MCQOPTIONS
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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.
| 4751. |
\[\cot x-\tan x=\] [MP PET 1986] |
| A. | \[\cot \,2x\] |
| B. | \[2{{\cot }^{2}}x\] |
| C. | \[2\,\,\cot \,2x\] |
| D. | \[{{\cot }^{2}}\,2x\] |
| Answer» D. \[{{\cot }^{2}}\,2x\] | |
| 4752. |
If \[\cos \theta =\frac{1}{2}\left( x+\frac{1}{x} \right)\], then \[\frac{1}{2}\left( {{x}^{2}}+\frac{1}{{{x}^{2}}} \right)=\] [AMU 1998] |
| A. | \[\sin 2\theta \] |
| B. | \[\cos \,2\theta \] |
| C. | \[\tan \,2\theta \] |
| D. | \[\sec \,2\theta \] |
| Answer» C. \[\tan \,2\theta \] | |
| 4753. |
If \[x+\frac{1}{x}=2\cos \alpha \], then \[{{x}^{n}}+\frac{1}{{{x}^{n}}}=\] [Karnataka CET 2004] |
| A. | \[{{2}^{n}}\cos \alpha \] |
| B. | \[{{2}^{n}}\cos n\alpha \] |
| C. | \[2i\,\sin \,n\,\alpha \] |
| D. | \[2\cos \,n\alpha \] |
| Answer» E. | |
| 4754. |
If \[x=\sec \theta +\tan \theta ,\]then \[x+\frac{1}{x}=\] [MP PET 1986] |
| A. | 1 |
| B. | \[2\sec \theta \] |
| C. | 2 |
| D. | \[2\tan \theta \] |
| Answer» C. 2 | |
| 4755. |
If \[\sec \theta +\tan \theta =p,\]then \[\tan \theta \]is equal to [MP PET 1994] |
| A. | \[\frac{2p}{{{p}^{2}}-1}\] |
| B. | \[\frac{{{p}^{2}}-1}{2p}\] |
| C. | \[\frac{{{p}^{2}}+1}{2p}\] |
| D. | \[\frac{2p}{{{p}^{2}}+1}\] |
| Answer» C. \[\frac{{{p}^{2}}+1}{2p}\] | |
| 4756. |
If \[\tan \theta +\sec \theta ={{e}^{x}},\]then \[\cos \theta \] equals [AMU 2002] |
| A. | \[\frac{({{e}^{x}}+{{e}^{-x}})}{2}\] |
| B. | \[\frac{2}{({{e}^{x}}+{{e}^{-x}})}\] |
| C. | \[\frac{({{e}^{x}}-{{e}^{-x}})}{2}\] |
| D. | \[\frac{({{e}^{x}}-{{e}^{-x}})}{({{e}^{x}}+{{e}^{-x}})}\] |
| Answer» C. \[\frac{({{e}^{x}}-{{e}^{-x}})}{2}\] | |
| 4757. |
\[\frac{\sin \theta }{1-\cot \theta }+\frac{\cos \theta }{1-\tan \theta }=\] [Karnataka CET 1998] |
| A. | 0 |
| B. | 1 |
| C. | \[\cos \theta -\sin \theta \] |
| D. | \[\cos \theta +\sin \theta \] |
| Answer» E. | |
| 4758. |
If \[\theta \] lies in the second quadrant, then the value of \[\sqrt{\left( \frac{1-\sin \theta }{1+\sin \theta } \right)}+\sqrt{\left( \frac{1+\sin \theta }{1-\sin \theta } \right)}\] |
| A. | \[2\sec \theta \] |
| B. | \[-2\sec \theta \] |
| C. | \[2\text{cosec}\theta \] |
| D. | None of these |
| Answer» C. \[2\text{cosec}\theta \] | |
| 4759. |
If \[\sin A,\cos A\] and \[\tan A\] are in G.P., then \[{{\cos }^{3}}A+{{\cos }^{2}}A\] is equal to |
| A. | 1 |
| B. | 2 |
| C. | 4 |
| D. | None of these |
| Answer» B. 2 | |
| 4760. |
If \[A\] lies in the second quadrant and \[3\tan A+4=0,\] the value of \[2\cot A-5\cos A+\sin A\]is equal to [Pb. CET 2000] |
| A. | \[\frac{-53}{10}\] |
| B. | \[\frac{-7}{10}\] |
| C. | \[\frac{7}{10}\] |
| D. | \[\frac{23}{10}\] |
| Answer» E. | |
| 4761. |
\[(m+2)\sin \theta +(2m-1)\cos \theta =2m+1,\]if |
| A. | \[\tan \theta =\frac{3}{4}\] |
| B. | \[\tan \theta =\frac{4}{3}\] |
| C. | \[\tan \theta =\frac{2m}{{{m}^{2}}+1}\] |
| D. | None of these |
| Answer» C. \[\tan \theta =\frac{2m}{{{m}^{2}}+1}\] | |
| 4762. |
If \[\sin (\alpha -\beta )=\frac{1}{2}\]and \[\cos (\alpha +\beta )=\frac{1}{2},\]where \[\alpha \] and \[\beta \] are positive acute angles, then |
| A. | \[\alpha =45{}^\circ ,\beta =15{}^\circ \] |
| B. | \[\alpha =15{}^\circ ,\beta =45{}^\circ \] |
| C. | \[\alpha =60{}^\circ ,\beta =15{}^\circ \] |
| D. | None of these |
| Answer» B. \[\alpha =15{}^\circ ,\beta =45{}^\circ \] | |
| 4763. |
If \[\sin \theta =\frac{-4}{5}\] and \[\theta \] lies in the third quadrant, then \[\cos \frac{\theta }{2}=\] |
| A. | \[\frac{1}{\sqrt{5}}\] |
| B. | \[-\frac{1}{\sqrt{5}}\] |
| C. | \[\sqrt{\frac{2}{5}}\] |
| D. | \[-\sqrt{\frac{2}{5}}\] |
| Answer» C. \[\sqrt{\frac{2}{5}}\] | |
| 4764. |
If \[\sin \theta =-\frac{1}{\sqrt{2}}\] and \[\tan \theta =1,\] then \[\theta \] lies in which quadrant |
| A. | First |
| B. | Second |
| C. | Third |
| D. | Fourth |
| Answer» D. Fourth | |
| 4765. |
If \[\tan \theta =\frac{-4}{3},\]then\[\sin \theta =\] [IIT 1979; Pb. CET 1995; Orissa JEE 2002] |
| A. | - 4/5 but not 4/5 |
| B. | - 4/5 or 4/5 |
| C. | 4/5 but not - 4/5 |
| D. | None of these |
| Answer» C. 4/5 but not - 4/5 | |
| 4766. |
.If \[\tan \theta =\frac{20}{21},\] cosq will be [MP PET 1994] |
| A. | \[\pm \frac{20}{41}\] |
| B. | \[\pm \frac{1}{21}\] |
| C. | \[\pm \frac{21}{29}\] |
| D. | \[\pm \frac{20}{21}\] |
| Answer» D. \[\pm \frac{20}{21}\] | |
| 4767. |
A circular wire of radius \[7\,cm\] is cut and bend again into an arc of a circle of radius \[12cm\]. The angle subtended by the arc at the centre is [Kerala (Engg.) 2002] |
| A. | \[{{50}^{o}}\] |
| B. | \[{{210}^{o}}\] |
| C. | \[{{100}^{o}}\] |
| D. | \[{{60}^{o}}\] |
| Answer» C. \[{{100}^{o}}\] | |
| 4768. |
If \[5\tan \theta =4,\] then \[\frac{5\sin \theta -3\cos \theta }{5\sin \theta +2\cos \theta }=\] [Karnataka CET 1998] |
| A. | 0 |
| B. | 1 |
| C. | 44348 |
| D. | 6 |
| Answer» D. 6 | |
| 4769. |
If \[\sin \theta =\frac{24}{25}\]and \[\theta \] lies in the second quadrant, then \[\sec \theta +\tan \theta =\] [MP PET 1997] |
| A. | -3 |
| B. | -5 |
| C. | -7 |
| D. | -9 |
| Answer» D. -9 | |
| 4770. |
If \[\sin \theta +\cos \theta =1\], then \[\sin \theta \cos \theta =\] [Karnataka CET 1998] |
| A. | 0 |
| B. | 1 |
| C. | 2 |
| D. | 44228 |
| Answer» B. 1 | |
| 4771. |
If \[\sin \theta +\cos \theta =m\]and \[\sec \theta +\text{cosec}\theta =n\], then \[n(m+1)(m-1)=\] [MP PET 1986] |
| A. | m |
| B. | n |
| C. | 2m |
| D. | 2n |
| Answer» D. 2n | |
| 4772. |
If \[\sin \theta +\text{cosec}\theta =\text{2}\], then \[{{\sin }^{2}}\theta +\text{cose}{{\text{c}}^{\text{2}}}\theta =\] [MP PET 1992; MNR 1990; UPSEAT 2002] |
| A. | 1 |
| B. | 4 |
| C. | 2 |
| D. | None of these |
| Answer» D. None of these | |
| 4773. |
\[\tan 1{}^\circ \tan 2{}^\circ \tan 3{}^\circ \tan 4{}^\circ ........\tan 89{}^\circ =\] [MP PET 1998, 2001; AMU 1999; Pb. CET 1994] |
| A. | 1 |
| B. | 0 |
| C. | \[\infty \] |
| D. | 44228 |
| Answer» B. 0 | |
| 4774. |
Which of the following relations is correct [WB JEE 1991] |
| A. | \[\sin 1<\sin 1{}^\circ \] |
| B. | \[\sin 1>\sin 1{}^\circ \] |
| C. | \[\sin 1=\sin 1{}^\circ \] |
| D. | \[\frac{\pi }{180}\sin \,\,\,1\,=\sin \,\,\,{{1}^{o}}\] |
| Answer» C. \[\sin 1=\sin 1{}^\circ \] | |
| 4775. |
The angle subtended at the centre of a circle of radius 3 metres by an arc of length 1 metre is equal to [MNR 1973] |
| A. | \[{{20}^{o}}\] |
| B. | \[{{60}^{o}}\] |
| C. | \[\frac{1}{3}\] radian |
| D. | 3 radians |
| Answer» D. 3 radians | |
| 4776. |
\[\int_{{}}^{{}}{\frac{dx}{\sqrt{{{x}^{2}}+{{a}^{2}}}}}\] is equal to [MP PET 2004] |
| A. | \[\frac{1}{2}x\sqrt{{{x}^{2}}+{{a}^{2}}}+\frac{1}{2}{{a}^{2}}\log (x+\sqrt{{{x}^{2}}+{{a}^{2}}})+c\] |
| B. | \[\frac{1}{2}\log ({{x}^{2}}+{{a}^{2}})+c\] |
| C. | \[\log (x+\sqrt{{{x}^{2}}+{{a}^{2}}})+c\] |
| D. | \[\log (x-\sqrt{{{x}^{2}}+{{a}^{2}}})+c\] |
| Answer» D. \[\log (x-\sqrt{{{x}^{2}}+{{a}^{2}}})+c\] | |
| 4777. |
\[\int{\frac{dx}{{{a}^{2}}-{{x}^{2}}}}\] is equal to [EAMCET 2002] |
| A. | \[\frac{1}{a}{{\tan }^{-1}}\left( \frac{x}{a} \right)\] |
| B. | \[\frac{1}{2a}{{\sin }^{-1}}\left( \frac{a-x}{a+x} \right)\] |
| C. | \[\frac{1}{2a}\log \,\left( \frac{a+x}{a-x} \right)\] |
| D. | \[\frac{1}{2a}\log \,\left( \frac{a-x}{a+x} \right)\] |
| Answer» D. \[\frac{1}{2a}\log \,\left( \frac{a-x}{a+x} \right)\] | |
| 4778. |
\[\int{\sqrt{{{x}^{2}}+{{a}^{2}}}\,\,dx}\] equals to [RPET 2001] |
| A. | \[\frac{x}{2}\sqrt{{{x}^{2}}+{{a}^{2}}}\,-\frac{{{a}^{2}}}{2}\log \{x+\sqrt{{{x}^{2}}+{{a}^{2}}}\}+c\] |
| B. | \[\frac{x}{2}\sqrt{{{x}^{2}}+{{a}^{2}}}\,+\frac{{{a}^{2}}}{2}\log \,\{x+\sqrt{{{x}^{2}}+{{a}^{2}}}\}+c\] |
| C. | \[\frac{x}{2}\sqrt{{{x}^{2}}+{{a}^{2}}}\,-\frac{{{a}^{2}}}{2}\log \{x-\sqrt{{{x}^{2}}+{{a}^{2}}}\}+c\] |
| D. | \[\frac{x}{2}\sqrt{{{x}^{2}}+{{a}^{2}}}\,+\frac{{{a}^{2}}}{2}\log \{x-\sqrt{{{x}^{2}}+{{a}^{2}}}\}+c\] |
| Answer» C. \[\frac{x}{2}\sqrt{{{x}^{2}}+{{a}^{2}}}\,-\frac{{{a}^{2}}}{2}\log \{x-\sqrt{{{x}^{2}}+{{a}^{2}}}\}+c\] | |
| 4779. |
\[\int{\frac{{{x}^{2}}}{{{x}^{2}}+4}\,\,dx}\] equals to [RPET 2001] |
| A. | \[x-2{{\tan }^{-1}}(x/2)+c\] |
| B. | \[x+2{{\tan }^{-1}}(x/2)+c\] |
| C. | \[x-4{{\tan }^{-1}}(x/2)+c\] |
| D. | \[x+4{{\tan }^{-1}}(x/2)+c\] |
| Answer» B. \[x+2{{\tan }^{-1}}(x/2)+c\] | |
| 4780. |
\[\int_{{}}^{{}}{\frac{dx}{\sqrt{{{x}^{2}}-{{a}^{2}}}}}\] equals [SCRA 1996] |
| A. | \[{{\sin }^{-1}}\left( \frac{x}{a} \right)+c\] |
| B. | \[{{\log }_{e}}|x+\sqrt{{{x}^{2}}-{{a}^{2}}}|+c\] |
| C. | \[{{\log }_{e}}|x-\sqrt{{{x}^{2}}-{{a}^{2}}}|+c\] |
| D. | \[\frac{x\sqrt{{{x}^{2}}-{{a}^{2}}}}{2+c}\] |
| Answer» C. \[{{\log }_{e}}|x-\sqrt{{{x}^{2}}-{{a}^{2}}}|+c\] | |
| 4781. |
\[\int_{{}}^{{}}{\sqrt{1+{{x}^{2}}}\ dx=}\] [MP PET 1987, 89] |
| A. | \[\frac{x}{2}\sqrt{1+{{x}^{2}}}+\frac{1}{2}\log (x+\sqrt{1+{{x}^{2}}})+c\] |
| B. | \[\frac{2}{3}{{(1+{{x}^{2}})}^{3/2}}+c\] |
| C. | \[\frac{2}{3}x{{(1+{{x}^{2}})}^{3/2}}+c\] |
| D. | None of these |
| Answer» B. \[\frac{2}{3}{{(1+{{x}^{2}})}^{3/2}}+c\] | |
| 4782. |
\[\int_{{}}^{{}}{\frac{dx}{1-{{x}^{2}}}=}\] [MP PET 1987, 92, 2000] |
| A. | \[{{\tan }^{-1}}x+c\] |
| B. | \[{{\sin }^{-1}}x+c\] |
| C. | \[\frac{1}{2}\ln \left| \frac{1+x}{1-x} \right|+c\] |
| D. | \[\frac{1}{2}\ln \left| \frac{1-x}{1+x} \right|+c\] |
| Answer» D. \[\frac{1}{2}\ln \left| \frac{1-x}{1+x} \right|+c\] | |
| 4783. |
The value of \[\int_{{}}^{{}}{\frac{dx}{\sqrt{1-x}}}\] is [Pb. CET 2001] |
| A. | \[2\sqrt{1-x}+c\] |
| B. | \[-2\sqrt{1-x}+c\] |
| C. | \[-{{\sin }^{-1}}\sqrt{x}+c\] |
| D. | \[{{\sin }^{-1}}\sqrt{x}+c\] |
| Answer» C. \[-{{\sin }^{-1}}\sqrt{x}+c\] | |
| 4784. |
\[\int_{{}}^{{}}{\frac{dx}{\sqrt{x}+\sqrt{x-2}}=}\] [MP PET 1990] |
| A. | \[\frac{1}{3}[{{x}^{3/2}}-{{(x-2)}^{3/2}}]+c\] |
| B. | \[\frac{2}{3}[{{x}^{3/2}}-{{(x-2)}^{3/2}}]+c\] |
| C. | \[\frac{1}{3}[{{(x-2)}^{3/2}}-{{x}^{3/2}}]+c\] |
| D. | \[\frac{2}{3}[{{(x-2)}^{3/2}}-{{x}^{3/2}}]+c\] |
| Answer» B. \[\frac{2}{3}[{{x}^{3/2}}-{{(x-2)}^{3/2}}]+c\] | |
| 4785. |
\[\int{\left( {{\sin }^{4}}x-{{\cos }^{4}}x \right)\,dx=}\] [RPET 2003] |
| A. | \[-\frac{\cos 2x}{2}+c\] |
| B. | \[-\frac{\sin 2x}{2}+c\] |
| C. | \[\frac{\sin 2x}{2}+c\] |
| D. | \[\frac{\cos 2x}{2}+c\] |
| Answer» C. \[\frac{\sin 2x}{2}+c\] | |
| 4786. |
\[\int{\sec x\tan x\,\,dx=}\] [RPET 2003] |
| A. | \[\sec x+\tan x+c\] |
| B. | \[\sec x+c\] |
| C. | \[\tan x+c\] |
| D. | \[-\sec x+c\] |
| Answer» C. \[\tan x+c\] | |
| 4787. |
\[\int{{{13}^{x}}dx}\] is [Kerala (Engg.) 2002] |
| A. | \[\frac{{{13}^{x}}}{\log 13}+c\] |
| B. | \[{{13}^{x+1}}+c\] |
| C. | \[14x+c\] |
| D. | \[{{14}^{x+1}}\]+ c |
| Answer» B. \[{{13}^{x+1}}+c\] | |
| 4788. |
\[\int{{{a}^{x}}\,\,dx=}\] [RPET 2003] |
| A. | \[\frac{{{a}^{x}}}{\log a}+c\] |
| B. | \[{{a}^{x}}\log a+c\] |
| C. | \[\log a+c\] |
| D. | \[{{a}^{x}}+c\] |
| Answer» B. \[{{a}^{x}}\log a+c\] | |
| 4789. |
If \[\int{f(x)\,dx=f(x)},\] then \[{{\int{\left[ f(x) \right]}}^{2}}\,\,dx\] is [DCE 2002] |
| A. | \[\frac{1}{2}{{\left[ f\left( x \right) \right]}^{2}}\] |
| B. | \[{{\left[ f\left( x \right) \right]}^{3}}\] |
| C. | \[\frac{{{\left[ f\left( x \right) \right]}^{3}}}{3}\] |
| D. | \[{{\left[ \,f\left( x \right) \right]}^{2}}\] |
| Answer» B. \[{{\left[ f\left( x \right) \right]}^{3}}\] | |
| 4790. |
The value of \[\int{\frac{1}{{{(x-5)}^{2}}}\,\,dx}\] is [Karnataka CET 2001; Pb. CET 2002] |
| A. | \[\frac{1}{x-5}+c\] |
| B. | \[-\frac{1}{x-5}+c\] |
| C. | \[\frac{2}{{{\left( x-5 \right)}^{3}}}+c\] |
| D. | \[-2{{\left( x-5 \right)}^{3}}+c\] |
| Answer» C. \[\frac{2}{{{\left( x-5 \right)}^{3}}}+c\] | |
| 4791. |
\[\int{\frac{a{{x}^{3}}+b{{x}^{2}}+c}{{{x}^{4}}}\,\,dx}\] equals to [RPET 2001] |
| A. | \[a\log x+\frac{b}{{{x}^{2}}}+\frac{c}{3{{x}^{3}}}+c\] |
| B. | \[a\log x+\frac{b}{x}-\frac{c}{3{{x}^{3}}}+c\] |
| C. | \[a\log x-\frac{b}{x}-\frac{c}{3{{x}^{3}}}+c\] |
| D. | None of these |
| Answer» D. None of these | |
| 4792. |
\[\int_{{}}^{{}}{{{\tan }^{-1}}\sqrt{\frac{1-\cos 2x}{1+\cos 2x}}}\ dx=\] |
| A. | \[2{{x}^{2}}+c\] |
| B. | \[{{x}^{2}}+c\] |
| C. | \[\frac{{{x}^{2}}}{2}+c\] |
| D. | \[2x+c\] |
| Answer» D. \[2x+c\] | |
| 4793. |
\[\int_{{}}^{{}}{\frac{1}{\sqrt{1+\cos x}}\ dx=}\] |
| A. | \[\sqrt{2}\log \left( \sec \frac{x}{2}+\tan \frac{x}{2} \right)+K\] |
| B. | \[\frac{1}{\sqrt{2}}\log \left( \sec \frac{x}{2}+\tan \frac{x}{2} \right)+K\] |
| C. | \[\log \left( \sec \frac{x}{2}+\tan \frac{x}{2} \right)+K\] |
| D. | None of these |
| Answer» B. \[\frac{1}{\sqrt{2}}\log \left( \sec \frac{x}{2}+\tan \frac{x}{2} \right)+K\] | |
| 4794. |
\[\int_{{}}^{{}}{{{e}^{x\log a}}.\ {{e}^{x}}\ dx}\]is equal to [Kerala (Engg.) 2005] |
| A. | \[{{(ae)}^{x}}+c\] |
| B. | \[\frac{{{(ae)}^{x}}}{\log (ae)}+c\] |
| C. | \[\frac{{{e}^{x}}}{1+\log a}+c\] |
| D. | None of these |
| Answer» C. \[\frac{{{e}^{x}}}{1+\log a}+c\] | |
| 4795. |
\[\int_{{}}^{{}}{{{e}^{\log (\sin x)}}dx=}\] [MP PET 1995] |
| A. | \[\sin x+c\] |
| B. | \[-\cos x+c\] |
| C. | \[{{e}^{\log (\cos x)}}+c\] |
| D. | None of these |
| Answer» C. \[{{e}^{\log (\cos x)}}+c\] | |
| 4796. |
\[\int_{{}}^{{}}{2\sin x}\cos x\ dx\]is equal to [SCRA 1996] |
| A. | \[\cos 2x+c\] |
| B. | \[\sin 2x+c\] |
| C. | \[{{\cos }^{2}}x+c\] |
| D. | \[{{\sin }^{2}}x+c\] |
| Answer» E. | |
| 4797. |
\[\int_{{}}^{{}}{\sqrt{1+\cos x}\ dx}\] equals [RPET 1996] |
| A. | \[2\sqrt{2}\sin \frac{x}{2}+c\] |
| B. | \[-2\sqrt{2}\sin \frac{x}{2}+c\] |
| C. | \[-2\sqrt{2}\cos \frac{x}{2}+c\] |
| D. | \[2\sqrt{2}\cos \frac{x}{2}+c\] |
| Answer» B. \[-2\sqrt{2}\sin \frac{x}{2}+c\] | |
| 4798. |
\[\int_{{}}^{{}}{\left( 2\sin x+\frac{1}{x} \right)\ dx}\] is equal to [MP PET 1999] |
| A. | \[-2\cos x+\log x+c\] |
| B. | \[2\cos x+\log x+c\] |
| C. | \[-2\sin x-\frac{1}{{{x}^{2}}}+c\] |
| D. | \[-2\cos x+\frac{1}{{{x}^{2}}}+c\] |
| Answer» B. \[2\cos x+\log x+c\] | |
| 4799. |
\[\int_{{}}^{{}}{\text{cose}{{\text{c}}^{2}}x\ dx}\] is equal to [MP PET 1999] |
| A. | \[\cot x+c\] |
| B. | \[-\cot x+c\] |
| C. | \[{{\tan }^{2}}x+c\] |
| D. | \[-{{\cot }^{2}}x+c\] |
| Answer» C. \[{{\tan }^{2}}x+c\] | |
| 4800. |
If \[f'(x)={{x}^{2}}+5\] and \[f(0)=-1\], then \[f(x)=\] |
| A. | \[{{x}^{3}}+5x-1\] |
| B. | \[{{x}^{3}}+5x+1\] |
| C. | \[\frac{1}{3}{{x}^{3}}+5x-1\] |
| D. | \[\frac{1}{3}{{x}^{3}}+5x+1\] |
| Answer» D. \[\frac{1}{3}{{x}^{3}}+5x+1\] | |