |
| B. | \[{{e}^{-12}}\] |
| C. | \[{{e}^{4}}\] |
| D. | \[{{e}^{3}}\] |
| Answer» C. \[{{e}^{4}}\] | |
| 3706. |
\[\underset{\theta \to \frac{\pi }{2}}{\mathop{\lim }}\,\frac{\frac{\pi }{2}-\theta }{\cot \theta }\]= [Karnataka CET 2004] |
| A. | 0 |
| B. | ?1 |
| C. | 1 |
| D. | \[\infty \] |
| Answer» D. \[\infty \] | |
| 3707. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{x}^{3}}\cot x}{1-\cos x}=\] [AI CBSE 1988; DSSE 1988] |
| A. | 0 |
| B. | 1 |
| C. | 2 |
| D. | ?2 |
| Answer» D. ?2 | |
| 3708. |
\[\underset{h\to 0}{\mathop{\lim }}\,\frac{\sqrt{x+h}-\sqrt{x}}{h}=\] [Roorkee 1983] |
| A. | \[\frac{1}{2\sqrt{x}}\] |
| B. | \[\frac{1}{\sqrt{x}}\] |
| C. | \[2\sqrt{x}\] |
| D. | \[\sqrt{x}\] |
| Answer» B. \[\frac{1}{\sqrt{x}}\] | |
| 3709. |
If \[\underset{x\to \infty }{\mathop{\lim }}\,{{\left( 1+\frac{a}{x}+\frac{b}{{{x}^{2}}} \right)}^{2x}}={{e}^{2}},\]then the values of a and b are [AIEEE 2004] |
| A. | \[a=1,\ b=2\] |
| B. | \[\cos (|x|)\,-|x|\] |
| C. | \[a\in R,\ b=2\] |
| D. | \[a\in R,\ b\in R\] |
| Answer» C. \[a\in R,\ b=2\] | |
| 3710. |
\[\underset{n\to \infty }{\mathop{\lim }}\,{{({{3}^{n}}+{{4}^{n}})}^{\frac{1}{n}}}=\] [Karnataka CET 2003] |
| A. | 3 |
| B. | 4 |
| C. | \[\infty \] |
| D. | e |
| Answer» C. \[\infty \] | |
| 3711. |
. \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\cos (\sin x)-1}{{{x}^{2}}}=\] [Orissa JEE 2003] |
| A. | 1 |
| B. | ? 1 |
| C. | ½ |
| D. | ?1/2 |
| Answer» E. | |
| 3712. |
\[\underset{x\to 0}{\mathop{\lim }}\,\,\frac{{{e}^{x}}-{{e}^{-x}}}{\sin x}\] is [Kurukshetra CEE 2002] |
| A. | 0 |
| B. | 1 |
| C. | 2 |
| D. | Non existent |
| Answer» D. Non existent | |
| 3713. |
Given that\[f'\](2)=6 and \[{f}'(1)=4)=\], then \[\underset{h\to 0}{\mathop{\lim }}\,\frac{f(2h+2+{{h}^{2}})-f(2)}{f(h-{{h}^{2}}+1)-f(1)}=\] [IIT Screening 2003] |
| A. | Does not exist |
| B. | Is equal to ? 3/2 |
| C. | Is equal to 3/2 |
| D. | Is equal to 3 |
| Answer» E. | |
| 3714. |
If \[\underset{x\to 0}{\mathop{\lim }}\,\frac{[(a-n)\,nx-\tan x]\sin nx}{{{x}^{2}}}=0,\] where n is non zero real number, then a is equal to [IIT Screening 2003] |
| A. | 0 |
| B. | \[\frac{n+1}{n}\] |
| C. | n |
| D. | \[n+\frac{1}{n}\] |
| Answer» E. | |
| 3715. |
If \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\log (3+x)\,-\log (3-x)}{x}=k,\,\] then the value of k is [AIEEE 2003] |
| A. | 0 |
| B. | \[-\frac{1}{3}\] |
| C. | \[\frac{2}{3}\] |
| D. | \[-\frac{2}{3}\] |
| Answer» D. \[-\frac{2}{3}\] | |
| 3716. |
\[\underset{x\to 0}{\mathop{\lim }}\,{{(1-ax)}^{\frac{1}{x}}}=\] [Karnataka CET 2003] |
| A. | e |
| B. | \[{{e}^{-a}}\] |
| C. | 1 |
| D. | \[{{e}^{a}}\] |
| Answer» C. 1 | |
| 3717. |
\[\underset{x\to \infty }{\mathop{\lim }}\,{{\left( \frac{x+3}{x+1} \right)}^{x+1}}=\] [RPET 2003, UPSEAT 2003] |
| A. | \[{{e}^{2}}\] |
| B. | \[{{e}^{3}}\] |
| C. | e |
| D. | \[{{e}^{-1}}\] |
| Answer» B. \[{{e}^{3}}\] | |
| 3718. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{|x|}{x}=\] [Roorkee 1982; UPSEAT 2001] |
| A. | 1 |
| B. | ?1 |
| C. | 0 |
| D. | Does not exist |
| Answer» E. | |
| 3719. |
\[\underset{x\to -2}{\mathop{\lim }}\,\frac{{{\sin }^{-1}}(x+2)}{{{x}^{2}}+2x}\] is equal to [Orissa JEE 2002] |
| A. | 0 |
| B. | \[\infty \] |
| C. | ?1/2 |
| D. | None of these |
| Answer» D. None of these | |
| 3720. |
If \[f(x)\,={{\cot }^{-1}}\left( \frac{3x-{{x}^{3}}}{1-3{{x}^{2}}} \right)\] and x\[g(x)={{\cos }^{-1}}\left( \frac{1-{{x}^{2}}}{1+{{x}^{2}}} \right)\], then \[\underset{x\to a}{\mathop{\lim }}\,\frac{f(x)-f(a)}{g(x)\,-g(a)},\] \[0 |
| A. | \[\frac{3}{2(1+{{a}^{2}})}\] |
| B. | \[\frac{3}{2(1+{{x}^{2}})}\] |
| C. | \[\frac{3}{2}\] |
| D. | \[-\frac{3}{2}\] |
| Answer» E. | |
| 3721. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{a}^{x}}-{{b}^{x}}}{{{e}^{x}}-1}\]= [Kerala (Engg.) 2002] |
| A. | \[\log \left( \frac{a}{b} \right)\] |
| B. | \[\log \left( \frac{b}{a} \right)\] |
| C. | \[\log (a\,b)\] |
| D. | \[\log \,(a+\,b)\] |
| Answer» B. \[\log \left( \frac{b}{a} \right)\] | |
| 3722. |
\[\underset{n\to \infty }{\mathop{\lim }}\,\,{{\left( \frac{{{n}^{2}}-n+1}{{{n}^{2}}-n-1} \right)}^{n(n-1)}}=\] [AMU 2002] |
| A. | e |
| B. | \[{{e}^{2}}\] |
| C. | \[{{e}^{-1}}\] |
| D. | 1 |
| Answer» C. \[{{e}^{-1}}\] | |
| 3723. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{4}^{x}}-{{9}^{x}}}{x({{4}^{x}}+{{9}^{x}})}=\] [EAMCET 2002] |
| A. | \[\log \left( \frac{2}{3} \right)\] |
| B. | \[\frac{1}{2}\log \left( \frac{3}{2} \right)\] |
| C. | \[\frac{1}{2}\log \left( \frac{2}{3} \right)\] |
| D. | \[\log \,\left( \frac{3}{2} \right)\] |
| Answer» B. \[\frac{1}{2}\log \left( \frac{3}{2} \right)\] | |
| 3724. |
\[\underset{x\to \infty }{\mathop{\lim }}\,\frac{\log {{x}^{n}}-[x]}{[x]},\,n\in N,\,\]\[\,(\,[x]\] denotes greatest integer less than or equal to x) [AIEEE 2002] |
| A. | Has value ?1 |
| B. | Has value 0 |
| C. | Has value 1 |
| D. | Does not exist |
| Answer» B. Has value 0 | |
| 3725. |
Let \[f(x)=4\] and \[f'(x)=4\], then \[\underset{x\to 2}{\mathop{\lim }}\,\,\frac{xf(2)-2f(x)}{x-2}\] equals [AIEEE 2002] |
| A. | 2 |
| B. | ? 2 |
| C. | ? 4 |
| D. | 3 |
| Answer» D. 3 | |
| 3726. |
\[\underset{x\to 0}{\mathop{\lim }}\,\,\,\cos \frac{1}{x}\] [UPSEAT 2002] |
| A. | Is continuous at \[x=0\] |
| B. | Is differentiable at \[(3,\,\,1)\] |
| C. | Does not exist |
| D. | None of these |
| Answer» D. None of these | |
| 3727. |
If \[f(9)=9\], \[f'(9)=4\], then \[\underset{x\to 9}{\mathop{\lim }}\,\frac{\sqrt{f(x)}-3}{\sqrt{x}-3}=\] [IIT 1988; Karnataka CET 1999] |
| A. | 2 |
| B. | 4 |
| C. | ?2 |
| D. | ?4 |
| Answer» C. ?2 | |
| 3728. |
\[x=1\] [MP PET 2002] |
| A. | \[{{\log }_{e}}3\] |
| B. | 0 |
| C. | 1 |
| D. | \[f(x)\] |
| Answer» E. | |
| 3729. |
If \[f(x)\,=\left| \,\begin{matrix} \sin x & \cos x & \tan x \\ {{x}^{3}} & {{x}^{2}} & x \\ 2x & 1 & 1 \\ \end{matrix}\, \right|\], then \[\underset{x\to 0}{\mathop{\lim }}\,\frac{f(x)}{{{x}^{2}}}\] is [Karnataka CET 2002] |
| A. | 3 |
| B. | ?1 |
| C. | 0 |
| D. | 1 |
| Answer» E. | |
| 3730. |
\[\underset{x\to 3}{\mathop{\lim }}\,\,[x]=\], (where [.] = greatest integer function) [DCE 2002] |
| A. | 2 |
| B. | 3 |
| C. | Does not exist |
| D. | None of these |
| Answer» D. None of these | |
| 3731. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin (\pi {{\cos }^{2}}x)}{{{x}^{2}}}=\] [IIT Screening 2001;UPSEAT 2001; MP PET 2002] |
| A. | \[(-1,1)\] |
| B. | \[\pi \] |
| C. | \[\pi /2\] |
| D. | 1 |
| Answer» C. \[\pi /2\] | |
| 3732. |
\[\underset{x\to \pi /2}{\mathop{\lim }}\,\frac{{{a}^{\cot x}}-{{a}^{\cos x}}}{\cot x-\cos x}=\] [Kerala (Engg.) 2001; J & K 2005] |
| A. | \[\log a\] |
| B. | \[\log 2\] |
| C. | a |
| D. | log x |
| Answer» B. \[\log 2\] | |
| 3733. |
\[\underset{x\to \infty }{\mathop{\lim }}\,\,{{\left( \frac{x+a}{x+b} \right)}^{x+b}}=\] [EAMCET 2001] |
| A. | 1 |
| B. | \[{{e}^{b-a}}\] |
| C. | \[{{e}^{a-b}}\] |
| D. | \[{{e}^{b}}\] |
| Answer» D. \[{{e}^{b}}\] | |
| 3734. |
\[\underset{m\to \infty }{\mathop{\lim }}\,\,{{\left( \cos \frac{x}{m} \right)}^{m}}=\] [AMU 2001] |
| A. | 0 |
| B. | e |
| C. | 1/e |
| D. | 1 |
| Answer» E. | |
| 3735. |
\[\underset{x\to 1}{\mathop{\lim }}\,\frac{1+\cos \pi \,x}{{{\tan }^{2}}\pi \,x}\] is equal to [AMU 2001] |
| A. | 0 |
| B. | 1/2 |
| C. | 1 |
| D. | 2 |
| Answer» C. 1 | |
| 3736. |
\[\underset{x\to 0}{\mathop{\lim }}\,\,\frac{{{(1+x)}^{1/x}}-e}{x}\] equals [UPSEAT 2001] |
| A. | \[\pi /2\] |
| B. | 0 |
| C. | \[2/e\] |
| D. | ?\[e/2\] |
| Answer» E. | |
| 3737. |
\[\underset{\alpha \to \beta }{\mathop{\lim }}\,\left[ \frac{{{\sin }^{2}}\alpha -{{\sin }^{2}}\beta }{{{\alpha }^{2}}-{{\beta }^{2}}} \right]=\] [MP PET 2001] |
| A. | 0 |
| B. | 1 |
| C. | \[\frac{\sin \beta }{\beta }\] |
| D. | \[\frac{\sin 2\beta }{2\beta }\] |
| Answer» E. | |
| 3738. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin 2x}{x}=\] [MNR 1990; UPSEAT 2000] |
| A. | 0 |
| B. | 1 |
| C. | ½ |
| D. | 2 |
| Answer» E. | |
| 3739. |
The value of \[\underset{x\to 0}{\mathop{\lim }}\,\,\left[ \frac{\sqrt{a+x}-\sqrt{a-x}}{x} \right]\] is [Karnataka CET 2001] |
| A. | 1 |
| B. | 0 |
| C. | \[\sqrt{a}\] |
| D. | \[1/\sqrt{a}\] |
| Answer» E. | |
| 3740. |
The value of \[\underset{x\to 0}{\mathop{\lim }}\,\,\left( \frac{{{e}^{x}}-1}{x} \right)\] is [Karnataka CET 2001] |
| A. | 1/2 |
| B. | \[\infty \] |
| C. | 1 |
| D. | 0 |
| Answer» D. 0 | |
| 3741. |
For \[x\in R,\,\,\,\underset{x\to \infty }{\mathop{\lim }}\,\,{{\left( \frac{x-3}{x+2} \right)}^{x}}\] is equal to [IIT Screening 2000] |
| A. | e |
| B. | \[{{e}^{-1}}\] |
| C. | \[{{e}^{-5}}\] |
| D. | \[{{e}^{5}}\] |
| Answer» D. \[{{e}^{5}}\] | |
| 3742. |
\[\underset{x\to 0}{\mathop{\lim }}\,\,\frac{\text{ln}\,(\cos x)}{{{x}^{2}}}\] is equal to [AMU 2000] |
| A. | 0 |
| B. | 1 |
| C. | \[\frac{1}{2}\] |
| D. | \[-\frac{1}{2}\] |
| Answer» E. | |
| 3743. |
The value of \[\underset{x\to 0}{\mathop{\lim }}\,\,\frac{(1-\cos 2x)\sin 5x}{{{x}^{2}}\sin 3x}\] is [MP PET 2000; UPSEAT 2000; Karnataka CET 2002] |
| A. | 10/3 |
| B. | 3/10 |
| C. | 6/5 |
| D. | 5/6 |
| Answer» B. 3/10 | |
| 3744. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{x\tan 2x-2x\tan x}{{{(1-\cos 2x)}^{2}}}\] is [IIT 1999] |
| A. | 2 |
| B. | ?2 |
| C. | \[\frac{1}{2}\] |
| D. | \[-\frac{1}{2}\] |
| Answer» D. \[-\frac{1}{2}\] | |
| 3745. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{\sin }^{-1}}x-{{\tan }^{-1}}x}{{{x}^{3}}}\] is equal to [RPET 2000] |
| A. | 0 |
| B. | 1 |
| C. | ?1 |
| D. | \[1/2\] |
| Answer» E. | |
| 3746. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{a}^{\sin x}}-1}{{{b}^{\sin x}}-1}=\] [Karnataka CET 2000] |
| A. | \[\frac{a}{b}\] |
| B. | \[\frac{b}{a}\] |
| C. | \[\frac{\log a}{\log b}\] |
| D. | \[\frac{\log b}{\log a}\] |
| Answer» D. \[\frac{\log b}{\log a}\] | |
| 3747. |
\[\underset{x\to 1}{\mathop{\lim }}\,\frac{1+\log x-x}{1-2x+{{x}^{2}}}=\] [Karnataka CET 2000; Pb. CET 2001] |
| A. | 1 |
| B. | ?1 |
| C. | 0 |
| D. | \[-\frac{1}{2}\] |
| Answer» E. | |
| 3748. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{\log \cos x}{x}=\] |
| A. | 0 |
| B. | 1 |
| C. | \[\infty \] |
| D. | None of these |
| Answer» B. 1 | |
| 3749. |
If \[f(x)=\left\{ \begin{align} & \sin x,x\ne n\pi ,n\in Z \\ & \,\,\,\,\,\,0,\,\,\text{otherwise} \\ \end{align} \right.\] and \[g(x)=\left\{ \begin{align} & {{x}^{2}}+1,x\ne 0,\,2 \\ & \,\,\,\,\,\,\,\,4,x=0 \\ & \,\,\,\,\,\,\,\,\,5,x=2 \\ \end{align} \right.\] then \[\underset{x\to 0}{\mathop{\lim }}\,g\{f(x)\}=\] [Karnataka CET 2000] |
| A. | 1 |
| B. | 0 |
| C. | \[\frac{1}{2}\] |
| D. | \[\frac{1}{4}\] |
| Answer» B. 0 | |
| 3750. |
If \[f(x)=\left\{ \begin{align} & x\ :\ x0 \\ \end{align} \right.,\]then \[\underset{x\to 0}{\mathop{\lim }}\,f(x)=\] [DCE 2000] |
| A. | 0 |
| B. | 1 |
| C. | 2 |
| D. | Does not exist |
| Answer» E. | |