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.
| 1701. |
For the function\[f(x)={{e}^{x}},a=0,b=1\], the value of c in mean value theorem will be [DCE 2002] |
| A. | log x |
| B. | \[\log (e-1)\] |
| C. | 0 |
| D. | 1 |
| Answer» C. 0 | |
| 1702. |
If \[f(x)=\cos x,0\le x\le \frac{\pi }{2}\], then the real number ?c? of the mean value theorem is [MP PET 1994] |
| A. | \[\frac{\pi }{6}\] |
| B. | \[\frac{\pi }{4}\] |
| C. | \[{{\sin }^{-1}}\left( \frac{2}{\pi } \right)\] |
| D. | \[{{\cos }^{-1}}\left( \frac{2}{\pi } \right)\] |
| Answer» D. \[{{\cos }^{-1}}\left( \frac{2}{\pi } \right)\] | |
| 1703. |
In the mean value theorem, \[f(b)-f(a)=(b-a)f'(c)\]if \[a=4\], \[b=9\] and \[f(x)=\sqrt{x}\] then the value of c is [J & K 2005] |
| A. | 8.00 |
| B. | 5.25 |
| C. | 4.00 |
| D. | 6.25 |
| Answer» E. | |
| 1704. |
The function \[f(x)={{(x-3)}^{2}}\] satisfies all the conditions of mean value theorem in [3, 4]. A point on\[y={{(x-3)}^{2}}\], where the tangent is parallel to the chord joining (3, 0) and (4, 1) is |
| A. | \[\left( \frac{7}{2},\frac{1}{2} \right)\] |
| B. | \[\left( \frac{7}{2},\frac{1}{4} \right)\] |
| C. | (1, 4) |
| D. | (4, 1) |
| Answer» C. (1, 4) | |
| 1705. |
The abscissa of the points of the curve \[y={{x}^{3}}\]in the interval [?2, 2], where the slope of the tangents can be obtained by mean value theorem for the interval [?2, 2], are [MP PET 1993] |
| A. | \[\pm \frac{2}{\sqrt{3}}\] |
| B. | \[\pm \sqrt{3}\] |
| C. | \[\pm \frac{\sqrt{3}}{2}\] |
| D. | 0 |
| Answer» B. \[\pm \sqrt{3}\] | |
| 1706. |
If the function \[f(x)={{x}^{3}}-6{{x}^{2}}+ax+b\] satisfies Rolle?s theorem in the interval \[[1,\,3]\] and \[f'\left( \frac{2\sqrt{3}+1}{\sqrt{3}} \right)=0\], then [MP PET 2002] |
| A. | \[a=-11\] |
| B. | \[a=-6\] |
| C. | \[a=6\] |
| D. | \[a=11\] |
| Answer» E. | |
| 1707. |
Let \[f(x)=\sqrt{x-1}+\sqrt{x+24-10\sqrt{x-1};}\] \[1 |
| A. | 0 |
| B. | \[\frac{1}{\sqrt{x-1}}\] |
| C. | \[2\sqrt{x-1}-5\] |
| D. | None of these |
| Answer» B. \[\frac{1}{\sqrt{x-1}}\] | |
| 1708. |
If \[f(x)\] satisfies the conditions of Rolle?s theorem in \[[1,\,2]\] and \[f(x)\] is continuous in \[[1,\,2]\] then \[\int_{1}^{2}{f'(x)dx}\] is equal to [DCE 2002] |
| A. | 3 |
| B. | 0 |
| C. | 1 |
| D. | 2 |
| Answer» C. 1 | |
| 1709. |
Consider the function \[f(x)={{e}^{-2x}}\]sin 2x over the interval \[\left( 0,\frac{\pi }{2} \right)\]. A real number \[c\in \left( 0,\frac{\pi }{2} \right)\,,\] as guaranteed by Rolle?s theorem, such that \[{f}'\,(c)=0\] is [AMU 1999] |
| A. | \[\pi /8\] |
| B. | \[\pi /6\] |
| C. | \[\pi /4\] |
| D. | \[\pi /3\] |
| Answer» B. \[\pi /6\] | |
| 1710. |
Let \[f(x)\] satisfy all the conditions of mean value theorem in [0, 2]. If f (0) = 0 and \[|f'(x)|\,\le \frac{1}{2}\] for all x, in [0, 2] then |
| A. | \[f(x)\le 2\] |
| B. | \[|f(x)|\le 1\] |
| C. | \[f(x)=2x\] |
| D. | \[f(x)=3\]for at least one x in [0, 2] |
| Answer» C. \[f(x)=2x\] | |
| 1711. |
For which interval, the function \[\frac{{{x}^{2}}-3x}{x-1}\] satisfies all the conditions of Rolle's theorem [MP PET 1993] |
| A. | [0, 3] |
| B. | [? 3, 0] |
| C. | [1.5, 3] |
| D. | For no interval |
| Answer» E. | |
| 1712. |
Let R be an equivalence relation on a finite set A having n elements. Then the number of ordered pairs in R is |
| A. | Less than n |
| B. | Greater than or equal to n |
| C. | Less than or equal to n |
| D. | None of these |
| Answer» C. Less than or equal to n | |
| 1713. |
Let \[P=\{(x,\,y)|{{x}^{2}}+{{y}^{2}}=1,\,x,\,y\in R\}\]. Then P is |
| A. | Reflexive |
| B. | Symmetric |
| C. | Transitive |
| D. | Anti-symmetric |
| Answer» C. Transitive | |
| 1714. |
The relation ?less than? in the set of natural numbers is [UPSEAT 1994, 98, 99; AMU 1999] |
| A. | Only symmetric |
| B. | Only transitive |
| C. | Only reflexive |
| D. | Equivalence relation |
| Answer» C. Only reflexive | |
| 1715. |
If A is the set of even natural numbers less than 8 and B is the set of prime numbers less than 7, then the number of relations from A to B is [NDA 2003] |
| A. | \[{{2}^{9}}\] |
| B. | \[{{9}^{2}}\] |
| C. | \[{{3}^{2}}\] |
| D. | \[{{2}^{9-1}}\] |
| Answer» B. \[{{9}^{2}}\] | |
| 1716. |
The relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)} on set A = {1, 2, 3} is |
| A. | Reflexive but not symmetric |
| B. | Reflexive but not transitive |
| C. | Symmetric and Transitive |
| D. | Neither symmetric nor transitive |
| Answer» B. Reflexive but not transitive | |
| 1717. |
Let S be the set of all real numbers. Then the relation R = {(a, b) : 1 + ab > 0} on S is [NDA 2003] |
| A. | Reflexive and symmetric but not transitive |
| B. | Reflexive and transitive but not symmetric |
| C. | Symmetric, transitive but not reflexive |
| D. | Reflexive, transitive and symmetric |
| E. | None of the above is true |
| Answer» B. Reflexive and transitive but not symmetric | |
| 1718. |
Let R = {(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)} be a relation on the set A = {1, 2, 3, 4}. The relation R is [AIEEE 2004] |
| A. | Reflexive |
| B. | Transitive |
| C. | Not symmetric |
| D. | A function |
| Answer» D. A function | |
| 1719. |
The number of reflexive relations of a set with four elements is equal to [UPSEAT 2004] |
| A. | \[{{2}^{16}}\] |
| B. | \[{{2}^{12}}\] |
| C. | \[{{2}^{8}}\] |
| D. | \[{{2}^{4}}\] |
| Answer» E. | |
| 1720. |
Let\[R=\{(3,\,3),\ (6,\ 6),\ (9,\,9),\ (12,\,12),\ (6,\,12),\ (3,\,9),(3,\,12),\,(3,\,6)\}\] be a relation on the set \[A=\{3,\,6,\,9,\,12\}\]. The relation is [AIEEE 2005] |
| A. | An equivalence relation |
| B. | Reflexive and symmetric only |
| C. | Reflexive and transitive only |
| D. | Reflexive only |
| Answer» D. Reflexive only | |
| 1721. |
Let n be a fixed positive integer. Define a relation R on the set Z of integers by, \[aRb\Leftrightarrow n|a-b\]|. Then R is |
| A. | Reflexive |
| B. | Symmetric |
| C. | Transitive |
| D. | Equivalence |
| Answer» D. Equivalence | |
| 1722. |
Let L denote the set of all straight lines in a plane. Let a relation R be defined by \[\alpha R\beta \Leftrightarrow \alpha \bot \beta ,\,\alpha ,\,\beta \in L\]. Then R is |
| A. | Reflexive |
| B. | Symmetric |
| C. | Transitive |
| D. | None of these |
| Answer» C. Transitive | |
| 1723. |
Let R = {(1, 3), (2, 2), (3, 2)} and S = {(2, 1), (3, 2), (2, 3)} be two relations on set A = {1, 2, 3}. Then RoS = |
| A. | {(1, 3), (2, 2), (3, 2), (2, 1), (2, 3)} |
| B. | {(3, 2), (1, 3)} |
| C. | {(2, 3), (3, 2), (2, 2)} |
| D. | {(2, 3), (3, 2)} |
| Answer» D. {(2, 3), (3, 2)} | |
| 1724. |
Let R and S be two equivalence relations on a set A. Then |
| A. | \[R\text{ }\cup \text{ }S\] is an equivalence relation on A |
| B. | \[R\text{ }\cap \text{ }S\] is an equivalence relation on A |
| C. | \[R-S\] is an equivalence relation on A |
| D. | None of these |
| Answer» C. \[R-S\] is an equivalence relation on A | |
| 1725. |
Solution set of \[x\equiv 3\] (mod 7), \[p\in Z,\] is given by |
| A. | {3} |
| B. | \[\{7p-3:p\in Z\}\] |
| C. | \[\{7p+3:p\in Z\}\] |
| D. | None of these |
| Answer» D. None of these | |
| 1726. |
The relation "congruence modulo m" is |
| A. | Reflexive only |
| B. | Transitive only |
| C. | Symmetric only |
| D. | An equivalence relation |
| Answer» E. | |
| 1727. |
In order that a relation R defined on a non-empty set A is an equivalence relation, it is sufficient, if R [Karnataka CET 1990] |
| A. | Is reflexive |
| B. | Is symmetric |
| C. | Is transitive |
| D. | Possesses all the above three properties |
| Answer» E. | |
| 1728. |
If R is an equivalence relation on a set A, then \[{{R}^{-1}}\] is |
| A. | Reflexive only |
| B. | Symmetric but not transitive |
| C. | Equivalence |
| D. | None of these |
| Answer» D. None of these | |
| 1729. |
The relation R defined on the set of natural numbers as {(a, b) : a differs from b by 3}, is given by |
| A. | {(1, 4, (2, 5), (3, 6),.....} |
| B. | {(4, 1), (5, 2), (6, 3),.....} |
| C. | {(1, 3), (2, 6), (3, 9),..} |
| D. | None of these |
| Answer» C. {(1, 3), (2, 6), (3, 9),..} | |
| 1730. |
Which one of the following relations on R is an equivalence relation |
| A. | \[a\,{{R}_{1}}\,b\Leftrightarrow |a|=|b|\] |
| B. | \[a{{R}_{2}}b\Leftrightarrow a\ge b\] |
| C. | \[a{{R}_{3}}b\Leftrightarrow a\text{ divides }b\] |
| D. | \[a{{R}_{4}}b\Leftrightarrow a<b\] |
| Answer» B. \[a{{R}_{2}}b\Leftrightarrow a\ge b\] | |
| 1731. |
Let A = {1, 2, 3, 4} and let R= {(2, 2), (3, 3), (4, 4), (1, 2)} be a relation on A. Then R is |
| A. | Reflexive |
| B. | Symmetric |
| C. | Transitive |
| D. | None of these |
| Answer» D. None of these | |
| 1732. |
The void relation on a set A is |
| A. | Reflexive |
| B. | Symmetric and transitive |
| C. | Reflexive and symmetric |
| D. | Reflexive and transitive |
| Answer» C. Reflexive and symmetric | |
| 1733. |
Let A be the non-void set of the children in a family. The relation 'x is a brother of y' on A is |
| A. | Reflexive |
| B. | Symmetric |
| C. | Transitive |
| D. | None of these |
| Answer» C. Transitive | |
| 1734. |
In the set A = {1, 2, 3, 4, 5}, a relation R is defined by R = {(x, y)| x, y \[\in \] A and x < y}. Then R is |
| A. | Reflexive |
| B. | Symmetric |
| C. | Transitive |
| D. | None of these |
| Answer» D. None of these | |
| 1735. |
The relation R defined on a set A is antisymmetric if \[(a,\,b)\in R\Rightarrow (b,\,a)\in R\] for |
| A. | Every (a, b) \[\in R\] |
| B. | No \[(a,\,b)\in R\] |
| C. | No \[(a,\,b),\,a\ne b,\,\in R\] |
| D. | None of these |
| Answer» D. None of these | |
| 1736. |
The relation "is subset of" on the power set P of a set A is |
| A. | Symmetric |
| B. | Anti-symmetric |
| C. | Equivalency relation |
| D. | None of these |
| Answer» C. Equivalency relation | |
| 1737. |
Let R = {(a, a)} be a relation on a set A. Then R is |
| A. | Symmetric |
| B. | Antisymmetric |
| C. | Symmetric and antisymmetric |
| D. | Neither symmetric nor anti-symmetric |
| Answer» D. Neither symmetric nor anti-symmetric | |
| 1738. |
Given two finite sets A and B such that n = 2, n = 3. Then total number of relations from A to B is |
| A. | 4 |
| B. | 8 |
| C. | 64 |
| D. | None of these |
| Answer» D. None of these | |
| 1739. |
The relation R defined in N as \[aRb\Leftrightarrow b\] is divisible by a is |
| A. | Reflexive but not symmetric |
| B. | Symmetric but not transitive |
| C. | Symmetric and transitive |
| D. | None of these |
| Answer» B. Symmetric but not transitive | |
| 1740. |
Let A = {1, 2, 3, 4} and R be a relation in A given by R = {(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 1), (3, 1), (1, 3)}. Then R is |
| A. | Reflexive |
| B. | Symmetric |
| C. | Transitive |
| D. | An equivalence relation |
| Answer» C. Transitive | |
| 1741. |
Let R be a reflexive relation on a set A and I be the identity relation on A. Then |
| A. | \[R\subset I\] |
| B. | \[I\subset R\] |
| C. | \[R=I\] |
| D. | None of these |
| Answer» C. \[R=I\] | |
| 1742. |
If \[R=\{(x,\,y)|x,\,y\in Z,\,{{x}^{2}}+{{y}^{2}}\le 4\}\] is a relation in Z, then domain of R is |
| A. | {0, 1, 2} |
| B. | {0, - 1, - 2} |
| C. | {- 2, - 1, 0, 1, 2} |
| D. | None of these |
| Answer» D. None of these | |
| 1743. |
R is a relation from {11, 12, 13} to {8, 10, 12} defined by \[y=x-3\]. Then \[{{R}^{-1}}\] is |
| A. | {(8, 11), (10, 13)} |
| B. | {(11, 18), (13, 10)} |
| C. | {(10, 13), (8, 11)} |
| D. | None of these |
| Answer» B. {(11, 18), (13, 10)} | |
| 1744. |
A relation R is defined from {2, 3, 4, 5} to {3, 6, 7, 10} by \[xRy\Leftrightarrow x\] is relatively prime to y. Then domain of R is |
| A. | {2, 3, 5} |
| B. | {3, 5} |
| C. | {2, 3, 4} |
| D. | {2, 3, 4, 5} |
| Answer» E. | |
| 1745. |
The relation R defined on the set A = {1, 2, 3, 4, 5} by R = {(x, y) : \[|{{x}^{2}}-{{y}^{2}}| |
| A. | {(1, 1), (2, 1), (3, 1), (4, 1), (2, 3)} |
| B. | {(2, 2), (3, 2), (4, 2), (2, 4)} |
| C. | {(3, 3), (3, 4), (5, 4), (4, 3), (3, 1)} |
| D. | None of these |
| Answer» E. | |
| 1746. |
If R is a relation from a finite set A having m elements to a finite set B having n elements, then the number of relations from A to B is |
| A. | \[{{2}^{mn}}\] |
| B. | \[{{2}^{mn}}-1\] |
| C. | \[2mn\] |
| D. | \[{{m}^{n}}\] |
| Answer» B. \[{{2}^{mn}}-1\] | |
| 1747. |
Let R be a reflexive relation on a finite set A having n-elements, and let there be m ordered pairs in R. Then |
| A. | \[m\ge n\] |
| B. | \[m\le n\] |
| C. | \[m=n\] |
| D. | None of these |
| Answer» B. \[m\le n\] | |
| 1748. |
Let A = {a, b, c} and B = {1, 2}. Consider a relation R defined from set A to set B. Then R is equal to set [Kurukshetra CEE 1995] |
| A. | A |
| B. | B |
| C. | A × B |
| D. | B × A |
| Answer» D. B × A | |
| 1749. |
A relation from P to Q is |
| A. | A universal set of P × Q |
| B. | P × Q |
| C. | An equivalent set of P × Q |
| D. | A subset of P × Q |
| Answer» E. | |
| 1750. |
If R is a relation from a set A to a set B and S is a relation from B to a set C, then the relation SoR |
| A. | Is from A to C |
| B. | Is from C to A |
| C. | Does not exist |
| D. | None of these |
| Answer» B. Is from C to A | |