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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.
| 1451. |
If \[{{a}_{n}}=2n+1\] and \[{{C}_{r}}={{\,}^{n}}{{C}_{r}}\] then\[{{a}_{0}}C_{0}^{2}+{{a}_{1}}C_{1}^{2}+{{a}_{2}}C_{2}^{2}+........{{a}_{n}}C_{n}^{2}=\] |
| A. | \[(n-1){{(}^{2n}}{{C}_{n}})\] |
| B. | \[n{{(}^{2n}}{{C}_{n}})\] |
| C. | \[(n+1){{(}^{2n}}{{C}_{n}})\] |
| D. | \[(n+1){{(}^{n}}{{C}_{n/2}})\] |
| Answer» D. \[(n+1){{(}^{n}}{{C}_{n/2}})\] | |
| 1452. |
\[\frac{1}{2}{{x}^{2}}+\frac{2}{3}{{x}^{3}}+\frac{3}{4}{{x}^{4}}+\frac{4}{5}{{x}^{5}}+\]................. is |
| A. | \[\frac{x}{1+x}+\log (1+x)\] |
| B. | \[\frac{x}{1-x}+\log (1+x)\] |
| C. | \[-\frac{x}{1-x}+\log (1+x)\] |
| D. | \[\frac{x}{1-x}+\log (1-x)\] |
| Answer» E. | |
| 1453. |
The value of \[^{20}{{C}_{0}}+{{\,}^{20}}{{C}_{1}}+{{\,}^{20}}{{C}_{2}}+{{\,}^{20}}{{C}_{3}}+{{\,}^{20}}{{C}_{4}}\]\[+{{\,}^{20}}{{C}_{12}}+{{\,}^{20}}{{C}_{13}}+{{\,}^{20}}{{C}_{14}}+{{\,}^{20}}{{C}_{15}}\] is |
| A. | \[{{2}^{19}}-\frac{\left( ^{20}{{C}_{10}}+{{\,}^{20}}{{C}_{9}} \right)}{2}\] |
| B. | \[{{2}^{19}}-\frac{\left( ^{20}{{C}_{10}}+\,2{{\times }^{20}}{{C}_{9}} \right)}{2}\] |
| C. | \[{{2}^{19}}-\frac{^{20}{{C}_{10}}}{2}\] |
| D. | None of these |
| Answer» C. \[{{2}^{19}}-\frac{^{20}{{C}_{10}}}{2}\] | |
| 1454. |
If the middle term in the expansion of \[{{\left( \frac{1}{x}+x\,\sin \,x \right)}^{10}}\] equals to \[7\frac{7}{8}\] then x is equal to; \[(n\in I)\] |
| A. | \[2n\pi \pm \frac{\pi }{6}\] |
| B. | \[n\pi +\frac{\pi }{6}\] |
| C. | \[n\pi +{{(-1)}^{n}}\frac{\pi }{6}\] |
| D. | \[n\pi +{{(-1)}^{n}}\frac{5\pi }{6}\] |
| Answer» D. \[n\pi +{{(-1)}^{n}}\frac{5\pi }{6}\] | |
| 1455. |
If the second term in the expansion \[{{\left( \sqrt[13]{a}+\frac{a}{\sqrt{{{a}^{-1}}}} \right)}^{n}}\] is \[14{{a}^{5/2}}\], then \[\frac{^{n}{{C}_{3}}}{^{n}{{C}_{2}}}=\] |
| A. | 4 |
| B. | 3 |
| C. | 12 |
| D. | 6 |
| Answer» B. 3 | |
| 1456. |
The remainder when \[{{27}^{40}}\] is divided by 12 is |
| A. | 3 |
| B. | 7 |
| C. | 9 |
| D. | 11 |
| Answer» D. 11 | |
| 1457. |
The sum of the series\[^{20}{{C}_{0}}-{{\,}^{20}}{{C}_{1}}+{{\,}^{20}}{{C}_{2}}-{{\,}^{20}}{{C}_{3}}+....\] \[-....+{{\,}^{20}}{{C}_{10}}\] is |
| A. | 0 |
| B. | \[^{20}{{C}_{10}}\] |
| C. | \[{{-}^{20}}{{C}_{10}}\] |
| D. | \[\frac{1}{2}{{\,}^{20}}{{C}_{10}}\] |
| Answer» E. | |
| 1458. |
If \[y=3x+6{{x}^{2}}+10{{x}^{3}}+........\infty \], then\[\frac{1}{3}y-\frac{1.4}{{{3}^{2}}2}{{y}^{2}}+\frac{1.4.7}{{{3}^{2}}3}{{y}^{3}}-.....\,\infty \] is equal to |
| A. | x |
| B. | \[1-x\] |
| C. | \[1 + x\] |
| D. | \[{{x}^{x}}\] |
| Answer» B. \[1-x\] | |
| 1459. |
If \[x+y=1\], then \[\sum\limits_{r=0}^{n}{{{r}^{n}}{{C}_{r}}{{x}^{r}}{{y}^{n-r}}}\] equals |
| A. | 1 |
| B. | n |
| C. | nx |
| D. | ny |
| Answer» D. ny | |
| 1460. |
If \[\pi (n)\] denotes product of all binomial coefficients in \[{{(1+x)}^{n}}\] then ratio of \[\pi (2002)\] to \[\pi (2001)\] is |
| A. | 2002 |
| B. | \[\frac{{{(2002)}^{2001}}}{(2001)!}\] |
| C. | \[\frac{{{(2001)}^{2002}}}{(2002)!}\] |
| D. | 2001 |
| Answer» C. \[\frac{{{(2001)}^{2002}}}{(2002)!}\] | |
| 1461. |
The coefficient of \[{{x}^{53}}\] in the expansion \[\sum\limits_{m=0}^{100}{^{100}{{C}_{m}}{{(x-3)}^{100-m}}{{2}^{m}}}\] is |
| A. | \[^{100}{{C}_{47}}\] |
| B. | \[^{100}{{C}_{53}}\] |
| C. | \[{{-}^{100}}{{C}_{53}}\] |
| D. | \[{{-}^{100}}{{C}_{100}}\] |
| Answer» D. \[{{-}^{100}}{{C}_{100}}\] | |
| 1462. |
The expression\[\frac{1}{\sqrt{3x+1}}\left[ {{\left( \frac{1+\sqrt{3x+1}}{2} \right)}^{7}}-{{\left( \frac{1-\sqrt{3x+1}}{2} \right)}^{7}} \right]\]is a polynomial in x of degree equal to |
| A. | 3 |
| B. | 4 |
| C. | 2 |
| D. | 5 |
| Answer» B. 4 | |
| 1463. |
The term independent of x in the expansion of \[{{[({{t}^{-1}}-1)x+{{({{t}^{-1}}+1)}^{-1}}{{x}^{-1}}]}^{8}}\] is |
| A. | \[56{{\left( \frac{1-t}{1+t} \right)}^{3}}\] |
| B. | \[56{{\left( \frac{1+t}{1-t} \right)}^{3}}\] |
| C. | \[70{{\left( \frac{1-t}{1+t} \right)}^{4}}\] |
| D. | \[70{{\left( \frac{1+t}{1-t} \right)}^{4}}\] |
| Answer» D. \[70{{\left( \frac{1+t}{1-t} \right)}^{4}}\] | |
| 1464. |
Sum of coefficients in the exansion of \[{{(x+2y+3z)}^{10}}\] is |
| A. | \[{{2}^{10}}\] |
| B. | \[{{3}^{10}}\] |
| C. | 1 |
| D. | \[{{6}^{10}}\] |
| Answer» E. | |
| 1465. |
The number of term in the expansion of \[{{[{{(x+4y)}^{3}}{{(x-4y)}^{3}}]}^{2}}\] is |
| A. | 6 |
| B. | 7 |
| C. | 8 |
| D. | 32 |
| Answer» C. 8 | |
| 1466. |
The coefficient of \[{{x}^{n}}\] in the polynomial\[(x+{{\,}^{n}}{{C}_{0}})(x+3.{{\,}^{n}}{{C}_{1}})(x+5.{{\,}^{n}}{{C}_{2}})...(x+{{(2n+1)}^{n}}{{C}_{n}})\] is |
| A. | \[n{{.2}^{n}}\] |
| B. | \[~n{{.2}^{n+1}}\] |
| C. | \[(n+1){{.2}^{n}}\] |
| D. | \[n{{.2}^{n}}+1\] |
| Answer» D. \[n{{.2}^{n}}+1\] | |
| 1467. |
If the sum of odd numbered terms and the sum of even numbered terms in the expansion of \[{{(x+a)}^{n}}\]are A and B respectively, then the value of \[{{({{x}^{2}}-{{a}^{2}})}^{n}}\] is |
| A. | \[{{A}^{2}}-{{B}^{2}}\] |
| B. | \[{{A}^{2}}+{{B}^{2}}\] |
| C. | 4AB |
| D. | None of these |
| Answer» B. \[{{A}^{2}}+{{B}^{2}}\] | |
| 1468. |
The positive integer just greater than \[{{(1+0.0001)}^{10000}}\] |
| A. | 4 |
| B. | 5 |
| C. | 2 |
| D. | 3 |
| Answer» E. | |
| 1469. |
If \[x={{\left( 2+\sqrt{3} \right)}^{n}}\], then find the value of \[x\,\left( 1-\left\{ x \right\} \right)\] where {x} denotes the fractional part of x |
| A. | 1 |
| B. | 2 |
| C. | \[{{2}^{2n}}\] |
| D. | \[{{2}^{n}}\] |
| Answer» B. 2 | |
| 1470. |
If the 7th term in the binomial expansion of \[{{\left( \frac{3}{\sqrt[3]{84}}+\sqrt{3}\,ln\,\,x \right)}^{9}},x>0\], is equal to 729, then x can be |
| A. | \[{{e}^{2}}\] |
| B. | e |
| C. | \[\frac{e}{2}\] |
| D. | 2e |
| Answer» C. \[\frac{e}{2}\] | |
| 1471. |
If x is very small in magnitude compared with a, then \[{{\left( \frac{a}{a+x} \right)}^{\frac{1}{2}}}+{{\left( \frac{a}{a-x} \right)}^{\frac{1}{2}}}\] can be approximately equal to |
| A. | \[1+\frac{1}{2}\frac{x}{a}\] |
| B. | \[\frac{x}{a}\] |
| C. | \[1+\frac{3}{4}\frac{{{x}^{2}}}{{{a}^{2}}}\] |
| D. | \[2+\frac{3}{4}\frac{{{x}^{2}}}{{{a}^{2}}}\] |
| Answer» E. | |
| 1472. |
The greatest value of the term independent of x in the expansion \[{{(x\,\,\sin \,\,p+{{x}^{-1}}\cos \,\,p)}^{10}},p\in R\] is |
| A. | \[{{2}^{5}}\] |
| B. | \[\frac{10!}{{{2}^{5}}{{(5!)}^{2}}}\] |
| C. | \[\frac{10!}{{{(5!)}^{2}}}\] |
| D. | None of these |
| Answer» C. \[\frac{10!}{{{(5!)}^{2}}}\] | |
| 1473. |
What are the values of k if the term independent of x in the expansion of \[{{\left( \sqrt{x}+\frac{k}{{{x}^{2}}} \right)}^{10}}\] is 405? |
| A. | \[\pm 3\] |
| B. | \[\pm 6\] |
| C. | \[\pm 5\] |
| D. | \[\pm 4\] |
| Answer» B. \[\pm 6\] | |
| 1474. |
For natural numbers \[m,n\,\,if{{(1-y)}^{m}}{{(1+y)}^{n}}\] \[=1+{{a}_{1}}y+{{a}_{2}}{{y}^{2}}+...\] and \[{{a}_{1}}={{a}_{2}}=10\], then \[(m,n)\] is |
| A. | (20, 45) |
| B. | (35, 20) |
| C. | (45, 35) |
| D. | (35, 45) |
| Answer» E. | |
| 1475. |
The value of \[\sum\limits_{r=0}^{n}{^{n}{{C}_{r}}\sin (rx)}\] is equal to |
| A. | \[{{2}^{n}}\cdot {{\cos }^{n}}\frac{x}{2}\cdot \sin \frac{nx}{2}\] |
| B. | \[{{2}^{n}}\cdot si{{n}^{n}}\frac{x}{2}\cdot \cos \frac{nx}{2}\] |
| C. | \[{{2}^{n+1}}\cdot {{\cos }^{n}}\frac{x}{2}\cdot \sin \frac{nx}{2}\] |
| D. | \[{{2}^{n+1}}\cdot si{{n}^{n}}\frac{x}{2}\cdot \cos \frac{nx}{2}\] |
| Answer» B. \[{{2}^{n}}\cdot si{{n}^{n}}\frac{x}{2}\cdot \cos \frac{nx}{2}\] | |
| 1476. |
The coefficient of \[{{a}^{3}}{{b}^{4}}c\] in the expansion of \[{{(1+a-b+c)}^{9}}\] is equal to |
| A. | \[\frac{9!}{3!6!}\] |
| B. | \[\frac{9!}{4!5!}\] |
| C. | \[\frac{9!}{3!5!}\] |
| D. | \[\frac{9!}{3!4!}\] |
| Answer» E. | |
| 1477. |
The sum of the series \[\frac{2}{1}.\frac{1}{3}+\frac{3}{2}.\frac{1}{9}+\frac{4}{3}.\frac{1}{27}+\frac{5}{4}.\frac{1}{81}+......\infty \] is equal to |
| A. | \[{{\log }_{e}}3-{{\log }_{e}}2\] |
| B. | \[\frac{1}{2}+{{\log }_{e}}3-{{\log }_{e}}2\] |
| C. | \[\frac{1}{2}+{{\log }_{e}}3+{{\log }_{e}}2\] |
| D. | \[{{\log }_{e}}3+{{\log }_{e}}2\] |
| Answer» C. \[\frac{1}{2}+{{\log }_{e}}3+{{\log }_{e}}2\] | |
| 1478. |
The sum of the rational terms in the expansion of \[{{(\sqrt{2}+{{3}^{1/5}})}^{10}}\] is equal to |
| A. | 40 |
| B. | 41 |
| C. | 42 |
| D. | 0 |
| Answer» C. 42 | |
| 1479. |
Area bounded by the curves \[y=\left[ \frac{{{x}^{2}}}{64}+2 \right]([\cdot ]\] denotes the greatest integer function), \[v=x-1\] and \[x=0\], above the x-axis is |
| A. | 2 sq. unit |
| B. | 3 sq. unit |
| C. | 4 sq. unit |
| D. | None of these |
| Answer» D. None of these | |
| 1480. |
The slope of the tangent to a curve \[y=f(x)\] at \[(x,f(x))\] is\[2x+1\]. If the curve passes through the point (1, 2), then the area of the region bounded by the curve, the x-axis and the line \[x=1\] is |
| A. | \[\frac{5}{6}\] sq. unit |
| B. | \[\frac{6}{5}\] sq. unit |
| C. | \[\frac{1}{6}\] sq. unit |
| D. | 6 sq. unit |
| Answer» B. \[\frac{6}{5}\] sq. unit | |
| 1481. |
If \[y=f(x)\] makes +ve intercept of 2 and 0 unit on x and y axes and encloses an area of 3/4 square unit with the axes then \[\int\limits_{0}^{2}{xf'(x)dx}\] is |
| A. | 44230 |
| B. | 1 |
| C. | 44291 |
| D. | -0.75 |
| Answer» E. | |
| 1482. |
The area enclosed between the curves \[y=a{{x}^{2}}\] and \[x=a{{y}^{2}}(a>0)\] is 1 sq. unit, then the value of a is |
| A. | \[\frac{1}{\sqrt{3}}\] |
| B. | \[\frac{1}{2}\] |
| C. | 1 |
| D. | \[\frac{1}{3}\] |
| Answer» B. \[\frac{1}{2}\] | |
| 1483. |
The area bounded by the curves \[{{x}^{2}}+{{y}^{2}}=25,\]\[4y=\left| 4-{{x}^{2}} \right|\] and \[x=0\], above x-axis is |
| A. | \[2+\frac{25}{2}{{\sin }^{-1}}\frac{4}{5}\] |
| B. | \[2+\frac{25}{4}{{\sin }^{-1}}\frac{4}{5}\] |
| C. | \[2+\frac{25}{2}{{\sin }^{-1}}\frac{1}{5}\] |
| D. | None of these |
| Answer» B. \[2+\frac{25}{4}{{\sin }^{-1}}\frac{4}{5}\] | |
| 1484. |
The area bounded by \[f(x)={{x}^{2}},0\le x\le 1,\] \[g(x)=-x+2,1\le x\le 2\] and \[x-axis\] is |
| A. | \[\frac{3}{2}\] |
| B. | \[\frac{4}{3}\] |
| C. | \[\frac{8}{3}\] |
| D. | None of these |
| Answer» E. | |
| 1485. |
The area of the region\[R=\{(x,y):\left| x \right|\le \left| y \right|\] and \[{{x}^{2}}+{{y}^{2}}\le 1\}\] is |
| A. | \[\frac{3\pi }{8}\] sq. unit |
| B. | \[\frac{5\pi }{8}\] sq. unit |
| C. | \[\frac{\pi }{2}\] sq. unit |
| D. | \[\frac{\pi }{8}\] sq. unit |
| Answer» D. \[\frac{\pi }{8}\] sq. unit | |
| 1486. |
The area bounded by the curve \[y=si{{n}^{-1}}x\] and the line \[x=0,\left| y \right|=\frac{\pi }{2}\] is |
| A. | 1 |
| B. | 2 |
| C. | \[\pi \] |
| D. | \[2\pi \] |
| Answer» C. \[\pi \] | |
| 1487. |
What is the area of the parabola \[{{x}^{2}}=y\] bounded by the line y = 1? |
| A. | \[\frac{1}{3}\] square unit |
| B. | \[\frac{2}{3}\] square unit |
| C. | \[\frac{4}{3}\] square units |
| D. | 2 square units |
| Answer» D. 2 square units | |
| 1488. |
If the area enclosed by \[{{y}^{2}}=4ax\] and line \[y=ax\]is 1/3 sq. units , then the area enclosed by \[y=4x\]with same parabola is |
| A. | 8 sq. units |
| B. | 4 sq. units |
| C. | 4/3 sq. units |
| D. | 8/3 sq. units |
| Answer» E. | |
| 1489. |
If the ordinate \[x=a\] divides the area bounded by x-axis, part of the curve \[y=1+\frac{8}{{{x}^{2}}}\] and the ordinates \[x=2,\text{ }x=4\] into two equal parts, then a is equal to |
| A. | \[\sqrt{2}\] |
| B. | \[2\sqrt{2}\] |
| C. | \[3\sqrt{2}\] |
| D. | None of these |
| Answer» C. \[3\sqrt{2}\] | |
| 1490. |
The figure shows as triangle AOB and the parabola\[y={{x}^{2}}\]. The ratio of the area of the triangle AOB to the area of the region AOB of the parabola \[y={{x}^{2}}\] is equal to |
| A. | \[\frac{3}{5}\] |
| B. | \[\frac{3}{4}\] |
| C. | \[\frac{7}{8}\] |
| D. | \[\frac{5}{6}\] |
| Answer» C. \[\frac{7}{8}\] | |
| 1491. |
Area bounded by the curve \[x{{y}^{2}}={{a}^{2}}(a-x)\]and y-axis is |
| A. | \[\pi {{a}^{2}}/2\] sq. units |
| B. | \[\pi {{a}^{2}}\] sq. units |
| C. | \[3\pi {{a}^{2}}\] sq. units |
| D. | None of these |
| Answer» C. \[3\pi {{a}^{2}}\] sq. units | |
| 1492. |
The area of the region bounded by the parabola \[{{(y-2)}^{2}}=x-1\], the tangent of the parabola at the point (2, 3) and the x-axis is: |
| A. | 6 |
| B. | 9 |
| C. | 12 |
| D. | 3 |
| Answer» C. 12 | |
| 1493. |
Area bounded by the curves \[y={{e}^{x}},y={{e}^{-x}}\] and the straight line \[x=1\] is (in sq. units) |
| A. | \[e+\frac{1}{e}\] |
| B. | \[e+\frac{1}{e}+2\] |
| C. | \[e+\frac{1}{e}-2\] |
| D. | \[e-\frac{1}{e}+2\] |
| Answer» D. \[e-\frac{1}{e}+2\] | |
| 1494. |
The area enclosed by the curve \[{{x}^{2}}y=36,\] the x-axis and the lines x = 6 and x = 9 is |
| A. | 6 |
| B. | 1 |
| C. | 4 |
| D. | 2 |
| Answer» E. | |
| 1495. |
The area of the region formed by\[{{x}^{2}}+{{y}^{2}}-6x-4y+12\le 0,y\le x\] and \[x\le \frac{5}{2}\] is |
| A. | \[\left( \frac{\pi }{6}-\frac{\sqrt{3}+1}{8} \right)sq\,\,unit\] |
| B. | \[\left( \frac{\pi }{6}+\frac{\sqrt{3}-1}{8} \right)sq\,\,unit\] |
| C. | \[\left( \frac{\pi }{6}-\frac{\sqrt{3}-1}{8} \right)sq\,\,unit\] |
| D. | None of theses |
| Answer» D. None of theses | |
| 1496. |
Which of the following is not the area of the region bounded by \[y={{e}^{x}}\] and \[x=0\] and y = e? |
| A. | \[e-1\] |
| B. | \[\int\limits_{1}^{e}{ln(e+1-y)dy}\] |
| C. | \[e-\int\limits_{0}^{1}{{{e}^{x}}dx}\] |
| D. | \[\int\limits_{1}^{e}{\,ln\,\,y\,\,dy}\] |
| Answer» D. \[\int\limits_{1}^{e}{\,ln\,\,y\,\,dy}\] | |
| 1497. |
What is the area enclosed between the curves\[{{y}^{2}}=12x\] and the lines \[x=0\] and \[y=6\]? |
| A. | 2 sq. unit |
| B. | 4 sq. unit |
| C. | 6 sq. unit |
| D. | 8 sq. unit |
| Answer» D. 8 sq. unit | |
| 1498. |
What is the area of the portion of the curve\[y=sin\text{ }x\], lying between \[x=0,\text{ }y=0\] and\[x=2\pi \]? |
| A. | 1 square unit |
| B. | 2 square units |
| C. | 4 square units |
| D. | 8 square units |
| Answer» C. 4 square units | |
| 1499. |
The area of the figure bounded by \[{{y}^{2}}=2x+1\]and \[x-y=1\] is |
| A. | \[\frac{2}{3}\] |
| B. | \[\frac{4}{3}\] |
| C. | \[\frac{8}{3}\] |
| D. | \[\frac{16}{3}\] |
| Answer» E. | |
| 1500. |
The area enclosed between the curve \[y=lo{{g}_{e}}\left( x+e \right)\] and the coordinate axes is |
| A. | 1 |
| B. | 2 |
| C. | 3 |
| D. | 4 |
| Answer» B. 2 | |