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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.
| 3651. |
\[\underset{x\to 0}{\mathop{\lim }}\,\left[ \frac{1}{x}-\frac{\log (1+x)}{{{x}^{2}}} \right]\]= |
| A. | ½ |
| B. | ?1/2 |
| C. | 1 |
| D. | ?1 |
| Answer» B. ?1/2 | |
| 3652. |
\[\underset{x\to \infty }{\mathop{\lim }}\,[x({{a}^{1/x}}-1)]\],\[(a>1)=\] |
| A. | \[\log x\] |
| B. | 1 |
| C. | 0 |
| D. | \[-\log \frac{1}{a}\] |
| Answer» E. | |
| 3653. |
\[\underset{x\to 0}{\mathop{\lim }}\,x\log (\sin x)=\] |
| A. | ?1 |
| B. | \[{{\log }_{e}}1\] |
| C. | 1 |
| D. | None of these |
| Answer» C. 1 | |
| 3654. |
\[\underset{x\to 0}{\mathop{\lim }}\,\left\{ \frac{\sin x-x+\frac{{{x}^{3}}}{6}}{{{x}^{5}}} \right\}=\] [MNR 1985] |
| A. | 1/120 |
| B. | ?1/120 |
| C. | 1/20 |
| D. | None of these |
| Answer» B. ?1/120 | |
| 3655. |
If \[f(r)=\pi {{r}^{2}}\], then \[\underset{h\to 0}{\mathop{\lim }}\,\frac{f(r+h)-f(r)}{h}=\] |
| A. | \[\pi {{r}^{2}}\] |
| B. | \[2\pi r\] |
| C. | \[2\pi \] |
| D. | \[2\pi {{r}^{2}}\] |
| Answer» C. \[2\pi \] | |
| 3656. |
The value of \[\underset{\theta \to 0}{\mathop{\lim }}\,\left( \frac{\sin \frac{\theta }{4}}{\theta } \right)\] is [MP PET 1993] |
| A. | 0 |
| B. | 1/4 |
| C. | 1 |
| D. | Not in existence |
| Answer» C. 1 | |
| 3657. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin 2x+\sin 6x}{\sin 5x-\sin 3x}=\] [AI CBSE 1988; AISSE 1988] |
| A. | ½ |
| B. | 1/4 |
| C. | 2 |
| D. | 4 |
| Answer» E. | |
| 3658. |
\[\underset{x\to 0+}{\mathop{\lim }}\,\frac{x{{e}^{1/x}}}{1+{{e}^{1/x}}}=\] |
| A. | 0 |
| B. | 1 |
| C. | \[\infty \] |
| D. | None of these |
| Answer» B. 1 | |
| 3659. |
\[\underset{x\to 1}{\mathop{\lim }}\,[x]=\] |
| A. | 0 |
| B. | 1 |
| C. | Does not exist |
| D. | None of these |
| Answer» D. None of these | |
| 3660. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{(1+x)}^{5}}-1}{{{(1+x)}^{3}}-1}=\] |
| A. | 0 |
| B. | 1 |
| C. | 5/3 |
| D. | 3/5 |
| Answer» D. 3/5 | |
| 3661. |
\[\underset{x\to \pi /6}{\mathop{\lim }}\,\frac{{{\cot }^{2}}\theta -3}{\text{cosec}\theta -2}=\] |
| A. | 2 |
| B. | 4 |
| C. | 6 |
| D. | 0 |
| Answer» C. 6 | |
| 3662. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{\cos ax-\cos bx}{{{x}^{2}}}=\] [AI CBSE 1988] |
| A. | \[\frac{{{a}^{2}}-{{b}^{2}}}{2}\] |
| B. | \[\frac{{{b}^{2}}-{{a}^{2}}}{2}\] |
| C. | \[{{a}^{2}}-{{b}^{2}}\] |
| D. | \[{{b}^{2}}-{{a}^{2}}\] |
| Answer» C. \[{{a}^{2}}-{{b}^{2}}\] | |
| 3663. |
\[\underset{x\to 1}{\mathop{\lim }}\,\frac{1}{|1-x|}=\] |
| A. | 0 |
| B. | 1 |
| C. | 2 |
| D. | \[\infty \] |
| Answer» E. | |
| 3664. |
If \[f(x)=\left\{ \begin{matrix} \frac{2}{5-x}, & \text{when }x3 \\ \end{matrix} \right.\], then |
| A. | \[\underset{x\to 3+}{\mathop{\lim }}\,f(x)=0\] |
| B. | \[\underset{x\to 3-}{\mathop{\lim }}\,f(x)=0\] |
| C. | \[\underset{x\to 3+}{\mathop{\lim }}\,f(x)\ne \underset{x\to 3-}{\mathop{\lim }}\,f(x)\] |
| D. | None of these |
| Answer» D. None of these | |
| 3665. |
\[\underset{x\to a}{\mathop{\lim }}\,\frac{{{(x+2)}^{5/3}}-{{(a+2)}^{5/3}}}{x-a}=\] [AI CBSE 1991] |
| A. | \[\frac{5}{3}{{(a+2)}^{2/3}}\] |
| B. | \[\frac{5}{3}{{(a+2)}^{5/3}}\] |
| C. | \[\frac{5}{3}{{a}^{2/3}}\] |
| D. | \[\frac{5}{3}{{a}^{5/3}}\] |
| Answer» B. \[\frac{5}{3}{{(a+2)}^{5/3}}\] | |
| 3666. |
\[\underset{x\to a}{\mathop{\lim }}\,\frac{{{x}^{2}}-{{a}^{2}}}{x-a}=\] [RPET 1995] |
| A. | 4a |
| B. | 1 |
| C. | 2a |
| D. | 0 |
| Answer» D. 0 | |
| 3667. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin {{x}^{o}}}{x}=\] |
| A. | 1 |
| B. | \[\pi /180\] |
| C. | Does not exist |
| D. | None of these |
| Answer» C. Does not exist | |
| 3668. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{x}{|x|+{{x}^{2}}}=\] |
| A. | 1 |
| B. | ?1 |
| C. | 0 |
| D. | Does not exist |
| Answer» E. | |
| 3669. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{\tan 2x-x}{3x-\sin x}=\] [IIT 1971] |
| A. | 0 |
| B. | 1 |
| C. | ½ |
| D. | 1/3 |
| Answer» D. 1/3 | |
| 3670. |
\[\underset{x\to \pi /2}{\mathop{\lim }}\,(\sec \theta -\tan \theta )=\] [IIT 1976; AMU 1999] |
| A. | 0 |
| B. | 1/2 |
| C. | 2 |
| D. | \[\infty \] |
| Answer» B. 1/2 | |
| 3671. |
If n is an integer, then \[\underset{x\to n+0}{\mathop{\lim }}\,(x-[n])=\] |
| A. | 0 |
| B. | 1 |
| C. | ?1 |
| D. | None of these |
| Answer» B. 1 | |
| 3672. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{x({{e}^{x}}-1)}{1-\cos x}=\] |
| A. | 0 |
| B. | \[\infty \] |
| C. | ?2 |
| D. | 2 |
| Answer» E. | |
| 3673. |
\[\underset{\alpha \to \pi /4}{\mathop{\lim }}\,\frac{\sin \alpha -\cos \alpha }{\alpha -\frac{\pi }{4}}=\] [IIT 1977] |
| A. | \[\sqrt{2}\] |
| B. | \[1/\sqrt{2}\] |
| C. | 1 |
| D. | None of these |
| Answer» B. \[1/\sqrt{2}\] | |
| 3674. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{2{{\sin }^{2}}3x}{{{x}^{2}}}=\] [Roorkee 1982; DCE 1999] |
| A. | 6 |
| B. | 9 |
| C. | 18 |
| D. | 3 |
| Answer» D. 3 | |
| 3675. |
\[\underset{x\to \infty }{\mathop{\lim }}\,\frac{\sin x}{x}=\] [IIT 1975; MP PET 2004] |
| A. | 1 |
| B. | 0 |
| C. | Does not exist |
| D. | None of these |
| Answer» C. Does not exist | |
| 3676. |
\[\underset{x\to 1}{\mathop{\lim }}\,\frac{x-1}{2{{x}^{2}}-7x+5}=\] [IIT 1976] |
| A. | 1/3 |
| B. | 1/11 |
| C. | ?1/3 |
| D. | None of these |
| Answer» D. None of these | |
| 3677. |
\[\underset{x\to \infty }{\mathop{\lim }}\,\sqrt{x}(\sqrt{x+5}-\sqrt{x})=\] |
| A. | 5 |
| B. | 3 |
| C. | 5/2 |
| D. | 3/2 |
| Answer» D. 3/2 | |
| 3678. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{e}^{\sin x}}-1}{x}=\] |
| A. | 1 |
| B. | e |
| C. | 1/e |
| D. | None of these |
| Answer» B. e | |
| 3679. |
The value of \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\int_{0}^{x}{\cos {{t}^{2}}}}{x}\,dt\] is |
| A. | 0 |
| B. | 1 |
| C. | \[-1\] |
| D. | None of these |
| Answer» C. \[-1\] | |
| 3680. |
The value of \[\underset{n\to \infty }{\mathop{\lim }}\,\frac{1+2+3+....n}{{{n}^{2}}+100}\]is equal [Pb. CET 2002] |
| A. | \[\infty \] |
| B. | \[\frac{1}{2}\] |
| C. | 2 |
| D. | 0 |
| Answer» C. 2 | |
| 3681. |
\[\underset{x\to \infty }{\mathop{\lim }}\,\frac{{{(x+1)}^{10}}+{{(x+2)}^{10}}+.....+{{(x+100)}^{10}}}{{{x}^{10}}+{{10}^{10}}}\] is equal to |
| A. | 0 |
| B. | 1 |
| C. | 10 |
| D. | 100 |
| Answer» E. | |
| 3682. |
If \[{{x}_{n}}=\frac{1-2+3-4+5-6+.....-2n}{\sqrt{{{n}^{2}}+1}+\sqrt{4{{n}^{2}}-1}},\] then \[\underset{n\to \infty }{\mathop{\lim }}\,{{x}_{n}}\] is equal to [AMU 2000] |
| A. | \[\frac{1}{3}\] |
| B. | \[-\frac{2}{3}\] |
| C. | \[\frac{2}{3}\] |
| D. | 1 |
| Answer» C. \[\frac{2}{3}\] | |
| 3683. |
The value of \[\underset{n\,\to \,\infty }{\mathop{\lim }}\,\frac{1-{{n}^{2}}}{\sum n}\] will be [UPSEAT 1999] |
| A. | ? 2 |
| B. | ? 1 |
| C. | 2 |
| D. | 1 |
| Answer» B. ? 1 | |
| 3684. |
\[\underset{n\to \infty }{\mathop{\lim }}\,\left\{ \frac{1}{{{n}^{2}}}+\frac{2}{{{n}^{2}}}+\frac{3}{{{n}^{2}}}+......+\frac{n}{{{n}^{2}}} \right\}\]is [SCRA 1996] |
| A. | ½ |
| B. | 0 |
| C. | 1 |
| D. | \[\infty \] |
| Answer» B. 0 | |
| 3685. |
\[\underset{n\to \infty }{\mathop{\lim }}\,\frac{1}{2}+\frac{1}{{{2}^{2}}}+\frac{1}{{{2}^{3}}}+...+\frac{1}{{{2}^{n}}}\]equals [RPET 1996] |
| A. | 2 |
| B. | ?1 |
| C. | 1 |
| D. | 3 |
| Answer» D. 3 | |
| 3686. |
The value of \[\underset{n\to \infty }{\mathop{\lim }}\,\cos \left( \frac{x}{2} \right)\cos \left( \frac{x}{4} \right)\cos \left( \frac{x}{8} \right)...\cos \left( \frac{x}{{{2}^{n}}} \right)\] is |
| A. | 1 |
| B. | \[\frac{\sin x}{x}\] |
| C. | \[\frac{x}{\sin x}\] |
| D. | None of these |
| Answer» C. \[\frac{x}{\sin x}\] | |
| 3687. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{1-\cos mx}{1-\cos nx}=\] [Kerala (Engg.)2002] |
| A. | \[m/n\] |
| B. | \[n/m\] |
| C. | \[\frac{{{m}^{2}}}{{{n}^{2}}}\] |
| D. | \[\frac{{{n}^{2}}}{{{m}^{2}}}\] |
| Answer» D. \[\frac{{{n}^{2}}}{{{m}^{2}}}\] | |
| 3688. |
If \[{{a}_{1}}=1\] and \[{{a}_{n+1}}=\frac{4+3{{a}_{n}}}{3+2{{a}_{n}}},\ n\ge 1\] and if \[-\frac{1}{3}\], then the value of a is |
| A. | \[\sqrt{2}\] |
| B. | \[-\sqrt{2}\] |
| C. | 2 |
| D. | None of these |
| Answer» B. \[-\sqrt{2}\] | |
| 3689. |
\[\underset{n\to \infty }{\mathop{\lim }}\,\left[ \frac{1}{{{n}^{3}}+1}+\frac{4}{{{n}^{3}}+1}+\frac{9}{{{n}^{3}}+1}+........+\frac{{{n}^{2}}}{{{n}^{3}}+1} \right]=\] |
| A. | \[1\] |
| B. | 2/3 |
| C. | 1/3 |
| D. | \[0\] |
| Answer» D. \[0\] | |
| 3690. |
\[[.]\]is equal to [IIT 1984; DCE 2000; Pb. CET 2000] |
| A. | 0 |
| B. | \[-\frac{1}{2}\] |
| C. | \[\log \left( \frac{2}{3} \right)\] |
| D. | None of these |
| Answer» C. \[\log \left( \frac{2}{3} \right)\] | |
| 3691. |
Let \[f:R\to R\]be a differentiable function having \[f(2)=6,f'(2)=\left( \frac{1}{48} \right).\] Then \[\underset{x\to 2}{\mathop{\lim }}\,\int\limits_{6}^{f(x)}{\frac{4{{t}^{3}}}{x-2}}\]dt equals [AIEEE 2005] |
| A. | 12 |
| B. | 18 |
| C. | 24 |
| D. | 36 |
| Answer» C. 24 | |
| 3692. |
The value of the constant \[\alpha \] and \[\beta \] such that \[\underset{x\to \infty }{\mathop{\lim }}\,\left( \frac{{{x}^{2}}+1}{x+1}-\alpha x-\beta \right)=0\] are respectively [Orissa JEE 2005] |
| A. | (1, 1) |
| B. | (?1, 1) |
| C. | (1, ?1) |
| D. | (0, 1) |
| Answer» D. (0, 1) | |
| 3693. |
The value of \[\underset{n\to \infty }{\mathop{\lim }}\,\frac{1}{1.3}+\frac{1}{3.5}+\frac{1}{5.7}+\frac{1}{7.9}+...+\frac{1}{(2n-1)(2n+1)}\] is equal to [DCE 2005] |
| A. | ½ |
| B. | 1/3 |
| C. | ¼ |
| D. | None of these |
| Answer» B. 1/3 | |
| 3694. |
The value of \[\underset{n\to \infty }{\mathop{\lim }}\,\frac{{{x}^{n}}}{{{x}^{n}}+1}\] where \[x |
| A. | ½ |
| B. | ?1/2 |
| C. | 1 |
| D. | None of these |
| Answer» D. None of these | |
| 3695. |
The value of \[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{27}^{x}}-{{9}^{x}}-{{3}^{x}}+1}{\sqrt{5}-\sqrt{4+\cos x}}\] is [J & K 2005] |
| A. | \[\sqrt{5}{{(\log 3)}^{2}}\] |
| B. | \[8\sqrt{5}\log 3\] |
| C. | \[16\sqrt{5}\log 3\] |
| D. | \[8\sqrt{5}{{(\log 3)}^{2}}\] |
| Answer» E. | |
| 3696. |
\[\underset{\theta \to 0}{\mathop{\lim }}\,\frac{4\theta (\tan \theta -2\theta \tan \theta )}{{{(1-\cos 2\theta )}^{2}}}\]is [Orissa JEE 2005] |
| A. | \[1/\sqrt{2}\] |
| B. | 1/2 |
| C. | 1 |
| D. | 2 |
| Answer» C. 1 | |
| 3697. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{2}^{x}}-1}{{{(1+x)}^{1/2}}-1}=\] [IIT 1983; Karnataka CET 1999] |
| A. | \[\log 2\] |
| B. | \[\log 4\] |
| C. | \[\log \sqrt{2}\] |
| D. | None of these |
| Answer» C. \[\log \sqrt{2}\] | |
| 3698. |
The value of \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\log [1+{{x}^{3}}]}{{{\sin }^{3}}x}=\] [AMU 2005] |
| A. | 0 |
| B. | 1 |
| C. | 3 |
| D. | None of these |
| Answer» C. 3 | |
| 3699. |
If \[\underset{n\to \infty }{\mathop{\lim }}\,\frac{1-{{(10)}^{n}}}{1+{{(10)}^{n+1}}}=\frac{-\alpha }{10}\], then give the value of \[\alpha \] is [Orissa JEE 2005] |
| A. | 0 |
| B. | ?1 |
| C. | 1 |
| D. | 2 |
| Answer» D. 2 | |
| 3700. |
If \[f(x)=\left\{ \begin{align} & \frac{\sin [x]}{[x]},\text{ when }[x]\ne 0 \\ & \,\,\,\,\,\,\,\,\,0,\text{ when }[x]=0 \\ \end{align} \right.\] where [x] is greatest integer function, then \[\underset{x\to 0}{\mathop{\lim }}\,f(x)=\] [IIT 1985; RPET 1995] |
| A. | ?1 |
| B. | 1 |
| C. | 0 |
| D. | None of these |
| Answer» E. | |