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.
| 8101. |
If \[\int_{{}}^{{}}{(\cos x-\sin x)\ dx=\sqrt{2}\sin (x+\alpha )+c}\], then \[\alpha =\] |
| A. | \[\frac{\pi }{3}\] |
| B. | \[-\frac{\pi }{3}\] |
| C. | \[\frac{\pi }{4}\] |
| D. | \[-\frac{\pi }{4}\] |
| Answer» D. \[-\frac{\pi }{4}\] | |
| 8102. |
\[\int_{{}}^{{}}{{{e}^{\sqrt{x}}}\ dx}\] is equal to [MP PET 1998] |
| A. | \[{{e}^{\sqrt{x}}}+A\] |
| B. | \[\frac{1}{2}{{e}^{\sqrt{x}}}+A\] |
| C. | \[2(\sqrt{x}-1){{e}^{\sqrt{x}}}+A\] |
| D. | \[2(\sqrt{x}+1){{e}^{\sqrt{x}}}+A\] (A is an arbitrary constant) |
| Answer» D. \[2(\sqrt{x}+1){{e}^{\sqrt{x}}}+A\] (A is an arbitrary constant) | |
| 8103. |
If \[f(x)={{e}^{-x}},\] then \[\frac{f(-a)}{f(b)}\] is equal to |
| A. | \[f(a+b)\] |
| B. | \[f(a-b)\] |
| C. | \[f(-a+b)\] |
| D. | \[f(-a-b)\] |
| Answer» E. | |
| 8104. |
The domain and range of the relation R given by \[R=\{(x,y):y=x+\frac{6}{x};\}\] where \[x,y\in N\] and \[x |
| A. | \[\{1,2,3\},\{7,5\}\] |
| B. | \[\{1,2\},\{7,5\}\] |
| C. | \[\{2,3\},\{5\}\] |
| D. | None of these |
| Answer» B. \[\{1,2\},\{7,5\}\] | |
| 8105. |
Let \[f(x)=\frac{\alpha {{x}^{2}}}{x+1},x\ne -1.\] The value of \[\alpha \] for which \[f(a)=a,(a\ne 0)\] is |
| A. | \[1-\frac{1}{a}\] |
| B. | \[\frac{1}{a}\] |
| C. | \[1+\frac{1}{a}\] |
| D. | \[\frac{1}{a}-1\] |
| Answer» D. \[\frac{1}{a}-1\] | |
| 8106. |
The domain of the function\[\sqrt{{{x}^{2}}-5x+6}+\sqrt{2x+8-{{x}^{2}}}\] is |
| A. | \[[2,3]\] |
| B. | \[[-2,\,\,4]\] |
| C. | \[[-2,2]\cup [3,4]\] |
| D. | \[[-2,1]\cup [2,4]\] |
| Answer» D. \[[-2,1]\cup [2,4]\] | |
| 8107. |
If \[{{\log }_{1/2}}\left( {{x}^{2}}-5x+7 \right)>0,\] then exhaustive range of values of x is |
| A. | \[(-\infty ,2)\cup (3,\infty )\] |
| B. | \[(2,3)\] |
| C. | \[(-\infty ,1)\cup (1,2)\cup (2,\infty )\] |
| D. | None of these |
| Answer» C. \[(-\infty ,1)\cup (1,2)\cup (2,\infty )\] | |
| 8108. |
The domain of the real valued function \[f(x)=\sqrt{5-4x-{{x}^{2}}}+{{x}^{2}}\log (x+4)\] is |
| A. | \[(-5,1)\] |
| B. | \[-5\le x\,\,and\,\,x\ge 1\] |
| C. | \[\left( -4,1 \right]\] |
| D. | \[\phi \] |
| Answer» D. \[\phi \] | |
| 8109. |
Let \[{{f}_{1}}(x)=\left\{ \begin{matrix} x,0\le x\le 1 \\ 1,x>1 \\ 0,otherwise \\ \end{matrix} \right.\] \[{{f}_{2}}(x)={{f}_{1}}(-x)\] for all x \[{{f}_{3}}(x)=-{{f}_{2}}(x)\] for all x \[{{f}_{4}}(x)={{f}_{3}}(-x)\] for all x Which of the following is necessarily true? |
| A. | \[{{f}_{4}}(x)={{f}_{1}}(x)\] for all x |
| B. | \[{{f}_{1}}(x)=-{{f}_{3}}(-x)\] for all x |
| C. | \[{{f}_{2}}(-x)={{f}_{4}}(x)\] for all x |
| D. | \[{{f}_{1}}(x)+{{f}_{3}}(x)=0\] for all x |
| Answer» C. \[{{f}_{2}}(-x)={{f}_{4}}(x)\] for all x | |
| 8110. |
If \[f(x)=\frac{{{2}^{x}}+{{2}^{-x}}}{2}\], then \[f(x+y).f(x-y)\] is equal to |
| A. | \[\frac{1}{2}[f(x+y)+f(x-y)]\] |
| B. | \[\frac{1}{2}[f(2x)+f(2y)]\] |
| C. | \[\frac{1}{2}[f(x+y).f(x-y)]\] |
| D. | None of these |
| Answer» C. \[\frac{1}{2}[f(x+y).f(x-y)]\] | |
| 8111. |
Which of the following relation is NOT a functions? |
| A. | \[f=\{(x,x)|x\in R\}\] |
| B. | \[g=\{(x,3)|x\in R\}\] |
| C. | \[h=\{(n,\frac{1}{n})|n\in I\}\] |
| D. | \[t=\{(n,\,\,{{n}^{2}})|n\in N\}\] |
| Answer» D. \[t=\{(n,\,\,{{n}^{2}})|n\in N\}\] | |
| 8112. |
Let \[R=\{x|x\in N,x\] is a multiple of 3 and \[x\le 100\}\] \[S=\{x|x\in N,\,\,x\] is a multiple of 5 and \[x\le 100\}\] What is the number of elements in \[(R\times S)\cap (S\times R)\] |
| A. | 36 |
| B. | 33 |
| C. | 20 |
| D. | 6 |
| Answer» B. 33 | |
| 8113. |
Let f and g be functions from R To R defined as \[f(x)=\left\{ \begin{matrix} 7{{x}^{2}}+x-8,x\le 1 \\ 4x+5,17 \\ \end{matrix},g(x)=\left\{ \begin{matrix} \left| x \right|,x |
| A. | \[(fog)(-3)=8\] |
| B. | \[(fog)(9)=683\] |
| C. | \[(gof)(0)=-8\] |
| D. | \[(gof)(6)=427\] |
| Answer» C. \[(gof)(0)=-8\] | |
| 8114. |
Let \[f:[4,\infty )\to [1,\infty )\]be a function defined by \[f(x)={{5}^{x(x-4)}},\] then \[{{f}^{-1}}(x)\]is |
| A. | \[2-\sqrt{4+{{\log }_{5}}x}\] |
| B. | \[2+\sqrt{4+{{\log }_{5}}x}\] |
| C. | \[{{\left( \frac{1}{5} \right)}^{x(x-4)}}\] |
| D. | None of these |
| Answer» C. \[{{\left( \frac{1}{5} \right)}^{x(x-4)}}\] | |
| 8115. |
If \[f:R\to R,g:R\to R\] and \[h:R\to R\]are such that \[f(x)={{x}^{2}},g(x)=tanx\]and \[h(x)=logx,\]then the value of \[(ho(gof))(x)ifx=\sqrt{\frac{\pi }{4}}\] will be |
| A. | 0 |
| B. | 1 |
| C. | -1 |
| D. | \[\pi \] |
| Answer» B. 1 | |
| 8116. |
If \[f(x)=\sqrt{3\left| x \right|-x-2}\] and \[g(x)=sinx,\] then domain of definition of fog (x) is |
| A. | \[\left\{ 2n\pi +\frac{\pi }{2} \right\},n\in I\] |
| B. | \[\underset{n\in I}{\mathop{\bigcup }}\,\]\[\left( 2n\pi +\frac{7\pi }{6},2n\pi +\frac{11\pi }{6} \right)\] |
| C. | \[\left( 2n\pi +\frac{7\pi }{6} \right),n\in I\] |
| D. | \[\{(4m+1)\frac{\pi }{2}:m\in I\}\underset{n\in I}{\mathop{\bigcup }}\,\left[ \left( 2n\pi +\frac{7\pi }{6},2n\pi +\frac{11\pi }{6} \right) \right]\] |
| Answer» E. | |
| 8117. |
Let \[f:\{2,3,4,5\}\to \{3,4,5,9\}\] and \[g:\{3,4,5,9\}\]\[\to \{7,11,15\}\] be functions defined as \[f(2)=3f(3)=4,f(4)=f(5)=5,g(3)=g(4)=7,\] and \[g(5)=g(9)=11.\] Then gof (5) is equal to |
| A. | 5 |
| B. | 7 |
| C. | 11 |
| D. | 1 |
| Answer» D. 1 | |
| 8118. |
If f(x) is an invertible function and \[g(x)=2f(x)+5,\] then the value of \[{{g}^{-1}}(x)\] is |
| A. | \[2{{f}^{-1}}(x)-5\] |
| B. | \[\frac{1}{2{{f}^{-1}}(x)+5}\] |
| C. | \[\frac{1}{2}{{f}^{-1}}(x)+5\] |
| D. | \[{{f}^{-1}}\left( \frac{x-5}{2} \right)\] |
| Answer» E. | |
| 8119. |
Let [x] denote the greatest integer \[\le x.\] If \[f(x)=[x]\] and \[g(x)=\left| x \right|,\]then the value of \[f\left( g\left( \frac{8}{5} \right) \right)-g\left( f\left( -\frac{8}{5} \right) \right)\]is |
| A. | 2 |
| B. | -2 |
| C. | 1 |
| D. | -1 |
| Answer» E. | |
| 8120. |
If \[f(x)=\left| x-2 \right|\] and \[g(x)=f[f(x)],\] Then for \[x>20,g'(x)\] is equal to |
| A. | \[-1\] |
| B. | 1 |
| C. | 2 |
| D. | None of these |
| Answer» C. 2 | |
| 8121. |
Let \[f(z)=sinz\] and \[g(z)=cosz.\] If * denotes a composition of functions, then the value of \[{{(f+ig)}^{*}}(f-ig)\] is: |
| A. | \[i{{e}^{-{{e}^{-iz}}}}\] |
| B. | \[i{{e}^{-{{e}^{iz}}}}\] |
| C. | \[-i{{e}^{-{{e}^{-iz}}}}\] |
| D. | None of these |
| Answer» C. \[-i{{e}^{-{{e}^{-iz}}}}\] | |
| 8122. |
Let R and S be two non-void relations in a set A. which of the following statements is not true? |
| A. | R and S transitive\[\Rightarrow \] \[R\cup S\] is transitive |
| B. | R and S transitive\[\Rightarrow R\cap S\] is transitive |
| C. | R and S transitive\[\Rightarrow R\cup S\] is symmetric |
| D. | R and S reflexive\[\Rightarrow R\cap S\] is reflexive |
| Answer» B. R and S transitive\[\Rightarrow R\cap S\] is transitive | |
| 8123. |
If \[f(x)=ax+b\] and \[g(x)=cx+d,\] then\[f\{f\{x\}=g\{f(x)\}\] is equivalent to? |
| A. | \[f(a)=g(c)\] |
| B. | \[f(b)=g(b)\] |
| C. | \[f(d)=g(b)\] |
| D. | \[f(c)=g(a)\] |
| Answer» D. \[f(c)=g(a)\] | |
| 8124. |
Let \[f(x)={{x}^{2}}+3x-3,x>0.\] If n points \[{{x}_{1}},{{x}_{2}},{{x}_{3}},...{{x}_{n}}\] are so chosen on the x-axis such that (i) \[\frac{1}{n}\sum\limits_{i=1}^{n}{{{f}^{-1}}({{x}_{i}})}=f\left( \frac{1}{n}\sum\limits_{i=1}^{n}{{{x}_{i}}} \right)\] (ii) \[\sum\limits_{i=1}^{n}{{{f}^{-1}}}({{x}_{i}})=\sum\limits_{i=1}^{n}{{{x}_{i}}},\] where \[{{f}^{-1}}\] denotes the inverse of f. The value of \[\frac{{{x}_{1}}+{{x}_{2}}+...+{{x}_{n}}}{n}=\] |
| A. | 1 |
| B. | 2 |
| C. | 3 |
| D. | 4 |
| Answer» B. 2 | |
| 8125. |
The domain of the function f(x)=\[\frac{1}{\sqrt{^{10}{{C}_{x-1}}-3{{\times }^{10}}{{C}_{x}}}}\] contains the points |
| A. | \[9,10,11\] |
| B. | \[9,10,12\] |
| C. | All natural numbers |
| D. | None of these |
| Answer» E. | |
| 8126. |
Let \[\rho \] be the relation on the set R of all real numbers defined by setting \[a\rho b\] if f\[\left| a-b \right|\le \frac{1}{2}.\]then, \[\rho \] is |
| A. | Reflexive and symmetric but not transitive |
| B. | Symmetric and transitive but not reflexive |
| C. | Transitive but neither reflexive nor symmetric |
| D. | None of these |
| Answer» B. Symmetric and transitive but not reflexive | |
| 8127. |
If \[f(x)=\frac{x}{x-1},\] then \[\frac{(fofo...of)(x)}{19times}\] is equal to: |
| A. | \[\frac{x}{x-1}\] |
| B. | \[{{\left( \frac{x}{x-1} \right)}^{19}}\] |
| C. | \[\frac{19x}{x-1}\] |
| D. | x |
| Answer» B. \[{{\left( \frac{x}{x-1} \right)}^{19}}\] | |
| 8128. |
If \[g(f(x))=\left| \sin x \right|\] and \[f(g(x))={{(sin\sqrt{x})}^{2}},\]then |
| A. | \[f(x)=si{{n}^{2}}x,g(x)=\sqrt{x}\] |
| B. | \[f(x)=sinx,g(x)=\left| x \right|\] |
| C. | \[f(x)={{x}^{2}},g(x)=sin\sqrt{x}\] |
| D. | f and g cannot be determined. |
| Answer» B. \[f(x)=sinx,g(x)=\left| x \right|\] | |
| 8129. |
The inverse of \[f(x)=\frac{2}{3}\frac{{{10}^{x}}-{{10}^{-x}}}{{{10}^{x}}+{{10}^{-x}}}\] is |
| A. | \[\frac{1}{3}{{\log }_{10}}\frac{1+x}{1-x}\] |
| B. | \[\frac{1}{2}{{\log }_{10}}\frac{2+3x}{2-3x}\] |
| C. | \[\frac{1}{3}{{\log }_{10}}\frac{2+3x}{2-3x}\] |
| D. | \[\frac{1}{6}{{\log }_{10}}\frac{2-3x}{2+3x}\] |
| Answer» C. \[\frac{1}{3}{{\log }_{10}}\frac{2+3x}{2-3x}\] | |
| 8130. |
The range of the function \[f(x){{=}^{7-x}}{{P}_{x-3}}\] is |
| A. | \[\{1,2,3\}\] |
| B. | \[\{1,2,3,4,5,6\}\] |
| C. | \[\{1,2,3,4\}\] |
| D. | \[\{1,2,3,4,5\}\] |
| Answer» B. \[\{1,2,3,4,5,6\}\] | |
| 8131. |
The number of surjection from \[A=\{1,2,...,n\},n\ge 2ontoB=\{a,b\}\] is |
| A. | \[^{n}{{P}_{2}}\] |
| B. | \[{{2}^{n}}-2\] |
| C. | \[{{2}^{n}}-1\] |
| D. | None of these |
| Answer» C. \[{{2}^{n}}-1\] | |
| 8132. |
If \[f:R\to R\] and \[g:R\to R\] are given by \[f(x)=\left| x \right|\] and \[g(x)=[x]\] for each \[x\in R,\] then \[[x\in R:g(f(x))\le f(g(x))\}=\] |
| A. | \[Z\cup (-\infty ,0)\] |
| B. | \[(-\infty ,0)\] |
| C. | \[Z\] |
| D. | R |
| Answer» E. | |
| 8133. |
Let \[f:R\to R\] be function defined by \[f(x)=sin(2x-3),\] then f is |
| A. | Injective |
| B. | surjective |
| C. | bijective |
| D. | None of these |
| Answer» E. | |
| 8134. |
The number of linear functions f satisfying \[f(x+f(x))=x+f(x)\forall x\in R\] is |
| A. | 0 |
| B. | 1 |
| C. | 2 |
| D. | 3 |
| Answer» D. 3 | |
| 8135. |
Let a relation R in the set R of real numbers be defined as (a, b) \[\in R\] if and only if \[1+ab>o\] for all \[a,b\in R\]. The relation R is |
| A. | Reflexive and symmetric |
| B. | Symmetric and transitive |
| C. | Only transitive |
| D. | An equivalence relation |
| Answer» B. Symmetric and transitive | |
| 8136. |
The domain of the function\[f(x){{=}^{24-x}}{{C}_{3x-1}}{{+}^{40-6x}}{{C}_{8x-10}}\] is, |
| A. | \[\{2,3\}\] |
| B. | \[\{1,2,3\}\] |
| C. | \[\{1,2,3,4\}\] |
| D. | None of these |
| Answer» B. \[\{1,2,3\}\] | |
| 8137. |
Let A and B be two finite sets having m and n elements respectively. Then, the total number of mapping from A and B is: |
| A. | \[mn\] |
| B. | \[{{2}^{mn}}\] |
| C. | \[{{m}^{n}}\] |
| D. | \[{{n}^{m}}\] |
| Answer» E. | |
| 8138. |
The relation R defined in \[A=\{1,2,3\}\] by aRb, if \[\left| {{a}^{2}}-{{b}^{2}} \right|\le 5.\] Which of the following is false? |
| A. | \[R=\{(1,1),(2,2),(3,3),(2,1),(1,2),(2,3),(3,2)\}\] |
| B. | \[{{R}^{-1}}=R\] |
| C. | Domain of \[R=\{1,2,3\}\] |
| D. | Range of \[R=\{5\}\] |
| Answer» E. | |
| 8139. |
Let \[f:R\to R\] be a function defined by \[f(x)=\frac{x-m}{x-n},\] where \[m\ne n,\] then |
| A. | f is one-one onto |
| B. | f is one-one into |
| C. | f is many-one onto |
| D. | f is many-one into |
| Answer» C. f is many-one onto | |
| 8140. |
Let \[n(A)=4\] and \[n(B)=6.\] The number of one to one functions from A to B is |
| A. | 24 |
| B. | 60 |
| C. | 120 |
| D. | 360 |
| Answer» E. | |
| 8141. |
Let n be a fixed positive integer. Define a relation R in the set Z of integers by aRb if and only if \[\frac{n}{a-b}.\]The relation R is |
| A. | reflexive |
| B. | Symmetric |
| C. | Transitive |
| D. | An equivalence relation |
| Answer» E. | |
| 8142. |
R is a relation from \[\{11,12,13\}\] to \[\{8,10,12\}\]defined by \[y=x-3.\] The relation \[{{R}^{-1}}\] is |
| A. | \[\{(11,8),(13,10)\}\] |
| B. | \[\{(8,11),(10,13)\}\] |
| C. | \[\{(8,11),(9,12),(10,13)\}\] |
| D. | None of these |
| Answer» C. \[\{(8,11),(9,12),(10,13)\}\] | |
| 8143. |
Let \[f:R\to R\] be given by \[f(x)={{(x+1)}^{2}}-1,x\ge -1.\] Then, \[{{f}^{-1}}(x),\] is |
| A. | \[-1+\sqrt{x+1}\] |
| B. | \[-1-\sqrt{x+1}\] |
| C. | Does not exist because f is not one-one |
| D. | Does not exist because f is not onto |
| Answer» B. \[-1-\sqrt{x+1}\] | |
| 8144. |
Let R be a relation over the \[N\times N\] and it is defined by (a, b) R (c, d) \[\Rightarrow a+d=b+c.\] Then R is |
| A. | Reflexive only |
| B. | Symmetric only |
| C. | Transitive only |
| D. | An equivalence relation |
| Answer» E. | |
| 8145. |
\[\underset{n\to \infty }{\mathop{\lim }}\,\frac{n{{(2n+1)}^{2}}}{(n+2)({{n}^{2}}+3n-1)}=\] |
| A. | 0 |
| B. | 2 |
| C. | 4 |
| D. | \[\infty \] |
| Answer» D. \[\infty \] | |
| 8146. |
Let \[f(x)=\left\{ \begin{align} & 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\forall x |
| A. | 1 |
| B. | ?1 |
| C. | \[\infty \] |
| D. | does not exist |
| Answer» E. | |
| 8147. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin x-x}{{{x}^{3}}}=\] [MNR 1980, 86] |
| A. | \[\frac{1}{3}\] |
| B. | \[-\frac{1}{3}\] |
| C. | \[\frac{1}{6}\] |
| D. | \[-\frac{1}{6}\] |
| Answer» E. | |
| 8148. |
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. | |
| 8149. |
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. \[R-\{-1,\ 1\}\] | |
| 8150. |
The value of \[\underset{x\to 2}{\mathop{\lim }}\,\frac{{{3}^{x/2}}-3}{{{3}^{x}}-9}\] is [MP PET 2000] |
| A. | 0 |
| B. | 1/3 |
| C. | \[1/6\] |
| D. | \[\ln 3\] |
| Answer» D. \[\ln 3\] | |