MCQOPTIONS
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MCQOPTIONS
This section includes 32 Mcqs, each offering curated multiple-choice questions to sharpen your Mathematics knowledge and support exam preparation. Choose a topic below to get started.
| 1. |
If\[S=\sum\limits_{n=0}^{\infty }{\frac{{{(\log x)}^{2n}}}{(2n)\,!},}\] then \[S\] = |
| A. | \[x+{{x}^{-1}}\] |
| B. | \[x-{{x}^{-1}}\] |
| C. | \[\frac{1}{2}(x+{{x}^{-1}})\] |
| D. | None of these |
| Answer» D. None of these | |
| 2. |
\[1.5+\frac{2.6}{1\,!}+\frac{3.7}{2\,!}+\frac{4.8}{3\,!}+.....\] is equal to |
| A. | \[13\,e\] |
| B. | \[15\,e\] |
| C. | \[9\,e+1\] |
| D. | \[5\,e\] |
| Answer» B. \[15\,e\] | |
| 3. |
\[1+\frac{1+2}{1\,!}+\frac{1+2+3}{2\,!}+\frac{1+2+3+4}{3\,!}+....\infty =\] |
| A. | 0 |
| B. | 1 |
| C. | \[\frac{7e}{2}\] |
| D. | \[2\,e\] |
| Answer» D. \[2\,e\] | |
| 4. |
The value of \[x\] satisfying\[{{\log }_{a}}x+{{\log }_{\sqrt{a}}}x+{{\log }_{3\sqrt{a}}}x+.........{{\log }_{a\sqrt{a}}}x=\frac{a+1}{2}\] will be |
| A. | \[x=a\] |
| B. | \[x={{a}^{a}}\] |
| C. | \[x={{a}^{-1/a}}\] |
| D. | \[x={{a}^{1/a}}\] |
| Answer» E. | |
| 5. |
The sum of the series \[\frac{1}{1+{{1}^{2}}+{{1}^{4}}}+\frac{2}{1+{{2}^{2}}+{{2}^{4}}}+\frac{3}{1+{{3}^{2}}+{{3}^{4}}}+.........\] to \[n\] terms is |
| A. | \[\frac{n({{n}^{2}}+1)}{{{n}^{2}}+n+1}\] |
| B. | \[\frac{n(n+1)}{2({{n}^{2}}+n+1)}\] |
| C. | \[\frac{n({{n}^{2}}-1)}{2({{n}^{2}}+n+1)}\] |
| D. | None of these |
| Answer» C. \[\frac{n({{n}^{2}}-1)}{2({{n}^{2}}+n+1)}\] | |
| 6. |
\[{{n}^{th}}\] term of the series\[1+\frac{4}{5}+\frac{7}{{{5}^{2}}}+\frac{10}{{{5}^{3}}}+........\]will be |
| A. | \[\frac{3n+1}{{{5}^{n-1}}}\] |
| B. | \[\frac{3n-1}{{{5}^{n}}}\] |
| C. | \[\frac{3n-2}{{{5}^{n-1}}}\] |
| D. | \[\frac{3n+2}{{{5}^{n-1}}}\] |
| Answer» D. \[\frac{3n+2}{{{5}^{n-1}}}\] | |
| 7. |
Suppose \[a,\,b,\,c\] are in A.P. and \[{{a}^{2}},{{b}^{2}},{{c}^{2}}\] are in G.P. If a < b < cand \[a+b+c=\frac{3}{2}\], then the value of a is [IIT Screening 2002] |
| A. | \[\frac{1}{2\sqrt{2}}\] |
| B. | \[\frac{1}{2\sqrt{3}}\] |
| C. | \[\frac{1}{2}-\frac{1}{\sqrt{3}}\] |
| D. | \[\frac{1}{2}-\frac{1}{\sqrt{2}}\] |
| Answer» E. | |
| 8. |
The A.M., H.M. and G.M. between two numbers are \[\frac{144}{15}\], 15 and 12, but not necessarily in this order. Then H.M., G.M. and A.M. respectively are |
| A. | \[15,\ 12,\ \frac{144}{15}\] |
| B. | \[\frac{144}{15},\ 12,\ 15\] |
| C. | \[12,\ 15,\ \frac{144}{15}\] |
| D. | \[\frac{144}{15},\ 15,\ 12\] |
| Answer» C. \[12,\ 15,\ \frac{144}{15}\] | |
| 9. |
If \[{{A}_{1}},\ {{A}_{2}};{{G}_{1}},\ {{G}_{2}}\] and \[{{H}_{1}},\ {{H}_{2}}\] be \[AM's,\ GM's\] and \[HM's\] between two quantities, then the value of \[\frac{{{G}_{1}}{{G}_{2}}}{{{H}_{1}}{{H}_{2}}}\] is [Roorkee 1983; AMU 2000] |
| A. | \[\frac{{{A}_{1}}+{{A}_{2}}}{{{H}_{1}}+{{H}_{2}}}\] |
| B. | \[\frac{{{A}_{1}}-{{A}_{2}}}{{{H}_{1}}+{{H}_{2}}}\] |
| C. | \[\frac{{{A}_{1}}+{{A}_{2}}}{{{H}_{1}}-{{H}_{2}}}\] |
| D. | \[\frac{{{A}_{1}}-{{A}_{2}}}{{{H}_{1}}-{{H}_{2}}}\] |
| Answer» B. \[\frac{{{A}_{1}}-{{A}_{2}}}{{{H}_{1}}+{{H}_{2}}}\] | |
| 10. |
If \[a,\ b,\ c\] are in H.P., then the value of \[\left( \frac{1}{b}+\frac{1}{c}-\frac{1}{a} \right)\,\left( \frac{1}{c}+\frac{1}{a}-\frac{1}{b} \right)\], is[MP PET 1998; Pb. CET 2000] |
| A. | \[\frac{2}{bc}+\frac{1}{{{b}^{2}}}\] |
| B. | \[\frac{3}{{{c}^{2}}}+\frac{2}{ca}\] |
| C. | \[\frac{3}{{{b}^{2}}}-\frac{2}{ab}\] |
| D. | None of these |
| Answer» D. None of these | |
| 11. |
If\[x,\,y,z\] are three consecutive positive integers, then \[\frac{1}{2}{{\log }_{e}}x+\frac{1}{2}{{\log }_{e}}z+\frac{1}{2xz+1}+\frac{1}{3}{{\left( \frac{1}{2xz+1} \right)}^{3}}+....=\] |
| A. | \[{{\log }_{e}}x\] |
| B. | \[{{\log }_{e}}y\] |
| C. | \[{{\log }_{e}}z\] |
| D. | None of these |
| Answer» C. \[{{\log }_{e}}z\] | |
| 12. |
If\[y=2{{x}^{2}}-1\], then\[\left[ \frac{1}{y}+\frac{1}{3{{y}^{3}}}+\frac{1}{5{{y}^{5}}}+.... \right]\] is equal to |
| A. | \[\frac{1}{2}\left[ \frac{1}{{{x}^{2}}}-\frac{1}{2{{x}^{4}}}+\frac{1}{3{{x}^{6}}}-..... \right]\] |
| B. | \[\frac{1}{2}\left[ \frac{1}{{{x}^{2}}}+\frac{1}{2{{x}^{4}}}+\frac{1}{3{{x}^{6}}}+..... \right]\] |
| C. | \[\frac{1}{2}\left[ \frac{1}{{{x}^{2}}}+\frac{1}{3{{x}^{6}}}+\frac{1}{5{{x}^{10}}}+..... \right]\] |
| D. | \[\frac{1}{2}\left[ \frac{1}{{{x}^{2}}}-\frac{1}{3{{x}^{6}}}+\frac{1}{5{{x}^{10}}}-..... \right]\] |
| Answer» C. \[\frac{1}{2}\left[ \frac{1}{{{x}^{2}}}+\frac{1}{3{{x}^{6}}}+\frac{1}{5{{x}^{10}}}+..... \right]\] | |
| 13. |
\[{{\log }_{e}}(1+x)=\sum\limits_{i=1}^{\infty }{\left[ \frac{{{(-1)}^{i+1}}{{x}^{i}}}{i} \right]}\] is defined for[Roorkee 1990] |
| A. | \[x\in (-1,\,1)\] |
| B. | Any positive (+) real x |
| C. | \[x\in (-1,\,1]\] |
| D. | Any positive (+) real \[x(x\ne 1)\] |
| Answer» D. Any positive (+) real \[x(x\ne 1)\] | |
| 14. |
\[\frac{1}{3}+\frac{1}{2\,.\,{{3}^{2}}}+\frac{1}{3\,.\,{{3}^{3}}}+\frac{1}{4\,.\,{{3}^{4}}}+.....\infty =\] [MNR 1975] |
| A. | \[{{\log }_{e}}2-{{\log }_{e}}3\] |
| B. | \[{{\log }_{e}}3-{{\log }_{e}}2\] |
| C. | \[{{\log }_{e}}6\] |
| D. | None of these |
| Answer» C. \[{{\log }_{e}}6\] | |
| 15. |
If \[m,\,n\] are the roots of the equation \[{{x}^{2}}-x-1=0\],then the value of \[\frac{\left( 1+m{{\log }_{e}}3+\frac{{{(m{{\log }_{e}}3)}^{2}}}{2\,!\,}+...\infty\right)\,\,\left( 1+n{{\log }_{e}}3+\frac{{{(n{{\log }_{e}}3)}^{2}}}{2\,!\,}+..\infty\right)\,}{\left( 1+mn{{\log }_{e}}3+\frac{{{(mn{{\log }_{e}}3)}^{2}}}{2\,!}+.....\infty\right)}\] |
| A. | 9 |
| B. | 3 |
| C. | 0 |
| D. | 1 |
| Answer» B. 3 | |
| 16. |
Jairam purchased a house in Rs. 15000 and paid Rs. 5000 at once. Rest money he promised to pay in annual installment of Rs. 1000 with 10% per annum interest. How much money is to be paid by Jairam [UPSEAT 1999] |
| A. | Rs. 21555 |
| B. | Rs. 20475 |
| C. | Rs. 20500 |
| D. | Rs. 20700 |
| Answer» D. Rs. 20700 | |
| 17. |
The sums of \[n\] terms of three A.P.'s whose first term is 1 and common differences are 1, 2, 3 are \[{{S}_{1}},\ {{S}_{2}},\ {{S}_{3}}\] respectively. The true relation is |
| A. | \[{{S}_{1}}+{{S}_{3}}={{S}_{2}}\] |
| B. | \[{{S}_{1}}+{{S}_{3}}=2{{S}_{2}}\] |
| C. | \[{{S}_{1}}+{{S}_{2}}=2{{S}_{3}}\] |
| D. | \[{{S}_{1}}+{{S}_{2}}={{S}_{3}}\] |
| Answer» C. \[{{S}_{1}}+{{S}_{2}}=2{{S}_{3}}\] | |
| 18. |
If \[a,\,b,\,c,\,d\] are positive real numbers such that \[a+b+c+d\] \[=2,\] then \[M=(a+b)(c+d)\] satisfies the relation [IIT Screening 2000] |
| A. | \[0<M\le 1\] |
| B. | \[1\le M\le 2\] |
| C. | \[2\le M\le 3\] |
| D. | \[3\le M\le 4\] |
| Answer» B. \[1\le M\le 2\] | |
| 19. |
\[{{2}^{\sin \theta }}+{{2}^{\cos \theta }}\] is greater than [AMU 2000] |
| A. | \[\frac{1}{2}\] |
| B. | \[\sqrt{2}\] |
| C. | \[{{2}^{\frac{1}{\sqrt{2}}}}\] |
| D. | \[{{2}^{\left( 1-\,\frac{1}{\sqrt{2}} \right)}}\] |
| Answer» E. | |
| 20. |
\[a,\,g,\,h\] are arithmetic mean, geometric mean and harmonic mean between two positive numbers x and y respectively. Then identify the correct statement among the following [Karnataka CET 2001] |
| A. | h is the harmonic mean between a and g |
| B. | No such relation exists between a, g and h |
| C. | g is the geometric mean between a and h |
| D. | A is the arithmetic mean between g and h |
| Answer» D. A is the arithmetic mean between g and h | |
| 21. |
If \[a\] be the arithmetic mean of \[b\] and \[c\] and \[{{G}_{1}},\ {{G}_{2}}\] be the two geometric means between them, then \[G_{1}^{3}+G_{2}^{3}=\] |
| A. | \[{{G}_{1}}{{G}_{2}}a\] |
| B. | \[2{{G}_{1}}{{G}_{2}}a\] |
| C. | \[3{{G}_{1}}{{G}_{2}}a\] |
| D. | None of these |
| Answer» C. \[3{{G}_{1}}{{G}_{2}}a\] | |
| 22. |
If the A.M. and G.M. of roots of a quadratic equations are 8 and 5 respectively, then the quadratic equation will be [Pb. CET 1990] |
| A. | \[{{x}^{2}}-16x-25=0\] |
| B. | \[{{x}^{2}}-8x+5=0\] |
| C. | \[{{x}^{2}}-16x+25=0\] |
| D. | \[{{x}^{2}}+16x-25=0\] |
| Answer» D. \[{{x}^{2}}+16x-25=0\] | |
| 23. |
If the A.M. of two numbers is greater than G.M. of the numbers by 2 and the ratio of the numbers is \[4:1\], then the numbers are [RPET 1988] |
| A. | 4, 1 |
| B. | 12, 3 |
| C. | 16, 4 |
| D. | None of these |
| Answer» D. None of these | |
| 24. |
The harmonic mean of two numbers is 4 and the arithmetic and geometric means satisfy the relation \[2A+{{G}^{2}}=27\], the numbers are [MNR 1987; UPSEAT 1999, 2000] |
| A. | \[6,\,3\] |
| B. | 5, 4 |
| C. | \[5,\ -2.5\] |
| D. | \[-3,\ 1\] |
| Answer» B. 5, 4 | |
| 25. |
If \[m\] is a root of the given equation \[(1-ab){{x}^{2}}-\] \[({{a}^{2}}+{{b}^{2}})x\] - \[(1+ab)=0\] and \[m\] harmonic means are inserted between \[a\] and \[b\], then the difference between the last and the first of the means equals |
| A. | \[b-a\] |
| B. | \[ab(b-a)\] |
| C. | \[a(b-a)\] |
| D. | \[ab(a-b)\] |
| Answer» C. \[a(b-a)\] | |
| 26. |
The first term of an infinite geometric progression is \[x\] and its sum is 5. Then [IIT Screening 2004] |
| A. | \[0\le x\le 10\] |
| B. | \[0<x<10\] |
| C. | \[-10<x<0\] |
| D. | \[x>10\] |
| Answer» C. \[-10<x<0\] | |
| 27. |
\[\alpha ,\ \beta \] are the roots of the equation \[{{x}^{2}}-3x+a=0\] and \[\gamma ,\ \delta \] are the roots of the equation \[{{x}^{2}}-12x+b=0\]. If \[\alpha ,\ \beta ,\ \gamma ,\ \delta \] form an increasing G.P., then \[(a,\ b)=\] [DCE 2000] |
| A. | (3, 12) |
| B. | (12, 3) |
| C. | (2, 32) |
| D. | (4, 16) |
| Answer» D. (4, 16) | |
| 28. |
If \[n\] geometric means between \[a\] and \[b\]be \[{{G}_{1}},\ {{G}_{2}},\ .....\]\[{{G}_{n}}\]and a geometric mean be \[G\], then the true relation is |
| A. | \[{{G}_{1}}.{{G}_{2}}........{{G}_{n}}=G\] |
| B. | \[{{G}_{1}}.{{G}_{2}}........{{G}_{n}}={{G}^{1/n}}\] |
| C. | \[{{G}_{1}}.{{G}_{2}}........{{G}_{n}}={{G}^{n}}\] |
| D. | \[{{G}_{1}}.{{G}_{2}}........{{G}_{n}}={{G}^{2/n}}\] |
| Answer» D. \[{{G}_{1}}.{{G}_{2}}........{{G}_{n}}={{G}^{2/n}}\] | |
| 29. |
If \[f(x)\] is a function satisfying \[f(x+y)=f(x)f(y)\] for all \[x,\ y\in N\] such that \[f(1)=3\] and \[\sum\limits_{x=1}^{n}{f(x)=120}\]. Then the value of \[n\] is [IIT 1992] |
| A. | 4 |
| B. | 5 |
| C. | 6 |
| D. | None of these |
| Answer» B. 5 | |
| 30. |
A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of the terms occupying odd places, then the common ratio will be equal to |
| A. | 2 |
| B. | 3 |
| C. | 4 |
| D. | 5 |
| Answer» C. 4 | |
| 31. |
Let \[n(>1)\] be a positive integer, then the largest integer \[m\] such that \[({{n}^{m}}+1)\] divides \[(1+n+{{n}^{2}}+.......+{{n}^{127}})\], is [IIT 1995] |
| A. | 32 |
| B. | 63 |
| C. | 64 |
| D. | 127 |
| Answer» D. 127 | |
| 32. |
If the sum of the \[n\]terms of G.P. is \[S\] product is \[P\] and sum of their inverse is \[R\], than \[{{P}^{2}}\] is equal to [IIT 1966; Roorkee 1981] |
| A. | \[\frac{R}{S}\] |
| B. | \[\frac{S}{R}\] |
| C. | \[{{\left( \frac{R}{S} \right)}^{n}}\] |
| D. | \[{{\left( \frac{S}{R} \right)}^{n}}\] |
| Answer» E. | |