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question_answer1)
Let \[{{A}_{n}}\] be the sum of the first n terms of the geometric series \[704+\frac{704}{2}+\frac{704}{4}+\frac{704}{8}+......\]and \[{{B}_{n}}\] be the sum of the first n terms of the geometric series \[1984+\frac{1984}{2}+\frac{1984}{4}+\frac{1984}{8}+......\] If \[{{A}_{n}}={{B}_{n}},\]then the value of n is (where \[n\in N\]).
A)
4 done
clear
B)
5 done
clear
C)
6 done
clear
D)
7 done
clear
View Solution play_arrow
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question_answer2)
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)
\[\frac{ab}{c}\] done
clear
B)
\[\frac{ac}{b}\] done
clear
C)
\[\frac{bc}{a}\] done
clear
D)
1 done
clear
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question_answer3)
ABC is a right angled triangle in which \[\angle B=90{}^\circ \]and \[BC=a\]. If n points \[{{L}_{1}},\,{{L}_{2}},....{{L}_{n}}\] on AB are such that AB is divided in n + 1 equal parts and \[{{L}_{1}}{{M}_{1}},\,\,{{L}_{2}}{{M}_{2}},....,\,\,{{L}_{n}}{{M}_{n}}\] are line segments parallel to BC and \[{{M}_{1}},\,{{M}_{2}},...{{M}_{n}}\] are on AC, then the sum of the lengths of \[{{L}_{1}}{{M}_{1}},{{L}_{2}}{{M}_{2}},....{{L}_{n}}{{M}_{n}}\] is
A)
\[\frac{a(n+1)}{2}\] done
clear
B)
\[\frac{a(n-1)}{2}\] done
clear
C)
\[\frac{an}{2}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer4)
The sum of \[i-2-3i+4...\] up to 100 terms, where \[i=\sqrt{-1}\] is
A)
\[50(1-i)\] done
clear
B)
\[25i\] done
clear
C)
\[.25\,(1+i)\] done
clear
D)
\[100(1-i)\] done
clear
View Solution play_arrow
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question_answer5)
Consider the sequence \[8A+2B,\text{ }6A+B,\text{ }4A,\,\,2A-B,........\] Which term of this sequence will have a coefficient of A which is twice the coefficient of B?
A)
10th done
clear
B)
14th done
clear
C)
16th done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer6)
If a, b, c are positive numbers, then least value of\[(a+b+c)\left( \frac{1}{a}+\frac{1}{b}+\frac{1}{c} \right)\] is
A)
1 done
clear
B)
6 done
clear
C)
9 done
clear
D)
None done
clear
View Solution play_arrow
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question_answer7)
If \[a=1+(\sqrt{3}-1)+{{(\sqrt{3}-1)}^{2}}+{{(\sqrt{3}-1)}^{3}}+....\] and\[ab=1\], then a and bare the roots of the equation
A)
\[{{x}^{2}}+4x-1=0\] done
clear
B)
C)
\[{{x}^{2}}+4x+1=0\] done
clear
D)
\[{{x}^{2}}-4x+1=0\] done
clear
View Solution play_arrow
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question_answer8)
Let \[{{a}_{n}}\] be the nth term of an A.P. If \[\sum\limits_{r=1}^{100}{{{a}_{2r}}=\alpha }\] and \[\sum\limits_{r=1}^{100}{{{a}_{2r-1}}=\beta }\], then the common difference of the A.P. is
A)
\[\alpha -\beta \] done
clear
B)
\[\beta -\alpha \] done
clear
C)
\[\frac{\alpha -\beta }{2}\] done
clear
D)
None done
clear
View Solution play_arrow
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question_answer9)
If \[x{{.}^{\ell n\left( \frac{y}{z} \right)}}.{{y}^{_{^{\ell n{{(XZ)}^{2}}}}}}.\,{{z}^{\ell n\left( \frac{x}{y} \right)}}={{y}^{4\,\ell n\,y}}\] for any x > 1, y >1 and z > 1, then which one of the following is correct?
A)
\[\ell n\,y\] is the GM of \[\ell n\,x,\,\ell n\,x,\,\ell n\,x\] and \[\ell n\,z\] done
clear
B)
\[\ell n\,y\] is the AM of \[\ell n\,x,\,\ell n\,x,\,\ell n\,x\]and \[\ell n\,z\] done
clear
C)
\[\ell n\,y\] is the HM of \[\ell n\,x,\,\ell n\,x,\,\ell n\,x\] and \[\ell n\,z\] done
clear
D)
\[\ell n\,y\] is the AM of \[\ell n,\,\,In\,\,x,\,\,\ell n\,z\] and \[\ell n\,z\] done
clear
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question_answer10)
What is the greatest value of the positive integer n satisfying the condition \[1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+....+\frac{1}{{{2}^{n-1}}}<2-\frac{1}{1000}\]?
A)
8 done
clear
B)
9 done
clear
C)
10 done
clear
D)
11 done
clear
View Solution play_arrow
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question_answer11)
If \[(1-p)(1+3x+9{{x}^{2}}+27{{x}^{3}}+81{{x}^{4}}+243{{x}^{5}})\]\[=1-{{p}^{6}},\] \[p\ne 1\] then the value of \[\frac{p}{x}\] is
A)
\[\frac{1}{3}\] done
clear
B)
3 done
clear
C)
\[\frac{1}{2}\] done
clear
D)
2 done
clear
View Solution play_arrow
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question_answer12)
Given that \[\alpha ,\,\gamma \]are root of the equation\[A{{x}^{2}}-4x+1=0\]. and \[\beta ,\,\delta \] the roots of the equation \[B{{x}^{2}}-6x+1=0,\] the values of A and B such that \[\alpha ,\beta ,\gamma \] and \[\delta \] are in H. P. are
A)
A = 3, B = 8 done
clear
B)
A = -3, B = 8 done
clear
C)
A = 3, B = - 8 done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer13)
If the pth term of an A.P. \[be\,\,\frac{1}{q}\] and qth term be \[\frac{1}{p},\] then the sum of its pqth terms will be
A)
\[\frac{pq-1}{2}\] done
clear
B)
\[\frac{1-pq}{2}\] done
clear
C)
\[\frac{pq+1}{2}\] done
clear
D)
\[-\frac{pq+1}{2}\] done
clear
View Solution play_arrow
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question_answer14)
Sum of n terms of series \[12+16+24+40+...\]will be
A)
\[2({{2}^{n}}-1)+8n\] done
clear
B)
\[2({{2}^{n}}-1)+6n\] done
clear
C)
\[3({{2}^{n}}-1)+8n\] done
clear
D)
\[4({{2}^{n}}-1)+8n\] done
clear
View Solution play_arrow
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question_answer15)
If mth terms of the series 63+65+67+69+........... and 3+10+17+24+..........be equal, then m=
A)
11 done
clear
B)
12 done
clear
C)
13 done
clear
D)
15 done
clear
View Solution play_arrow
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question_answer16)
\[{{\left( x+\frac{1}{x} \right)}^{2}}+{{\left( {{x}^{2}}+\frac{1}{{{x}^{2}}} \right)}^{2}}+{{\left( {{x}^{{}}}+\frac{1}{{{x}^{3}}} \right)}^{2}}....\] upto n terms is
A)
\[\frac{{{x}^{2n}}-1}{{{x}^{2}}-1}\times \frac{{{x}^{2n+2}}+1}{{{x}^{2n}}}+2n\] done
clear
B)
\[\frac{{{x}^{2n}}+1}{{{x}^{2}}+1}\times \frac{{{x}^{2n+2}}-1}{{{x}^{2n}}}-2n\] done
clear
C)
\[\frac{{{x}^{2n}}-1}{{{x}^{2}}-1}\times \frac{{{x}^{2n}}-1}{{{x}^{2n}}}-2n\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer17)
Let \[{{S}_{n}}(1\le n\le 9)\] denotes the sum of n terms of series 1+22+333+.................+9999999999, then for \[2\le n\le 9\]
A)
\[{{S}_{n}}-{{S}_{n-1}}=\frac{1}{9}({{10}^{n}}-{{n}^{2}}+n)\] done
clear
B)
\[{{S}_{n}}=\frac{1}{9}({{10}^{n}}-{{n}^{2}}+2n-2)\] done
clear
C)
\[9\,({{S}_{n}}-{{S}_{n-1}})=n\,({{10}^{n}}-1)\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer18)
If \[{{a}^{2}},{{b}^{2}},{{c}^{2}}\] are in A.P. consider two statements
(i) \[\frac{1}{b+c},\frac{1}{c+a},\frac{1}{a+b}\] are in A.P. |
(ii) \[\frac{a}{b+c},\frac{b}{c+a},\frac{c}{a+b}\] are in A.P. |
A)
(i) and (ii) both correct done
clear
B)
(i) and (ii) both incorrect done
clear
C)
(i) correct (ii) incorrect done
clear
D)
(i) incorrect (ii) correct done
clear
View Solution play_arrow
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question_answer19)
The value of the infinite product \[{{6}^{\frac{1}{2}}}\times {{6}^{\frac{1}{2}}}\times {{6}^{\frac{3}{8}}}\times {{6}^{\frac{1}{4}}}....\] is
A)
6 done
clear
B)
36 done
clear
C)
216 done
clear
D)
\[\infty \] done
clear
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question_answer20)
a, b, c are in G.P. with 1<a<b<n, and n>1 is an integer, \[lo{{g}_{a}}\text{ }n,\text{ }lo{{g}_{b}}\text{ }n,\text{ }lo{{g}_{c}}\,n\] form a sequence. This sequence is which one of the following?
A)
Harmonic progression done
clear
B)
Arithmetic progression done
clear
C)
Geometric progression done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer21)
If S, P and R are the sum, product and sum of the reciprocals of n terms of an increasing GP respectively and \[{{S}^{n}}={{R}^{n}}.{{P}^{k}},\] then k is equal to
A)
1 done
clear
B)
2 done
clear
C)
3 done
clear
D)
None of these done
clear
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question_answer22)
If a, b and 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:
A)
\[\frac{2}{bc}+\frac{1}{{{b}^{2}}}\] done
clear
B)
\[\frac{3}{{{c}^{2}}}+\frac{2}{ca}\] done
clear
C)
\[\frac{3}{{{b}^{2}}}-\frac{2}{ab}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer23)
A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of terms occupying odd places, then the common ratio is
A)
5 done
clear
B)
1 done
clear
C)
4 done
clear
D)
3 done
clear
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question_answer24)
The sum to n terms of the series 2+5+14+41+........is
A)
\[{{3}^{n-1}}+8n-3\] done
clear
B)
\[{{8.3}^{n}}+4n-8\] done
clear
C)
\[{{3}^{n+1}}+\frac{8}{3}n+1\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer25)
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)
\[{{m}^{2}}-m\left( 4r-1 \right)+4{{r}^{2}}-2=0\] done
clear
B)
\[{{m}^{2}}-m\left( 4r+1 \right)+4{{r}^{2}}+2=0\] done
clear
C)
\[{{m}^{2}}-m(4r+1)+4{{r}^{2}}-2=0\] done
clear
D)
\[{{m}^{2}}-m\left( 4r-1 \right)+4{{r}^{2}}+2=0\] done
clear
View Solution play_arrow
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question_answer26)
If a, b, c are in A. P., then\[(a+2b-c)(2b+c-a)(c+a-b)\] equals
A)
\[\frac{1}{2}abc\] done
clear
B)
abc done
clear
C)
2 abc done
clear
D)
4 abc done
clear
View Solution play_arrow
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question_answer27)
If \[{{S}_{n}}=\]\[(1+{{3}^{-1}})(1+{{3}^{-2}})(1+{{3}^{-4}})(1+{{3}^{-8}})....(1+{{3}^{-{{2}^{n}}}}),\] then \[{{S}_{\infty }}\] is equal to
A)
1 done
clear
B)
\[\frac{1}{2}\] done
clear
C)
\[\frac{3}{2}\] done
clear
D)
None done
clear
View Solution play_arrow
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question_answer28)
If, 8, -4 and 13 be three (not necessarily consecutive term) of an A.P., how many such A.P. s are possible?
A)
1 done
clear
B)
2 done
clear
C)
Infinitely many done
clear
D)
No such A.P. is possible done
clear
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question_answer29)
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)
\[\frac{2}{\sqrt{3}}\] done
clear
B)
\[2\sqrt{3}\] done
clear
C)
\[\frac{\sqrt{3}}{2}\] done
clear
D)
\[\frac{1}{2\sqrt{3}}\] done
clear
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question_answer30)
The value of \[0.0\overline{37}\] where \[0.0\overline{37}\] stands for the number .0373737........... is:
A)
37/1000 done
clear
B)
37/990 done
clear
C)
1/37 done
clear
D)
1/27 done
clear
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question_answer31)
If a, b, and c are in A.P ., and p and p' are, respectively, A.M. and G.M. between a and b while q, q' are, respectively, the A,M. and G M. between b and c, then
A)
\[{{p}^{2}}+{{q}^{2}}=p{{'}^{2}}+q{{'}^{2}}\] done
clear
B)
\[pq=p'q'\] done
clear
C)
\[{{p}^{2}}-{{q}^{2}}=p{{'}^{2}}-q{{'}^{2}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer32)
ABCD is a square of lengths \[a,\,a\in N,a>1\]. Let \[{{L}_{1}},{{L}_{2}},{{L}_{3}},...\] be points BC such that \[B{{L}_{1}}={{L}_{1}}{{L}_{2}}={{L}_{2}}{{L}_{3}}=...=1\] and \[{{M}_{1}},{{M}_{2}},{{M}_{3}},...\] be points on CD such that \[C{{M}_{1}}={{M}_{1}}{{M}_{2}}={{M}_{2}}{{M}_{3}}=...=1\]. Then, \[\sum\limits_{n=1}^{a-1}{(AL_{n}^{2}+{{L}_{n}}M_{n}^{2})}\] is equal to
A)
\[\frac{1}{2}a{{(a-1)}^{2}}\] done
clear
B)
\[\frac{1}{2}a(a-1)(4a-1)\] done
clear
C)
\[\frac{1}{2}(a-1)(2a-1)(4a-1)\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer33)
If \[{{S}_{n}}\] denotes the sum of first n terms of an A.P. whose first term is a and \[\frac{{{S}_{nx}}}{{{S}_{x}}}\] is independent of x, then \[{{S}_{p}}=\]
A)
\[{{P}^{3}}\] done
clear
B)
\[{{P}^{2}}a\] done
clear
C)
\[P{{a}^{2}}\] done
clear
D)
\[{{a}^{3}}\] done
clear
View Solution play_arrow
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question_answer34)
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)
\[\left\{ \frac{\pi }{2},\frac{3\pi }{4} \right\}\] done
clear
B)
\[\left\{ \frac{\pi }{4},\frac{3\pi }{4} \right\}\] done
clear
C)
\[\left\{ \frac{\pi }{3},\frac{2\pi }{3} \right\}\] done
clear
D)
\[\left\{ \frac{\pi }{6},\frac{5\pi }{6} \right\}\] done
clear
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question_answer35)
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
A)
\[n{{(2c)}^{1/n}}\] done
clear
B)
\[(n+1){{c}^{1/n}}\] done
clear
C)
\[2n{{c}^{1/n}}\] done
clear
D)
\[(n+1){{(2c)}^{1/n}}\] done
clear
View Solution play_arrow
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question_answer36)
If \[\frac{1}{\sqrt{b}+\sqrt{c}},\frac{1}{\sqrt{c}+\sqrt{a}},\frac{1}{\sqrt{a}+\sqrt{b}}\] are in A.P. then \[{{9}^{ax+1}},{{9}^{bx+1}},{{9}^{cx+1}},x\ne 0\] are in:
A)
G.P done
clear
B)
G.P. only if x<0 done
clear
C)
G.P. only if x>0 done
clear
D)
none of these done
clear
View Solution play_arrow
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question_answer37)
If \[{{\log }_{10}}2,\,{{\log }_{10}}({{2}^{x}}-1)\] and \[{{\log }_{10}}({{2}^{x}}+3)\] are three consecutive terms of an A.P, then the value of x is
A)
1 done
clear
B)
\[lo{{g}_{5}}2\] done
clear
C)
\[lo{{g}_{2}}5\] done
clear
D)
\[lo{{g}_{10}}5\] done
clear
View Solution play_arrow
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question_answer38)
What is the 15th term of the series 3, 7, 13, 21, 31, 43, ....?
A)
205 done
clear
B)
225 done
clear
C)
238 done
clear
D)
241 done
clear
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question_answer39)
If a, b, c are in GP., then
A)
\[{{a}^{2}},\text{ }{{b}^{2}},\text{ }{{c}^{2}}\] are in GP. done
clear
B)
\[{{a}^{2}}(b+c),{{c}^{2}}(a+b),{{b}^{2}}(a+c)\] are in G.P. done
clear
C)
\[\frac{a}{b+c},\frac{b}{c+a},\frac{c}{a+b}\] are in G.P. done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer40)
If \[x=\sum\limits_{n=0}^{\infty }{{{a}^{n}}},y=\sum\limits_{n=0}^{\infty }{{{b}^{n}}},\,z=\sum\limits_{n=0}^{\infty }{{{c}^{n}}}\] where a, b, c are in A.P and \[\left| a \right|<1,\left| b \right|<1,\left| c \right|<1\] then \[x,y,z\] are in
A)
G.P. done
clear
B)
A.P. done
clear
C)
H.P. done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer41)
The sum of the series \[\frac{2}{3}+\frac{8}{9}+\frac{26}{27}+\frac{80}{81}+....n\] terms is:
A)
\[n-\frac{1}{2}({{3}^{n}}-1)\] done
clear
B)
\[n+\frac{1}{2}({{3}^{n}}-1)\] done
clear
C)
\[n-\frac{1}{2}(1-{{3}^{-n}})\] done
clear
D)
\[n+\frac{1}{2}({{3}^{-n}}-1)\] done
clear
View Solution play_arrow
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question_answer42)
The value of \[\sum\limits_{n=1}^{10}{\sum\limits_{m=1}^{10}{({{m}^{2}}+{{n}^{2}})}}\] equals
A)
4235 done
clear
B)
5050 done
clear
C)
7700 done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer43)
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)
\[1000\,\pi \] done
clear
B)
\[5050\,\pi \] done
clear
C)
\[4950\,\pi \] done
clear
D)
\[5151\text{ }\pi \] done
clear
View Solution play_arrow
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question_answer44)
If \[{{a}_{1}},{{a}_{2}},{{a}_{3}}.....\] are in A.P. and \[a_{1}^{2}-a_{2}^{2}+a_{3}^{2}-a_{4}^{2}+......+a_{2k-1}^{2}-a_{2k}^{2}\] \[=M(a_{1}^{2}-a_{2k}^{2}).\] Then M =
A)
\[\frac{k-1}{k+1}\] done
clear
B)
\[\frac{k}{2k-1}\] done
clear
C)
\[\frac{k+1}{2k+1}\] done
clear
D)
None done
clear
View Solution play_arrow
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question_answer45)
The maximum sum of the series \[20+19\frac{1}{3}\]\[+18\frac{2}{3}+18+....\] is
A)
300 done
clear
B)
310 done
clear
C)
\[311\frac{2}{3}\] done
clear
D)
\[333\frac{1}{3}\] done
clear
View Solution play_arrow
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question_answer46)
What is the sum of the series 0.5 + 0.55 + 0.555 +... to n terms?
A)
\[\frac{5}{9}\left[ n-\frac{2}{9}\left( 1-\frac{1}{{{10}^{n}}} \right) \right]\] done
clear
B)
\[\frac{1}{9}\left[ 5-\frac{2}{9}\left( 1-\frac{1}{{{10}^{n}}} \right) \right]\] done
clear
C)
\[\frac{1}{9}\left[ n-\frac{5}{9}\left( 1-\frac{1}{{{10}^{n}}} \right) \right]\] done
clear
D)
\[\frac{5}{9}\left[ n-\frac{1}{9}\left( 1-\frac{1}{{{10}^{n}}} \right) \right]\] done
clear
View Solution play_arrow
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question_answer47)
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 done
clear
B)
21 done
clear
C)
45 done
clear
D)
54 done
clear
View Solution play_arrow
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question_answer48)
Three numbers are in G.P. such that their sum is 38 and their product is 1728. The greatest number among them is:
A)
18 done
clear
B)
16 done
clear
C)
14 done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer49)
The minimum value of \[\frac{{{x}^{4}}+{{y}^{4}}+{{z}^{2}}}{xyz}\] for positive real number x, y, z is
A)
\[\sqrt{2}\] done
clear
B)
\[2\sqrt{2}\] done
clear
C)
\[4\sqrt{2}\] done
clear
D)
\[8\sqrt{2}\] done
clear
View Solution play_arrow
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question_answer50)
\[\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+....+\frac{1}{n(n+1)}\] equals
A)
\[\frac{1}{n(n+1)}\] done
clear
B)
\[\frac{n}{n+1}\] done
clear
C)
\[\frac{2n}{n+1}\] done
clear
D)
\[\frac{2}{n(n+1)}\] done
clear
View Solution play_arrow
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question_answer51)
For \[-\pi <x<\pi ,\] the values of x which satisfy the relation \[{{11}^{1+\left| \cos \,x \right|+co{{s}^{2}}x+\left| {{\cos }^{3}}x \right|+...upt{{o}^{\infty }}}}=121\] are given by
A)
\[\pm \frac{\pi }{3},\pm \frac{2\pi }{3}\] done
clear
B)
\[\frac{\pi }{3},\frac{2\pi }{4}\] done
clear
C)
\[\frac{\pi }{4},\frac{3\pi }{4}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer52)
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)
\[\pm 1\] done
clear
B)
\[\pm 2\] done
clear
C)
\[\pm 3\] done
clear
D)
\[\pm 4\] done
clear
View Solution play_arrow
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question_answer53)
There are four numbers of which the first three are in G.P. and the last three are in A.R, whose common difference is 6. If the first and the last numbers are equal then two other numbers are
A)
-2, 4 done
clear
B)
-4, 2 done
clear
C)
2, 6 done
clear
D)
none done
clear
View Solution play_arrow
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question_answer54)
It is given that \[\frac{1}{{{2}^{n}}\sin \,\alpha },1,\,{{2}^{n}}\sin \] a are in A.P. for some value of \[\alpha \]. Let say for n = 1, the \[\alpha \] satisfying the above A.P. is \[{{\alpha }_{1}},\] for n = 2, the value is \[{{\alpha }_{2}},\] and so on. If \[S=\sum\limits_{i=1}^{\infty }{\sin \,{{\alpha }_{i}},}\]then the value of S is
A)
1 done
clear
B)
\[\frac{1}{2}\] done
clear
C)
2 done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer55)
The sum of an infinite GP is x and the common ratio r is such that \[\left| r \right|<1\]. If the first term of the GP is 2, then which one of the following is correct?
A)
\[-1<x<1\] done
clear
B)
\[-\infty <x<1\] done
clear
C)
\[1<x<\infty \] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer56)
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 done
clear
B)
2 only done
clear
C)
Both 1 and 2 done
clear
D)
Neither 1 and 2 done
clear
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question_answer57)
lf \[x=1+a+{{a}^{2}}+....................\]to infinity and \[y=1+b+{{b}^{2}}+....................\] to infinity, where a, b are proper fractions, then \[1+ab+{{a}^{2}}{{b}^{2}}+.....\] to infinity is equal:
A)
\[\frac{xy}{x+y-1}\] done
clear
B)
\[\frac{xy}{x-y-1}\] done
clear
C)
\[\frac{xy}{x-y+1}\] done
clear
D)
\[\frac{xy}{x+y+1}\] done
clear
View Solution play_arrow
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question_answer58)
If \[a+b+c=3\] and \[a>0,b>0,c>0,\] then the greatest value of \[{{a}^{2}}{{b}^{3}}{{c}^{2}}\] is
A)
\[\frac{{{3}^{10}}{{.2}^{4}}}{{{7}^{7}}}\] done
clear
B)
\[\frac{{{3}^{9}}{{.2}^{4}}}{{{7}^{7}}}\] done
clear
C)
\[\frac{{{3}^{8}}{{.2}^{4}}}{{{7}^{7}}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer59)
If the sum to infinity of the series, \[1+4x+7{{x}^{2}}+10{{x}^{3}}+........,\] is 35/16, where \[\left| x \right|<1\], then x equals to
A)
19/7 done
clear
B)
1/5 done
clear
C)
¼ done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer60)
The sum of the series 3.6 + 4.7 + 5.8 + ......upto (n - 2) terms
A)
\[{{n}^{3}}+{{n}^{2}}+n+2\] done
clear
B)
\[\frac{1}{6}(2{{n}^{3}}+12{{n}^{2}}+10\,n-84)\] done
clear
C)
\[{{n}^{3}}+{{n}^{2}}+n\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer61)
If the nth term of an arithmetic progression is\[3n+7\], then what is the sum of its first 50 terms?
A)
3925 done
clear
B)
4100 done
clear
C)
4175 done
clear
D)
8200 done
clear
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question_answer62)
A series is such that its every even term is 'a' times the term before it and every odd term is c times the term before it. The sum of 2n term of the series is (the first term is unity)
A)
\[\frac{(1-{{c}^{n}})(1-{{a}^{n}})}{1-ac}\] done
clear
B)
\[\frac{(1+a)(1-{{c}^{n}}{{a}^{n}})}{1-ac}\] done
clear
C)
\[\frac{(1+{{c}^{n}})(1+{{a}^{n}})}{1-ac}\] done
clear
D)
\[\frac{(1+a)(1+{{c}^{n}}{{a}^{n}})}{1+ac}\] done
clear
View Solution play_arrow
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question_answer63)
Let a = 111... 1(55 digits), \[b=1+10+{{10}^{2}}+...+{{10}^{4}},\] \[c=1+{{10}^{5}}+{{10}^{10}}+{{10}^{15}}+....+{{10}^{50}},\]then
A)
\[a=b+c\] done
clear
B)
\[a=bc\] done
clear
C)
\[b=ac\] done
clear
D)
\[c=ab\] done
clear
View Solution play_arrow
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question_answer64)
The value of x + y + z is 15 if a, x, y, z, b are in A.P. while the value of \[\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\] is \[\frac{5}{3}\] if a, x, y, z, b are in H.P. Then the value of a and b are
A)
2 and 8 done
clear
B)
1 and 9 done
clear
C)
3 and 7 done
clear
D)
None done
clear
View Solution play_arrow
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question_answer65)
If the positive integers a, b, c, d are in AP, then the numbers abc, abd, acd, bcd are in
A)
HP done
clear
B)
AP done
clear
C)
GP done
clear
D)
None of the above done
clear
View Solution play_arrow
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question_answer66)
Which one of the following options is correct?
A)
\[si{{n}^{2}}30{}^\circ ,\text{ }si{{n}^{2}}45{}^\circ ,\text{ }si{{n}^{2}}60{}^\circ \] are in GP done
clear
B)
\[co{{s}^{2}}30{}^\circ ,\text{ }co{{s}^{2}}45{}^\circ ,\text{ }co{{s}^{2}}60{}^\circ \] are in GP done
clear
C)
\[co{{t}^{2}}30{}^\circ ,\text{ }co{{t}^{2}}45{}^\circ ,\text{ }co{{t}^{2}}60{}^\circ \] are in GP done
clear
D)
\[ta{{n}^{2}}30{}^\circ ,\text{ }ta{{n}^{2}}45{}^\circ ,\text{ }ta{{n}^{2}}60{}^\circ \] are in GP done
clear
View Solution play_arrow
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question_answer67)
a, b, c are three distinct real numbers and they are in a GP. if a + b + c = xb, then
A)
\[x\le -1\,\,or\,\,x\ge 3\] done
clear
B)
\[x<-1\,\,or\,\,x>3\] done
clear
C)
\[x\le -1\,\,or\,\,x>3\] done
clear
D)
\[x<-3\,\,or\,\,x>2\] done
clear
View Solution play_arrow
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question_answer68)
If a, b, c are the sides of a triangle, then the minimum value of \[\frac{a}{b+c-a}+\frac{b}{c+a-b}+\]\[\frac{c}{a+b-c}\] is equal to
A)
3 done
clear
B)
6 done
clear
C)
9 done
clear
D)
12 done
clear
View Solution play_arrow
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question_answer69)
The 20th terms of the series 2 + 3 + 5 +9+16 +....... is
A)
950 done
clear
B)
975 done
clear
C)
990 done
clear
D)
1010 done
clear
View Solution play_arrow
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question_answer70)
The 100th term of the sequence 1, 2, 2, 3, 3, 3, 4, 4, 4, 4,... is
A)
12 done
clear
B)
13 done
clear
C)
14 done
clear
D)
15 done
clear
View Solution play_arrow
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question_answer71)
. An A.P. whose first term is unity and in which the sum of first half of any even number of terms to that of second half of the same number of terms is a constant ratio, then the common difference is:
A)
2 done
clear
B)
1 done
clear
C)
3 done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer72)
The perimeter of a triangle whose sides are in A.P. is 21 cm and the product of lengths of the shortest side and the longest side exceeds the length of the other side by 6 cm. The longest side of the triangle is
A)
1 cm done
clear
B)
7 cm done
clear
C)
13 cm done
clear
D)
None done
clear
View Solution play_arrow
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question_answer73)
The expression \[\frac{{{a}^{n+1}}+{{b}^{n+1}}}{{{a}^{n}}+{{b}^{n}}}\] is \[[a\ne b\ne 0]\] is (where a and b are unequal non-zero numbers)
A)
A.M. between a and b if \[n=-1\] done
clear
B)
G.M. between a and b if \[n=-\frac{1}{2}\] done
clear
C)
H.M. between a and b if n = 0 done
clear
D)
All are correct done
clear
View Solution play_arrow
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question_answer74)
The sum of \[\frac{\frac{1}{2}.\frac{2}{2}}{{{1}^{3}}}+\frac{\frac{2}{2}.\frac{3}{2}}{{{1}^{3}}+{{2}^{3}}}+\frac{\frac{3}{2}.\frac{4}{2}}{{{1}^{3}}+{{2}^{3}}+{{3}^{3}}}+.....\] upto n terms is equal to
A)
\[\frac{n-1}{n}\] done
clear
B)
\[\frac{n}{n+1}\] done
clear
C)
\[\frac{n+1}{n+2}\] done
clear
D)
\[\frac{n+1}{n}\] done
clear
View Solution play_arrow
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question_answer75)
What is the product of first 2n + 1 terms of a geometric progression?
A)
The (n + 1)th power of the nth term of the GP done
clear
B)
The (2n + 1)th power of the nth term of the GP done
clear
C)
The (2n + 1)th power of the (n + 1)th term of the GP done
clear
D)
The nth power of the (n + 1)th terms of the GP done
clear
View Solution play_arrow
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question_answer76)
If \[{{\log }_{10}}2,\,{{\log }_{10}}({{2}^{x}}-1),lo{{g}_{10}}({{2}^{x}}+3)\] are three consecutive terms of an AP, then which one of the following is correct?
A)
\[x=0\] done
clear
B)
\[x=1\] done
clear
C)
\[x={{\log }_{2}}5\] done
clear
D)
\[x={{\log }_{5}}2\] done
clear
View Solution play_arrow
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question_answer77)
Let S be the sum, P be the product and R be the sum of the reciprocals of 3 terms of a G.P. Then \[{{P}^{2}}{{R}^{3}}:{{S}^{3}}\] is equal to
A)
1 : 1 done
clear
B)
Common ratio: 1 done
clear
C)
\[{{\left( \text{first term} \right)}^{\text{2}}}\text{: }{{\left( \text{common ratio} \right)}^{\text{2}}}\] done
clear
D)
\[{{\left( common\text{ }ratio \right)}^{n}}:1\] done
clear
View Solution play_arrow
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question_answer78)
If \[{{t}_{n}}\] denotes the nth term of a G.P. whose common ratio is r, then the progression whose nth term is \[\frac{1}{t_{n}^{2}+t_{n+1}^{2}}\] is
A)
A.P. done
clear
B)
G.P. done
clear
C)
H.P. done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer79)
If \[{{\log }_{e}}5,\,{{\log }_{e}}({{5}^{x}}-1)\] and \[{{\log }_{e}}\left( {{5}^{x}}-\frac{11}{5} \right)\]are in A.P then the values of x are
A)
\[{{\log }_{5}}4\,\,and\,\,{{\log }_{5}}3\] done
clear
B)
\[{{\log }_{3}}4\,\,and\,\,{{\log }_{4}}3\] done
clear
C)
\[{{\log }_{3}}4\,\,and\,\,{{\log }_{3}}5\] done
clear
D)
\[{{\log }_{5}}6\,\,and\,\,{{\log }_{5}}7\] done
clear
View Solution play_arrow
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question_answer80)
2+4+7+11+16+........ to n terms =
A)
\[\frac{1}{6}({{n}^{2}}+3n+8)\] done
clear
B)
\[\frac{n}{6}({{n}^{2}}+3n+8)\] done
clear
C)
\[\frac{1}{6}({{n}^{2}}-3n+8)\] done
clear
D)
\[\frac{n}{6}({{n}^{2}}-3n+8)\] done
clear
View Solution play_arrow
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question_answer81)
If the angles A< B<C of a triangle are in A.P., then
A)
\[{{c}^{2}}={{a}^{2}}+{{b}^{2}}-ab\] done
clear
B)
\[{{b}^{2}}={{a}^{2}}+{{c}^{2}}-ac\] done
clear
C)
\[{{c}^{2}}={{a}^{2}}+{{b}^{2}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer82)
Let a, b, c, be in A.P. with a common difference d. Then \[{{e}^{1/c}},{{e}^{b/ac}},{{e}^{1/a}}\] are in:
A)
G.P. with common ratio \[{{e}^{d}}\] done
clear
B)
G.P. with common ratio \[{{e}^{1/d}}\] done
clear
C)
G.P. with common ratio \[{{e}^{d/({{b}^{2}}-{{d}^{2}})}}\] done
clear
D)
A.P. done
clear
View Solution play_arrow
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question_answer83)
The least value of n (a natural number), for which the sum S of the series \[1+\frac{1}{2}+\frac{1}{{{2}^{2}}}+\frac{1}{{{2}^{3}}}+.....\]differs from \[{{S}_{n}}\] by a quantity \[<{{10}^{-6}}\], is
A)
21 done
clear
B)
20 done
clear
C)
19 done
clear
D)
None done
clear
View Solution play_arrow
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question_answer84)
\[\sum\limits_{k=1}^{n}{k{{(1+1/n)}^{k-1}}=}\]
A)
\[n(n-1)\] done
clear
B)
\[n(n+1)\] done
clear
C)
\[{{n}^{2}}\] done
clear
D)
\[{{(n+1)}^{2}}\] done
clear
View Solution play_arrow
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question_answer85)
The equation \[\left( {{a}^{2}}+{{b}^{2}} \right){{x}^{2}}-2b\left( a+c \right)x+\] \[\left( {{b}^{2}}+{{c}^{2}} \right)=0\] has equal roots. Which one of the following is correct about a, b, and c?
A)
They are in AP done
clear
B)
They are in GP done
clear
C)
They are in HP done
clear
D)
They are neither in AP, nor in GP, nor in HP done
clear
View Solution play_arrow
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question_answer86)
What does the series \[1+{{3}^{-\frac{1}{2}}}+3+\frac{1}{3\sqrt{3}}+...\] represents?
A)
AP done
clear
B)
GP done
clear
C)
HP done
clear
D)
None of the above series done
clear
View Solution play_arrow
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question_answer87)
In a, GP. of 3n terms, \[{{S}_{1}}\] denotes the sum of first n terms, \[{{S}_{2}}\] the sum of the second block of n terms and \[{{S}_{3}}\] the sum of last n terms. Then \[{{S}_{1}},{{S}_{2}},{{S}_{3}},\] are in
A)
A.P. done
clear
B)
G.P. done
clear
C)
H.P. done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer88)
If \[x>0,\frac{{{x}^{n}}}{1+x+{{x}^{2}}+...+{{x}^{2n}}}\] is
A)
\[\cdot \le \frac{1}{2n+1}\] done
clear
B)
\[<\frac{1}{2n+1}\] done
clear
C)
\[^{3}\frac{1}{2n+1}\] done
clear
D)
\[>\frac{2}{2n+1}\] done
clear
View Solution play_arrow
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question_answer89)
If \[{{a}_{1}},{{a}_{2}},{{a}_{3}},.....{{a}_{n}}\] are in A.P., and\[\frac{1}{{{a}_{1}}{{a}_{n}}}+\frac{1}{{{a}_{2}}{{a}_{n-1}}}+\frac{1}{{{a}_{3}}{{a}_{n-2}}}+.....+\frac{1}{{{a}_{n}}{{a}_{1}}}\]\[=K\left( \frac{1}{{{a}_{1}}}+\frac{1}{{{a}_{2}}}+\frac{1}{{{a}_{3}}}+....+\frac{1}{{{a}_{n}}} \right)\]. Then K is
A)
\[\frac{2}{{{a}_{1}}+{{a}_{n}}}\] done
clear
B)
\[\frac{n}{{{a}_{1}}+{{a}_{n}}}\] done
clear
C)
\[\frac{1}{{{a}_{1}}+{{a}_{n}}}\] done
clear
D)
\[\frac{n-1}{{{a}_{1}}+{{a}_{n}}}\] done
clear
View Solution play_arrow
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question_answer90)
If the sum of the first ten terms of the series \[{{\left( 1\frac{3}{5} \right)}^{2}}+{{\left( 2\frac{2}{5} \right)}^{2}}+{{\left( 3\frac{1}{5} \right)}^{2}}+{{4}^{2}}+{{\left( 4\frac{4}{5} \right)}^{2}}+.....,\] is \[\frac{16}{5}m,\] then m is equal to:
A)
100 done
clear
B)
99 done
clear
C)
102 done
clear
D)
101 done
clear
View Solution play_arrow
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question_answer91)
If \[\frac{1}{a},\frac{1}{b},\frac{1}{c}\] are A. P., then \[\left( \frac{1}{a}+\frac{1}{b}-\frac{a}{c} \right)\] \[\left( \frac{1}{b}+\frac{1}{c}-\frac{1}{a} \right)\] is equal to
A)
\[\frac{4}{ac}-\frac{3}{{{b}^{2}}}\] done
clear
B)
\[\frac{{{b}^{2}}-ac}{{{a}^{2}}{{b}^{2}}{{c}^{2}}}\] done
clear
C)
\[\frac{4}{ac}-\frac{1}{{{b}^{2}}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer92)
The roots of the equation \[{{(x-1)}^{2}}-4\left| x-1 \right|+3=0\]
A)
Form an A.P. done
clear
B)
Form a G.P. done
clear
C)
Form an H.P. done
clear
D)
Do not form any progression done
clear
View Solution play_arrow
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question_answer93)
x and y are positive number. Let g and a be G. M. and AM of these numbers. Also let G be G.M. of x + 1 and y + 1. If G and g are roots of equation\[{{x}^{2}}-5x+6=0\], then
A)
\[x=2,\,y=\frac{3}{4}\] done
clear
B)
\[x=\frac{3}{4},y=12\] done
clear
C)
\[x=\frac{5}{2},y=\frac{8}{5}\] done
clear
D)
\[x=y=2\] done
clear
View Solution play_arrow
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question_answer94)
The sum of the infinite series \[\frac{{{2}^{2}}}{2!}+\frac{{{2}^{4}}}{4!}+\frac{{{2}^{6}}}{6!}+....\] is equal to
A)
\[\frac{{{e}^{2}}+1}{2e}\] done
clear
B)
\[\frac{{{e}^{4}}+1}{2{{e}^{2}}}\] done
clear
C)
\[\frac{{{({{e}^{2}}-1)}^{2}}}{2{{e}^{2}}}\] done
clear
D)
\[\frac{{{({{e}^{2}}+1)}^{2}}}{2{{e}^{2}}}\] done
clear
View Solution play_arrow
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question_answer95)
If \[\left| x \right|<\frac{1}{2},\] what is the value of \[1+n\left[ \frac{x}{1-x} \right]+\left[ \frac{n(n+1)}{2!} \right]{{\left[ \frac{x}{1-x} \right]}^{2}}+.......\infty \]?
A)
\[{{\left[ \frac{1-x}{1-2x} \right]}^{n}}\] done
clear
B)
\[{{(1-x)}^{n}}\] done
clear
C)
\[{{\left[ \frac{1-2x}{1-x} \right]}^{n}}\] done
clear
D)
\[{{\left( \frac{1}{1-x} \right)}^{n}}\] done
clear
View Solution play_arrow
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question_answer96)
The harmonic mean H of two numbers is 4 and the arithmetic mean A and geometric mean G satisfy the equation \[2A+{{G}^{2}}=27\]. The two numbers are
A)
6, 3 done
clear
B)
9, 5 done
clear
C)
12, 7 done
clear
D)
3, 1 done
clear
View Solution play_arrow
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question_answer97)
The sum of an infinite geometric series is 2 and the sum of the geometric series made from the cubes of this infinite series is 24. Then the series is
A)
\[3+\frac{3}{2}-\frac{3}{4}+\frac{3}{8}-....\] done
clear
B)
\[3+\frac{3}{2}-\frac{3}{4}+\frac{3}{8}+....\] done
clear
C)
\[3-\frac{3}{2}+\frac{3}{4}-\frac{3}{8}+....\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer98)
a, b, c are the first three terms of a geometric series. If the harmonic mean of a and b is 12 and that of b and c is 36, then the first five terms of the series are
A)
9, 18, 27, 36, 45 done
clear
B)
8, 24, 72, 216, 648 done
clear
C)
4, 22, 38, 46 done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer99)
The fourth term of an A.P. is three times of the first term and the seventh term exceeds the twice of the third term by one, then the common difference of the progression is
A)
2 done
clear
B)
3 done
clear
C)
\[\frac{3}{2}\] done
clear
D)
-1 done
clear
View Solution play_arrow
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question_answer100)
The sum to n terms of the series \[\frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+..........\] is
A)
\[n-1-{{2}^{-n}}\] done
clear
B)
1 done
clear
C)
\[n-1+{{2}^{-n}}\] done
clear
D)
\[1+{{2}^{-n}}\] done
clear
View Solution play_arrow