MCQOPTIONS
Saved Bookmarks
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.
| 5051. |
If \[\varphi (x)={{x}^{2}}+1\] and \[\psi (x)={{3}^{x}}\], then \[\varphi \{\psi (x)\}\] and \[\psi \{\varphi (x)\}=\] |
| A. | \[{{3}^{2x+1}},\ {{3}^{{{x}^{2}}+1}}\] |
| B. | \[{{3}^{2x+1}},\ {{3}^{{{x}^{2}}}}+1\] |
| C. | \[{{3}^{2x}}+1,\ {{3}^{{{x}^{2}}+1}}\] |
| D. | None of these |
| Answer» D. None of these | |
| 5052. |
If \[f(x)={{x}^{2}}+1\],then \[fof(x)\] is equal to |
| A. | \[{{x}^{2}}+1\] |
| B. | \[{{x}^{2}}+2x+2\] |
| C. | \[{{x}^{4}}+2{{x}^{2}}+2\] |
| D. | None of these |
| Answer» D. None of these | |
| 5053. |
If \[f(x)=|\cos x|\]and \[g(x)=[x]\], then \[gof(x)\] is equal to |
| A. | \[|\cos \ [x]|\] |
| B. | \[|\cos x|\] |
| C. | \[[|\cos x|]\] |
| D. | \[|[\cos x]|\] |
| Answer» D. \[|[\cos x]|\] | |
| 5054. |
If \[f(x)={{e}^{2x}}\] and \[g(x)=\log \sqrt{x}\]\[(x>0)\], then \[fog(x)\] is equal to |
| A. | \[{{e}^{2x}}\] |
| B. | \[\log \sqrt{x}\] |
| C. | \[{{e}^{2x}}\log \sqrt{x}\] |
| D. | x |
| Answer» E. | |
| 5055. |
If f is an exponential function and g is a logarithmic function, then \[fog(1)\] will be |
| A. | e |
| B. | \[{{\log }_{e}}e\] |
| C. | 0 |
| D. | 2e |
| Answer» C. 0 | |
| 5056. |
If \[f(x)={{x}^{2}}-1\] and \[g(x)=3x+1\], then \[(gof)(x)=\] |
| A. | \[{{x}^{2}}-1\] |
| B. | \[2{{x}^{2}}-1\] |
| C. | \[3{{x}^{2}}-2\] |
| D. | \[2{{x}^{2}}+2\] |
| Answer» D. \[2{{x}^{2}}+2\] | |
| 5057. |
If \[f(x)=2x\] and g is identity function, then |
| A. | \[(fog)(x)=g(x)\] |
| B. | \[(g+g)(x)=g(x)\] |
| C. | \[(fog)(x)=(g+g)(x)\] |
| D. | None of these |
| Answer» D. None of these | |
| 5058. |
If f be the greatest integer function and g be the modulus function, then \[(gof)\left( -\frac{5}{3} \right)-(fog)\left( -\frac{5}{3} \right)=\] |
| A. | 1 |
| B. | ?1 |
| C. | 2 |
| D. | 4 |
| Answer» B. ?1 | |
| 5059. |
Let \[f(x)=\sin x+\cos x,\ g(x)={{x}^{2}}-1\]. Thus \[g(f(x))\] is invertible for \[x\in \] [IIT Screening 2004] |
| A. | \[\left[ -\frac{\pi }{2},\ 0 \right]\] |
| B. | \[\left[ -\frac{\pi }{2},\ \pi \right]\] |
| C. | \[\left[ -\frac{\pi }{2},\ \frac{\pi }{4} \right]\] |
| D. | \[\left[ 0,\ \frac{\pi }{2} \right]\] |
| Answer» D. \[\left[ 0,\ \frac{\pi }{2} \right]\] | |
| 5060. |
If \[f(x)={{x}^{2}}+1\], then \[{{f}^{-1}}(17)\] and \[{{f}^{-1}}(-3)\]will be [UPSEAT 2003] |
| A. | 4, 1 |
| B. | 4, 0 |
| C. | 3, 2 |
| D. | None of these |
| Answer» E. | |
| 5061. |
Let the function f be defined by \[f(x)=\frac{2x+1}{1-3x}\], then \[{{f}^{-1}}(x)\] is [Kerala (Engg.) 2002] |
| A. | \[\frac{x-1}{3x+2}\] |
| B. | \[\frac{3x+2}{x-1}\] |
| C. | \[\frac{x+1}{3x-2}\] |
| D. | \[\frac{2x+1}{1-3x}\] |
| Answer» B. \[\frac{3x+2}{x-1}\] | |
| 5062. |
Inverse of the function \[y=2x-3\] is [UPSEAT 2002] |
| A. | \[\frac{x+3}{2}\] |
| B. | \[\frac{x-3}{2}\] |
| C. | \[\frac{1}{2x-3}\] |
| D. | None of these |
| Answer» B. \[\frac{x-3}{2}\] | |
| 5063. |
Let \[f(\theta )=\sin \theta (\sin \theta +\sin 3\theta )\], then \[f(\theta )\] [IIT Screening 2000] |
| A. | \[\ge 0\] only when \[\theta \ge 0\] |
| B. | \[\le 0\] for all real \[\theta \] |
| C. | \[\ge 0\] for all real \[\theta \] |
| D. | \[\le 0\]only when \[\theta \le 0\] |
| Answer» D. \[\le 0\]only when \[\theta \le 0\] | |
| 5064. |
Which of the following function is inverse function [AMU 2000] |
| A. | \[f(x)=\frac{1}{x-1}\] |
| B. | \[f(x)={{x}^{2}}\] for all\[x\] |
| C. | \[f(x)={{x}^{2}}\], \[x\ge 0\] |
| D. | \[f(x)={{x}^{2}},\ x\le 0\] |
| Answer» B. \[f(x)={{x}^{2}}\] for all\[x\] | |
| 5065. |
If \[f(x)=\frac{x}{1+x}\], then \[{{f}^{-1}}(x)\] is equal to [AMU 1999] |
| A. | \[\frac{(1+x)}{x}\] |
| B. | \[\frac{1}{(1+x)}\] |
| C. | \[\frac{(1+x)}{(1-x)}\] |
| D. | \[\frac{x}{(1-x)}\] |
| Answer» E. | |
| 5066. |
If \[f:IR\to IR\] is defined by \[f(x)=3x-4\], then \[{{f}^{-1}}:IR\to IR\] is [SCRA 1996] |
| A. | \[4-3x\] |
| B. | \[\frac{x+4}{3}\] |
| C. | \[\frac{1}{3x-4}\] |
| D. | \[\frac{3}{x+4}\] |
| Answer» C. \[\frac{1}{3x-4}\] | |
| 5067. |
If \[f(x)=3x-5\], then \[{{f}^{-1}}(x)\] [IIT 1998] |
| A. | Is given by \[\frac{1}{3x-5}\] |
| B. | Is given by \[\frac{x+5}{3}\] |
| C. | Does not exist because f is not one-one |
| D. | Does not exist because f is not onto |
| Answer» C. Does not exist because f is not one-one | |
| 5068. |
If the function \[f:[1,\ \infty )\to [1,\ \infty )\] is defined by \[f(x)={{2}^{x(x-1)}},\] then \[{{f}^{-1}}\](x) is [IIT 1999] |
| A. | \[{{\left( \frac{1}{2} \right)}^{x(x-1)}}\] |
| B. | \[\frac{1}{2}(1+\sqrt{1+4{{\log }_{2}}x})\] |
| C. | \[\frac{1}{2}(1-\sqrt{1+4{{\log }_{2}}x})\] |
| D. | Not defined |
| Answer» C. \[\frac{1}{2}(1-\sqrt{1+4{{\log }_{2}}x})\] | |
| 5069. |
The inverse of the function \[f(x)=\frac{{{e}^{x}}-{{e}^{-x}}}{{{e}^{x}}+{{e}^{-x}}}+2\] is given by [Kurukshetra CEE 1996] |
| A. | \[{{\log }_{e}}{{\left( \frac{x-2}{x-1} \right)}^{1/2}}\] |
| B. | \[{{\log }_{e}}{{\left( \frac{x-1}{3-x} \right)}^{1/2}}\] |
| C. | \[{{\log }_{e}}{{\left( \frac{x}{2-x} \right)}^{1/2}}\] |
| D. | \[{{\log }_{e}}{{\left( \frac{x-1}{x+1} \right)}^{-2}}\] |
| Answer» C. \[{{\log }_{e}}{{\left( \frac{x}{2-x} \right)}^{1/2}}\] | |
| 5070. |
If \[y=f(x)=\frac{x+2}{x-1}\], then \[x=\] [IIT 1984] |
| A. | \[f(y)\] |
| B. | \[2f(y)\] |
| C. | \[\frac{1}{f(y)}\] |
| D. | None of these |
| Answer» B. \[2f(y)\] | |
| 5071. |
Which of the following function is invertible [AMU 2001] |
| A. | \[f(x)={{2}^{x}}\] |
| B. | \[f(x)={{x}^{3}}-x\] |
| C. | \[f(x)={{x}^{2}}\] |
| D. | None of these |
| Answer» B. \[f(x)={{x}^{3}}-x\] | |
| 5072. |
If \[f(x)=\frac{x-3}{x+1}\], then \[f[f\{f(x)\}]\] equals [RPET 1996] |
| A. | x |
| B. | ?x |
| C. | \[\frac{x}{2}\] |
| D. | \[-\frac{1}{x}\] |
| Answer» B. ?x | |
| 5073. |
The function \[f(x)=\log (x+\sqrt{{{x}^{2}}+1})\], is [AIEEE 2003; MP PET 2003; UPSEAT 2003] |
| A. | An even function |
| B. | An odd function |
| C. | A Periodic function |
| D. | Neither an even nor odd function |
| Answer» C. A Periodic function | |
| 5074. |
The function \[f(x)=\sin \left( \log (x+\sqrt{{{x}^{2}}+1}) \right)\] is [Orissa JEE 2002] |
| A. | Even function |
| B. | Odd function |
| C. | Neither even nor odd |
| D. | Periodic function |
| Answer» C. Neither even nor odd | |
| 5075. |
If \[f(x)=\log \frac{1+x}{1-x}\], then \[f(x)\] is [Kerala (Engg.) 2002] |
| A. | Even function |
| B. | \[f({{x}_{1}})f({{x}_{2}})=f({{x}_{1}}+{{x}_{2}})\] |
| C. | \[\frac{f({{x}_{1}})}{f({{x}_{2}})}=f({{x}_{1}}-{{x}_{2}})\] |
| D. | Odd function |
| Answer» E. | |
| 5076. |
Which of the following function is even function [RPET 2000] |
| A. | \[f(x)=\frac{{{a}^{x}}+1}{{{a}^{x}}-1}\] |
| B. | \[f(x)=x\left( \frac{{{a}^{x}}-1}{{{a}^{x}}+1} \right)\] |
| C. | \[f(x)=\frac{{{a}^{x}}-{{a}^{-x}}}{{{a}^{x}}+{{a}^{-x}}}\] |
| D. | \[f(x)=\sin x\] |
| Answer» C. \[f(x)=\frac{{{a}^{x}}-{{a}^{-x}}}{{{a}^{x}}+{{a}^{-x}}}\] | |
| 5077. |
For \[\theta >\frac{\pi }{3}\], the value of \[f(\theta )={{\sec }^{2}}\theta +{{\cos }^{2}}\theta \] always lies in the interval [Orissa JEE 2002] |
| A. | (0, 2) |
| B. | [0, 1] |
| C. | (1, 2) |
| D. | \[[2,\ \infty )\] |
| Answer» E. | |
| 5078. |
The function \[f:R\to R\] is defined by \[f(x)={{\cos }^{2}}x+{{\sin }^{4}}x\] for \[x\in R\], then \[f(R)=\] [EAMCET 2002] |
| A. | \[\left( \frac{3}{4},\ 1 \right]\] |
| B. | \[\left[ \frac{3}{4},\ 1 \right)\] |
| C. | \[\left[ \frac{3}{4},\ 1 \right]\] |
| D. | \[\left( \frac{3}{4},\ 1 \right)\] |
| Answer» D. \[\left( \frac{3}{4},\ 1 \right)\] | |
| 5079. |
Function \[{{\sin }^{-1}}\sqrt{x}\] is defined in the interval |
| A. | (?1, 1) |
| B. | [0, 1] |
| C. | [?1, 0] |
| D. | (?1, 2) |
| Answer» C. [?1, 0] | |
| 5080. |
The interval for which \[{{\sin }^{-1}}\sqrt{x}+{{\cos }^{-1}}\sqrt{x}=\frac{\pi }{2}\] holds [IIT Screening] |
| A. | \[[0,\ \infty )\] |
| B. | \[[0,\ 3]\] |
| C. | [0, 1] |
| D. | [0, 2] |
| Answer» D. [0, 2] | |
| 5081. |
Range of \[f(x)=\frac{{{x}^{2}}+34x-71}{{{x}^{2}}+2x-7}\] is [Roorkee 1983] |
| A. | [5, 9] |
| B. | \[(-\infty ,\ 5]\cup [9,\ \infty )\] |
| C. | (5, 9) |
| D. | None of these |
| Answer» C. (5, 9) | |
| 5082. |
If \[\varphi (x)={{a}^{x}}\], then \[{{\{\varphi (p)\}}^{3}}\]is equal to [MP PET 1999] |
| A. | \[\varphi (3p)\] |
| B. | \[3\varphi (p)\] |
| C. | \[6\varphi (p)\] |
| D. | \[2\varphi (p)\] |
| Answer» B. \[3\varphi (p)\] | |
| 5083. |
sIf \[f(x)=\cos (\log x)\], then \[f(x)f(y)-\frac{1}{2}[f(x/y)+f(xy)]=\] [IIT 1983; RPET 1995; MP PET 1995; Karnataka CET 1999; UPSEAT 2001] |
| A. | \[-1\] |
| B. | \[\frac{1}{2}\] |
| C. | \[-2\] |
| D. | None of these |
| Answer» E. | |
| 5084. |
The differential equation of the family of curves represented by the equation \[{{x}^{2}}y=a\], is |
| A. | \[\frac{dy}{dx}+\frac{2y}{x}=0\] |
| B. | \[\frac{dy}{dx}+\frac{2x}{y}=0\] |
| C. | \[\frac{dy}{dx}-\frac{2y}{x}=0\] |
| D. | \[\frac{dy}{dx}-\frac{2x}{y}=0\] |
| Answer» B. \[\frac{dy}{dx}+\frac{2x}{y}=0\] | |
| 5085. |
If \[y=c{{e}^{{{\sin }^{-1}}x}}\], then corresponding to this the differential equation is |
| A. | \[\frac{dy}{dx}=\frac{y}{\sqrt{1-{{x}^{2}}}}\] |
| B. | \[\frac{dy}{dx}=\frac{1}{\sqrt{1-{{x}^{2}}}}\]\[\] |
| C. | \[\frac{dy}{dx}=\frac{x}{\sqrt{1-{{x}^{2}}}}\] |
| D. | None of these |
| Answer» B. \[\frac{dy}{dx}=\frac{1}{\sqrt{1-{{x}^{2}}}}\]\[\] | |
| 5086. |
The differential equation for all the straight lines which are at a unit distance from the origin is [MP PET 1993] |
| A. | \[{{\left( y-x\frac{dy}{dx} \right)}^{2}}=1-{{\left( \frac{dy}{dx} \right)}^{2}}\] |
| B. | \[{{\left( y+x\frac{dy}{dx} \right)}^{2}}=1+{{\left( \frac{dy}{dx} \right)}^{2}}\] |
| C. | \[{{\left( y-x\frac{dy}{dx} \right)}^{2}}=1+{{\left( \frac{dy}{dx} \right)}^{2}}\] |
| D. | \[{{\left( y+x\frac{dy}{dx} \right)}^{2}}=1-{{\left( \frac{dy}{dx} \right)}^{2}}\] |
| Answer» D. \[{{\left( y+x\frac{dy}{dx} \right)}^{2}}=1-{{\left( \frac{dy}{dx} \right)}^{2}}\] | |
| 5087. |
The differential equation of the family of curves \[y=a\cos (x+b)\] is [MP PET 1993] |
| A. | \[\frac{{{d}^{2}}y}{d{{x}^{2}}}-y=0\] |
| B. | \[\frac{{{d}^{2}}y}{d{{x}^{2}}}+y=0\] |
| C. | \[\frac{{{d}^{2}}y}{d{{x}^{2}}}+2y=0\] |
| D. | None of these |
| Answer» C. \[\frac{{{d}^{2}}y}{d{{x}^{2}}}+2y=0\] | |
| 5088. |
The differential equation whose solution is \[y=A\sin x+B\cos x,\] is [CEE 1993; Kerala (Engg.) 2002] |
| A. | \[\frac{{{d}^{2}}y}{d{{x}^{2}}}+y=0\] |
| B. | \[\frac{{{d}^{2}}y}{d{{x}^{2}}}-y=0\] |
| C. | \[\frac{dy}{dx}+y=0\] |
| D. | None of these |
| Answer» B. \[\frac{{{d}^{2}}y}{d{{x}^{2}}}-y=0\] | |
| 5089. |
\[y=\frac{x}{x+1}\] is a solution of the differential equation |
| A. | \[{{y}^{2}}\frac{dy}{dx}={{x}^{2}}\] |
| B. | \[{{x}^{2}}\frac{dy}{dx}={{y}^{2}}\] |
| C. | \[y\frac{dy}{dx}=x\] |
| D. | \[x\frac{dy}{dx}=y\] |
| Answer» C. \[y\frac{dy}{dx}=x\] | |
| 5090. |
The differential equation for which \[{{\sin }^{-1}}x+{{\sin }^{-1}}y=c\] is given by [Karnataka CET 2003] |
| A. | \[\sqrt{1-{{x}^{2}}}\,\,dx\,\,+\sqrt{1-{{y}^{2}}}\,\,dy=0\] |
| B. | \[\sqrt{1-{{x}^{2}}}\,\,dy\,\,+\sqrt{1-{{y}^{2}}}\,\,dx=0\] |
| C. | \[\sqrt{1-{{x}^{2}}}\,\,dy\,\,-\sqrt{1-{{y}^{2}}}\,\,dx=0\] |
| D. | \[\sqrt{1-{{x}^{2}}}\,\,dx\,-\sqrt{1-{{y}^{2}}}\,\,dy=0\] |
| Answer» C. \[\sqrt{1-{{x}^{2}}}\,\,dy\,\,-\sqrt{1-{{y}^{2}}}\,\,dx=0\] | |
| 5091. |
If \[x=\sin t\], \[y=\cos pt\], then [Karnataka CET 2005] |
| A. | \[(1-{{x}^{2}}){{y}_{2}}+x{{y}_{1}}+{{p}^{2}}y=0\] |
| B. | \[(1-{{x}^{2}}){{y}_{2}}+x{{y}_{1}}-{{p}^{2}}y=0\] |
| C. | \[(1+{{x}^{2}}){{y}_{2}}-x{{y}_{1}}+{{p}^{2}}y=0\] |
| D. | \[(1-{{x}^{2}}){{y}_{2}}-x{{y}_{1}}+{{p}^{2}}y=0\] |
| Answer» E. | |
| 5092. |
The differential equation of the family of curves represented by the equation \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\] is |
| A. | \[x+y\frac{dy}{dx}=0\] |
| B. | \[y\frac{dy}{dx}=x\] |
| C. | \[y\frac{{{d}^{2}}y}{d{{x}^{2}}}+{{\left( \frac{dy}{dx} \right)}^{2}}=0\] |
| D. | None of these |
| Answer» B. \[y\frac{dy}{dx}=x\] | |
| 5093. |
Differential equation of \[y=\sec ({{\tan }^{-1}}x)\] is [UPSEAT 2002] |
| A. | \[(1+{{x}^{2}})\frac{dy}{dx}=y+x\] |
| B. | \[(1+{{x}^{2}})\frac{dy}{dx}=y-x\] |
| C. | \[(1+{{x}^{2}})\frac{dy}{dx}=xy\] |
| D. | \[(1+{{x}^{2}})\frac{dy}{dx}=\frac{x}{y}\] |
| Answer» D. \[(1+{{x}^{2}})\frac{dy}{dx}=\frac{x}{y}\] | |
| 5094. |
The differential equation satisfied by the family of curves \[y=ax\cos \,\left( \frac{1}{x}+b \right)\], where a, b are parameters, is [MP PET 2003] |
| A. | \[{{x}^{2}}{{y}_{2}}+y=0\] |
| B. | \[{{x}^{4}}{{y}_{2}}+y=0\] |
| C. | \[x{{y}_{2}}-y=0\] |
| D. | \[{{x}^{4}}{{y}_{2}}-y=0\] |
| Answer» C. \[x{{y}_{2}}-y=0\] | |
| 5095. |
If \[{{x}^{2}}+{{y}^{2}}=1\] then \[\left( {y}'=\frac{dy}{dx},{y}''=\frac{{{d}^{2}}y}{d{{x}^{2}}} \right)\] [IIT Screening 2000] |
| A. | \[y{y}''-2{{({y}')}^{2}}+1=0\] |
| B. | \[y{y}''+{{({y}')}^{2}}+1=0\] |
| C. | \[y{y}''-{{({y}')}^{2}}-1=0\] |
| D. | \[y{y}''+2{{({y}')}^{2}}+1=0\] |
| Answer» C. \[y{y}''-{{({y}')}^{2}}-1=0\] | |
| 5096. |
\[y=a{{e}^{mx}}+b{{e}^{-mx}}\] satisfies which of the following differential equations [Karnataka CET 2002] |
| A. | \[\frac{dy}{dx}-my=0\] |
| B. | \[\frac{dy}{dx}+my=0\] |
| C. | \[\frac{{{d}^{2}}y}{d{{x}^{2}}}+{{m}^{2}}y=0\] |
| D. | \[\frac{{{d}^{2}}y}{d{{x}^{2}}}-{{m}^{2}}y=0\] |
| Answer» E. | |
| 5097. |
The differential equation of all straight lines passing through the origin is [DCE 2002; Kerala (Engg.) 2002; UPSEAT 2004] |
| A. | \[y=\sqrt{x\frac{dy}{dx}}\] |
| B. | \[\frac{dy}{dx}=y+x\] |
| C. | \[\frac{dy}{dx}=\frac{y}{x}\] |
| D. | None of these |
| Answer» D. None of these | |
| 5098. |
If \[y=a{{x}^{n+1}}+b{{x}^{-n}},\] then \[{{x}^{2}}\frac{{{d}^{2}}y}{d{{x}^{2}}}\] equals to [RPET 2001] |
| A. | \[n(n-1)y\] |
| B. | \[n(n+1)y\] |
| C. | ny |
| D. | n2y |
| Answer» C. ny | |
| 5099. |
The differential equation obtained on eliminating A and B from the equation \[y=A\cos \omega t+B\sin \omega t\] is [Karnataka CET 2000; Pb. CET 2001] |
| A. | \[{y}''=-{{\omega }^{2}}y\] |
| B. | \[{y}''+y=0\] |
| C. | \[{y}''+{y}'=0\] |
| D. | \[{y}''-{{\omega }^{2}}y=0\] |
| Answer» B. \[{y}''+y=0\] | |
| 5100. |
The elimination of the arbitrary constants A, B and C from \[y=A+Bx+C{{e}^{-x}}\]leads to the differential equation [AMU 1999] |
| A. | \[{{{y}'}'}'-{y}'=0\] |
| B. | \[{{{y}'}'}'-{{y}'}'+{y}'=0\] |
| C. | \[{{{y}'}'}'+{{y}'}'=0\] |
| D. | \[{{y}'}'+{{y}'}'-{y}'=0\] |
| Answer» D. \[{{y}'}'+{{y}'}'-{y}'=0\] | |