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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.
| 1401. |
If \[|z-2|=\min \{|z-1|,|z-5|\},\] where z is a complex number, then |
| A. | \[\operatorname{Re}(z)=\frac{3}{2}\] |
| B. | \[\operatorname{Re}(z)=\frac{7}{2}\] |
| C. | \[\operatorname{Re}(z)\in \left\{ \frac{3}{2},\frac{7}{2} \right\}\] |
| D. | None of these |
| Answer» D. None of these | |
| 1402. |
If the roots of \[a{{x}^{2}}+bx+c=0\] are \[\sin \alpha \] and \[\cos \alpha \] for some \[\alpha ,\] then which one of the following is correct? |
| A. | \[{{a}^{2}}+{{b}^{2}}=2ac\] |
| B. | \[{{b}^{2}}-{{c}^{2}}=2ab\] |
| C. | \[{{b}^{2}}-{{a}^{2}}=2ac\] |
| D. | \[{{b}^{2}}+{{c}^{2}}=2ab\] |
| Answer» D. \[{{b}^{2}}+{{c}^{2}}=2ab\] | |
| 1403. |
If the point \[{{z}_{1}}=1+i\] where \[i=\sqrt{-1}\] is the reflection of a point \[{{z}_{2}}=x+iy\] in the line \[i\bar{z}-iz=5,\] then the point \[{{z}_{2}}\] is |
| A. | \[1+4i\] |
| B. | \[4+i\] |
| C. | \[1-i\] |
| D. | \[-1-i\] |
| Answer» B. \[4+i\] | |
| 1404. |
If \[\alpha \] and \[\beta \] be the values of x in \[{{m}^{2}}({{x}^{2}}-x)+2mx+3=0\] and \[{{m}_{1}}\] and \[{{m}_{2}}\] be two values of m for which \[\alpha \] and \[\beta \] are connected by the relation \[\frac{\alpha }{\beta }+\frac{\beta }{\alpha }=\frac{4}{3}.\] Then the value of \[\frac{m_{1}^{2}}{{{m}_{2}}}+\frac{m_{2}^{2}}{{{m}_{1}}}\] is |
| A. | 6 |
| B. | 68 |
| C. | \[\frac{3}{68}\] |
| D. | \[-\frac{68}{3}\] |
| Answer» E. | |
| 1405. |
The solution of \[2\sqrt{2}\,\,{{x}^{4}}=(\sqrt{3}-1)+i(\sqrt{3}+1)\] is |
| A. | \[\pm \,\,\left( \cos \frac{5\pi }{48}+i\sin \frac{5\pi }{48} \right)\] |
| B. | \[\pm \,\,\left( \cos \frac{7\pi }{48}+i\sin \frac{7\pi }{48} \right)\] |
| C. | \[\pm \,\,\left( \cos \frac{19\pi }{48}-i\sin \frac{19\pi }{48} \right)\] |
| D. | None of these. |
| Answer» B. \[\pm \,\,\left( \cos \frac{7\pi }{48}+i\sin \frac{7\pi }{48} \right)\] | |
| 1406. |
For the complex numbers \[{{z}_{1}}\] and \[{{z}_{2}}\] if \[|1-{{\bar{z}}_{1}}{{z}_{2}}{{|}^{2}}-|{{z}_{1}}-{{z}_{2}}{{|}^{2}}=k(1-|{{z}_{1}}{{|}^{2}})(1-|{{z}_{2}}{{|}^{2}})\] then ?k? equals to |
| A. | 1 |
| B. | \[-1\] |
| C. | 2 |
| D. | \[-2\] |
| Answer» B. \[-1\] | |
| 1407. |
If both the roots of \[k(6{{x}^{2}}+3)+rx+2{{x}^{2}}-1=0\]and \[6k(2{{x}^{2}}+1)+px+4{{x}^{2}}-2=0\] are common, then \[2r-p\]is equal to |
| A. | \[-1\] |
| B. | 0 |
| C. | 1 |
| D. | 2 |
| Answer» C. 1 | |
| 1408. |
If \[z=\frac{\pi }{4}{{(1+i)}^{4}}\left( \frac{1-\sqrt{\pi }i}{\sqrt{\pi }+i}+\frac{\sqrt{\pi }-i}{1+\sqrt{\pi }i} \right),\] then \[\left( \frac{|z|}{am{{p}^{(z)}}} \right)\] equals |
| A. | 1 |
| B. | \[\pi \] |
| C. | \[3\pi \] |
| D. | 4 |
| Answer» E. | |
| 1409. |
The value of a for which the sum of the squares of the roots of the equation \[2{{x}^{2}}-2(a-2)x-(a+1)=0\] is least, is |
| A. | 1 |
| B. | \[3/2\] |
| C. | 2 |
| D. | None |
| Answer» C. 2 | |
| 1410. |
If \[A=\left| x\in IR:{{x}^{2}}+6x-70\},\] then which of the following is/ are correct? 1. \[(A\cap B)=(-2,1)\] 2. \[(A\backslash B)=(-7,-2)\] Select the correct answer using the code given below: |
| A. | 1 only |
| B. | 2 Only |
| C. | Both 1 and 2 |
| D. | Neither 1 nor 2 |
| Answer» D. Neither 1 nor 2 | |
| 1411. |
If \[|z|=\max \,\{|z-1|,\,|z+1|\}\] then |
| A. | \[|z+\bar{z}|=\frac{1}{2}\] |
| B. | \[z+\bar{z}=1\] |
| C. | \[|z+\bar{z}|=1\] |
| D. | None of these |
| Answer» E. | |
| 1412. |
If \[\left| {{z}_{1}} \right|=\left| {{z}_{2}} \right|=.......\left| {{z}_{n}} \right|=1,\] then the value of \[\left| {{z}_{1}}+{{z}_{2}}+....{{z}_{n}} \right|-\left| \frac{1}{{{z}_{1}}}+\frac{1}{{{z}_{2}}}+......+\frac{1}{{{z}_{n}}} \right|\] is, |
| A. | 0 |
| B. | 1 |
| C. | \[-1\] |
| D. | None |
| Answer» B. 1 | |
| 1413. |
The greatest and the least absolute value of \[z+1,\] where \[|z+4|\le 3\] are respectively |
| A. | 6 and 0 |
| B. | 10 and 6 |
| C. | 4 and 3 |
| D. | None of these |
| Answer» B. 10 and 6 | |
| 1414. |
If \[(a+ib)(c+id)(e+if)(g+ih)=A+iB,\] then \[({{a}^{2}}+{{b}^{2}})({{c}^{2}}+{{d}^{2}})({{e}^{2}}+{{f}^{2}})({{g}^{2}}+{{h}^{2}})=\] |
| A. | \[{{A}^{2}}+{{B}^{2}}\] |
| B. | \[{{A}^{2}}-{{B}^{2}}\] |
| C. | \[{{A}^{2}}\] |
| D. | \[{{B}^{2}}\] |
| Answer» B. \[{{A}^{2}}-{{B}^{2}}\] | |
| 1415. |
What is \[\frac{{{(1+i)}^{4n+5}}}{{{(1-i)}^{4n+3}}}\] equal to, where n is a natural number and \[i=\sqrt{-1}\]? |
| A. | 2 |
| B. | \[2i\] |
| C. | \[-2\] |
| D. | i |
| Answer» B. \[2i\] | |
| 1416. |
If one root of the equation \[(1-m){{x}^{2}}+1x+1=0\]is double the other and 1 is real, then what is the greatest value of m? |
| A. | \[-\frac{9}{8}\] |
| B. | \[\frac{9}{8}\] |
| C. | \[-\frac{8}{9}\] |
| D. | \[\frac{8}{9}\] |
| Answer» C. \[-\frac{8}{9}\] | |
| 1417. |
If \[\alpha ,\beta \] are real and \[{{\alpha }^{2}},{{\beta }^{2}}\] are the roots of the equation \[{{a}^{2}}{{x}^{2}}-x+1-{{a}^{2}}=0\left( \frac{1}{\sqrt{2}} |
| A. | \[{{a}^{2}}\] |
| B. | \[\frac{1-{{a}^{2}}}{{{a}^{2}}}\] |
| C. | \[1-{{a}^{2}}\] |
| D. | \[1+{{a}^{2}}\] |
| Answer» C. \[1-{{a}^{2}}\] | |
| 1418. |
If \[z=x+iy,\,\,{{z}^{1/3}}=a-ib,\] then \[\frac{x}{a}-\frac{y}{b}=k({{a}^{2}}-{{b}^{2}})\] where k is equal to |
| A. | 1 |
| B. | 2 |
| C. | 3 |
| D. | 4 |
| Answer» E. | |
| 1419. |
Let \[{{A}_{0}}{{A}_{1}}{{A}_{2}}{{A}_{3}}{{A}_{4}}{{A}_{5}}\] be a regular hexagon inscribed in a circle of unit radius. Then the product of the lengths of the line segments \[{{A}_{0}}{{A}_{1}},{{A}_{0}}{{A}_{2}}\] and \[{{A}_{0}}{{A}_{4}}\] is |
| A. | \[\frac{3}{4}\] |
| B. | \[3\sqrt{3}\] |
| C. | 3 |
| D. | \[\frac{3\sqrt{3}}{2}\] |
| Answer» D. \[\frac{3\sqrt{3}}{2}\] | |
| 1420. |
If \[\omega =\frac{z}{z-\frac{1}{3}i}\] and \[|\omega |=1,\] then z lies on |
| A. | an ellipse |
| B. | a circle |
| C. | a straight line |
| D. | a parabola |
| Answer» D. a parabola | |
| 1421. |
A value of b for which the equations \[{{x}^{2}}+bx-1=0\] \[{{x}^{2}}+x+b=0\] have one root in common is |
| A. | \[-\sqrt{2}\] |
| B. | \[-i\sqrt{3}\] |
| C. | \[i\sqrt{5}\] |
| D. | \[\sqrt{2}\] |
| Answer» C. \[i\sqrt{5}\] | |
| 1422. |
If the roots of the quadratic equation\[{{x}^{2}}+px+q=0\] are \[\tan 30{}^\circ \] and \[\tan 15{}^\circ \] respectively, then the value of \[2+q-p\] is |
| A. | 2 |
| B. | 3 |
| C. | 0 |
| D. | 1 |
| Answer» C. 0 | |
| 1423. |
The fractional part of \[\frac{{{2}^{4n}}}{15}\] is |
| A. | \[\frac{1}{15}\] |
| B. | \[\frac{2}{15}\] |
| C. | \[\frac{4}{15}\] |
| D. | None of these |
| Answer» B. \[\frac{2}{15}\] | |
| 1424. |
The greatest integer less than or equal to; \[{{(\sqrt{2}+1)}^{6}}\] is |
| A. | 196 |
| B. | 197 |
| C. | 198 |
| D. | 199 |
| Answer» C. 198 | |
| 1425. |
If \[{{P}_{n}}\] denotes the product of the binomial coefficients in the expansion of \[{{(1+x)}^{n}}\], then \[\frac{{{P}_{n+1}}}{{{P}_{n}}}\] equals |
| A. | \[\frac{n+1}{n!}\] |
| B. | \[\frac{{{n}^{n}}}{n!}\] |
| C. | \[\frac{{{(n+1)}^{n}}}{(n+1)!}\] |
| D. | \[\frac{{{(n+1)}^{n+1}}}{(n+1)!}\] |
| Answer» E. | |
| 1426. |
The coefficient of \[{{x}^{n}}\] in the expansion of \[{{e}^{{{e}^{x}}}}\] is |
| A. | \[\frac{{{e}^{x}}}{n!}\] |
| B. | \[\frac{{{n}^{n}}}{n!}\] |
| C. | \[\frac{1}{n!}\] |
| D. | None of these |
| Answer» E. | |
| 1427. |
If the third term in the expansion of \[{{[x+{{x}^{{{\log }_{\,10}}\,x}}]}^{5}}\] is \[{{10}^{6}}\], then x may be |
| A. | 1 |
| B. | \[\sqrt{10}\] |
| C. | 10 |
| D. | \[{{10}^{-2/5}}\] |
| Answer» D. \[{{10}^{-2/5}}\] | |
| 1428. |
If x is so small that \[{{x}^{3}}\] and higher powers of x may be neglected, then \[\frac{{{(1+x)}^{\frac{3}{2}}}-{{\left( 1+\frac{1}{2}x \right)}^{3}}}{{{(1-x)}^{\frac{1}{2}}}}\] may be approximated as |
| A. | \[1-\frac{3}{8}{{x}^{2}}\] |
| B. | \[3x+\frac{3}{8}{{x}^{2}}\] |
| C. | \[-\frac{3}{8}{{x}^{2}}\] |
| D. | \[\frac{x}{2}-\frac{3}{8}{{x}^{2}}\] |
| Answer» D. \[\frac{x}{2}-\frac{3}{8}{{x}^{2}}\] | |
| 1429. |
What is the coefficient of \[{{x}^{3}}\] in \[\frac{(3-2x)}{{{(1+3x)}^{3}}}?\] |
| A. | -272 |
| B. | -540 |
| C. | -870 |
| D. | -918 |
| Answer» E. | |
| 1430. |
\[\frac{{{C}_{0}}}{1}+\frac{{{C}_{2}}}{3}+\frac{{{C}_{4}}}{5}+\frac{{{C}_{6}}}{7}+....=\] |
| A. | \[\frac{{{2}^{n+1}}}{n+1}\] |
| B. | \[\frac{{{2}^{n+1}}-1}{n+1}\] |
| C. | \[\frac{{{2}^{n}}}{n+1}\] |
| D. | None of these |
| Answer» D. None of these | |
| 1431. |
\[\frac{1}{1.2}+\frac{1.3}{1.2.3.4}+\frac{1.3.5}{1.2.3.4.5.6}+....\infty =\] |
| A. | \[\sqrt{e}\] |
| B. | \[\sqrt{e}+1\] |
| C. | \[\sqrt{e}-1\] |
| D. | \[e-1\] |
| Answer» D. \[e-1\] | |
| 1432. |
The value of \[{{(}^{7}}{{C}_{0}}+{{\,}^{7}}{{C}_{1}})+{{(}^{7}}{{C}_{1}}+{{\,}^{7}}{{C}_{2}})+...+\]\[{{(}^{7}}{{C}_{6}}+{{\,}^{7}}{{C}_{7}})\] is |
| A. | \[{{2}^{8}}-2\] |
| B. | \[{{2}^{8}}-1\] |
| C. | \[{{2}^{8}}+1\] |
| D. | \[{{2}^{8}}\] |
| Answer» B. \[{{2}^{8}}-1\] | |
| 1433. |
If \[\sum\limits_{r=0}^{n}{\frac{r+2}{r+1}{{\,}^{n}}{{C}_{r}}=\frac{{{2}^{8}}-1}{6}}\], then n is |
| A. | 8 |
| B. | 4 |
| C. | 6 |
| D. | 5 |
| Answer» E. | |
| 1434. |
The coefficient of \[{{x}^{-7}}\] in the expansion of \[{{\left[ ax-\frac{1}{b{{x}^{2}}} \right]}^{11}}\] will be: |
| A. | \[\frac{462}{{{b}^{5}}}{{a}^{6}}\] |
| B. | \[\frac{462{{a}^{5}}}{{{b}^{6}}}\] |
| C. | \[\frac{-462{{a}^{5}}}{{{b}^{6}}}\] |
| D. | \[\frac{-462{{a}^{6}}}{{{b}^{5}}}\] |
| Answer» C. \[\frac{-462{{a}^{5}}}{{{b}^{6}}}\] | |
| 1435. |
The coefficient of \[{{x}^{m}}\] in \[{{(1+x)}^{m}}+{{(1+x)}^{m+1}}+......+{{(1+x)}^{n}},m\le n\] is |
| A. | \[^{n+1}{{C}_{m+1}}\] |
| B. | \[^{n-1}{{C}_{m-1}}\] |
| C. | \[^{n}{{C}_{m}}\] |
| D. | \[^{n}{{C}_{m+1}}\] |
| Answer» B. \[^{n-1}{{C}_{m-1}}\] | |
| 1436. |
Find the 7th term from the end in the expansion of \[{{\left( x-\frac{2}{{{x}^{2}}} \right)}^{10}}\]. |
| A. | \[^{10}{{C}_{4}}\] |
| B. | \[^{10}{{C}_{4}}{{.2}^{4}}x\] |
| C. | \[{{2}^{4}}.{{x}^{2}}\] |
| D. | \[^{10}{{C}_{4}}{{.2}^{4}}\left( \frac{1}{{{x}^{2}}} \right)\] |
| Answer» E. | |
| 1437. |
If the fourth term in the expansion of \[{{\left( \sqrt{{{x}^{\left( \frac{1}{\log \,x+1} \right)}}}+{{x}^{1/12}} \right)}^{6}}\] is equal to 200 and \[\operatorname{x} > 1\], then x is equal to \[(log=lo{{g}_{10}})\] |
| A. | \[{{10}^{\sqrt{2}}}\] |
| B. | 10 |
| C. | \[{{10}^{4}}\] |
| D. | None of these |
| Answer» C. \[{{10}^{4}}\] | |
| 1438. |
If the sum of the coefficients in the expansion of \[{{(1-3x+10{{x}^{2}})}^{n}}\] is a and if the sum of the coefficients in the expansion of \[{{(1+{{x}^{2}})}^{n}}\] is b, then |
| A. | \[a=3b\] |
| B. | \[a={{b}^{3}}\] |
| C. | \[b={{a}^{3}}\] |
| D. | None of these |
| Answer» C. \[b={{a}^{3}}\] | |
| 1439. |
If \[\frac{{{e}^{x}}}{1-x}={{B}_{0}}+{{B}_{1}}x+{{B}_{2}}{{x}^{2}}+...+{{B}_{n}}{{x}^{n}}\] then \[{{B}_{n}}-{{B}_{n-1}}\] is |
| A. | \[\frac{1}{n!}-\frac{1}{(n-1)!}\] |
| B. | \[\frac{1}{n!}\] |
| C. | \[\frac{1}{(n-1)!}\] |
| D. | \[\frac{1}{n!}+\frac{1}{(n-1)!}\] |
| Answer» C. \[\frac{1}{(n-1)!}\] | |
| 1440. |
If \[x\ne 0\], then the sum of the series\[1+\frac{x}{2!}+\frac{2{{x}^{2}}}{3!}+\frac{3{{x}^{3}}}{4!}+.......\infty \] is |
| A. | \[\frac{{{e}^{x}}+1}{x}\] |
| B. | \[\frac{{{e}^{x}}\,(x-1)}{x}\] |
| C. | \[\frac{{{e}^{x}}\,(x-1)+1}{x}\] |
| D. | \[\frac{{{e}^{x}}\,(x-1)+1+x}{x}\] |
| Answer» E. | |
| 1441. |
The value of\[\left( \begin{matrix} 30 \\ 0 \\ \end{matrix} \right)\left( \begin{matrix} 30 \\ 10 \\ \end{matrix} \right)-\left( \begin{matrix} 30 \\ 1 \\ \end{matrix} \right)\left( \begin{matrix} 30 \\ 11 \\ \end{matrix} \right)+\left( \begin{matrix} 30 \\ 2 \\ \end{matrix} \right)\left( \begin{matrix} 30 \\ 12 \\ \end{matrix} \right)...\]\[+\left( \begin{matrix} 30 \\ 20 \\ \end{matrix} \right)\left( \begin{matrix} 30 \\ 30 \\ \end{matrix} \right)\] is where \[\left( \begin{matrix} n \\ r \\ \end{matrix} \right)={{\,}^{n}}{{C}_{r}}\] |
| A. | \[\left( \begin{matrix} 30 \\ 10 \\ \end{matrix} \right)\] |
| B. | \[\left( \begin{matrix} 30 \\ 15 \\ \end{matrix} \right)\] |
| C. | \[\left( \begin{matrix} 60 \\ 30 \\ \end{matrix} \right)\] |
| D. | \[\left( \begin{matrix} 31 \\ 10 \\ \end{matrix} \right)\] |
| Answer» B. \[\left( \begin{matrix} 30 \\ 15 \\ \end{matrix} \right)\] | |
| 1442. |
The number of integral terms in the expansion of \[{{(\sqrt{3}+\sqrt[8]{5})}^{256}}\] is |
| A. | 35 |
| B. | 32 |
| C. | 33 |
| D. | 34 |
| Answer» D. 34 | |
| 1443. |
If number of terms in the expansion of\[{{(x-2y+3z)}^{n}}\] is 45, then n= |
| A. | 7 |
| B. | 8 |
| C. | 9 |
| D. | \[{{6}^{10}}\] |
| Answer» C. 9 | |
| 1444. |
The sum \[1+\frac{1+a}{2!}+\frac{1+a+{{a}^{2}}}{3!}+.....\infty \] is equal to |
| A. | \[{{e}^{a}}\] |
| B. | \[\frac{{{e}^{a}}-e}{a-1}\] |
| C. | \[(a-1){{e}^{a}}\] |
| D. | \[(a+1){{e}^{a}}\] |
| Answer» C. \[(a-1){{e}^{a}}\] | |
| 1445. |
If \[{{C}_{0}},{{C}_{1}},\,{{C}_{2}}{{,}^{.}}.......,\,\,{{C}_{15}}\] are binomial coefficients in \[{{(1+x)}^{15}}\], then\[\frac{{{C}_{1}}}{{{C}_{0}}}+2\frac{{{C}_{2}}}{{{C}_{1}}}+3\frac{{{C}_{3}}}{{{C}_{2}}}+....+15\frac{{{C}_{15}}}{{{C}_{14}}}=\] |
| A. | 60 |
| B. | 120 |
| C. | 64 |
| D. | 124 |
| Answer» C. 64 | |
| 1446. |
The co-efficient of \[{{x}^{n}}\] in the expansion of\[\frac{{{e}^{7x}}+{{e}^{x}}}{{{e}^{3x}}}\] is |
| A. | \[\frac{{{4}^{n-1}}+{{(-2)}^{n}}}{n!}\] |
| B. | \[\frac{{{4}^{n-1}}+{{2}^{n}}}{n!}\] |
| C. | \[\frac{{{4}^{n}}+{{(-2)}^{n}}}{n!}\] |
| D. | \[\frac{{{4}^{n-1}}+{{(-2)}^{n-1}}}{n!}\] |
| Answer» D. \[\frac{{{4}^{n-1}}+{{(-2)}^{n-1}}}{n!}\] | |
| 1447. |
If \[{{(1+x)}^{15}}={{C}_{0}}+{{C}_{1}}x+{{C}_{2}}{{x}^{2}}+...+{{C}_{15}}{{x}^{15}}\] then\[{{C}_{2}}+2{{C}_{3}}+3{{C}_{4}}+.....+14{{C}_{15}}\] is equal to |
| A. | \[{{14.2}^{14}}\] |
| B. | \[{{13.2}^{14}}+1\] |
| C. | \[{{13.2}^{14}}-1\] |
| D. | None of these |
| Answer» C. \[{{13.2}^{14}}-1\] | |
| 1448. |
The coefficient of \[{{x}^{83}}\] in \[{{(1+x+{{x}^{2}}+{{x}^{3}}+{{x}^{4}})}^{n}}\]\[{{(1-x)}^{n+3}},is-{{\,}^{n}}{{C}_{2\lambda }}\], then find the value of \[\lambda \] |
| A. | 12 |
| B. | 10 |
| C. | 9 |
| D. | 8 |
| Answer» E. | |
| 1449. |
The approximate value of \[{{(1.0002)}^{3000}}\] is |
| A. | 1.6 |
| B. | 1.4 |
| C. | 1.8 |
| D. | 1.2 |
| Answer» B. 1.4 | |
| 1450. |
The ninth term in the expansion of\[{{\left\{ {{3}^{{{\log }_{3}}\sqrt{{{25}^{x-1}}+7}}}+{{3}^{-1/8\,\,{{\log }_{3}}\left( {{5}^{x-1}}+1 \right)}} \right\}}^{10}}\]is equal to 180, then x is |
| A. | A prime number |
| B. | An irrational number |
| C. | Has non-zero fractional part |
| D. | None of these |
| Answer» C. Has non-zero fractional part | |