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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.
| 6651. |
If \[f(x)=\left\{ \begin{align} & x,\ \ \text{when}0 |
| A. | \[\underset{x\to 1/2+}{\mathop{\lim }}\,f(x)=2\] |
| B. | \[\underset{x\to 1/2-}{\mathop{\lim }}\,f(x)=2\] |
| C. | \[f(x)\]is continuous at \[x=\frac{1}{2}\] |
| D. | \[f(x)\]is discontinuous at \[x=\frac{1}{2}\] |
| Answer» E. | |
| 6652. |
If \[f(x)=\left\{ \begin{align} & \frac{1-\cos x}{x},\,x\ne 0 \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,k,\,x=0 \\ \end{align} \right.\]is continuous at \[x=0\] then \[k=\] [Karnataka CET 2004] |
| A. | 0 |
| B. | \[\frac{1}{2}\] |
| C. | \[\frac{1}{4}\] |
| D. | \[-\frac{1}{2}\] |
| Answer» B. \[\frac{1}{2}\] | |
| 6653. |
The function \[f(x)=\frac{1-\sin x+\cos x}{1+\sin x+\cos x}\] is not defined at \[x=\pi .\] The value of \[f(\pi ),\] so that \[f(x)\] is continuous at \[x=\pi \], is [Orissa JEE 2003] |
| A. | \[-\frac{1}{2}\] |
| B. | \[\frac{1}{2}\] |
| C. | ? 1 |
| D. | 1 |
| Answer» D. 1 | |
| 6654. |
If \[f(x)\,=\,\left\{ \begin{matrix} \frac{\sqrt{1+kx}-\sqrt{1-kx}}{x} & \text{,for}-1\le x |
| A. | ? 4 |
| B. | ? 3 |
| C. | ? 2 |
| D. | ? 1 |
| Answer» D. ? 1 | |
| 6655. |
For the function \[f(x)\,=\frac{{{\log }_{e}}(1+x)-{{\log }_{e}}(1-x)}{x}\] to be continuous at \[x=0,\] the value of \[f(0),\] should be [MP PET 2003] |
| A. | ?1 |
| B. | 0 |
| C. | ?2 |
| D. | 2 |
| Answer» E. | |
| 6656. |
The function defined by \[f(x)\,=\,\left\{ \begin{matrix} {{\left( {{x}^{2}}+{{e}^{\frac{1}{2-x}}} \right)}^{-1}} & , & x\ne 2 \\ k & , & x=2 \\ \end{matrix} \right.\], is continuous from right at the point x = 2, then k is equal to [Orissa JEE 2002] |
| A. | 0 |
| B. | 1/4 |
| C. | ?1/4 |
| D. | None of these |
| Answer» C. ?1/4 | |
| 6657. |
If \[f(x)=\left\{ \begin{matrix} \frac{{{x}^{2}}-9}{x-3}\,, & \text{if }x\ne 3 \\ 2x+k\,, & \text{otherwise} \\ \end{matrix} \right.\], is continuous at \[x=3,\] then \[k=\] [Kerala (Engg.) 2002] |
| A. | 3 |
| B. | 0 |
| C. | ?6 |
| D. | 1/6 |
| Answer» C. ?6 | |
| 6658. |
If function \[f(x)=\left\{ \begin{matrix} x\,\,\,\,\,, & \text{if}\,x\,\text{is rational} \\ 1-x, & \text{if}\,x\,\text{is irrational} \\ \end{matrix}, \right.\] then \[f(x)\] is continuous at ...... number of points [UPSEAT 2002] |
| A. | \[\infty \] |
| B. | 1 |
| C. | 0 |
| D. | None of these |
| Answer» D. None of these | |
| 6659. |
If \[f(x)\,=\frac{2-\sqrt{x+4}}{\sin 2x},\,\,(x\ne 0),\] is continuous function at \[x=0\], then \[f(0)\] equals [MP PET 2002] |
| A. | \[\frac{1}{4}\] |
| B. | \[-\frac{1}{4}\] |
| C. | \[\frac{1}{8}\] |
| D. | \[-\frac{1}{8}\] |
| Answer» E. | |
| 6660. |
Let \[f(x)=\,\left\{ \begin{matrix} \frac{\sin \pi x}{5x}, & x\ne 0 \\ k, & x=0 \\ \end{matrix} \right.\]. If \[f(x)\] is continuous at \[x=0,\] then \[k=\] [Karnataka CET 2002] |
| A. | \[\frac{\pi }{5}\] |
| B. | \[\frac{5}{\pi }\] |
| C. | 1 |
| D. | 0 |
| Answer» B. \[\frac{5}{\pi }\] | |
| 6661. |
If \[f(x)=\left\{ \begin{matrix} \frac{1-\sin x}{\pi -2x}, & x\ne \frac{\pi }{2} \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\lambda \,, & x=\frac{\pi }{2} \\ \end{matrix} \right.\], be continuous at \[x=\pi /2,\] then value of \[\lambda \] is [RPET 2002] |
| A. | ?1 |
| B. | 1 |
| C. | 0 |
| D. | 2 |
| Answer» D. 2 | |
| 6662. |
In order that the function\[f(x)={{(x+1)}^{1/x}}\]is continuous at \[x=0\], \[f(0)\]must be defined as [MNR 1989] |
| A. | \[f(0)=0\] |
| B. | \[f(0)=e\] |
| C. | \[f(0)=1/e\] |
| D. | \[f(0)=1\] |
| Answer» C. \[f(0)=1/e\] | |
| 6663. |
If \[f(x)=\,|x|\], then \[f(x)\] is [DCE 2002] |
| A. | Continuous for all x |
| B. | Differentiable at \[x=0\] |
| C. | Neither continuous nor differentiable at \[x=0\] |
| D. | None of these |
| Answer» B. Differentiable at \[x=0\] | |
| 6664. |
The function \[f(x)=\sin |x|\] is [DCE 2002] |
| A. | Continuous for all x |
| B. | Continuous only at certain points |
| C. | Differentiable at all points |
| D. | None of these |
| Answer» B. Continuous only at certain points | |
| 6665. |
In order that the function \[f(x)={{(x+1)}^{\cot \,x}}\] is continuous at \[x=0\], \[f(0)\] must be defined as [UPSEAT 2000; Kurukshetra CEE 2001; Pb. CET 2004] |
| A. | \[f(0)=\frac{1}{e}\] |
| B. | \[f(0)=0\] |
| C. | \[f(0)=e\] |
| D. | None of these |
| Answer» D. None of these | |
| 6666. |
If \[f(x)=\frac{{{x}^{2}}-10x+25}{{{x}^{2}}-7x+10}\] for \[x\]\[\ne \,\]5 and f is continuous at \[x=5,\] then \[f(5)=\] [EAMCET 2001] |
| A. | 0 |
| B. | 5 |
| C. | 10 |
| D. | 25 |
| Answer» B. 5 | |
| 6667. |
The values of A and B such that the function \[f(x)=\left\{ \begin{matrix} -2\sin x, & x\le -\frac{\pi }{2} \\ A\sin x+B, & -\frac{\pi }{2} |
| A. | \[A=0,\,B=1\] |
| B. | \[A=1,\,B=1\] |
| C. | \[A=-1,\,B=1\] |
| D. | \[A=-1,\,B=0\] |
| Answer» D. \[A=-1,\,B=0\] | |
| 6668. |
If the function \[f(x)=\,\left\{ \begin{matrix} 5x-4 & , & \text{if} & 0 |
| A. | ? 1 |
| B. | 0 |
| C. | 1 |
| D. | None of these |
| Answer» B. 0 | |
| 6669. |
The function \[f(x)\,=\left\{ \begin{align} & x+2\,\,\,\,,\,\,\,1\le x\le 2 \\ & 4\,\,\,\,\,\,\,\,\,\,\,,\,\,\,x=2 \\ & 3x-2\,\,,\,\,\,x>2 \\ \end{align} \right.\] is continuous at [DCE 1999] |
| A. | \[x=2\] only |
| B. | \[x\le 2\] |
| C. | \[x\ge 2\] |
| D. | None of these |
| Answer» D. None of these | |
| 6670. |
If \[f(x)=\left\{ \begin{align} & \,\,\,\,\,\,\,\frac{\sin [x]}{[x]+1},\,\,\text{for}\,x>0 \\ & \frac{\cos \frac{\pi }{2}[x]}{[x]},\,\,\text{for}\,x |
| A. | Equal to 0 |
| B. | Equal to 1 |
| C. | Equal to ?1 |
| D. | Indeterminate |
| Answer» B. Equal to 1 | |
| 6671. |
If the function \[f(x)=\left\{ \begin{align} & 1+\sin \frac{\pi x}{2}\,\,,\,\text{for}\,-\infty |
| A. | 0, 2 |
| B. | 1, 1 |
| C. | 2, 0 |
| D. | 2, 1 |
| Answer» D. 2, 1 | |
| 6672. |
Function \[f(x)=\left\{ \begin{align} & \,\,\,x-1,\ x |
| A. | For all real values of x |
| B. | For \[x=2\]only |
| C. | For all real values of x such that \[x\ne 2\] |
| D. | For all integral values of x only |
| Answer» B. For \[x=2\]only | |
| 6673. |
If the function \[f(x)=\left\{ \begin{align} & {{(\cos x)}^{1/x}},\ x\ne 0 \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,k,\,x=0 \\ \end{align} \right.\]is continuous at \[x=0\], then the value of k is [Kurukshetra CEE 1996] |
| A. | 1 |
| B. | ?1 |
| C. | 0 |
| D. | e |
| Answer» B. ?1 | |
| 6674. |
Let \[f(x)=\left\{ \begin{align} & \frac{{{x}^{4}}-5{{x}^{2}}+4}{|(x-1)(x-2)|},\ \ x\ne 1,\ 2 \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,6,\,\,\,x=1 \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,12,\,\,\,x=2 \\ \end{align} \right.\] Then \[f(x)\]is continuous on the set |
| A. | R |
| B. | \[R-\{1\}\] |
| C. | \[R-\{2\}\] |
| D. | \[f:R\to R\] |
| Answer» E. | |
| 6675. |
Let \[f(x)=\left\{ \begin{align} & \frac{x-4}{|x-4|}+a,\ x4 \\ \end{align} \right.\]. Then \[f(x)\]is continuous at \[x=4\] when |
| A. | \[a=0,\ b=0\] |
| B. | \[a=1,\ b=1\] |
| C. | \[a=-1,\ b=1\] |
| D. | \[a=1,\ b=-1\] |
| Answer» E. | |
| 6676. |
If \[f(x)=\left\{ \begin{align} & \sin x,\ x\ne n\pi ,\ \ n\in Z \\ & \,\,\,\,\,\,2,\,\text{otherwise} \\ \end{align} \right.\] and \[g(x)=\left\{ \begin{align} & {{x}^{2}}+1,\ x\ne 0,\,2 \\ & \,\,\,\,\,\,\,\,\,4,\,x=0 \\ & \,\,\,\,\,\,\,\,\,\,5,x=2 \\ \end{align} \right.,\] then \[\underset{x\to 0}{\mathop{\lim }}\,g\,\{f(x)\}\] is [Kurukshetra CEE 1996] |
| A. | 5 |
| B. | 6 |
| C. | 7 |
| D. | 1 |
| Answer» E. | |
| 6677. |
The value of k which makes \[f(x)=\left\{ \begin{align} & \sin \frac{1}{x},\ x\ne 0 \\ & \,\,\,\,\,\,\,\,k,\,x=0 \\ \end{align} \right.\] continuous at \[x=0\]is [MNR 1995] |
| A. | 8 |
| B. | 1 |
| C. | ?1 |
| D. | None of these |
| Answer» E. | |
| 6678. |
If \[f(x)=\left\{ \begin{align} & x+\lambda ,\ x\,3 \\ \end{align} \right.\]is continuous at\[x=3\], then \[\lambda =\] [MP PET 1994, 2001; RPET 1999] |
| A. | 4 |
| B. | 3 |
| C. | 2 |
| D. | 1 |
| Answer» E. | |
| 6679. |
If \[f(x)=\left\{ \begin{align} & \frac{{{x}^{2}}+3x-10}{{{x}^{2}}+2x-15},\ \ \text{when }x\ne -5 \\ & \,\,a\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,,\text{when }x=-5 \\ \end{align} \right.\]is continuous at \[x=-5\], then the value of 'a' will be [MP PET 1987] |
| A. | \[\frac{3}{2}\] |
| B. | \[\frac{7}{8}\] |
| C. | \[\frac{8}{7}\] |
| D. | \[\frac{2}{3}\] |
| Answer» C. \[\frac{8}{7}\] | |
| 6680. |
If \[f(x)=\left\{ \begin{align} & \,\,\,\,{{x}^{2}},\,\text{when}\,x\le 1 \\ & x+5,\text{when }x>\text{1} \\ \end{align} \right.\], then [AISSE 1983] |
| A. | \[f(x)\]is continuous at \[x=1\] |
| B. | \[f(x)\]is discontinuous at\[x=1\] |
| C. | \[\underset{x\to 1}{\mathop{\lim }}\,f(x)=1\] |
| D. | None of these |
| Answer» C. \[\underset{x\to 1}{\mathop{\lim }}\,f(x)=1\] | |
| 6681. |
Let \[f(x)=\left\{ \begin{align} & \frac{{{x}^{3}}+{{x}^{2}}-16x+20}{{{(x-2)}^{2}}},\text{if}\ x\ne 2 \\ & \ \ \ \ \ \,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \,\ k,\ \text{if}\ x=2 \\ \end{align} \right.\]. If \[f(x)\] be continuous for all x, then k = [IIT 1981] |
| A. | 7 |
| B. | ?7 |
| C. | \[\pm 7\] |
| D. | None of these |
| Answer» B. ?7 | |
| 6682. |
If \[f(x)=\left\{ \begin{align} & \frac{x-|x|}{x},\text{when}\,x\ne 0 \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,2,\,\text{when}\,x=0 \\ \end{align} \right.\], then [AI CBSE 1982] |
| A. | \[f(x)\]is continuous at \[x=0\] |
| B. | \[\left[ 0,\frac{\pi }{2} \right]\]is discontinuous at \[x=0\] |
| C. | \[\underset{x\to 0}{\mathop{\lim }}\,f(x)=2\] |
| D. | None of these |
| Answer» C. \[\underset{x\to 0}{\mathop{\lim }}\,f(x)=2\] | |
| 6683. |
If \[f(x)=\left\{ \begin{align} & a{{x}^{2}}-b,\,\,\text{when }0\le x |
| A. | \[a=2,\ b=0\] |
| B. | \[a=1,\ b=-1\] |
| C. | \[a=4,\ b=2\] |
| D. | All the above |
| Answer» E. | |
| 6684. |
If \[f(x)=\left\{ \begin{align} & \,\,\,\,\,\,\,\,\,\frac{1-\cos 4x}{{{x}^{2}}},\ \ \text{when}\,x0 \\ \end{align} \right.\], is continuous at \[x=0\], then the value of 'a' will be [IIT 1990; AMU 2000] |
| A. | 8 |
| B. | ?8 |
| C. | 4 |
| D. | None of these |
| Answer» B. ?8 | |
| 6685. |
If \[f(x)=\left\{ \begin{align} & \,\,\,\,\,\,\,\,\,\,\,\,x\sin x,\,\text{when }0 |
| A. | \[f(x)\]is discontinuous at \[x=\pi /2\] |
| B. | \[f(x)\]is continuous at \[x=\pi /2\] |
| C. | \[f(x)\]is continuous at \[x=0\] |
| D. | None of these |
| Answer» B. \[f(x)\]is continuous at \[x=\pi /2\] | |
| 6686. |
If \[f(x)=\left\{ \begin{align} & 1+x,\ \text{when }x\le 2 \\ & 5-x,\,\text{when }\,x\le 3 \\ \end{align} \right.\], then |
| A. | \[f(x)\]is continuous at \[x=2\] |
| B. | \[f(x)\]is discontinuous at \[A=0,\,B=1\] |
| C. | \[f(x)\]is continuous at\[x=3\] |
| D. | None of these |
| Answer» B. \[f(x)\]is discontinuous at \[A=0,\,B=1\] | |
| 6687. |
If \[f(x)=\left\{ \begin{align} & {{x}^{2}},\,\,\text{when}\,\,\,x\ne 1 \\ & \,\,\,2,\text{when}\,\,x=1 \\ \end{align} \right.\]then |
| A. | \[\underset{x\to 1}{\mathop{\lim }}\,f(x)=2\] |
| B. | \[f(x)\]is continuous at \[x=1\] |
| C. | \[f(x)\]is discontinuous at \[x=1\] |
| D. | None of these |
| Answer» D. None of these | |
| 6688. |
If\[f(x)=\left\{ \begin{align} & \frac{|x-a|}{x-a},\text{when}\,x\ne a \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,1,\text{when}\,x=a \\ \end{align} \right.\],then [AI CBSE 1983] |
| A. | \[f(x)\]is continuous at \[x=a\] |
| B. | \[f(x)\]is discontinuous at \[x=a\] |
| C. | \[\underset{x\to a}{\mathop{\lim }}\,f(x)=1\] |
| D. | None of these |
| Answer» C. \[\underset{x\to a}{\mathop{\lim }}\,f(x)=1\] | |
| 6689. |
If \[f(x)=|x-b|,\]then function [AI CBSE 1984] |
| A. | Is continuous at \[x=1\] |
| B. | Is continuous at \[x=b\] |
| C. | Is discontinuous at\[x=b\] |
| D. | None of these |
| Answer» C. Is discontinuous at\[x=b\] | |
| 6690. |
The function \[f(x)=\frac{\log (1+ax)-\log (1-bx)}{x}\]is not defined at \[x=0\]. The value which should be assigned to f at x =0 so that it is continuos at \[x=0\], is [IIT 1983; MP PET 1995; Karnataka CET 1999; Kurukshetra CEE 2002; AMU 2002] |
| A. | \[a-b\] |
| B. | \[a+b\] |
| C. | \[\log a+\log b\] |
| D. | \[\log a-\log b\] |
| Answer» C. \[\log a+\log b\] | |
| 6691. |
If \[f(x)=\left\{ \begin{align} & \frac{5}{2}-x\,,\,\text{when}\,x2 \\ \end{align} \right.\], then |
| A. | \[f(x)\]is continuous at \[x=2\] |
| B. | \[f(x)\]is discontinuous at \[x=2\] |
| C. | \[\underset{x\to 2}{\mathop{\lim }}\,f(x)=1\] |
| D. | None of these |
| Answer» C. \[\underset{x\to 2}{\mathop{\lim }}\,f(x)=1\] | |
| 6692. |
If \[f(x)=\left\{ \begin{align} & \frac{{{x}^{2}}-1}{x+1},\,\text{when }x\ne -1 \\ & \,\,\,\,\,\,\,\,-2,\,\text{when }x=-1 \\ \end{align} \right.\],then |
| A. | \[\underset{x\to {{(-1)}^{-}}}{\mathop{\lim }}\,f(x)=-2\] |
| B. | \[\underset{x\to {{(-1)}^{+}}}{\mathop{\lim }}\,f(x)=-2\] |
| C. | \[f(x)\]is continuous at \[x=-1\] |
| D. | All the above are correct |
| Answer» E. | |
| 6693. |
If \[f(x)=\left\{ \begin{align} & 1+{{x}^{2}},\,\,\,\text{when}\,0\le x\le 1 \\ & 1-x\,\,\,,\text{when}\,\,x>1 \\ \end{align} \right.\], then |
| A. | \[\underset{x\to {{1}^{+}}}{\mathop{\lim }}\,f(x)\ne 0\] |
| B. | \[\underset{x\to {{1}^{-}}}{\mathop{\lim }}\,f(x)\ne 2\] |
| C. | \[f(x)\]is discontinuous at \[x=1\] |
| D. | None of these |
| Answer» D. None of these | |
| 6694. |
If \[f(x)=\left\{ \begin{align} & \frac{\sin 2x}{5x},\text{when}\,x\ne 0 \\ & \,\,\,\,\,\,\,\,\,\,\,\,k,\text{when }x=0 \\ \end{align} \right.\] is continuous at\[x=0\], then the value of k will be [AI CBSE 1991] |
| A. | 1 |
| B. | \[\frac{2}{5}\] |
| C. | \[-\frac{2}{5}\] |
| D. | None of these |
| Answer» C. \[-\frac{2}{5}\] | |
| 6695. |
If \[f(x)=\left\{ \begin{align} & {{\sin }^{-1}}|x|,\text{when}\,x\ne 0 \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0,\,\text{when }x=0 \\ \end{align} \right.\] then |
| A. | \[\underset{x\to 0+}{\mathop{\lim }}\,f(x)\ne 0\] |
| B. | \[\underset{x\to 0-}{\mathop{\lim }}\,f(x)\ne 0\] |
| C. | \[f(x)\]is continuous at\[x=0\] |
| D. | None of these |
| Answer» D. None of these | |
| 6696. |
If \[f(x)=\left\{ \begin{align} & \,\,\,\,\,\,\,\,\,-{{x}^{2}},\,\text{when }x\le 0 \\ & \,\,\,\,\,5x-4,\,\text{when}0 |
| A. | \[f:R\to R\]is continuous at \[x=0\] |
| B. | \[f(x)\] is continuous \[x=2\] |
| C. | \[f(x)\]is discontinuous at\[x=1\] |
| D. | None of these |
| Answer» C. \[f(x)\]is discontinuous at\[x=1\] | |
| 6697. |
For the function \[f(x)=\left\{ \begin{align} & \frac{{{\sin }^{2}}ax}{{{x}^{2}}},\,\text{when}\,x\ne 0 \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,1,\text{when}\,x=0 \\ \end{align} \right.\] which one is a true statement |
| A. | \[f(x)\]is continuous at \[x=0\] |
| B. | \[f(x)\]is discontinuous at \[x=0\], when \[a\ne \pm 1\] |
| C. | \[\underset{x\to 1}{\mathop{\lim }}\,(1-x+[x-1]+[1-x])\] is continuous at \[x=a\] |
| D. | None of these |
| Answer» C. \[\underset{x\to 1}{\mathop{\lim }}\,(1-x+[x-1]+[1-x])\] is continuous at \[x=a\] | |
| 6698. |
At which points the function\[f(x)=\frac{x}{[x]}\], where\[[.]\] is greatest integer function, is discontinuous |
| A. | Only positive integers |
| B. | All positive and negative integers and (0, 1) |
| C. | All rational numbers |
| D. | None of these |
| Answer» C. All rational numbers | |
| 6699. |
Which of the following statements is true for graph \[f(x)=\log x\] |
| A. | Graph shows that function is continuous |
| B. | Graph shows that function is discontinuous |
| C. | Graph finds for negative and positive values of x |
| D. | Graph is symmetric along x-axis |
| Answer» B. Graph shows that function is discontinuous | |
| 6700. |
If the function \[f(x)=\left\{ \begin{align} & \frac{k\cos x}{\pi -2x},\text{when }x\ne \frac{\pi }{2} \\ & 3,\ \ \ \ \ \ \ \ \ \text{when }x=\frac{\pi }{2} \\ \end{align} \right.\] be continuous at \[x=\frac{\pi }{2}\], then k = |
| A. | 3 |
| B. | 6 |
| C. | 12 |
| D. | None of these |
| Answer» C. 12 | |