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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.
| 7751. |
Let a, b, c be in AP. Consider the following statements: 1. \[\frac{1}{ab},\frac{1}{ca}\] and \[\frac{1}{bc}\] are in A.P. 2. \[\frac{1}{\sqrt{b}+\sqrt{c}},\frac{1}{\sqrt{c}+\sqrt{a}}\] and \[\frac{1}{\sqrt{a}+\sqrt{b}}\] are in A.P. Which of the statements given above is/are correct? |
| A. | 1 only |
| B. | 2 only |
| C. | Both 1 and 2 |
| D. | Neither 1 and 2 |
| Answer» D. Neither 1 and 2 | |
| 7752. |
If the roots of the equation \[{{x}^{3}}-12{{x}^{2}}+39x-28=0\] are in A.P., then their common difference will be: |
| A. | \[\pm 1\] |
| B. | \[\pm 2\] |
| C. | \[\pm 3\] |
| D. | \[\pm 4\] |
| Answer» D. \[\pm 4\] | |
| 7753. |
Concentric circles of radii 1, 2, 3,...100 cm are drawn. The interior of the smallest circle is coloured red and the angular regions are coloured alternately green and red, so that no two adjacent regions are of the same colour. The total area of the green regions on sq cm is equal to |
| A. | \[1000\,\pi \] |
| B. | \[5050\,\pi \] |
| C. | \[4950\,\pi \] |
| D. | \[5151\text{ }\pi \] |
| Answer» C. \[4950\,\pi \] | |
| 7754. |
If \[(1+3+5+...+p)+(1+3+5+...+q)\]\[=(1+3+5+...+r)\]where each set of parentheses contains the sum of consecutive odd integers as shown, what is the smallest possible value of (p + q + r) where\[p>6\]? |
| A. | 12 |
| B. | 21 |
| C. | 45 |
| D. | 54 |
| Answer» C. 45 | |
| 7755. |
What is the 15th term of the series 3, 7, 13, 21, 31, 43, ....? |
| A. | 205 |
| B. | 225 |
| C. | 238 |
| D. | 241 |
| Answer» E. | |
| 7756. |
The value of x in \[(0,\pi )\] which satisfy the equation \[{{8}^{1+\left| \cos \,x \right|+xo{{s}^{2}}x+\left| {{\cos }^{3}}x \right|+.......to\,\,\infty \,}}={{4}^{3}}\] is |
| A. | \[\left\{ \frac{\pi }{2},\frac{3\pi }{4} \right\}\] |
| B. | \[\left\{ \frac{\pi }{4},\frac{3\pi }{4} \right\}\] |
| C. | \[\left\{ \frac{\pi }{3},\frac{2\pi }{3} \right\}\] |
| D. | \[\left\{ \frac{\pi }{6},\frac{5\pi }{6} \right\}\] |
| Answer» D. \[\left\{ \frac{\pi }{6},\frac{5\pi }{6} \right\}\] | |
| 7757. |
The value of \[0.0\overline{37}\] where \[0.0\overline{37}\] stands for the number .0373737........... is: |
| A. | 37/1000 |
| B. | 37/990 |
| C. | 13516 |
| D. | 46388 |
| Answer» C. 13516 | |
| 7758. |
What is the sum of the series\[1+\frac{1}{8}+\frac{1.3}{8.16}+\frac{1.3.5}{8.16.24}+....\infty \]? |
| A. | \[\frac{2}{\sqrt{3}}\] |
| B. | \[2\sqrt{3}\] |
| C. | \[\frac{\sqrt{3}}{2}\] |
| D. | \[\frac{1}{2\sqrt{3}}\] |
| Answer» B. \[2\sqrt{3}\] | |
| 7759. |
If the coefficients of rth, (r + 1)th, and (r + 2)th terms in the binomial expansion of \[{{(1+y)}^{m}}\] are in A.P, then m and r satisfy the equation |
| A. | \[{{m}^{2}}-m\left( 4r-1 \right)+4{{r}^{2}}-2=0\] |
| B. | \[{{m}^{2}}-m\left( 4r+1 \right)+4{{r}^{2}}+2=0\] |
| C. | \[{{m}^{2}}-m(4r+1)+4{{r}^{2}}-2=0\] |
| D. | \[{{m}^{2}}-m\left( 4r-1 \right)+4{{r}^{2}}+2=0\] |
| Answer» D. \[{{m}^{2}}-m\left( 4r-1 \right)+4{{r}^{2}}+2=0\] | |
| 7760. |
If \[\left| \,r\, \right|>1\] and \[x=a+\frac{a}{r}+\frac{a}{{{r}^{2}}}+....to\,\,\infty \],\[y=b-\frac{b}{r}+\frac{b}{{{r}^{2}}}-....\,to\,\,\infty \]and \[z=c+\frac{c}{{{r}^{2}}}+\frac{c}{{{r}^{4}}}+....to\,\,\infty \]then \[\frac{xy}{z}=\] |
| A. | \[\frac{ab}{c}\] |
| B. | \[\frac{ac}{b}\] |
| C. | \[\frac{bc}{a}\] |
| D. | 1 |
| Answer» B. \[\frac{ac}{b}\] | |
| 7761. |
\[{{\left( x+\frac{1}{x} \right)}^{2}}+{{\left( {{x}^{2}}+\frac{1}{{{x}^{2}}} \right)}^{2}}+{{\left( {{x}^{{}}}+\frac{1}{{{x}^{3}}} \right)}^{2}}....\] upto n terms is |
| A. | \[\frac{{{x}^{2n}}-1}{{{x}^{2}}-1}\times \frac{{{x}^{2n+2}}+1}{{{x}^{2n}}}+2n\] |
| B. | \[\frac{{{x}^{2n}}+1}{{{x}^{2}}+1}\times \frac{{{x}^{2n+2}}-1}{{{x}^{2n}}}-2n\] |
| C. | \[\frac{{{x}^{2n}}-1}{{{x}^{2}}-1}\times \frac{{{x}^{2n}}-1}{{{x}^{2n}}}-2n\] |
| D. | None of these |
| Answer» B. \[\frac{{{x}^{2n}}+1}{{{x}^{2}}+1}\times \frac{{{x}^{2n+2}}-1}{{{x}^{2n}}}-2n\] | |
| 7762. |
If \[{{a}_{1}},{{a}_{2}},....{{a}_{n}}\] are positive real numbers whose product is a fixed number c, then the minimum value of \[{{a}_{1}}+{{a}_{2}}+...\] \[+{{a}_{n-1}}+2{{a}_{n}}\]is [IIT Screening 2002] |
| A. | \[n{{(2c)}^{1/n}}\] |
| B. | \[(n+1)\,{{c}^{1/n}}\] |
| C. | \[2n{{c}^{1/n}}\] |
| D. | \[(n+1){{(2c)}^{1/n}}\] |
| Answer» B. \[(n+1)\,{{c}^{1/n}}\] | |
| 7763. |
If \[a,\,b,\,c\] are three unequal numbers such that \[a,\,b,\,c\] are in A.P. and b - a, c - b, a are in G.P., then a : b : c is [UPSEAT 2001] |
| A. | 1 : 2 : 3 |
| B. | 2: 3 : 1 |
| C. | 1 : 3 : 2 |
| D. | 3 : 2 : 1 |
| Answer» B. 2: 3 : 1 | |
| 7764. |
If a,b,c are in A.P., then \[{{2}^{ax+1}},{{2}^{bx+1}},\,{{2}^{cx+1}},x\ne 0\] are in [DCE 2000; Pb. CET 2000] |
| A. | A.P. |
| B. | G.P. only when \[x>\text{0}\] |
| C. | G.P. if \[x<0\] |
| D. | G.P. for all \[x\ne 0\] |
| Answer» E. | |
| 7765. |
The sum of three decreasing numbers in A.P. is 27. If \[-1,\,-1,\,3\] are added to them respectively, the resulting series is in G.P. The numbers are [AMU 1999] |
| A. | 5, 9, 13 |
| B. | 15, 9, 3 |
| C. | 13, 9, 5 |
| D. | 17, 9, 1 |
| Answer» E. | |
| 7766. |
If \[a,\ b,\ c\], d are any four consecutive coefficients of any expanded binomial, then \[\frac{a+b}{a},\ \frac{b+c}{b},\ \frac{c+d}{c}\] are in |
| A. | A.P. |
| B. | G.P. |
| C. | H.P. |
| D. | None of the above |
| Answer» D. None of the above | |
| 7767. |
If \[{{\log }_{a}}x,\ {{\log }_{b}}x,\ {{\log }_{c}}x\] be in H.P., then \[a,\ b,\ c\] are in |
| A. | A.P. |
| B. | H.P. |
| C. | G.P. |
| D. | None of these |
| Answer» D. None of these | |
| 7768. |
If \[{{x}^{a}}={{x}^{b/2}}{{z}^{b/2}}={{z}^{c}}\], then \[a,b,c\] are in |
| A. | A.P. |
| B. | G.P. |
| C. | H.P. |
| D. | None of these |
| Answer» D. None of these | |
| 7769. |
If the arithmetic mean of two numbers be \[A\] and geometric mean be\[G\], then the numbers will be |
| A. | \[A\pm ({{A}^{2}}-{{G}^{2}})\] |
| B. | \[\sqrt{A}\pm \sqrt{{{A}^{2}}-{{G}^{2}}}\] |
| C. | \[A\pm \sqrt{(A+G)(A-G)}\] |
| D. | \[\frac{A\pm \sqrt{(A+G)(A-G)}}{2}\] |
| Answer» D. \[\frac{A\pm \sqrt{(A+G)(A-G)}}{2}\] | |
| 7770. |
If \[a,\ b,\ c\] are in A.P. and \[a,\ c-b,\ b-a\] are in G.P. \[(a\ne b\ne c)\], then \[a:b:c\] is |
| A. | \[1:3:5\] |
| B. | \[1:2:4\] |
| C. | \[1:2:3\] |
| D. | None of these |
| Answer» D. None of these | |
| 7771. |
If \[\frac{a+b}{1-ab},\ b,\ \frac{b+c}{1-bc}\] are in A.P., then \[a,\ \frac{1}{b},\ c\] are in |
| A. | A.P. |
| B. | G.P. |
| C. | H.P. |
| D. | None of these |
| Answer» D. None of these | |
| 7772. |
If the first and \[{{(2n-1)}^{th}}\] terms of an A.P., G.P. and H.P. are equal and their \[{{n}^{th}}\] terms are respectively \[a,\ b\] and \[c\], then [IIT 1985, 88] |
| A. | \[a\ge b\ge c\] |
| B. | \[a+c=b\] |
| C. | \[ac-{{b}^{2}}=0\] |
| D. | (a) and (c) both |
| Answer» E. | |
| 7773. |
If \[a,\,b,\ c\] are in A.P. and \[{{a}^{2}},\ {{b}^{2}},\ {{c}^{2}}\] are in H.P., then [MNR 1986, 1988; IIT 1977, 2003] |
| A. | \[a=b=c\] |
| B. | \[2b=3a+c\] |
| C. | \[{{b}^{2}}=\sqrt{(ac/8)}\] |
| D. | None of these |
| Answer» B. \[2b=3a+c\] | |
| 7774. |
If \[a,\ b,\ c\] are in A.P., then \[\frac{1}{bc},\ \frac{1}{ca},\ \frac{1}{ab}\] will be in [MP PET 1985; Roorkee 1975; DCE 2002] |
| A. | A.P. |
| B. | G.P. |
| C. | H.P. |
| D. | None of these |
| Answer» B. G.P. | |
| 7775. |
If the A.M. and H.M. of two numbers is 27 and 12 respectively, then G.M. of the two numbers will be [RPET 1987] |
| A. | 9 |
| B. | 18 |
| C. | 24 |
| D. | 36 |
| Answer» C. 24 | |
| 7776. |
The sum of the series \[3.6+4.7+5.8+........\]upto \[(n-2)\] terms [EAMCET 1980] |
| A. | \[{{n}^{3}}+{{n}^{2}}+n+2\] |
| B. | \[\frac{1}{6}(2{{n}^{3}}+12{{n}^{2}}+10n-84)\] |
| C. | \[{{n}^{3}}+{{n}^{2}}+n\] |
| D. | None of these |
| Answer» C. \[{{n}^{3}}+{{n}^{2}}+n\] | |
| 7777. |
The value of \[\frac{1}{(1+a)(2+a)}+\frac{1}{(2+a)(3+a)}\] \[\frac{1}{(3+a)(4+a)}\]+ ..... +\[\infty \] is, (where a is a constant) [AMU 2005] |
| A. | \[\frac{1}{1+a}\] |
| B. | \[\frac{2}{1+a}\] |
| C. | \[\infty \] |
| D. | None of these |
| Answer» B. \[\frac{2}{1+a}\] | |
| 7778. |
First term of the \[{{11}^{th}}\]group in the following groups (1), (2, 3, 4), (5, 6, 7, 8, 9),???.is |
| A. | 89 |
| B. | 97 |
| C. | 101 |
| D. | 123 |
| Answer» D. 123 | |
| 7779. |
The sum to \[n\] terms of the infinite series \[{{1.3}^{2}}+{{2.5}^{2}}+{{3.7}^{2}}+..........\infty \] is [AMU 1982] |
| A. | \[\frac{n}{6}(n+1)(6{{n}^{2}}+14n+7)\] |
| B. | \[\frac{n}{6}(n+1)(2n+1)(3n+1)\] |
| C. | \[4{{n}^{3}}+4{{n}^{2}}+n\] |
| D. | None of these |
| Answer» B. \[\frac{n}{6}(n+1)(2n+1)(3n+1)\] | |
| 7780. |
The sum of the series \[1.3.5+.2.5.8+3.7.11+.........\]upto \['n'\] terms is [Dhanbad Engg. 1972] |
| A. | \[\frac{n\,(n+1)(9{{n}^{2}}+23n+13)}{6}\] |
| B. | \[\frac{n\,(n-1)(9{{n}^{2}}+23n+12)}{6}\] |
| C. | \[\frac{(n+1)(9{{n}^{2}}+23n+13)}{6}\] |
| D. | \[\frac{n\,(9{{n}^{2}}+23n+13)}{6}\] |
| Answer» B. \[\frac{n\,(n-1)(9{{n}^{2}}+23n+12)}{6}\] | |
| 7781. |
The sum of the series \[{{1.3}^{2}}+{{2.5}^{2}}+{{3.7}^{2}}+..........\]upto \[20\] terms is [IIT 1973] |
| A. | 188090 |
| B. | 189080 |
| C. | 199080 |
| D. | None of these |
| Answer» B. 189080 | |
| 7782. |
The sum of the series \[{{1}^{2}}.2+{{2}^{2}}.3+{{3}^{2}}.4+........\] to n terms is |
| A. | \[\frac{{{n}^{3}}{{(n+1)}^{3}}(2n+1)}{24}\] |
| B. | \[\frac{n(n+1)(3{{n}^{2}}+7n+2)}{12}\] |
| C. | \[\frac{n(n+1)}{6}[n(n+1)+(2n+1)]\] |
| D. | \[\frac{n(n+1)}{12}[6n(n+1)+2(2n+1)]\] |
| Answer» C. \[\frac{n(n+1)}{6}[n(n+1)+(2n+1)]\] | |
| 7783. |
The sum of \[(n+1)\] terms of \[\frac{1}{1}+\frac{1}{1+2}+\frac{1}{1+2+3}+......\,\,\text{is }\] [RPET 1999] |
| A. | \[\frac{n}{n+1}\] |
| B. | \[\frac{2n}{n+1}\] |
| C. | \[\frac{2}{n\,(n+1)}\] |
| D. | \[\frac{2\,(n+1)}{n+2}\] |
| Answer» E. | |
| 7784. |
If x, 2y, 3z are in A.P. where the distinct numbers x, y, z are in G.P., then the common ratio of the G.P. is |
| A. | 3 |
| B. | \[\frac{1}{3}\] |
| C. | 2 |
| D. | \[\frac{1}{2}\] |
| Answer» C. 2 | |
| 7785. |
Consider the ten numbers \[ar,a{{r}^{2}},a{{r}^{3}},...a{{r}^{10}}\].If their sum is 18 and the sum the reciprocals is 6, then the product of these ten numbers is |
| A. | 81 |
| B. | 243 |
| C. | 343 |
| D. | 324 |
| Answer» C. 343 | |
| 7786. |
The sum of the series \[1+\frac{1}{4.2!}+\frac{1}{16.4!}+\frac{1}{64.6!}+...\]is |
| A. | \[\frac{e-1}{\sqrt{e}}\] |
| B. | \[\frac{e+1}{\sqrt{e}}\] |
| C. | \[\frac{e-1}{2\sqrt{e}}\] |
| D. | \[\frac{e+1}{2\sqrt{e}}\] |
| Answer» E. | |
| 7787. |
The 15th terms of the series \[2\frac{1}{2}+1\frac{7}{13}+1\frac{1}{9}+\frac{20}{23}+...\]is |
| A. | \[\frac{10}{39}\] |
| B. | \[\frac{10}{21}\] |
| C. | \[\frac{10}{23}\] |
| D. | none of these |
| Answer» B. \[\frac{10}{21}\] | |
| 7788. |
The \[{{9}^{th}}\] term of the series \[27+9+5\frac{2}{5}+3\frac{6}{7}+........\] will be [MP PET 1983] |
| A. | \[1\frac{10}{17}\] |
| B. | \[\frac{10}{17}\] |
| C. | \[\frac{16}{27}\] |
| D. | \[\frac{17}{27}\] |
| Answer» B. \[\frac{10}{17}\] | |
| 7789. |
\[1+\frac{{{4}^{2}}}{3\,!}+\frac{{{4}^{4}}}{5\,!}+......\infty =\] |
| A. | \[\frac{{{e}^{4}}+{{e}^{-4}}}{4}\] |
| B. | \[\frac{{{e}^{4}}-{{e}^{-4}}}{4}\] |
| C. | \[\frac{{{e}^{4}}+{{e}^{-4}}}{8}\] |
| D. | \[\frac{{{e}^{4}}-{{e}^{-4}}}{8}\] |
| Answer» E. | |
| 7790. |
\[1-x+\frac{{{x}^{2}}}{2\,!}-\frac{{{x}^{3}}}{3\,!}+....\infty =\] [MP PET 1986] |
| A. | \[{{e}^{x}}\] |
| B. | \[{{e}^{-x}}\] |
| C. | \[e\] |
| D. | \[{{e}^{{{x}^{2}}}}\] |
| Answer» C. \[e\] | |
| 7791. |
The sum of infinity of a geometric progression is \[\frac{4}{3}\] and the first term is \[\frac{3}{4}\]. The common ratio is [MP PET 1994] |
| A. | 7/16 |
| B. | 9/16 |
| C. | 1/9 |
| D. | 7/9 |
| Answer» B. 9/16 | |
| 7792. |
If the 9th term of an A.P. be zero, then the ratio of its 29th and 19th term is |
| A. | 1 : 2 |
| B. | 2 : 1 |
| C. | 1 : 3 |
| D. | 3 : 1 |
| Answer» C. 1 : 3 | |
| 7793. |
A number is the reciprocal of the other. If the arithmetic mean of the two numbers be \[\frac{13}{12}\], then the numbers are |
| A. | \[\frac{1}{4},\ \frac{4}{1}\] |
| B. | \[\frac{3}{4},\ \frac{4}{3}\] |
| C. | \[\frac{2}{5},\ \frac{5}{2}\] |
| D. | \[\frac{3}{2},\ \frac{2}{3}\] |
| Answer» E. | |
| 7794. |
The value of \[\sum\limits_{r=1}^{n}{\log \left( \frac{{{a}^{r}}}{{{b}^{r-1}}} \right)}\] is |
| A. | \[\frac{n}{2}\log \left( \frac{{{a}^{n}}}{{{b}^{n}}} \right)\] |
| B. | \[\frac{n}{2}\log \left( \frac{{{a}^{n+1}}}{{{b}^{n}}} \right)\] |
| C. | \[\frac{n}{2}\log \left( \frac{{{a}^{n+1}}}{{{b}^{n-1}}} \right)\] |
| D. | \[\frac{n}{2}\log \left( \frac{{{a}^{n+1}}}{{{b}^{n+1}}} \right)\] |
| Answer» D. \[\frac{n}{2}\log \left( \frac{{{a}^{n+1}}}{{{b}^{n+1}}} \right)\] | |
| 7795. |
\[\frac{{{x}^{2}}-{{y}^{2}}}{1\,!}+\frac{{{x}^{4}}-{{y}^{4}}}{2\,!}+\frac{{{x}^{6}}-{{y}^{6}}}{3\,!}+......\infty =\] |
| A. | \[{{e}^{x}}-{{e}^{y}}\] |
| B. | \[{{e}^{{{x}^{2}}}}-{{e}^{{{y}^{2}}}}\] |
| C. | \[2+{{e}^{{{x}^{2}}}}-{{e}^{{{y}^{2}}}}\] |
| D. | \[\frac{{{e}^{x}}-{{e}^{y}}}{2}\] |
| Answer» C. \[2+{{e}^{{{x}^{2}}}}-{{e}^{{{y}^{2}}}}\] | |
| 7796. |
Fifth term of a G.P. is 2, then the product of its 9 terms is [Pb. CET 1990, 94; AIEEE 2002] |
| A. | 256 |
| B. | 512 |
| C. | 1024 |
| D. | None of these |
| Answer» C. 1024 | |
| 7797. |
\[\frac{1}{2}+\frac{3}{2}\,.\,\frac{1}{4}+\frac{5}{3}.\frac{1}{8}+\frac{7}{4}.\frac{1}{16}+.....\infty =\] |
| A. | \[2-{{\log }_{e}}2\] |
| B. | \[2+{{\log }_{e}}2\] |
| C. | \[{{\log }_{e}}4\] |
| D. | None of these |
| Answer» B. \[2+{{\log }_{e}}2\] | |
| 7798. |
If \[g(x)={{x}^{2}}+x-2\] and \[\frac{1}{2}gof(x)=2{{x}^{2}}-5x+2,\]then which is not a possible f(x)? |
| A. | \[2x-3\] |
| B. | \[-2x+2\] |
| C. | \[x-3\] |
| D. | None of these |
| Answer» D. None of these | |
| 7799. |
Which pair of functions is identical? |
| A. | \[{{\sin }^{-1}}(sinx)\,\,and\,\,sin\,(si{{n}^{-1}}x)\] |
| B. | \[{{\log }_{e}}{{e}^{x}},{{e}^{{{\log }_{e}}x}}\] |
| C. | \[{{\log }_{e}}{{x}^{2}},2lo{{g}_{e}}x\] |
| D. | None of these |
| Answer» E. | |
| 7800. |
A function F(x) satisfies the functional equation \[{{x}^{2}}F(x)+F(1-x)=2x-{{x}^{4}}\]for all real x. F(x) must be |
| A. | \[{{x}^{2}}\] |
| B. | \[1-{{x}^{2}}\] |
| C. | \[1+{{x}^{2}}\] |
| D. | \[{{x}^{2}}+x+1\] |
| Answer» D. \[{{x}^{2}}+x+1\] | |