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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.
| 5751. |
\[\frac{d}{dx}\left[ {{\sin }^{2}}{{\cot }^{-1}}\left\{ \sqrt{\frac{1-x}{1+x}} \right\} \right]\] equals [MP PET 2002] |
| A. | \[-1\] |
| B. | \[\frac{1}{2}\] |
| C. | \[-\frac{1}{2}\] |
| D. | 1 |
| Answer» C. \[-\frac{1}{2}\] | |
| 5752. |
If \[y={{\cos }^{-1}}\left( \frac{3\cos x+4\sin x}{5} \right)\], then \[\frac{dy}{dx}=\] |
| A. | 0 |
| B. | 1 |
| C. | \[-1\] |
| D. | \[\frac{1}{2}\] |
| Answer» C. \[-1\] | |
| 5753. |
Let \[3f(x)-2f(1/x)=x,\] then \[f'(2)\]is equal to [MP PET 2000] |
| A. | \[2/7\] |
| B. | \[1/2\] |
| C. | 2 |
| D. | \[7/2\] |
| Answer» C. 2 | |
| 5754. |
If \[f(x)={{\cot }^{-1}}\left( \frac{{{x}^{x}}-{{x}^{-x}}}{2} \right)\,,\]then \[f'(1)\] is equal to [RPET 2000] |
| A. | ? 1 |
| B. | 1 |
| C. | \[\log \,\,2\] |
| D. | \[-\log \,2\] |
| Answer» B. 1 | |
| 5755. |
If \[y={{\sin }^{-1}}\frac{\sqrt{(1+x)}+\sqrt{(1-x)}}{2}\], then \[\frac{dy}{dx}=\] |
| A. | \[\frac{1}{\sqrt{(1-{{x}^{2}})}}\] |
| B. | \[-\frac{1}{\sqrt{(1-{{x}^{2}})}}\] |
| C. | \[-\frac{1}{2\sqrt{(1-{{x}^{2}})}}\] |
| D. | None of these |
| Answer» D. None of these | |
| 5756. |
If \[y={{\tan }^{-1}}\sqrt{\frac{a-x}{a+x}}\], then \[\frac{dy}{dx}=\] |
| A. | \[{{\cos }^{-1}}\frac{x}{a}\] |
| B. | \[-{{\cos }^{-1}}\frac{x}{a}\] |
| C. | \[\frac{1}{2}{{\cos }^{-1}}\frac{x}{a}\] |
| D. | None of these |
| Answer» E. | |
| 5757. |
The differential coefficient of \[{{\cos }^{-1}}\left\{ \sqrt{\frac{1+x}{2}} \right\}\]with respect to x is [MP PET 1993] |
| A. | \[-\frac{1}{2\sqrt{1-{{x}^{2}}}}\] |
| B. | \[\frac{1}{2\sqrt{1-{{x}^{2}}}}\] |
| C. | \[\frac{1}{\sqrt{1-x}}\] |
| D. | \[{{\sin }^{-1}}\left\{ \sqrt{\frac{1+x}{2}} \right\}\] |
| Answer» B. \[\frac{1}{2\sqrt{1-{{x}^{2}}}}\] | |
| 5758. |
If \[y={{\sin }^{-1}}\sqrt{1-{{x}^{2}}}\], then \[dy/dx=\] [AISSE 1987] |
| A. | \[\frac{1}{\sqrt{1-{{x}^{2}}}}\] |
| B. | \[\frac{1}{\sqrt{1+{{x}^{2}}}}\] |
| C. | \[-\frac{1}{\sqrt{1-{{x}^{2}}}}\] |
| D. | \[-\frac{1}{\sqrt{{{x}^{2}}-1}}\] |
| Answer» D. \[-\frac{1}{\sqrt{{{x}^{2}}-1}}\] | |
| 5759. |
\[\frac{d}{dx}{{\sin }^{-1}}(2ax\sqrt{1-{{a}^{2}}{{x}^{2}}})=\] |
| A. | \[\frac{2a}{\sqrt{{{a}^{2}}-{{x}^{2}}}}\] |
| B. | \[\frac{a}{\sqrt{{{a}^{2}}-{{x}^{2}}}}\] |
| C. | \[\frac{2a}{\sqrt{1-{{a}^{2}}{{x}^{2}}}}\] |
| D. | \[\frac{a}{\sqrt{1-{{a}^{2}}{{x}^{2}}}}\] |
| Answer» D. \[\frac{a}{\sqrt{1-{{a}^{2}}{{x}^{2}}}}\] | |
| 5760. |
\[\frac{d}{dx}\left\{ {{\cos }^{-1}}\left( \frac{1-{{x}^{2}}}{1+{{x}^{2}}} \right) \right\}=\] [AISSE 1984] |
| A. | \[\frac{1}{1+{{x}^{2}}}\] |
| B. | \[-\frac{1}{1+{{x}^{2}}}\] |
| C. | \[-\frac{2}{1+{{x}^{2}}}\] |
| D. | \[\frac{2}{1+{{x}^{2}}}\] |
| Answer» E. | |
| 5761. |
If \[y=\sin (2{{\sin }^{-1}}x),\]then \[\frac{dy}{dx}=\] [AI CBSE 1983] |
| A. | \[\frac{2-4{{x}^{2}}}{\sqrt{1-{{x}^{2}}}}\] |
| B. | \[\frac{2+4{{x}^{2}}}{\sqrt{1-{{x}^{2}}}}\] |
| C. | \[\frac{2-4{{x}^{2}}}{\sqrt{1+{{x}^{2}}}}\] |
| D. | \[\frac{2+4{{x}^{2}}}{\sqrt{1+{{x}^{2}}}}\] |
| Answer» B. \[\frac{2+4{{x}^{2}}}{\sqrt{1-{{x}^{2}}}}\] | |
| 5762. |
If \[\sqrt{1-{{x}^{2}}}+\sqrt{1-{{y}^{2}}}=a(x-y)\], then \[\frac{dy}{dx}=\] [MNR 1983; ISM Dhanbad 1987; RPET 1991] |
| A. | \[\sqrt{\frac{1-{{x}^{2}}}{1-{{y}^{2}}}}\] |
| B. | \[\sqrt{\frac{1-{{y}^{2}}}{1-{{x}^{2}}}}\] |
| C. | \[\sqrt{\frac{{{x}^{2}}-1}{1-{{y}^{2}}}}\] |
| D. | \[\sqrt{\frac{{{y}^{2}}-1}{1-{{x}^{2}}}}\] |
| Answer» C. \[\sqrt{\frac{{{x}^{2}}-1}{1-{{y}^{2}}}}\] | |
| 5763. |
Consider \[f(x)=\left\{ \begin{align} & \frac{{{x}^{2}}}{|x|},\,x\ne 0 \\ & \,\,\,\,\,\,\,0,\,x=0 \\ \end{align} \right.\] [EAMCET 1994] |
| A. | \[f(x)\]is discontinuous everywhere |
| B. | \[f(x)\]is continuous everywhere |
| C. | \[f'(x)\]exists in \[(-1,1)\] |
| D. | \[f'(x)\]exists in \[(-2,2)\] |
| Answer» C. \[f'(x)\]exists in \[(-1,1)\] | |
| 5764. |
The function \[f(x)=|x|\] at \[x=0\] is [MP PET 1993] |
| A. | Continuous but non-differentiable |
| B. | Discontinuous and differentiable |
| C. | Discontinuous and non-differentiable |
| D. | Continuous and differentiable |
| Answer» B. Discontinuous and differentiable | |
| 5765. |
The function \[f(x)=\left\{ \begin{align} & x,\,\,\text{if 0}\le x\le \text{1} \\ & \text{1,}\,\text{ if}\,1 |
| A. | Continuous at all x, \[0\le x\le 2\]and differentiable at all x, except \[2/3\]in the interval [0,2] |
| B. | Continuous and differentiable at all x in [0,2] |
| C. | Not continuous at any point in [0,2] |
| D. | Not differentiable at any point [0,2] |
| Answer» B. Continuous and differentiable at all x in [0,2] | |
| 5766. |
There exists a function \[f(x)\]satisfying \[f(0)=1\], \[f'(0)=-1,\ f(x)>0\]for all x and [Kurukshetra CEE 1998] |
| A. | \[f(x)<0\],\[\forall x\] |
| B. | \[-1<f''(x)<0,\,\forall x\] |
| C. | \[-2<f''(x)\le -1,\,\forall x\] |
| D. | \[f''(x)<-2,\,\forall x\] |
| Answer» E. | |
| 5767. |
Let \[h(x)=\min \{x,\,{{x}^{2}}\},\]for every real number of x. Then [IIT 1998] |
| A. | h is continuous for all x |
| B. | h is differentiable for all x |
| C. | \[h'(x)=1\], for all \[b=1\] |
| D. | h is not differentiable at two values of x |
| Answer» B. h is differentiable for all x | |
| 5768. |
Function \[y={{\sin }^{-1}}\left( \frac{2x}{1+{{x}^{2}}} \right)\] is not differentiable for [IIT Screening] |
| A. | \[|x|\,<1\] |
| B. | \[x=1,-1\] |
| C. | \[|x|\,>1\] |
| D. | None of these |
| Answer» C. \[|x|\,>1\] | |
| 5769. |
If \[f(x)=x(\sqrt{x}-\sqrt{x+1}),\] then [IIT 1985] |
| A. | \[f(x)\] is continuous but non- differentiable at \[x=0\] |
| B. | \[f(x)\] is differentiable at \[x=0\] |
| C. | \[f(x)\] is not differentiable at \[x=0\] |
| D. | None of these |
| Answer» D. None of these | |
| 5770. |
The set of all those points, where the function \[f(x)=\frac{x}{1+|x|}\]is differentiable, is |
| A. | \[(-\infty ,\infty )\] |
| B. | \[[0,\infty ]\] |
| C. | \[(-\infty ,\,0)\cup (0,\infty )\] |
| D. | \[(0,\infty )\] |
| Answer» B. \[[0,\infty ]\] | |
| 5771. |
If \[f(x)=\left\{ \begin{align} & a{{x}^{2}}+b;\,\,x\le 0 \\ & \,\,\,\,\,\,\,\,\,{{x}^{2}};x>0\, \\ \end{align} \right.\] possesses derivative at \[x=0\], then |
| A. | \[a=0,b=0\] |
| B. | \[a>0,=0\] |
| C. | \[a\in R,=0\] |
| D. | None of these |
| Answer» D. None of these | |
| 5772. |
If \[f(x)=\left\{ \begin{align} & \,\,\,\,\,\,\,\,\,\,\,\,1,\,\,x |
| A. | 1 |
| B. | 0 |
| C. | \[\infty \] |
| D. | Does not exist |
| Answer» E. | |
| 5773. |
If \[f(x)=\left\{ \begin{align} & x+2\,,-1 |
| A. | 1 |
| B. | ? 1 |
| C. | 0 |
| D. | Does not exist |
| Answer» E. | |
| 5774. |
If \[f(x)=\left\{ \begin{align} & x,\,\,\,\,\,\,\,\,\,\,\,0\le x\le 1 \\ & 2x-1,\,\,\,1 |
| A. | f is discontinuous at\[x=1\] |
| B. | f is differentiable at \[x=1\] |
| C. | f is continuous but not differentiable at \[x=1\] |
| D. | None of these |
| Answer» D. None of these | |
| 5775. |
If \[f(x)=|x-3|,\]then f is [SCRA 1996; RPET 1997] |
| A. | Discontinuous at \[x=2\] |
| B. | Not differentiable \[x=2\] |
| C. | Differentiable at \[x=3\] |
| D. | Continuous but not differentiable at \[x=3\] |
| Answer» E. | |
| 5776. |
Which of the following is not true [Kurukshetra CEE 1996] |
| A. | Every differentiable function is continuous |
| B. | If derivative of a function is zero at all points, then the function is constant |
| C. | If a function has maximum or minima at a point, then the function is differentiable at that point and its derivative is zero |
| D. | If a function is constant, then its derivative is zero at all points |
| Answer» D. If a function is constant, then its derivative is zero at all points | |
| 5777. |
The function which is continuous for all real values of x and differentiable at \[x=0\]is [MP PET 1996] |
| A. | \[|x|\] |
| B. | \[\log x\] |
| C. | sin x |
| D. | \[{{x}^{\frac{1}{2}}}\] |
| Answer» D. \[{{x}^{\frac{1}{2}}}\] | |
| 5778. |
The function \[f(x)=({{x}^{2}}-1)|{{x}^{2}}-3x+2|+\cos (|x|)\] is not differentiable at [IIT 1999] |
| A. | ?1 |
| B. | 0 |
| C. | 1 |
| D. | 2 |
| Answer» E. | |
| 5779. |
Let \[f(x+y)=f(x)+f(y)\]and \[f(x)={{x}^{2}}g(x)\] for all \[x,y\in R\], where \[g(x)\] is continuous function. Then \[f'(x)\] is equal to |
| A. | \[g'(x)\] |
| B. | \[g(0)\] |
| C. | \[g(0)+g'(x)\] |
| D. | 0 |
| Answer» E. | |
| 5780. |
If \[f(x)={{x}^{2}}-2x+4\] and \[\frac{f(5)-f(1)}{5-1}=f'(c)\] then value of c will be [AMU 2005] |
| A. | 0 |
| B. | 1 |
| C. | 2 |
| D. | 3 |
| Answer» E. | |
| 5781. |
Let \[f(x+y)=f(x)f(y)\] and \[f(x)=1+\sin (3x)g(x)\] where \[g(x)\] is continuous then \[f'(x)\] is [Kerala (Engg.) 2005] |
| A. | \[f(x)g(0)\] |
| B. | \[3g(0)\] |
| C. | \[f(x)\cos 3x\] |
| D. | \[3f(x)g(0)\] |
| E. | \[3f(x)g(x)\] |
| Answer» D. \[3f(x)g(0)\] | |
| 5782. |
Let f be continuous on [1, 5] and differentiable in (1, 5). If \[f(1)\]=?3 and \[f'(x)\ge 9\] for all \[x\in (1,\ 5)\], then [Kerala (Engg.) 2005] |
| A. | \[f(5)\ge 33\] |
| B. | \[f(5)\ge 36\] |
| C. | \[f(5)\le 36\] |
| D. | \[f(5)\ge 9\] |
| E. | \[f(5)\le 9\] |
| Answer» C. \[f(5)\le 36\] | |
| 5783. |
If \[f(x)\] is a differentiable function such that \[f:R\to r\] and \[f\left( \frac{1}{n} \right)=0\ \forall \ n\ge 1,n\in I\] then [IIT Screening 2005] |
| A. | \[f(x)=0\ \forall \ x\in (0,\,1)\] |
| B. | \[f(x)=0\forall x\in (0,\,1)\] |
| C. | \[f(0)=0\] but \[f'(0)\] may or may not be 0 |
| D. | \[|f(x)|\,\le 1\ \forall \ x\in (0,\,1)\] |
| Answer» C. \[f(0)=0\] but \[f'(0)\] may or may not be 0 | |
| 5784. |
If \[f(x)\] is twice differentiable polynomial function such that \[f(1)=1,f(2)=-4,f(3)=9\], then [IIT Screening 2005] |
| A. | \[f''(x)=2,\forall x\in R\] |
| B. | There exist at least one \[x\in (1,\,3)\] such that \[f''(x)=2\] |
| C. | There exist at least one \[x\in (2,\,3)\] such that \[f'(x)=5=f''(x)\] |
| D. | There exist at least one \[x\in (1,\,2)\] such that \[f(x)=3\] |
| Answer» C. There exist at least one \[x\in (2,\,3)\] such that \[f'(x)=5=f''(x)\] | |
| 5785. |
\[f(x)=\left| \left| x \right|-1 \right|\] is not differentiable at [IIT Screening 2005] |
| A. | 0 |
| B. | \[\pm 1,\,0\] |
| C. | 1 |
| D. | \[\pm \,1\] |
| Answer» C. 1 | |
| 5786. |
If \[f(x)=\left\{ \begin{align} & x\frac{{{e}^{(1/x)}}-{{e}^{(-1/x)}}}{{{e}^{(1/x)}}+{{e}^{(-1/x)}}},\,x\ne 0 \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0,\,x=0 \\ \end{align} \right.\] then which of the following is true [Kurukshetra CEE 1998] |
| A. | f is continuous and differentiable at every point |
| B. | f is continuous at every point but is not differentiable |
| C. | f is differentiable at every point |
| D. | f is differentiable only at the origin |
| Answer» C. f is differentiable at every point | |
| 5787. |
Let \[f\] be differentiable for all \[x\] . If \[f(1)=-2\] and \[f'(x)\ge 2\] for \[x\in [1,6]\] , then [AIEEE 2005] |
| A. | \[f(6)<5\] |
| B. | \[f(6)=5\] |
| C. | \[f(6)\ge 8\] |
| D. | \[f(6)<8\] |
| Answer» D. \[f(6)<8\] | |
| 5788. |
If f is a real- valued differentiable function satisfying \[|f(x)-f(y)|\le {{(x-y)}^{2}},x,y\in R\] and \[f(0)=0\] , then \[f(1)\] equal [AIEEE 2005] |
| A. | 2 |
| B. | 1 |
| C. | ?1 |
| D. | 0 |
| Answer» E. | |
| 5789. |
Suppose \[f(x)\] is differentiable at \[x=1\] and \[\underset{h\to 0}{\mathop{\lim }}\,\frac{1}{h}f(1+h)=5\] , then \[f'(1)\] equals [AIEEE 2005] |
| A. | 5 |
| B. | 6 |
| C. | 3 |
| D. | 4 |
| Answer» B. 6 | |
| 5790. |
The value of m for which the function \[f(x)=\left\{ \begin{align} & m{{x}^{2}},\,x\le 1 \\ & \,\,\,\,2x,\,x>1 \\ \end{align} \right.\] is differentiable at \[x=1\] ,is [MP PET 1998] |
| A. | 0 |
| B. | 1 |
| C. | 2 |
| D. | Does not exist |
| Answer» E. | |
| 5791. |
If \[f(x)\,=\frac{x}{1+|x|}\] for \[x\in R,\] then \[f'(0)=\] [EAMCET 2003] |
| A. | 0 |
| B. | 1 |
| C. | 2 |
| D. | 3 |
| Answer» C. 2 | |
| 5792. |
If \[f(x)\,=\,\,\left\{ \begin{matrix} \frac{x-1}{2{{x}^{2}}-7x+5} & \text{for }x\ne 1 \\ -\frac{1}{3} & \text{for }x=1 \\ \end{matrix}\,\,, \right.\] then \[f'(1)=\] [EAMCET 2003] |
| A. | ?1/9 |
| B. | ?2/9 |
| C. | ?1/3 |
| D. | 1/3 |
| Answer» C. ?1/3 | |
| 5793. |
Which of the following is not true [Kerala (Engg.) 2002] |
| A. | A polynomial function is always continuous |
| B. | A continuous function is always differentiable |
| C. | A differentiable function is always continuous |
| D. | is continuous for all x |
| Answer» C. A differentiable function is always continuous | |
| 5794. |
The function \[f(x)={{x}^{2}}\,\,\sin \frac{1}{x},\,x\ne \,0,\,\,f(0)\,=0\] at \[x=0\] [MP PET 2003] |
| A. | Is continuous but not differentiable |
| B. | Is discontinuous |
| C. | Is having continuous derivative |
| D. | Is continuous and differentiable |
| Answer» E. | |
| 5795. |
sThe function \[f(x)=\left\{ \begin{matrix} {{e}^{2x}}-1 & , & x\le 0 \\ ax+\frac{b{{x}^{2}}}{2}-1 & , & x>0 \\ \end{matrix} \right.\] is continuous and differentiable for [AMU 2002] |
| A. | \[[.]\] |
| B. | \[a=2,\,b=4\] |
| C. | \[a=2,\,\]any \[b\] |
| D. | Any \[a,\,\,\,b=4\] |
| Answer» D. Any \[a,\,\,\,b=4\] | |
| 5796. |
If \[f(x)=\left\{ \begin{align} & \ x+1,\ \text{when}\,x |
| A. | 0 |
| B. | 1 |
| C. | 2 |
| D. | Does not exist |
| Answer» E. | |
| 5797. |
Let \[f(x)=\left\{ \begin{matrix} 0, & x |
| A. | f is continuous but not differentiable |
| B. | f is differentiable but not continuous |
| C. | \[{f}'\] is continuous but not differentiable |
| D. | \[{f}'\] is continuous and differentiable |
| Answer» D. \[{f}'\] is continuous and differentiable | |
| 5798. |
The left-hand derivative of \[f(x)=[x]\sin (\pi x)\] at \[x=k,\,\,k\]is an integer and \[[x]\]= greatest integer \[\le x,\,\] is [IIT Screening 2001] |
| A. | \[{{(-1)}^{k}}\,\,(k-1)\,\pi \] |
| B. | \[{{(-1)}^{k-1}}(k-1)\,\pi \] |
| C. | \[{{(-1)}^{k}}k\pi \] |
| D. | \[{{(-1)}^{k-1}}k\,\pi \] |
| Answer» B. \[{{(-1)}^{k-1}}(k-1)\,\pi \] | |
| 5799. |
A function \[f(x)\,=\left\{ \begin{matrix} 1+x, & x\le 2 \\ 5-x, & x>2 \\ \end{matrix} \right.\,\] is [AMU 2001] |
| A. | Not continuous at \[x=2\] |
| B. | Differentiable at \[x=2\] |
| C. | Continuous but not differentiable at \[x=2\] |
| D. | None of these |
| Answer» D. None of these | |
| 5800. |
The function \[y={{e}^{-|x|}}\] is [AMU 2000] |
| A. | Continuous and differentiable at \[x=0\] |
| B. | Neither continuous nor differentiable at \[x=0\] |
| C. | Continuous but not differentiable at \[x=0\] |
| D. | Not continuous but differentiable at \[x=0\] |
| Answer» D. Not continuous but differentiable at \[x=0\] | |