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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.
| 3801. |
\[\underset{x\to \infty }{\mathop{\lim }}\,{{\left( \frac{x+2}{x+1} \right)}^{x+3}}\] is [MNR 1994] |
| A. | \[1\] |
| B. | \[e\] |
| C. | \[{{e}^{2}}\] |
| D. | \[{{e}^{3}}\] |
| Answer» C. \[{{e}^{2}}\] | |
| 3802. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{e}^{\alpha \ x}}-{{e}^{\beta \ x}}}{x}=\] [MP PET 1994; DCE 2005] |
| A. | \[\alpha +\beta \] |
| B. | \[\frac{1}{\alpha }+\beta \] |
| C. | \[{{\alpha }^{2}}-{{\beta }^{2}}\] |
| D. | \[\alpha -\beta \] |
| Answer» E. | |
| 3803. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{x}{\tan x}\]is equal to [MNR 1995] |
| A. | 0 |
| B. | 1 |
| C. | 4 |
| D. | Not defined |
| Answer» C. 4 | |
| 3804. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sqrt{\frac{1}{2}(1-\cos 2x)}}{x}=\] [IIT 1991; AIEEE 2002; RPET 2001, 02] |
| A. | 1 |
| B. | ?1 |
| C. | 0 |
| D. | None of these |
| Answer» E. | |
| 3805. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{\log (a+x)-\log a}{x}+k\underset{x\to e}{\mathop{\lim }}\,\frac{\log x-1}{x-e}=1,\]then [IIT Screening] |
| A. | \[k=e\left( 1-\frac{1}{a} \right)\] |
| B. | \[k=e(1+a)\] |
| C. | \[k=e(2-a)\] |
| D. | The equality is not possible |
| Answer» B. \[k=e(1+a)\] | |
| 3806. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sqrt{3+x}-\sqrt{3-x}}{x}=\] [MP PET 1987] |
| A. | ?1 |
| B. | 0 |
| C. | \[\sqrt{3}\] |
| D. | \[\frac{1}{\sqrt{3}}\] |
| Answer» E. | |
| 3807. |
\[\underset{x\to \pi /2}{\mathop{\lim }}\,\frac{\tan 3x}{x}=\] [MP PET 1987] |
| A. | \[\infty \] |
| B. | \[3\] |
| C. | \[\frac{1}{3}\] |
| D. | 0 |
| Answer» B. \[3\] | |
| 3808. |
If \[f(x)=\left\{ \begin{align} & x,\ \ \ \,\,\text{when}\ x>1 \\ & {{x}^{2}},\,\,\,\text{when}\,\,x |
| A. | \[{{x}^{2}}\] |
| B. | \[x\] |
| C. | \[-1\] |
| D. | 1 |
| Answer» E. | |
| 3809. |
If \[\underset{x\to 2}{\mathop{\lim }}\,\frac{{{x}^{n}}-{{2}^{n}}}{x-2}=80\], where n is a positive integer, then \[n=\] |
| A. | 3 |
| B. | 5 |
| C. | 2 |
| D. | None of these |
| Answer» C. 2 | |
| 3810. |
If \[f(x)=\left\{ \begin{align} & x\sin \frac{1}{x},\ \ \ \ \ x\ne 0 \\ & \ \ \ \ \ \ 0,\ \ \ \ \ x=0 \\ \end{align} \right.\], then \[\underset{x\to 0}{\mathop{\lim }}\,f(x)=\] [IIT 1988; MNR 1988; SCRA 1996; UPSEAT 2000, 01] |
| A. | 1 |
| B. | 0 |
| C. | ?1 |
| D. | None of these |
| Answer» C. ?1 | |
| 3811. |
If \[{{\sin }^{-1}}x+{{\cot }^{-1}}\left( \frac{1}{2} \right)=\frac{\pi }{2},\]then x is [Roorkee 1999; Karnataka CET 1999] |
| A. | 0 |
| B. | \[\frac{1}{\sqrt{5}}\] |
| C. | \[\frac{2}{\sqrt{5}}\] |
| D. | \[\frac{\sqrt{3}}{2}\] |
| Answer» C. \[\frac{2}{\sqrt{5}}\] | |
| 3812. |
If \[{{\tan }^{-1}}x+2{{\cot }^{-1}}x=\frac{2\pi }{3},\]then x = [Karnataka CET 1999] |
| A. | \[\sqrt{2}\] |
| B. | 3 |
| C. | \[\sqrt{3}\] |
| D. | \[\frac{\sqrt{3}-1}{\sqrt{3}+1}\] |
| Answer» D. \[\frac{\sqrt{3}-1}{\sqrt{3}+1}\] | |
| 3813. |
\[{{\tan }^{-1}}\left( \frac{1}{11} \right)+{{\tan }^{-1}}\left( \frac{2}{12} \right)=\] [DCE 1999] |
| A. | \[{{\tan }^{-1}}\left( \frac{33}{132} \right)\] |
| B. | \[{{\tan }^{-1}}\left( \frac{1}{2} \right)\] |
| C. | \[{{\tan }^{-1}}\left( \frac{132}{33} \right)\] |
| D. | None of these |
| Answer» E. | |
| 3814. |
\[{{\tan }^{-1}}\frac{1}{2}+{{\tan }^{-1}}\frac{1}{3}\]= [MP PET 1997, 2003; UPSEAT 2000; Karnataka CET 2001; Pb. CET 2004] |
| A. | 0 |
| B. | \[\pi /4\] |
| C. | \[\pi /2\] |
| D. | \[\pi \] |
| Answer» C. \[\pi /2\] | |
| 3815. |
\[{{\sin }^{-1}}x+{{\cos }^{-1}}x\] is equal to [Pb. CET 1997; DCE 2002] |
| A. | \[\frac{\pi }{4}\] |
| B. | \[\frac{\pi }{2}\] |
| C. | -1 |
| D. | 1 |
| Answer» C. -1 | |
| 3816. |
\[{{\sin }^{-1}}\frac{4}{5}+2{{\tan }^{-1}}\frac{1}{3}=\] [ISM Dhanbad 1971] |
| A. | \[\frac{\pi }{2}\] |
| B. | \[\frac{\pi }{3}\] |
| C. | \[\frac{\pi }{4}\] |
| D. | None of these |
| Answer» B. \[\frac{\pi }{3}\] | |
| 3817. |
\[\sin \left\{ {{\sin }^{-1}}\frac{1}{2}+{{\cos }^{-1}}\frac{1}{2} \right\}=\] [EAMCET 1985] |
| A. | 0 |
| B. | -1 |
| C. | 2 |
| D. | 1 |
| Answer» E. | |
| 3818. |
\[{{\tan }^{-1}}\frac{{{c}_{1}}x-y}{{{c}_{1}}y+x}+{{\tan }^{-1}}\frac{{{c}_{2}}-{{c}_{1}}}{1+{{c}_{2}}{{c}_{1}}}+\]\[{{\tan }^{-1}}\frac{{{c}_{3}}-{{c}_{2}}}{1+{{c}_{3}}{{c}_{2}}}+...+{{\tan }^{-1}}\frac{1}{{{c}_{n}}}=\] |
| A. | \[{{\tan }^{-1}}\frac{y}{x}\] |
| B. | \[{{\tan }^{-1}}yx\] |
| C. | \[{{\tan }^{-1}}\frac{x}{y}\] |
| D. | \[{{\tan }^{-1}}(x-y)\] |
| Answer» D. \[{{\tan }^{-1}}(x-y)\] | |
| 3819. |
If \[{{\tan }^{-1}}\frac{x-1}{x+1}+{{\tan }^{-1}}\frac{2x-1}{2x+1}={{\tan }^{-1}}\frac{23}{36},\]then x = [ISM Dhanbad 1973] |
| A. | \[\frac{3}{4},\frac{-3}{8}\] |
| B. | \[\frac{3}{4},\frac{3}{8}\] |
| C. | \[\frac{4}{3},\frac{3}{8}\] |
| D. | None of these |
| Answer» E. | |
| 3820. |
If \[\tan (x+y)=33\]and \[x={{\tan }^{-1}}3,\]then y will be |
| A. | \[0.3\] |
| B. | \[{{\tan }^{-1}}(1.3)\] |
| C. | \[{{\tan }^{-1}}(0.3)\] |
| D. | \[{{\tan }^{-1}}\left( \frac{1}{18} \right)\] |
| Answer» D. \[{{\tan }^{-1}}\left( \frac{1}{18} \right)\] | |
| 3821. |
During \[\cos ({{\tan }^{-1}}x)=\] [MP PET 1988; MNR 1981] |
| A. | \[\sqrt{1+{{x}^{2}}}\] |
| B. | \[\frac{1}{\sqrt{1+{{x}^{2}}}}\] |
| C. | \[1+{{x}^{2}}\] |
| D. | None of these |
| Answer» C. \[1+{{x}^{2}}\] | |
| 3822. |
If \[{{({{\tan }^{-1}}x)}^{2}}+{{({{\cot }^{-1}}x)}^{2}}=\frac{5{{\pi }^{2}}}{8},\]then \[x\] equals |
| A. | -1 |
| B. | 1 |
| C. | 0 |
| D. | None of these |
| Answer» B. 1 | |
| 3823. |
If \[a |
| A. | 0 |
| B. | 1 |
| C. | 2 |
| D. | Infinite |
| Answer» B. 1 | |
| 3824. |
If \[k\le {{\sin }^{-1}}x+{{\cos }^{-1}}x+{{\tan }^{-1}}x\le K,\]then |
| A. | \[k=0,\,K=\pi \] |
| B. | \[k=0,K=\frac{\pi }{2}\] |
| C. | \[k=\frac{\pi }{2},K=\pi \] |
| D. | None of these |
| Answer» B. \[k=0,K=\frac{\pi }{2}\] | |
| 3825. |
The greatest and the least value of \[{{({{\sin }^{-1}}x)}^{3}}+{{({{\cos }^{-1}}x)}^{3}}\]are |
| A. | \[-\frac{\pi }{2},\,\frac{\pi }{2}\] |
| B. | \[-\frac{{{\pi }^{3}}}{8},\,\frac{{{\pi }^{3}}}{8}\] |
| C. | \[\frac{7{{\pi }^{3}}}{8},\,\,\frac{{{\pi }^{3}}}{32}\] |
| D. | None of these |
| Answer» D. None of these | |
| 3826. |
A solution of the equation \[{{\tan }^{-1}}(1+x)\] \[+{{\tan }^{-1}}(1-x)\] \[=\frac{\pi }{2}\] is [Karnataka CET 1993] |
| A. | \[x=1\] |
| B. | \[x=-1\] |
| C. | \[x=0\] |
| D. | \[x=\pi \] |
| Answer» D. \[x=\pi \] | |
| 3827. |
\[{{\sin }^{-1}}\left( \frac{3}{5} \right)+{{\tan }^{-1}}\left( \frac{1}{7} \right)=\] [Karnataka CET 1994] |
| A. | \[\frac{\pi }{4}\] |
| B. | \[\frac{\pi }{2}\] |
| C. | \[{{\cos }^{-1}}\left( \frac{4}{5} \right)\] |
| D. | \[\pi \] |
| Answer» B. \[\frac{\pi }{2}\] | |
| 3828. |
\[{{\cos }^{-1}}\left( \frac{15}{17} \right)+2{{\tan }^{-1}}\left( \frac{1}{5} \right)=\] [EAMCET 1981] |
| A. | \[\frac{\pi }{2}\] |
| B. | \[{{\cos }^{-1}}\left( \frac{171}{221} \right)\] |
| C. | \[\frac{\pi }{4}\] |
| D. | None of these |
| Answer» E. | |
| 3829. |
\[2{{\tan }^{-1}}\left( \frac{1}{3} \right)+{{\tan }^{-1}}\left( \frac{1}{7} \right)=\] [EAMCET 1983] |
| A. | \[{{\tan }^{-1}}\left( \frac{49}{29} \right)\] |
| B. | \[\frac{\pi }{2}\] |
| C. | 0 |
| D. | \[\frac{\pi }{4}\] |
| Answer» E. | |
| 3830. |
If \[{{\sin }^{-1}}\frac{x}{5}+\text{cose}{{\text{c}}^{-1}}\left( \frac{5}{4} \right)=\frac{\pi }{2},\]then \[x=\] [EAMCET 1983; Karnataka CET 2004] |
| A. | 4 |
| B. | 5 |
| C. | 1 |
| D. | 3 |
| Answer» E. | |
| 3831. |
\[{{\tan }^{-1}}\frac{x}{\sqrt{{{a}^{2}}-{{x}^{2}}}}=\] |
| A. | \[\frac{1}{a}{{\sin }^{-1}}\left( \frac{x}{a} \right)\] |
| B. | \[a{{\sin }^{-1}}\left( \frac{x}{a} \right)\] |
| C. | \[{{\sin }^{-1}}\left( \frac{x}{a} \right)\] |
| D. | \[{{\sin }^{-1}}\left( \frac{a}{x} \right)\] |
| Answer» D. \[{{\sin }^{-1}}\left( \frac{a}{x} \right)\] | |
| 3832. |
\[{{\tan }^{-1}}\left( \frac{x}{y} \right)-{{\tan }^{-1}}\,\left( \frac{x-y}{x+y} \right)\] is [EAMCET 1992] |
| A. | \[\frac{\pi }{2}\] |
| B. | \[\frac{\pi }{3}\] |
| C. | \[\frac{\pi }{4}\] |
| D. | \[\frac{\pi }{4}\]or \[-\frac{3\pi }{4}\] |
| Answer» D. \[\frac{\pi }{4}\]or \[-\frac{3\pi }{4}\] | |
| 3833. |
\[{{\tan }^{-1}}\left( \frac{1}{4} \right)+{{\tan }^{-1}}\left( \frac{2}{9} \right)=\] [EAMCET 1994] |
| A. | \[\frac{1}{2}{{\cos }^{-1}}\left( \frac{3}{5} \right)\] |
| B. | \[\frac{1}{2}{{\sin }^{-1}}\left( \frac{3}{5} \right)\] |
| C. | \[\frac{1}{2}{{\tan }^{-1}}\left( \frac{3}{5} \right)\] |
| D. | \[{{\tan }^{-1}}\left( \frac{1}{2} \right)\] |
| Answer» B. \[\frac{1}{2}{{\sin }^{-1}}\left( \frac{3}{5} \right)\] | |
| 3834. |
\[4{{\tan }^{-1}}\frac{1}{5}-{{\tan }^{-1}}\frac{1}{70}+{{\tan }^{-1}}\frac{1}{99}=\] [Roorkee 1981] |
| A. | \[\frac{\pi }{2}\] |
| B. | \[\frac{\pi }{3}\] |
| C. | \[\frac{\pi }{4}\] |
| D. | None of these |
| Answer» D. None of these | |
| 3835. |
\[{{\tan }^{-1}}\frac{3}{4}+{{\tan }^{-1}}\frac{3}{5}-{{\tan }^{-1}}\frac{8}{19}=\] [AMU 1976, 77] |
| A. | \[\frac{\pi }{4}\] |
| B. | \[\frac{\pi }{3}\] |
| C. | \[\frac{\pi }{6}\] |
| D. | None of these |
| Answer» B. \[\frac{\pi }{3}\] | |
| 3836. |
\[{{\cos }^{-1}}\frac{1}{2}+2{{\sin }^{-1}}\frac{1}{2}\]is equal to [MP PET 1998; UPSEAT 2004] |
| A. | \[\frac{\pi }{4}\] |
| B. | \[\frac{\pi }{6}\] |
| C. | \[\frac{\pi }{3}\] |
| D. | \[\frac{2\pi }{3}\] |
| Answer» E. | |
| 3837. |
If \[{{\sin }^{-1}}\left( \frac{2a}{1+{{a}^{2}}} \right)+{{\sin }^{-1}}\left( \frac{2b}{1+{{b}^{2}}} \right)=2{{\tan }^{-1}}x,\]then \[x=\] [MNR 1984; UPSEAT 1999; Pb. CET 2004] |
| A. | \[\frac{a-b}{1+ab}\] |
| B. | \[\frac{b}{1+ab}\] |
| C. | \[\frac{b}{1-ab}\] |
| D. | \[\frac{a+b}{1-ab}\] |
| Answer» E. | |
| 3838. |
If \[{{\cot }^{-1}}\alpha +{{\cot }^{-1}}\beta ={{\cot }^{-1}}x,\]then \[x=\] [MP PET 1992] |
| A. | \[\alpha +\beta \] |
| B. | \[\alpha -\beta \] |
| C. | \[\frac{1+\alpha \beta }{\alpha +\beta }\] |
| D. | \[\frac{\alpha \beta -1}{\alpha +\beta }\] |
| Answer» E. | |
| 3839. |
\[{{\sin }^{-1}}\frac{1}{\sqrt{5}}+{{\cot }^{-1}}3\]is equal to [MP PET 1993; Karnataka CET 1995] |
| A. | \[\frac{\pi }{6}\] |
| B. | \[\frac{\pi }{4}\] |
| C. | \[\frac{\pi }{3}\] |
| D. | \[\frac{\pi }{2}\] |
| Answer» C. \[\frac{\pi }{3}\] | |
| 3840. |
If \[{{\cos }^{-1}}x+{{\cos }^{-1}}y=2\pi ,\]then \[{{\sin }^{-1}}x+{{\sin }^{-1}}y\]is equal to |
| A. | \[\pi \] |
| B. | \[-\pi \] |
| C. | \[\frac{\pi }{2}\] |
| D. | None of these |
| Answer» C. \[\frac{\pi }{2}\] | |
| 3841. |
\[{{\tan }^{-1}}\left( \frac{\sqrt{1+{{x}^{2}}}-1}{x} \right)=\] |
| A. | \[{{\tan }^{-1}}x\] |
| B. | \[\frac{1}{2}{{\tan }^{-1}}x\] |
| C. | \[2{{\tan }^{-1}}x\] |
| D. | None of these |
| Answer» C. \[2{{\tan }^{-1}}x\] | |
| 3842. |
If \[{{\tan }^{-1}}(x-1)+{{\tan }^{-1}}x+{{\tan }^{-1}}(x+1)={{\tan }^{-1}}3x\],then x = |
| A. | \[\pm \frac{1}{2}\] |
| B. | \[0,\,\frac{1}{2}\] |
| C. | \[0,\,-\frac{1}{2}\] |
| D. | \[0,\,\pm \frac{1}{2}\] |
| Answer» E. | |
| 3843. |
\[{{\tan }^{-1}}\frac{1-{{x}^{2}}}{2x}+{{\cos }^{-1}}\frac{1-{{x}^{2}}}{1+{{x}^{2}}}=\] |
| A. | \[\frac{\pi }{4}\] |
| B. | \[\frac{\pi }{2}\] |
| C. | \[\pi \] |
| D. | 0 |
| Answer» C. \[\pi \] | |
| 3844. |
\[{{\cot }^{-1}}3+\text{cose}{{\text{c}}^{-1}}\sqrt{5}\]= |
| A. | \[\frac{\pi }{3}\] |
| B. | \[\frac{\pi }{4}\] |
| C. | \[\frac{\pi }{6}\] |
| D. | \[\frac{\pi }{2}\] |
| Answer» C. \[\frac{\pi }{6}\] | |
| 3845. |
If \[{{\tan }^{-1}}\frac{a+x}{a}+{{\tan }^{-1}}\frac{a-x}{a}=\frac{\pi }{6}\],then \[{{x}^{2}}\]= |
| A. | \[2\sqrt{3}a\] |
| B. | \[\sqrt{3}a\] |
| C. | \[2\sqrt{3}{{a}^{2}}\] |
| D. | None of these |
| Answer» D. None of these | |
| 3846. |
\[{{\cot }^{-1}}\frac{xy+1}{x-y}+{{\cot }^{-1}}\frac{yz+1}{y-z}+{{\cot }^{-1}}\frac{zx+1}{z-x}=\] |
| A. | 0 |
| B. | 1 |
| C. | \[{{\cot }^{-1}}x+{{\cot }^{-1}}y+{{\cot }^{-1}}z\] |
| D. | None of these |
| Answer» B. 1 | |
| 3847. |
\[{{\tan }^{-1}}x+{{\cot }^{-1}}(x+1)=\] |
| A. | \[{{\tan }^{-1}}({{x}^{2}}+1)\] |
| B. | \[{{\tan }^{-1}}({{x}^{2}}+x)\] |
| C. | \[{{\tan }^{-1}}(x+1)\] |
| D. | \[{{\tan }^{-1}}({{x}^{2}}+x+1)\] |
| Answer» E. | |
| 3848. |
\[\cos \left[ {{\tan }^{-1}}\frac{1}{3}+{{\tan }^{-1}}\frac{1}{2} \right]=\] [MP PET 1991; MNR 1990] |
| A. | \[\frac{1}{\sqrt{2}}\] |
| B. | \[\frac{\sqrt{3}}{2}\] |
| C. | \[\frac{1}{2}\] |
| D. | \[\frac{\pi }{4}\] |
| Answer» B. \[\frac{\sqrt{3}}{2}\] | |
| 3849. |
\[2{{\sin }^{-1}}\frac{3}{5}+{{\cos }^{-1}}\frac{24}{25}=\] |
| A. | \[\frac{\pi }{2}\] |
| B. | \[\frac{2\pi }{3}\] |
| C. | \[\frac{5\pi }{3}\] |
| D. | None of these |
| Answer» B. \[\frac{2\pi }{3}\] | |
| 3850. |
If \[{{\cot }^{-1}}x+{{\tan }^{-1}}3=\frac{\pi }{2}\], then x = |
| A. | 44256 |
| B. | 44287 |
| C. | 3 |
| D. | 4 |
| Answer» D. 4 | |