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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.
| 951. |
The number of real values of parameter k for which \[{{(lo{{g}_{16}}x)}^{2}}-{{\log }_{16}}x+{{\log }_{16}}k=0\] will have exactly one solution is |
| A. | 0 |
| B. | 2 |
| C. | 1 |
| D. | 4 |
| Answer» D. 4 | |
| 952. |
The solution set of the inequality \[\left| x+2 \right|-\left| x-1 \right| |
| A. | \[\left( \frac{9}{2},\infty \right)\] |
| B. | \[\left( -\infty ,\frac{3}{2} \right)\] |
| C. | \[\left( -2,-\frac{3}{2} \right)\] |
| D. | \[\left( -1,\frac{3}{2} \right)\] |
| Answer» B. \[\left( -\infty ,\frac{3}{2} \right)\] | |
| 953. |
The solution set of the inequality \[{{5}^{x+2}}>{{\left( \frac{1}{25} \right)}^{1/x}}\]is |
| A. | \[(-2,0)\] |
| B. | \[(-2,2)\] |
| C. | \[(-5,5)\] |
| D. | \[(0,\infty )\] |
| Answer» E. | |
| 954. |
If x satisfies the inequalities \[x+7 |
| A. | \[(-\infty ,3)\] |
| B. | \[(1,3)\] |
| C. | \[(4,\infty )\] |
| D. | \[(-\infty ,-1)\] |
| Answer» D. \[(-\infty ,-1)\] | |
| 955. |
The solution set of \[\frac{2x-1}{3}\ge \left( \frac{3x-2}{4} \right)-\left( \frac{2-x}{5} \right)\] is\[(-\infty ,a]\]. The value of ?a? is |
| A. | 2 |
| B. | 3 |
| C. | 4 |
| D. | 5 |
| Answer» B. 3 | |
| 956. |
The solution set of \[{{(x)}^{2}}+{{(x+1)}^{2}}=25,\] where (x) is the least integer greater than or equal to x, is |
| A. | \[(2,4)\] |
| B. | \[(-5,-4]\cup (2,3]\] |
| C. | \[[-4,-3)\cup [3,4)\] |
| D. | None of these |
| Answer» C. \[[-4,-3)\cup [3,4)\] | |
| 957. |
A man wants to cut three lengths from a single piece of board of length 91 cm. The second length is to be 3 cm longer than the shortest and the third length is to be twice as long as the shortest. The possible length of the shortest board, if the third piece is to be at least 5 cm longer than the second, is |
| A. | Less than 8 cm |
| B. | Greater than or equal to 8 cm but less then or equal to 22 cm |
| C. | Less than 22 cm |
| D. | Greater than 22 cm |
| Answer» C. Less than 22 cm | |
| 958. |
Which of the following linear inequalities satisfy the shaded region of the given figure? |
| A. | \[2x+3y\ge 3\] |
| B. | \[3x+4y\le 18\] |
| C. | \[x-6y\le 3\] |
| D. | All of these |
| Answer» E. | |
| 959. |
If \[y=(1+{{x}^{1/4}})(1+{{x}^{1/2}})(1-{{x}^{1/4}}),\] then \[\frac{dy}{dx}\] is equal to |
| A. | 1 |
| B. | -1 |
| C. | x |
| D. | \[\sqrt{x}\] |
| Answer» C. x | |
| 960. |
The value of \[\underset{n\to \infty }{\mathop{\lim }}\,\left[ \sqrt[3]{{{(n+1)}^{2}}}-\sqrt[3]{{{(n-1)}^{2}}} \right]\] is |
| A. | 1 |
| B. | -1 |
| C. | 0 |
| D. | \[-\infty \] |
| Answer» D. \[-\infty \] | |
| 961. |
If \[\underset{x\to a}{\mathop{\lim }}\,\left[ \frac{f(x)}{g(x)} \right]\]exist, then which one of the following correct? |
| A. | Both \[\underset{x\to a}{\mathop{\lim }}\,f(x)\] and \[\underset{x\to a}{\mathop{\lim }}\,g(x)\] must exist |
| B. | \[\underset{x\to a}{\mathop{\lim }}\,f(x)\] need not exist but \[\underset{x\to a}{\mathop{\lim }}\,g(x)\] must exist |
| C. | Both \[\underset{x\to a}{\mathop{\lim }}\,f(x)\] and \[\underset{x\to a}{\mathop{\lim }}\,g(x)\] need not exist |
| D. | None of these |
| Answer» E. | |
| 962. |
The value of \[\underset{x\to \pi /2}{\mathop{\lim }}\,{{\tan }^{2}}x(\sqrt{2{{\sin }^{2}}x+3\sin x+4}\] \[-\sqrt{{{\sin }^{2}}x+6\sin x+2)}\] is equal to |
| A. | \[\frac{1}{10}\] |
| B. | \[\frac{1}{11}\] |
| C. | \[\frac{1}{12}\] |
| D. | \[\frac{1}{8}\] |
| Answer» D. \[\frac{1}{8}\] | |
| 963. |
If \[\underset{x\to 0}{\mathop{\lim }}\,{{(1+asinx)}^{\cos ecx}}=3.\]then a is |
| A. | ln 2 |
| B. | ln 3 |
| C. | ln 4 |
| D. | \[{{e}^{3}}\] |
| Answer» C. ln 4 | |
| 964. |
What is \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin 2x+4x}{2x+\sin 4x}\] equal to? |
| A. | 0 |
| B. | \[\frac{1}{2}\] |
| C. | 1 |
| D. | 2 |
| Answer» D. 2 | |
| 965. |
What is \[\underset{n\to \infty }{\mathop{\lim }}\,\frac{1+2+3+...+n}{{{1}^{2}}+{{2}^{2}}+{{3}^{2}}+...{{n}^{2}}}\] equal to? |
| A. | 5 |
| B. | 2 |
| C. | 1 |
| D. | 0 |
| Answer» E. | |
| 966. |
If \[\underset{x\to \infty }{\mathop{\lim }}\,{{\left( 1+\frac{a}{x}+\frac{b}{{{x}^{2}}} \right)}^{2x}}={{e}^{2}},\] then the values of a and b, are |
| A. | \[a=1\] and \[b=2\] |
| B. | \[a=1,b\in R\] |
| C. | \[a\in R,b=2\] |
| D. | \[a\in R,b\in R\] |
| Answer» C. \[a\in R,b=2\] | |
| 967. |
For the function \[f(x)=\frac{{{x}^{100}}}{100}+\frac{{{x}^{99}}}{99}+...\frac{{{x}^{2}}}{2}+x+1.\] \[f'(1)=mf'(0),\] Where m is equal to |
| A. | 50 |
| B. | 0 |
| C. | 100 |
| D. | 200 |
| Answer» D. 200 | |
| 968. |
If \[{{z}_{r}}=\cos \frac{r\alpha }{{{n}^{2}}}+i\sin \frac{r\alpha }{{{n}^{2}}},\] where \[r=1,2,3,...n,\] then \[\underset{x\to \infty }{\mathop{\lim }}\,{{z}_{1}}{{z}_{2}}{{z}_{3}}...{{z}_{n}}\] is equal to |
| A. | \[\cos \alpha +i\sin \alpha \] |
| B. | \[\cos (\alpha /2)-i\,\,sin(\alpha /2)\] |
| C. | \[{{e}^{i\alpha /2}}\] |
| D. | \[\sqrt[3]{{{e}^{i\alpha }}}\] |
| Answer» D. \[\sqrt[3]{{{e}^{i\alpha }}}\] | |
| 969. |
If \[m,\,\,\,n\in {{I}_{0}}\] and \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\tan 2x-n\sin x}{{{x}^{3}}}=\] some integer, then value of this limit is |
| A. | 3 |
| B. | 2 |
| C. | \[\frac{16+n}{12}\] |
| D. | None of these |
| Answer» B. 2 | |
| 970. |
Let \[f:R\to R\] be such that \[f(1)=3\] and \[f'(1)=6.\] Then \[\underset{x\to 0}{\mathop{\lim }}\,{{\left( \frac{f(1+x)}{f(1)} \right)}^{1/x}}\] equals |
| A. | 1 |
| B. | \[{{e}^{1/2}}\] |
| C. | \[{{e}^{2}}\] |
| D. | \[{{e}^{3}}\] |
| Answer» D. \[{{e}^{3}}\] | |
| 971. |
If \[f(x)=\underset{n\,\to \,\infty }{\mathop{\lim }}\,n({{x}^{1/n}}-1),\] then for \[x>0,\,\,y>0,\]\[f(xy)\] is equal to |
| A. | \[f(x)f(y)\] |
| B. | \[f(x)+f(y)\] |
| C. | \[f(x)-f(y)\] |
| D. | None of these |
| Answer» C. \[f(x)-f(y)\] | |
| 972. |
Let \[\alpha \] and \[\beta \] be the roots of \[a{{x}^{2}}+bx+c=0.\]Then \[\underset{x\to \alpha }{\mathop{\lim }}\,\frac{1-\cos (a{{x}^{2}}+bx+c)}{{{(x-\alpha )}^{2}}}\] is equal to: |
| A. | 0 |
| B. | \[\frac{1}{2}{{(\alpha -\beta )}^{2}}\] |
| C. | \[\frac{{{a}^{2}}}{2}{{(\alpha -\beta )}^{2}}\] |
| D. | None of these |
| Answer» D. None of these | |
| 973. |
\[\underset{x\to 0}{\mathop{\lim }}\,\left( \frac{10\sin 9x}{9\sin 10x} \right)\left( \frac{8\sin 7x}{7\sin 8x} \right)\left( \frac{6\sin 5x}{5\sin 6x} \right)\left( \frac{4\sin 3x}{3\sin 4x} \right)\] \[\left( \frac{\sin x}{\sin 2x} \right)=\] |
| A. | \[\frac{63}{256}\] |
| B. | \[\frac{1}{6}\] |
| C. | \[\frac{6}{5}\] |
| D. | \[\frac{1}{2}\] |
| Answer» E. | |
| 974. |
\[\underset{x\to 0}{\mathop{\lim }}\,\left[ \frac{\sin (sgn(x))}{(sgn(x))} \right],\] where [.] denotes the greatest integer function, is equal to |
| A. | 0 |
| B. | 1 |
| C. | -1 |
| D. | Does not exist |
| Answer» B. 1 | |
| 975. |
The value of \[\underset{x\to \infty }{\mathop{\lim }}\,\frac{({{2}^{{{x}^{n}}}}){{e}^{\frac{1}{^{x}}}}-({{3}^{{{x}^{n}}}}){{e}^{\frac{1}{x}}}}{{{x}^{n}}}\](where \[n\in N\]) is |
| A. | \[\log n\left( \frac{2}{3} \right)\] |
| B. | 0 |
| C. | \[n\log n\left( \frac{2}{3} \right)\] |
| D. | Not defined |
| Answer» C. \[n\log n\left( \frac{2}{3} \right)\] | |
| 976. |
\[\underset{x\to \infty }{\mathop{\lim }}\,\left( \frac{{{x}^{2}}}{3x-2}-\frac{x}{3} \right)=\] |
| A. | \[\frac{1}{3}\] |
| B. | \[\frac{2}{3}\] |
| C. | \[\frac{-2}{3}\] |
| D. | \[\frac{2}{9}\] |
| Answer» E. | |
| 977. |
If \[a=\min \{{{x}^{2}}+4x+5,x\in R\}\]and \[b=\underset{\theta \to 0}{\mathop{\lim }}\,\frac{1-\cos 2\theta }{{{\theta }^{2}}},\] then the value of \[\sum\limits_{r=0}^{n}{{{a}^{r}}.{{b}^{n-r}}}\] is |
| A. | \[\frac{{{2}^{n+1}}-1}{{{4.2}^{n}}}\] |
| B. | \[{{2}^{n+1}}-1\] |
| C. | \[\frac{{{2}^{n+1}}-1}{{{3.2}^{n}}}\] |
| D. | None of these |
| Answer» C. \[\frac{{{2}^{n+1}}-1}{{{3.2}^{n}}}\] | |
| 978. |
\[\underset{x\to 0}{\mathop{\lim }}\,\left[ \min ({{y}^{2}}-4y+11)\frac{\sin x}{x} \right]\] (where [.] denotes the greatest integer function) is |
| A. | 5 |
| B. | 6 |
| C. | 7 |
| D. | Does not exist |
| Answer» C. 7 | |
| 979. |
If \[m,n\in {{I}_{0}}\] and \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\tan 2x-n\sin x}{{{x}^{3}}}=\]some integer, then value of this limit is |
| A. | 3 |
| B. | 2 |
| C. | \[\frac{16+n}{12}\] |
| D. | None of these |
| Answer» B. 2 | |
| 980. |
The limit \[\underset{n\,\to \,\infty }{\mathop{\lim }}\,\,\underset{r=3}{\overset{n}{\mathop{\prod }}}\,\,\,\frac{{{r}^{3}}-8}{{{r}^{3}}+8}\] is equal to |
| A. | \[\frac{2}{7}\] |
| B. | \[\frac{1}{12}\] |
| C. | \[\frac{19}{52}\] |
| D. | None of these |
| Answer» B. \[\frac{1}{12}\] | |
| 981. |
Let f(x) be a polynomial function satisfying \[f(x).f\left( \frac{1}{x} \right)=f(x)+f\left( \frac{1}{x} \right).\] if \[f(4)=65\] and \[{{l}_{1}},{{l}_{2}},{{l}_{3}}\]are in \[GP,\] then \[f'({{l}_{1}}),f'({{l}_{2}}),f'({{l}_{3}})\] are in |
| A. | AP |
| B. | GP |
| C. | HP |
| D. | None of these |
| Answer» C. HP | |
| 982. |
If \[\{x\}\]denotes the fractional part of x, then \[\underset{x\to [a]}{\mathop{\lim }}\,\frac{{{e}^{\{x\}}}-\{x\}-1}{{{\{x\}}^{2}}},\] Where [a] denotes the integral part of a, is equal to |
| A. | 0 |
| B. | \[\frac{1}{2}\] |
| C. | \[e-2\] |
| D. | None of these |
| Answer» E. | |
| 983. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin {{x}^{4}}-{{x}^{4}}\cos {{x}^{4}}+{{x}^{20}}}{{{x}^{4}}({{e}^{2{{x}^{4}}}}1-2{{x}^{4}})}\] is equal to |
| A. | 0 |
| B. | \[-1/6\] |
| C. | \[1/6\] |
| D. | Does not exist |
| Answer» D. Does not exist | |
| 984. |
Let \[f(x)=x{{(-1)}^{[1/x]}},x\ne 0,\] where [x] denotes the greatest integer less than or equal to x then, \[\underset{x\to 0}{\mathop{\lim }}\,f(x)=\] |
| A. | Does not exist |
| B. | 2 |
| C. | 0 |
| D. | -1 |
| Answer» D. -1 | |
| 985. |
If \[f(x)=\left\{ \begin{matrix} \frac{{{[x]}^{2}}+\sin [x]}{[x]}for[x]\ne 0 \\ 0for[x]=0 \\ \end{matrix} \right.\], where [x] denotes the greatest integer less than or equal to\[x,\] Then \[\underset{x\to 0}{\mathop{\lim }}\,f(x)\] equals |
| A. | 1 |
| B. | 0 |
| C. | -1 |
| D. | None of these |
| Answer» E. | |
| 986. |
\[\underset{x\to 0}{\mathop{\lim }}\,{{\left| x \right|}^{[cosx]}}\] is |
| A. | 1 |
| B. | Does not exist |
| C. | 0 |
| D. | None of these |
| Answer» B. Does not exist | |
| 987. |
\[\underset{n\,\to \,\infty }{\mathop{\lim }}\,\frac{{{5}^{n+1}}+{{3}^{n}}-{{2}^{2n}}}{{{5}^{n}}+{{2}^{n}}+{{3}^{2n+3}}}\] is equal to |
| A. | 5 |
| B. | 3 |
| C. | 1 |
| D. | 0 |
| Answer» E. | |
| 988. |
Let \[f(x)=x-[x],\] where [x] denotes the greatest integer \[\le x\] and \[g(x)=\underset{n\,\to \,\infty }{\mathop{\lim }}\,\frac{{{\{f(x)\}}^{2n}}-1}{{{\{f(x)\}}^{2n}}+1},\] then g(x) is equal to |
| A. | 0 |
| B. | 1 |
| C. | -1 |
| D. | None of these |
| Answer» D. None of these | |
| 989. |
Let \[f(x)=\left\{ \begin{matrix} x\sin \left( \frac{1}{x} \right)+\sin \left( \frac{1}{{{x}^{2}}} \right),x\ne 0 \\ 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x=0 \\ \end{matrix} \right.\]then \[\underset{x\to \infty }{\mathop{\lim }}\,f(x)\] equals |
| A. | 0 |
| B. | \[-1/2\] |
| C. | 1 |
| D. | None of these |
| Answer» D. None of these | |
| 990. |
If f be a function given by \[f(x)=2{{x}^{2}}+3x-5.\] Then, \[f'(0)=mf'(-1),\] where m is equal to |
| A. | -1 |
| B. | -2 |
| C. | -3 |
| D. | -4 |
| Answer» D. -4 | |
| 991. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin [cosx]}{1+[cosx]}\] (\[[\,\cdot \,]\] denotes the greatest integer function) |
| A. | Equal to 1 |
| B. | Equal to 0 |
| C. | Does not exist |
| D. | None of these |
| Answer» C. Does not exist | |
| 992. |
Let the sequence \[\] of real numbers satisfies the recurrence relation \[{{b}_{n+1}}=\frac{1}{3}\left( 2{{b}_{n}}+\frac{125}{{{b}^{2}}_{n}} \right),{{b}_{n}}\ne 0.\] Then find \[\underset{n\,\to \,\infty }{\mathop{\lim }}\,{{b}_{n}}.\] |
| A. | 10 |
| B. | 15 |
| C. | 5 |
| D. | 25 |
| Answer» D. 25 | |
| 993. |
If \[x>0\] and \[g\] is a bounded function, then \[\underset{n\,\to \,\infty }{\mathop{\lim }}\,\frac{f(x){{e}^{nx}}+g(x)}{{{e}^{nx}}+1}\] is |
| A. | 0 |
| B. | \[f(x)\] |
| C. | \[g(x)\] |
| D. | None of these |
| Answer» C. \[g(x)\] | |
| 994. |
If \[f(x)=\left\{ \begin{matrix} {{x}^{n}}\sin (1/{{x}^{2}}),x\ne 0 \\ 0,x=0 \\ \end{matrix} \right.\], \[(n\in I)\], then |
| A. | \[\underset{x\to 0}{\mathop{\lim }}\,f(x)\] exists for \[n>1\] |
| B. | \[\underset{x\to 0}{\mathop{\lim }}\,f(x)\] exists for \[n<0\] |
| C. | \[\underset{x\to 0}{\mathop{\lim }}\,f(x)\] Does not exist for any value of n |
| D. | \[\underset{x\to 0}{\mathop{\lim }}\,f(x)\] cannot be determined |
| Answer» B. \[\underset{x\to 0}{\mathop{\lim }}\,f(x)\] exists for \[n<0\] | |
| 995. |
If \[{{\cos }^{-1}}x+{{\cos }^{-1}}y+{{\cos }^{-1}}z=\pi ,\]then |
| A. | \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}+xyz=0\] |
| B. | \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}+2xyz=0\] |
| C. | \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}+xyz=1\] |
| D. | \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}+2xyz=1\] |
| Answer» E. | |
| 996. |
The set of values of k for which \[{{x}^{2}}-kx+{{\sin }^{-1}}(sin4)>0\] for all real x is |
| A. | \[\phi \] |
| B. | \[(-2,2)\] |
| C. | \[R\] |
| D. | \[(-\infty ,-2)\cup (2,\infty )\] |
| Answer» B. \[(-2,2)\] | |
| 997. |
The formula \[{{\sin }^{-1}}\{2x(1-{{x}^{2}})\}=2si{{n}^{-1}}x\] is true for all values of x lying in the interval |
| A. | \[[-1,1]\] |
| B. | \[[0,1]\] |
| C. | \[[-1,0]\] |
| D. | \[\left[ -1/\sqrt{2},1/\sqrt{2} \right]\] |
| Answer» E. | |
| 998. |
If \[{{a}_{1}},{{a}_{2}},{{a}_{3}},....{{a}_{n}}\] is an \[A.P.\] with common difference d; \[(d>0)\] then \[\tan \left[ {{\tan }^{-1}}\left( \frac{d}{1+{{a}_{1}}{{a}_{2}}} \right)+{{\tan }^{-1}}\left( \frac{d}{1+{{a}_{2}}{{a}_{3}}} \right)+...+ta{{n}^{-1}}\left( \frac{d}{1+{{a}_{n-1}}{{a}_{n}}} \right) \right]\]is equal to |
| A. | \[\frac{(n-1)d}{{{a}_{1}}+{{a}_{n}}}\] |
| B. | \[\frac{(n-1)d}{1+{{a}_{1}}{{a}_{n}}}\] |
| C. | \[\frac{nd}{1+{{a}_{1}}{{a}_{n}}}\] |
| D. | \[\frac{{{a}_{n}}-{{a}_{1}}}{{{a}_{n}}+{{a}_{1}}}\] |
| Answer» C. \[\frac{nd}{1+{{a}_{1}}{{a}_{n}}}\] | |
| 999. |
If \[{{\sin }^{-1}}x+{{\sin }^{-1}}y+{{\sin }^{-1}}z=\pi ,\] then \[{{x}^{4}}+{{y}^{4}}+{{z}^{4}}+4{{x}^{2}}{{y}^{2}}{{z}^{2}}=k({{x}^{2}}{{y}^{2}}+{{y}^{2}}{{z}^{2}}+{{z}^{2}}{{x}^{2}}).\]where k = |
| A. | 1 |
| B. | 2 |
| C. | 4 |
| D. | None of these |
| Answer» C. 4 | |
| 1000. |
If \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}={{r}^{2}},\] then\[{{\tan }^{-1}}\frac{xy}{zr}+{{\tan }^{-1}}\frac{yz}{xr}+{{\tan }^{-1}}\frac{xz}{yr}=\] |
| A. | \[\pi \] |
| B. | \[\frac{\pi }{2}\] |
| C. | 0 |
| D. | None of these |
| Answer» C. 0 | |