MCQOPTIONS
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MCQOPTIONS
This section includes 54 Mcqs, each offering curated multiple-choice questions to sharpen your Mathematics knowledge and support exam preparation. Choose a topic below to get started.
| 1. |
The domain of the derivative of the function \[f(x)=\left\{ \begin{align} & {{\tan }^{-1}}x\ \ \ \ \ ,\ |x|\ \le 1 \\ & \frac{1}{2}(|x|\ -1)\ ,\ |x|\ >1 \\ \end{align} \right.\] is [IIT Screening 2002] |
| A. | \[R-\{0\}\] |
| B. | \[R-\{1\}\] |
| C. | \[R-\{-1\}\] |
| D. | \[R-\{-1,\ 1\}\] |
| Answer» D. \[R-\{-1,\ 1\}\] | |
| 2. |
The function \[f(x)=\frac{{{\sec }^{-1}}x}{\sqrt{x-[x]}},\] where [.] denotes the greatest integer less than or equal to x is defined for all x belonging to |
| A. | R |
| B. | \[R-\{(-1,\ 1)\cup (n|n\in Z)\}\] |
| C. | \[{{R}^{+}}-(0,\ 1)\] |
| D. | \[{{R}^{+}}-\{n|n\in N\}\] |
| Answer» C. \[{{R}^{+}}-(0,\ 1)\] | |
| 3. |
The function f satisfies the functional equation \[3f(x)+2f\left( \frac{x+59}{x-1} \right)=10x+30\] for all real \[x\ne 1\]. The value of \[f(7)\] is [Kerala (Engg.) 2005] |
| A. | 8 |
| B. | 4 |
| C. | ?8 |
| D. | 11 |
| E. | 44 |
| Answer» C. ?8 | |
| 4. |
If \[f(x)=\frac{x-|x|}{|x|}\], then \[f(-1)=\] [SCRA 1996] |
| A. | 1 |
| B. | ?2 |
| C. | 0 |
| D. | 2 |
| Answer» C. 0 | |
| 5. |
If x is real, then value of the expression \[\frac{{{x}^{2}}+14x+9}{{{x}^{2}}+2x+3}\] lies between [UPSEAT 2002] |
| A. | 5 and 4 |
| B. | 5 and ?4 |
| C. | ? 5 and 4 |
| D. | None of these |
| Answer» D. None of these | |
| 6. |
If \[f(x)=\left\{ \begin{align} & \,\,\,\,\,\,\,\,\,\,\,\,1,\,\,x<0 \\ & 1+\sin x,\,\,0\le x<\frac{\pi }{2} \\ \end{align} \right.\]then \[f'(0)=\] [MP PET 1994] |
| A. | 1 |
| B. | 0 |
| C. | \[\infty \] |
| D. | Does not exist |
| Answer» E. | |
| 7. |
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 | |
| 8. |
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. | |
| 9. |
The domain of the function \[f(x)=\frac{1}{{{\log }_{10}}(1-x)}+\sqrt{x+2}\] is [DCE 2000] |
| A. | \[]-3,\ -2.5[\cup ]-2.5,\ -2[\] |
| B. | \[[-2,\ 0[\cup ]0,\ 1[\] |
| C. | ]0,1[ |
| D. | None of these |
| Answer» C. ]0,1[ | |
| 10. |
If function \[f(x)=\frac{1}{2}-\tan \left( \frac{\pi x}{2} \right)\]; \[(-1 |
| A. | \[(-1,\ 1)\] |
| B. | \[\left[ -\frac{1}{2},\ \frac{1}{2} \right]\] |
| C. | \[\left[ -1,\ \frac{1}{2} \right]\] |
| D. | \[\left[ -\frac{1}{2},\ -1 \right]\] |
| Answer» B. \[\left[ -\frac{1}{2},\ \frac{1}{2} \right]\] | |
| 11. |
Let \[g(x)=x.\,f(x),\]where \[f(x)=\left\{ \begin{align} & x\sin \frac{1}{x},\,x\ne 0 \\ & \,\,\,\,\,\,\,\,\,\,\,\,0,\,x=0 \\ \end{align} \right.\] at \[x=0\] [IIT Screening 1994; UPSEAT 2004] |
| A. | g is differentiable but g' is not continuous |
| B. | g is differentiable while f is not |
| C. | Both f and g are differentiable |
| D. | g is differentiable and g' is continuous |
| Answer» B. g is differentiable while f is not | |
| 12. |
The value of \[p\] for which the function \[f(x)=\left\{ \begin{align} & \frac{{{({{4}^{x}}-1)}^{3}}}{\sin \frac{x}{p}\log \left[ 1+\frac{{{x}^{2}}}{3} \right]},\,x\ne 0 \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,12{{(\log 4)}^{3}},\,\,x=0 \\ \end{align} \right.\]may be continuous at \[x=0\], is[Orissa JEE 2004] |
| A. | 1 |
| B. | 2 |
| C. | 3 |
| D. | None of these |
| Answer» E. | |
| 13. |
Let \[f(x)\]be defined for all \[x>0\]and be continuous. Let \[f(x)\]satisfy \[f\left( \frac{x}{y} \right)=f(x)-f(y)\]for all x, y and \[f(e)=1,\]then [IIT 1995] |
| A. | \[f(x)=\ln x\] |
| B. | \[f(x)\]is bounded |
| C. | \[f\left( \frac{1}{x} \right)\to 0\]as\[x\to 0\] |
| D. | \[x\,f(x)\to 1\]as \[x\to 0\] |
| Answer» B. \[f(x)\]is bounded | |
| 14. |
The function\[f(x)=[x]\cos \left[ \frac{2x-1}{2} \right]\pi ,\,\]where\[[.]\] denotes the greatest integer function, is discontinuous at [IIT 1995] |
| A. | All x |
| B. | No x |
| C. | All integer points |
| D. | x which is not an integer |
| Answer» D. x which is not an integer | |
| 15. |
The values of a and b such that \[\underset{x\to 0}{\mathop{\lim }}\,\frac{x(1+a\cos x)-b\sin x}{{{x}^{3}}}=1\], are [Roorkee 1996] |
| A. | \[\frac{5}{2},\ \frac{3}{2}\] |
| B. | \[\frac{5}{2},\ -\frac{3}{2}\] |
| C. | \[-\frac{5}{2},\ -\frac{3}{2}\] |
| D. | None of these |
| Answer» D. None of these | |
| 16. |
If \[\underset{x\to a}{\mathop{\lim }}\,\frac{{{a}^{x}}-{{x}^{a}}}{{{x}^{x}}-{{a}^{a}}}=-1\], then[EAMCET 2003] |
| A. | \[a=1\] |
| B. | \[a=0\] |
| C. | \[a=e\] |
| D. | None of these |
| Answer» B. \[a=0\] | |
| 17. |
\[\underset{x\to \infty }{\mathop{\lim }}\,\frac{{{x}^{n}}}{{{e}^{x}}}=0\] for [IIT 1992] |
| A. | No value of n |
| B. | n is any whole number |
| C. | \[n=0\] only |
| D. | \[n=2\] only |
| Answer» C. \[n=0\] only | |
| 18. |
True statement for \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt{2+3x}-\sqrt{2-3x}}\] is [BIT Ranchi 1982] |
| A. | Does not exist |
| B. | Lies between 0 and \[\frac{1}{2}\] |
| C. | Lies between \[\frac{1}{2}\] and 1 |
| D. | Greater then 1 |
| Answer» C. Lies between \[\frac{1}{2}\] and 1 | |
| 19. |
\[\underset{x\to 1}{\mathop{\lim }}\,(1-x)\tan \left( \frac{\pi x}{2} \right)=\] [IIT 1978, 84; RPET 1997, 2001; UPSEAT 2003; Pb. CET 2003] |
| A. | \[\frac{\pi }{2}\] |
| B. | \[\pi \] |
| C. | \[\frac{2}{\pi }\] |
| D. | 0 |
| Answer» D. 0 | |
| 20. |
Let \[f(x)=\left\{ \begin{align} & {{x}^{p}}\sin \frac{1}{x},x\ne 0 \\ & 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,,x=0 \\ \end{align} \right.\] then \[f(x)\]is continuous but not differential at \[x=0\] if [DCE 2005] |
| A. | \[0<p\le 1\] |
| B. | \[1\le p<\infty \] |
| C. | \[-\infty <p<0\] |
| D. | p = 0 |
| Answer» B. \[1\le p<\infty \] | |
| 21. |
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 | |
| 22. |
If \[f(x)=\left\{ \begin{align} & x\sin \frac{1}{x},\,\,x\ne 0 \\ & \,\,\,\,\,\,\,\,\,\,\,\,k,\,\,x=0 \\ \end{align} \right.\]is continuous at \[x=0\], then the value of k is [MP PET 1999; AMU 1999; RPET 2003] |
| A. | 1 |
| B. | ?1 |
| C. | 0 |
| D. | 2 |
| Answer» D. 2 | |
| 23. |
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. \[-\frac{2}{5}\] | |
| 24. |
If \[f(x)=\left\{ \begin{align} & \frac{1}{x}\sin {{x}^{2}},\,x\ne 0 \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,0,\,x=0 \\ \end{align} \right.\], 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 | |
| 25. |
If \[f(x)=\left\{ \begin{align} & {{x}^{2}}\sin \frac{1}{x},\ \ \ \text{when }x\ne 0 \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,0,\,\,\,\,\text{when}\,x=0 \\ \end{align} \right.\], then |
| A. | \[f(0+0)=1\] |
| B. | \[f(0-0)=1\] |
| C. | f is continuous at\[x=0\] |
| D. | None of these |
| Answer» D. None of these | |
| 26. |
The natural domain of the real valued function defined by \[f(x)=\sqrt{{{x}^{2}}-1}+\sqrt{{{x}^{2}}+1}\] is [SCRA 1996] |
| A. | \[1<x<\infty \] |
| B. | \[-\infty <x<\infty \] |
| C. | \[-\infty <x<-1\] |
| D. | \[(-\infty ,\ \infty )-(-1,\ 1)\] |
| Answer» E. | |
| 27. |
If in greatest integer function, the domain is a set of real numbers, then range will be set of |
| A. | Real numbers |
| B. | Rational numbers |
| C. | Imaginary numbers |
| D. | Integers |
| Answer» E. | |
| 28. |
Which one of the following is a objective function on the set of real numbers [Kerala (Engg.) 2002] |
| A. | \[2x-5\] |
| B. | \[|x|\] |
| C. | \[{{x}^{2}}\] |
| D. | \[{{x}^{2}}+1\] |
| Answer» B. \[|x|\] | |
| 29. |
If \[f:R\to S\] defined by \[f(x)=\sin x-\sqrt{3}\cos x+1\]is onto, then the interval of S is [AIEEE 2004; IIT Screening 2004] |
| A. | [?1, 3] |
| B. | [1, 1] |
| C. | [0, 1] |
| D. | [0, ?1] |
| Answer» B. [1, 1] | |
| 30. |
Set A has 3 elements and set B has 4 elements. The number of injection that can be defined from A to B is[UPSEAT 2001] |
| A. | 144 |
| B. | 12 |
| C. | 24 |
| D. | 64 |
| Answer» D. 64 | |
| 31. |
The graph of the function \[y=f(x)\] is symmetrical about the line \[x=2\], then [AIEEE 2004] |
| A. | \[f(x)=-f(-x)\] |
| B. | \[f(2+x)=f(2-x)\] |
| C. | \[f(x)=f(-x)\] |
| D. | \[f(x+2)=f(x-2)\] |
| Answer» C. \[f(x)=f(-x)\] | |
| 32. |
The Domain of function \[f(x)={{\log }_{e}}(x-[x])\] is [AMU 2005] |
| A. | R |
| B. | R-Z |
| C. | \[(0,+\infty )\] |
| D. | Z |
| Answer» B. R-Z | |
| 33. |
If \[f(x)=2{{x}^{6}}+3{{x}^{4}}+4{{x}^{2}}\] then \[f'(x)\] is [DCE 2005] |
| A. | Even function |
| B. | An odd function |
| C. | Neither even nor odd |
| D. | None of these |
| Answer» C. Neither even nor odd | |
| 34. |
Let \[g(x)=1+x-[x]\] and \[f(x)=\left\{ \begin{align} & -1,\ x<0 \\ & 0,\ \ x=0,\\\ & \text{1,}\ \ \ \text{x}>\text{0} \\ \end{align} \right.\]then for all \[x,\ f(g(x))\] is equal to [IIT Screening 2001; UPSEAT 2001] |
| A. | x |
| B. | 1 |
| C. | \[f(x)\] |
| D. | \[g(x)\] |
| Answer» C. \[f(x)\] | |
| 35. |
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 | |
| 36. |
The number of points at which the function \[f(x)=|x-0.5|+|x-1|+\tan x\] does not have a derivative in the interval (0, 2), is [MNR 1995] |
| A. | 1 |
| B. | 2 |
| C. | 3 |
| D. | 4 |
| Answer» D. 4 | |
| 37. |
Let \[f(x)=\left\{ \begin{align} & 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\forall x<0 \\ & 1+\sin x\,\,\,\forall 0\le x\le \pi /2 \\ \end{align} \right.\], then what is the value of \[f'(x)\] at \[x=0\] [Orissa JEE 2005] |
| A. | 1 |
| B. | ?1 |
| C. | \[\infty \] |
| D. | does not exist |
| Answer» E. | |
| 38. |
\[f(x)=\left| \left| x \right|-1 \right|\] is not differentiable at [IIT Screening 2005] |
| A. | 0 |
| B. | \[\pm 1,\,0\] |
| C. | 1 |
| D. | \[\pm \,1\] |
| Answer» C. 1 | |
| 39. |
Let \[f(x)=\left\{ \begin{matrix} 0, & x<0\\ {{x}^{2}}, & x\ge 0\\ \end{matrix} \right.\] , then for all values of \[x\] [IIT 1984; MP PET 2002] |
| A. | f is continuous but not differentiable |
| B. | fis differentiable but not continuous |
| C. | \[{f}'\] is continuous but not differentiable |
| D. | \[{f}'\] is continuous and differentiable |
| Answer» D. \[{f}'\] is continuous and differentiable | |
| 40. |
The function \[y\,=\,|\sin x|\] is continuous for any x but it is not differentiable at [AMU 2000] |
| A. | \[x=0\] only |
| B. | \[x=\pi \] only |
| C. | \[x=k\,\pi \,(k\] is an integer) only |
| D. | \[x=0\] and \[x=k\,\pi \,(k\] is an integer) |
| Answer» E. | |
| 41. |
The domain of definition of the function \[y(x)\] given by \[{{2}^{x}}+{{2}^{y}}=2\] is [IIT Screening 2000; DCE 2001] |
| A. | (0, 1] |
| B. | [0, 1] |
| C. | \[(-\infty ,\ 0]\] |
| D. | \[(-\infty ,\ 1)\] |
| Answer» E. | |
| 42. |
The function \[f(x)=\max [(1-x),\,(1+x),\,2],\] \[x\in (-\infty ,\,\infty ),\]is [IIT 1995] |
| A. | Continuous at all points |
| B. | Differentiable at all points |
| C. | Differentiable at all points except at \[x=1\]and \[x=-1\] |
| D. | Continuous at all points except at \[x=1\]and \[x=-1\]where it is discontinuous |
| Answer» D. Continuous at all points except at \[x=1\]and \[x=-1\]where it is discontinuous | |
| 43. |
If \[f(x)=sgn ({{x}^{3}})\], then [DCE 2001] |
| A. | f is continuous but not derivable at \[x=0\] |
| B. | \[f'({{0}^{+}})=2\] |
| C. | \[f'({{0}^{-}})=1\] |
| D. | f is not derivable at \[x=0\] |
| Answer» E. | |
| 44. |
Suppose \[f:[2,\ 2]\to R\] is defined by\[f(x)=\left\{ \begin{align} & -1\,\,\,\,\,\,\,\,\,\,\,\,\,\text{for}\ -2\le x\le 0 \\ & x-1\ \ \ \ \ \text{for}\ 0\le x\le 2 \\ \end{align} \right.\], then \[\{x\in (-2,\ 2):x\le 0\] and \[f(|x|)=x\}=\] [EAMCET 2003] |
| A. | \[\{-1\}\] |
| B. | {0} |
| C. | \[\{-1/2\}\] |
| D. | \[\varphi \] |
| Answer» D. \[\varphi \] | |
| 45. |
If \[f:R\to R\] satisfies \[f(x+y)=f(x)+f(y)\], for all \[x,\ y\in R\] and \[f(1)=7\], then \[\sum\limits_{r=1}^{n}{f(r)}\] is [AIEEE 2003] |
| A. | \[\frac{7n}{2}\] |
| B. | \[\frac{7(n+1)}{2}\] |
| C. | \[7n(n+1)\] |
| D. | \[\frac{7n(n+1)}{2}\] |
| Answer» E. | |
| 46. |
If \[{{x}_{1}}=3\]and\[x>0\]then \[\underset{n\to \infty }{\mathop{\lim }}\,{{x}_{n}}\] is equal to |
| A. | -1 |
| B. | 2 |
| C. | \[\sqrt{5}\] |
| D. | 3 |
| Answer» C. \[\sqrt{5}\] | |
| 47. |
Let \[f(x)={{(x+1)}^{2}}-1,\ \ (x\ge -1)\]. Then the set \[S=\{x:f(x)={{f}^{-1}}(x)\}\] is [IIT 1995] |
| A. | Empty |
| B. | {0, -1} |
| C. | {0, 1, -1} |
| D. | \[\left\{ 0,\ -1,\ \frac{-3+i\sqrt{3}}{2},\ \frac{-3-i\sqrt{3}}{2} \right\}\] |
| Answer» E. | |
| 48. |
Let \[2{{\sin }^{2}}x+3\sin x-2>0\] and \[{{x}^{2}}-x-2<0\] (x is measured in radians). Then x lies in the interval[IIT 1994] |
| A. | \[\left( \frac{\pi }{6},\ \frac{5\pi }{6} \right)\] |
| B. | \[\left( -1,\ \frac{5\pi }{6} \right)\] |
| C. | \[(-1,\ 2)\] |
| D. | \[\left( \frac{\pi }{6},\ 2 \right)\] |
| Answer» E. | |
| 49. |
Therange of the function \[f(x){{=}^{7-x}}{{P}_{x-3}}\] is [AIEEE 2004] |
| A. | {1, 2, 3, 4, 5} |
| B. | (1, 2, 3, 4, 5, 6) |
| C. | {1, 2, 3, 4} |
| D. | {1, 2, 3} |
| Answer» E. | |
| 50. |
If \[f(x)=\left\{ \begin{align} & \,\,\,\,\,\,\,{{e}^{x}};\,\,\,\,x\le 0 \\ & |1-x|;\,\,x>0 \\ \end{align} \right.\], then [Roorkee 1995] |
| A. | \[f(x)\] is differentiable at \[x=0\] |
| B. | \[f(x)\] is continuous at \[x=0\] |
| C. | \[f(x)\] is differentiable at \[x=1\] |
| D. | \[f(x)\] is continuous at \[x=1\] |
| Answer» C. \[f(x)\] is differentiable at \[x=1\] | |