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.
| 8001. |
The number of points in (1, 3), where \[f(x)={{a}^{[{{x}^{2}}]}},a>1\], is not differentiable, where [x] denotes the integral part of x. |
| A. | 5 |
| B. | 7 |
| C. | 9 |
| D. | 11 |
| Answer» C. 9 | |
| 8002. |
If \[f(x)=\left\{ \begin{align} & \left( {{x}^{2}}/a \right)-a,\,\,when\,\,\,xa \\ \end{align} \right.\] |
| A. | \[\underset{x\to a}{\mathop{\lim }}\,f(x)=a\] |
| B. | \[f(x)\] is continuous at x = a |
| C. | \[f(x)\] is discontinuous at x = a |
| D. | None of these |
| Answer» C. \[f(x)\] is discontinuous at x = a | |
| 8003. |
Consider the function\[f(x)=\left\{ \begin{matrix} {{x}^{2}}, & x>2 \\ 3x-2, & x\le 2 \\ \end{matrix} \right.\]. Which one of the following statements is correct in respect of the above function? |
| A. | f(x) is derivable but not continuous at x = 2. |
| B. | f(x) is continuous but not derivable at x = 2. |
| C. | f(x) is neither continuous nor derivable at x = 2. |
| D. | f(x) is continuous as well as derivable at x = 2. |
| Answer» C. f(x) is neither continuous nor derivable at x = 2. | |
| 8004. |
Let \[f:[2,7]\to [0,\infty )\] be a continuous and differentiable function. Then, \[(f(7)-f(2))\frac{{{(f(7))}^{2}}+{{(f(2))}^{2}}+f(2)f(7)}{3}\] is, where \[c\in [2,7]\] [2, 7]. |
| A. | \[5{{f}^{2}}(c)f'(c)\] |
| B. | \[5f'(c)\] |
| C. | \[f(c)f'(c)\] |
| D. | None of these |
| Answer» B. \[5f'(c)\] | |
| 8005. |
If\[f(x)=\left\{ \begin{matrix} mx+1x\le \frac{\pi }{2} \\ \sin x+nx>\frac{\pi }{2} \\ \end{matrix}\,\,\,\text{is}\,\,\text{continuous}\,\,\text{at} \right.\]\[x=\frac{\pi }{2}\], then which one of the following is correct? |
| A. | m = 1, n = 0 |
| B. | \[m=\frac{n\pi }{2}+1\] |
| C. | \[n=m\left( \frac{\pi }{2} \right)\] |
| D. | \[m=n=\frac{\pi }{2}\] |
| Answer» D. \[m=n=\frac{\pi }{2}\] | |
| 8006. |
The value of p for which the function\[f(x)=\left\{ \begin{matrix} \frac{{{({{4}^{x}}-1)}^{3}}}{\sin \frac{x}{p}\log \left[ 1+\frac{{{x}^{2}}}{3} \right]},x\ne 0 \\ 12{{(log\,4)}^{3}},x=0 \\ \end{matrix} \right.\]may be continuous at \[x=0\], is |
| A. | 1 |
| B. | 2 |
| C. | 3 |
| D. | None of these |
| Answer» E. | |
| 8007. |
If \[f''(x) |
| A. | Exactly once in (a, b) |
| B. | At most once in (a, b) |
| C. | At least once in (a, b) |
| D. | None of these |
| Answer» C. At least once in (a, b) | |
| 8008. |
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 |
| A. | 3 |
| B. | 4 |
| C. | 5 |
| D. | 6 |
| Answer» D. 6 | |
| 8009. |
If f(x) is differentiable everywhere, then which one of the following is correct? |
| A. | \[\left| f \right|\] is differentiable everywhere |
| B. | \[{{\left| f \right|}^{2}}\]is differentiable everywhere |
| C. | \[f\left| f \right|\]is not differentiable at some points |
| D. | None of the above |
| Answer» D. None of the above | |
| 8010. |
If \[f(0)=0,f'(0)=2\], then the derivative of \[y=f(f(f(f(x)))\] at \[x=0\] is |
| A. | 2 |
| B. | 8 |
| C. | 16 |
| D. | 4 |
| Answer» D. 4 | |
| 8011. |
Let \[f\] be a function which is continuous and differentiable for all real x. If \[f(2)=-4\] and \[f'(x)\ge 6\] for all \[x\text{ }\in [2,\text{ }4],\] then |
| A. | \[f(4)<8\] |
| B. | \[f(4)\ge 8\] |
| C. | \[f(4)\ge 12\] |
| D. | None of these |
| Answer» C. \[f(4)\ge 12\] | |
| 8012. |
If \[f(x)={{x}^{\alpha }}log\text{ }x\] and \[f(0)=0\], then the value of a for which Rolle's theorem can be applied in [0, 1] is |
| A. | -2 |
| B. | -1 |
| C. | 0 |
| D. | ½ |
| Answer» E. | |
| 8013. |
Let \[y={{t}^{10}}+1\] and \[x={{t}^{8}}+1\], then \[\frac{{{d}^{2}}y}{d{{x}^{2}}}\] is equal to: |
| A. | \[\frac{5}{2}t\] |
| B. | \[20{{t}^{8}}\] |
| C. | \[\frac{5}{16{{t}^{6}}}\] |
| D. | None of these |
| Answer» D. None of these | |
| 8014. |
If \[f(xy)=f(x).f(y)\] for all x, y and f(x) is continuous at x = 2, then f(x) is not necessarily continuous in: |
| A. | \[(-\infty ,\infty )\] |
| B. | \[(0,\infty )\] |
| C. | \[(-\infty ,0)\] |
| D. | \[(2,\infty )\] |
| Answer» B. \[(0,\infty )\] | |
| 8015. |
Given \[f:[-2a,2a]\to R\] is an odd function such that the left hand derivative at x = a is zero and \[f(x)=f(2a-x)\forall x\in (a,2a),\] then its left had derivative at \[x=-a\] is |
| A. | 0 |
| B. | a |
| C. | #NAME? |
| D. | Does not exist |
| Answer» B. a | |
| 8016. |
The number of points at which the function \[f(x)=\left| x-0.5 \right|+\left| x-1 \right|+\tan x\] does not have a derivative in the interval (0, 2) is |
| A. | 0 |
| B. | 1 |
| C. | 2 |
| D. | 3 |
| Answer» E. | |
| 8017. |
What is \[\underset{x\to 0}{\mathop{\lim }}\,\frac{2(1-\cos x)}{{{x}^{2}}}\] equal to? |
| A. | 0 |
| B. | 44228 |
| C. | ¼ |
| D. | 1 |
| Answer» E. | |
| 8018. |
Consider the following statements: 1. The function f (x) = greatest integer \[\le x,\text{ }x\in R\]is a continuous function. 2. All trigonometric functions are continuous on R. Which of the statements given above is/are correct? |
| A. | 1 only |
| B. | 2 only |
| C. | Both 1 and 2 |
| D. | Neither 1 nor 2 |
| Answer» E. | |
| 8019. |
The function f(x)=|2 sgn 2x|+2 has |
| A. | jump discontinuity |
| B. | removal discontinuity |
| C. | infinite discontinuity |
| D. | no discontinuity at x=0 |
| Answer» B. removal discontinuity | |
| 8020. |
Let \[f(x)=\underset{n\to \infty }{\mathop{\lim }}\,\frac{\log (2+x)-{{x}^{2n}}\sin x}{1+{{x}^{2n}}}\], then |
| A. | f is continuous at x=1 |
| B. | \[\underset{x\to {{1}^{+}}}{\mathop{\lim }}\,f(x)=log3\] |
| C. | \[\underset{x\to {{1}^{+}}}{\mathop{\lim }}\,f(x)=-\sin 1\] |
| D. | \[\underset{x\to {{1}^{-}}}{\mathop{\lim }}\,f(x)\]does not exist |
| Answer» D. \[\underset{x\to {{1}^{-}}}{\mathop{\lim }}\,f(x)\]does not exist | |
| 8021. |
\[f(x)=\left\{ \begin{matrix} \frac{x}{2{{x}^{2}}+\left| x \right|,}\,\,x\ne 0 \\ 1.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x=0 \\ \end{matrix} \right.\]. Then \[f(x)\] is |
| A. | continuous but non-differentiable at x=0 |
| B. | differentiable at x=0 |
| C. | discontinuous at x=0 |
| D. | none of these |
| Answer» D. none of these | |
| 8022. |
The value of \[{{\cos }^{-1}}x+{{\cos }^{-1}}\left( \frac{x}{2}+\frac{1}{2}\sqrt{3-3{{x}^{2}}} \right);\frac{1}{2}\le x\le 1\] is |
| A. | \[-\frac{\pi }{3}\] |
| B. | \[\frac{\pi }{3}\] |
| C. | \[\frac{3}{\pi }\] |
| D. | \[-\frac{3}{\pi }\] |
| Answer» C. \[\frac{3}{\pi }\] | |
| 8023. |
If \[\sin \left( {{\sin }^{-1}}\frac{1}{5}+{{\cos }^{-1}}x \right)=1\], then what is x equal to? |
| A. | 0 |
| B. | 1 |
| C. | \[\frac{4}{5}\] |
| D. | \[\frac{1}{5}\] |
| Answer» E. | |
| 8024. |
In a triangle ABC. If \[A={{\tan }^{-1}}2\] and \[B={{\tan }^{-1}}3,\]then C is equal to |
| A. | \[\frac{\pi }{3}\] |
| B. | \[\frac{\pi }{4}\] |
| C. | \[\frac{\pi }{6}\] |
| D. | \[\frac{\pi }{2}\] |
| Answer» C. \[\frac{\pi }{6}\] | |
| 8025. |
Let \[-1\le x\le 1.\] If \[\cos (si{{n}^{-1}}x)=\frac{1}{2},\] then how many value does \[\tan (co{{s}^{-1}}x)\] assume? |
| A. | One |
| B. | Two |
| C. | Four |
| D. | Infinite |
| Answer» C. Four | |
| 8026. |
If \[{{\sin }^{-1}}1+{{\sin }^{-1}}\frac{4}{5}={{\sin }^{-1}}x,\] then what is x equal to? |
| A. | 44319 |
| B. | 44320 |
| C. | 1 |
| D. | 0 |
| Answer» B. 44320 | |
| 8027. |
If \[{{\sin }^{-1}}\left( \frac{2a}{1+{{a}^{2}}} \right)-{{\cos }^{-1}}\left( \frac{1-{{b}^{2}}}{1+{{b}^{2}}} \right)={{\tan }^{-1}}\left( \frac{2x}{1-{{x}^{2}}} \right),\]then what is the value of x? |
| A. | \[a/b\] |
| B. | \[ab\] |
| C. | \[b/a\] |
| D. | \[\frac{a-b}{1+ab}\] |
| Answer» E. | |
| 8028. |
The value of\[{{\tan }^{-1}}\left( \frac{1}{2}\tan 2A)+{{\tan }^{-1}}(cotA)+ta{{n}^{-1}}(co{{t}^{3}}A) \right)\] is |
| A. | 0 if \[\frac{\pi }{4}<A<\frac{\pi }{2}\] |
| B. | \[\pi \], if \[0<A<\frac{\pi }{4}\] |
| C. | Both a and b |
| D. | None of these |
| Answer» D. None of these | |
| 8029. |
If the equation \[{{(si{{n}^{-1}}x)}^{3}}+{{(co{{s}^{-1}}x)}^{3}}=a{{\pi }^{2}}\] has no real root then |
| A. | \[a>0\] |
| B. | \[a<\frac{1}{32}\] |
| C. | \[a<3\] |
| D. | None of these |
| Answer» C. \[a<3\] | |
| 8030. |
What is the value of:\[\cos \left[ {{\tan }^{-1}}\left\{ \tan \left( \frac{15\pi }{4} \right) \right\} \right]?\] |
| A. | \[-\frac{1}{\sqrt{2}}\] |
| B. | 0 |
| C. | \[\frac{1}{\sqrt{2}}\] |
| D. | \[\frac{1}{2\sqrt{2}}\] |
| Answer» D. \[\frac{1}{2\sqrt{2}}\] | |
| 8031. |
If \[{{\tan }^{-1}}(2x)+ta{{n}^{-1}}(3x)=\frac{\pi }{4}\]then x is equal to |
| A. | -1 |
| B. | -2 |
| C. | 1 |
| D. | 2 |
| Answer» B. -2 | |
| 8032. |
Solving \[2{{\cos }^{-1}}x={{\sin }^{-1}}(2x\sqrt{1-{{x}^{2}}}),\]we get |
| A. | \[x\in \left[ \frac{\sqrt{2}}{2},1 \right]\] |
| B. | \[x=3\] |
| C. | \[x\in [3,4]\] |
| D. | \[x=0\] |
| Answer» B. \[x=3\] | |
| 8033. |
The complete solution set of \[{{[co{{t}^{-1}}x]}^{2}}-6[co{{t}^{-1}}x]+9\le 0,\] Where [.] denotes the greatest integer function, is |
| A. | \[(-\infty ,\cot 3]\] |
| B. | \[[\cot 3,\cot 2)\] |
| C. | \[[\cot 3,\infty )\] |
| D. | None of these |
| Answer» B. \[[\cot 3,\cot 2)\] | |
| 8034. |
The sum of the infinite series \[{{\cot }^{-1}}2+{{\cot }^{-1}}8+{{\cot }^{-1}}18+{{\cot }^{-1}}32+...\] is, |
| A. | \[\pi \] |
| B. | \[\frac{\pi }{2}\] |
| C. | \[\frac{\pi }{4}\] |
| D. | None of these |
| Answer» D. None of these | |
| 8035. |
\[{{\sin }^{-1}}\left( a-\frac{{{a}^{2}}}{3}+\frac{{{a}^{3}}}{9}+... \right)+{{\cos }^{-1}}(1+b+{{b}^{2}}+...)=\frac{\pi }{2}\]when |
| A. | \[a=-3\] and \[b=1\] |
| B. | \[a=1\] and \[b=-\frac{1}{3}\] |
| C. | \[a=\frac{1}{6}\] and \[b=\frac{1}{2}\] |
| D. | None of these |
| Answer» C. \[a=\frac{1}{6}\] and \[b=\frac{1}{2}\] | |
| 8036. |
If \[{{\tan }^{-1}}\frac{x}{\pi } |
| A. | 2 |
| B. | 5 |
| C. | 7 |
| D. | None of these |
| Answer» C. 7 | |
| 8037. |
\[\tan \left\{ \frac{1}{2}{{\sin }^{-1}}\frac{2x}{1+{{x}^{2}}}+\frac{1}{2}{{\cos }^{-1}}\frac{1-{{y}^{2}}}{1+{{y}^{2}}} \right\}=\] |
| A. | \[\frac{x-y}{1+xy}\] |
| B. | \[\frac{x+y}{1-xy}\] |
| C. | \[\frac{x-y}{x+y}\] |
| D. | \[\frac{1-xy}{1+xy}\] |
| Answer» C. \[\frac{x-y}{x+y}\] | |
| 8038. |
If \[ax+b(sec(ta{{n}^{-1}}x))=c\] and \[ay+b\]\[(sec\,.\,(ta{{n}^{-1}}y))=c,\] then \[\frac{x+y}{1-xy}=\] |
| A. | \[\frac{ac}{{{a}^{2}}+{{c}^{2}}}\] |
| B. | \[\frac{2ac}{a-c}\] |
| C. | \[\frac{2ac}{{{a}^{2}}-{{c}^{2}}}\] |
| D. | \[\frac{a+c}{1-ac}\] |
| Answer» D. \[\frac{a+c}{1-ac}\] | |
| 8039. |
If \[{{\sin }^{-1}}x={{\tan }^{-1}}y,\] what is the value of\[\frac{1}{{{x}^{2}}}-\frac{1}{{{y}^{2}}}?\] |
| A. | 1 |
| B. | -1 |
| C. | 0 |
| D. | 2 |
| Answer» B. -1 | |
| 8040. |
Complete solution set of \[{{\tan }^{2}}(si{{n}^{-1}}x)>1\]is |
| A. | \[\left( -1,-\frac{1}{\sqrt{2}} \right)\cup \left( \frac{1}{\sqrt{2}},1 \right)\] |
| B. | \[\left( -\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}} \right)\tilde{\ }\{0\}\] |
| C. | \[(-1,1)\tilde{\ }\{0\}\] |
| D. | None of these |
| Answer» B. \[\left( -\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}} \right)\tilde{\ }\{0\}\] | |
| 8041. |
If \[{{\cos }^{-1}}\sqrt{p}+{{\cos }^{-1}}\sqrt{1-p}+{{\cos }^{-1}}\sqrt{1-q}=\frac{3\pi }{4}\] then the value of q is equal to |
| A. | 1 |
| B. | \[\frac{1}{\sqrt{2}}\] |
| C. | \[\frac{1}{3}\] |
| D. | \[\frac{1}{2}\] |
| Answer» E. | |
| 8042. |
If \[[si{{n}^{-1}}co{{s}^{-1}}si{{n}^{-1}}ta{{n}^{-1}}x]=1,\] where \[[.]\] denotes the greatest integer function, then x belongs to the interval |
| A. | \[[tan\,sin\,cos1,tan\,sin\,cos\,sin1]\] |
| B. | \[(tan\,sin\,cos1,tan\,sin\,cos\,sin1)\] |
| C. | \[[-1,1]\] |
| D. | \[[sin\,cos\,tan1,sin\,cos\,tan1]\] |
| Answer» B. \[(tan\,sin\,cos1,tan\,sin\,cos\,sin1)\] | |
| 8043. |
What is the value of x that satisfies the equation\[{{\cos }^{-1}}x=2{{\sin }^{-1}}x\]? |
| A. | \[\frac{1}{2}\] |
| B. | \[-1\] |
| C. | \[1\] |
| D. | \[-\frac{1}{2}\] |
| Answer» B. \[-1\] | |
| 8044. |
The range of \[y=(co{{t}^{-1}}x)(co{{t}^{-1}}(-x))\] is |
| A. | \[(\left. 0,\frac{{{\pi }^{2}}}{4} \right]\] |
| B. | \[(0,\pi )\] |
| C. | \[(0,2\pi ]\] |
| D. | \[(0,1]\] |
| Answer» C. \[(0,2\pi ]\] | |
| 8045. |
The value of \[3{{\tan }^{-1}}\frac{1}{2}+2{{\tan }^{-1}}\frac{1}{5}+{{\sin }^{-1}}\frac{142}{65\sqrt{5}}\]is |
| A. | \[\frac{\pi }{4}\] |
| B. | \[\frac{\pi }{2}\] |
| C. | \[\pi \] |
| D. | None of these |
| Answer» D. None of these | |
| 8046. |
If \[{{\sin }^{-1}}a+{{\sin }^{-1}}b+{{\sin }^{-1}}c=\pi ,\] then find the value of\[a\sqrt{1-{{a}^{2}}}+b\sqrt{1-{{b}^{2}}}+c\sqrt{1-{{c}^{2}}}.\] |
| A. | \[abc\] |
| B. | \[a+b+c\] |
| C. | \[\frac{1}{a}\times \frac{1}{b}\times \frac{1}{c}\] |
| D. | \[2abc\] |
| Answer» E. | |
| 8047. |
If \[\sum\limits_{i=1}^{2n}{{{\cos }^{-1}}{{x}_{i}}=0}\] then \[\sum\limits_{i=1}^{2n}{{{x}_{i}}}\]is |
| A. | n |
| B. | 2n |
| C. | \[\frac{n\left( n+1 \right)}{2}\] |
| D. | None of these |
| Answer» C. \[\frac{n\left( n+1 \right)}{2}\] | |
| 8048. |
If \[{{\sin }^{-1}}\frac{1}{x}={{\sin }^{-1}}\frac{1}{a}+{{\sin }^{-1}}\frac{1}{b},\] then the value of x is |
| A. | \[\frac{ab}{\sqrt{{{a}^{2}}-1}+\sqrt{{{b}^{2}}-1}}\] |
| B. | \[\frac{ab}{\sqrt{{{a}^{2}}-1}-\sqrt{{{b}^{2}}-1}}\] |
| C. | \[\frac{2ab}{\sqrt{{{a}^{2}}-1}+\sqrt{{{b}^{2}}-1}}\] |
| D. | None of these |
| Answer» B. \[\frac{ab}{\sqrt{{{a}^{2}}-1}-\sqrt{{{b}^{2}}-1}}\] | |
| 8049. |
The solutions set of the equations \[{{\sin }^{-1}}x=2{{\tan }^{-1}}x\] is |
| A. | \[\left\{ 1,2 \right\}\] |
| B. | \[\left\{ -1,2 \right\}\] |
| C. | \[\left\{ -1,1,0 \right\}\] |
| D. | \[\left\{ 1,1/2,0 \right\}\] |
| Answer» D. \[\left\{ 1,1/2,0 \right\}\] | |
| 8050. |
What is \[\sin [co{{t}^{-1}}\{cos(ta{{n}^{-1}}x)]\] where \[x>0\], equal to? |
| A. | \[\sqrt{\frac{({{x}^{2}}+1)}{({{x}^{2}}+2)}}\] |
| B. | \[\sqrt{\frac{({{x}^{2}}+2)}{({{x}^{2}}+1)}}\] |
| C. | \[\frac{({{x}^{2}}+1)}{({{x}^{2}}+2)}\] |
| D. | \[\frac{({{x}^{2}}+2)}{({{x}^{2}}+1)}\] |
| Answer» B. \[\sqrt{\frac{({{x}^{2}}+2)}{({{x}^{2}}+1)}}\] | |