JEE Main & Advanced Mathematics Sequence & Series Question Bank

done Relation between AP., GP. and HP. Total Questions - 96

• question_answer1)
  If the arithmetic, geometric and harmonic means between two distinct positive real numbers be \[A,\ G\] and \[H\] respectively, then the relation between them is [MP PET 1984; Roorkee 1995]
  A)
  \[A>G>H\] done clear
  B)
  \[A>G<H\] done clear
  C)
  \[H>G>A\] done clear
  D)
  \[G>A>H\] done clear
  View Solution play_arrow

• question_answer2)
  If the arithmetic, geometric and harmonic means between two positive real numbers be \[A,\ G\] and \[H\], then [AMU 1979, 1982; MP PET 1993]
  A)
  \[{{A}^{2}}=GH\] done clear
  B)
  \[{{H}^{2}}=AG\] done clear
  C)
  \[G=AH\] done clear
  D)
  \[{{G}^{2}}=AH\] done clear
  View Solution play_arrow

• question_answer3)
  If \[a,\ b,\ c\] be in A.P. and \[b,\ c,\ d\] be in H.P., then
  A)
  \[ab=cd\] done clear
  B)
  \[ad=bc\] done clear
  C)
  \[ac=bd\] done clear
  D)
  \[abcd=1\] done clear
  View Solution play_arrow

• question_answer4)
  If \[a,\ b,\ c\] are in A.P., then\[\frac{a}{bc},\ \frac{1}{c},\ \frac{2}{b}\] are in [MNR 1982; MP PET 2002]
  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

• question_answer5)
  If the roots of\[a\,(b-c){{x}^{2}}+b\,(c-a)x+c\,(a-b)=0\] be equal, then \[a,\ b,\ c\]are in [RPET 1997]
  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

• question_answer6)
  If  \[{{\log }_{a}}x,\ {{\log }_{b}}x,\ {{\log }_{c}}x\] be in H.P., then \[a,\ b,\ c\] are in
  A)
  A.P. done clear
  B)
  H.P. done clear
  C)
  G.P. done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer7)
  If three numbers be in G.P., then their logarithms will be in [BIT 1992]
  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

• question_answer8)
  If \[{{p}^{th}},\ {{q}^{th}},\ {{r}^{th}}\] and \[{{s}^{th}}\] terms of an A.P. be in G.P., then \[(p-q),\ (q-r),\ (r-s)\] will be in [MP PET 1993]
  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

• question_answer9)
  If the arithmetic and geometric means of a and b be \[A\] and \[G\] respectively, then the value of \[A-G\] will be
  A)
  \[\frac{a-b}{a}\] done clear
  B)
  \[\frac{a+b}{2}\] done clear
  C)
  \[{{\left[ \frac{\sqrt{a}-\sqrt{b}}{\sqrt{2}} \right]}^{2}}\] done clear
  D)
  \[\frac{2ab}{a+b}\] done clear
  View Solution play_arrow

• question_answer10)
  If \[\frac{1}{b-c},\ \frac{1}{c-a},\ \frac{1}{a-b}\]be consecutive terms of an A.P., then \[{{(b-c)}^{2}},\ {{(c-a)}^{2}},\ {{(a-b)}^{2}}\] will be 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

• question_answer11)
  If \[{{a}^{1/x}}={{b}^{1/y}}={{c}^{1/z}}\]and \[a,\ b,\ c\] are in G.P., then \[x,\ y,\ z\]  will be in       [IIT 1969; UPSEAT 2001]
  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

• question_answer12)
  If the arithmetic mean of two numbers be \[A\] and geometric mean be\[G\], then the numbers will be
  A)
  \[A\pm ({{A}^{2}}-{{G}^{2}})\] done clear
  B)
  \[\sqrt{A}\pm \sqrt{{{A}^{2}}-{{G}^{2}}}\] done clear
  C)
  \[A\pm \sqrt{(A+G)(A-G)}\] done clear
  D)
  \[\frac{A\pm \sqrt{(A+G)(A-G)}}{2}\] done clear
  View Solution play_arrow

• question_answer13)
  If \[\frac{1}{b-a}+\frac{1}{b-c}=\frac{1}{a}+\frac{1}{c}\], then \[a,\ b,\ c\]  are in [MNR 1984; MP PET 1997; UPSEAT 2000]
  A)
  A.P. done clear
  B)
  G.P. done clear
  C)
  H.P. done clear
  D)
  In G.P. and H.P. both done clear
  View Solution play_arrow

• question_answer14)
  If \[a\] and \[b\] are two different positive real numbers, then which of the following relations is true [MP PET 1982; MP PET 2002]
  A)
  \[2\sqrt{ab}>(a+b)\] done clear
  B)
  \[2\sqrt{ab}<(a+b)\] done clear
  C)
  \[2\sqrt{ab}=(a+b)\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer15)
  If \[{{b}^{2}},\,{{a}^{2}},\,{{c}^{2}}\] are in A.P., then \[a+b,\,b+c,\,c+a\] will be in [AMU 1974]
  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

• question_answer16)
  If \[a,\ b,\ c\] are in A.P. as well as in G.P., then [MNR 1981]
  A)
  \[a=b\ne c\] done clear
  B)
  \[a\ne b=c\] done clear
  C)
  \[a\ne b\ne c\] done clear
  D)
  \[a=b=c\] done clear
  View Solution play_arrow

• question_answer17)
  If \[a,\ b,\ c\] are in G.P. and \[x,\,y\] are the arithmetic means between \[a,\ b\] and \[b,\ c\] respectively, then \[\frac{a}{x}+\frac{c}{y}\]is equal to Roorkee 1969]
  A)
  0 done clear
  B)
  1 done clear
  C)
  2 done clear
  D)
  \[\frac{1}{2}\] done clear
  View Solution play_arrow

• question_answer18)
  If \[a,\ b,\ c\] are in A.P. and \[a,\ b,\ d\] in G.P., then \[a,\ a-b,\ d-c\] will be in  [Ranchi BIT 1968]
  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

• question_answer19)
  If \[{{a}^{2}},\ {{b}^{2}},\ {{c}^{2}}\] are in A.P., then \[{{(b+c)}^{-1}},\ {{(c+a)}^{-1}}\] and \[{{(a+b)}^{-1}}\] will be in    [Roorkee 1968; RPET 1996]
  A)
  H.P. done clear
  B)
  G.P. done clear
  C)
  A.P. done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer20)
  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. done clear
  B)
  G.P. done clear
  C)
  H.P. done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer21)
  If \[x,\ 1,\ z\] are in A.P. and \[x,\ 2,\ z\] are in G.P., then \[x,\ 4,\ z\] will be in [IIT 1965]
  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

• question_answer22)
  If \[a,\ b,\ c\] are in A.P. and \[|a|,\ |b|,\ |c|\ <1\] and \[x=1+a+{{a}^{2}}+........\infty \]\[y=1+b+{{b}^{2}}+.......\infty \]\[z=1+c+{{c}^{2}}........\infty \] Then\[x,\ y,\ z\] shall be in [Karnataka CET 1995; AIEEE 2005]
  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

• question_answer23)
  If three unequal non-zero real numbers \[a,\ b,\ c\]are in G.P. and \[b-c,\ c-a,\ a-b\]are in H.P., then the value of \[a+b+c\] is independent of
  A)
  \[a\] done clear
  B)
  \[b\] done clear
  C)
  \[c\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer24)
  If \[a,\ b,\ c\]are in A.P., \[b,\ c,\ d\] are in  G.P. and \[c,\ d,\ e\]are in H.P., then \[a,\ c,\ e\] are in  [AMU 1988, 2001; MP PET 1993]
  A)
  No particular order done clear
  B)
  A.P. done clear
  C)
  G.P. done clear
  D)
  H.P. done clear
  View Solution play_arrow

• question_answer25)
  If \[a,\ b,\ c\] are in G.P., \[a-b,\ c-a,\ b-c\]are in H.P., then \[a+4b+c\]is equal to
  A)
  0 done clear
  B)
  \[1\] done clear
  C)
  \[-1\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer26)
  Given \[{{a}^{x}}={{b}^{y}}={{c}^{z}}={{d}^{u}}\] and \[a,\ b,\ c,\ d\] are in G.P., then \[x,y,z,u\]  are in [ISM Dhanbad 1972; Roorkee 1984; RPET 2001]
  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

• question_answer27)
  If \[{{A}_{1}},\ {{A}_{2}}\] are the two A.M.'s between two numbers \[a\]and \[b\]and \[{{G}_{1}},\ {{G}_{2}}\] be two G.M.'s between same two numbers, then \[\frac{{{A}_{1}}+{{A}_{2}}}{{{G}_{1}}.{{G}_{2}}}=\]    [Roorkee 1983; DCE 1998]
  A)
  \[\frac{a+b}{ab}\] done clear
  B)
  \[\frac{a+b}{2ab}\] done clear
  C)
  \[\frac{2ab}{a+b}\] done clear
  D)
  \[\frac{ab}{a+b}\] done clear
  View Solution play_arrow

• question_answer28)
  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 done clear
  B)
  18 done clear
  C)
  24 done clear
  D)
  36 done clear
  View Solution play_arrow

• question_answer29)
  If \[a,\ b,\ c\] are in A.P., then \[{{3}^{a}},\ {{3}^{b}},\ {{3}^{c}}\] shall be in [Pb. CET 1990]
  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

• question_answer30)
  If the \[{{(m+1)}^{th}},\ {{(n+1)}^{th}}\] and \[{{(r+1)}^{th}}\] terms of an A.P. are in G.P. and \[m,\ n,\ r\] are in H.P., then the value of the ratio of the common difference to the first term of the A.P. is    [MNR 1989; Roorkee 1994]
  A)
  \[-\frac{2}{n}\] done clear
  B)
  \[\frac{2}{n}\] done clear
  C)
  \[-\frac{n}{2}\] done clear
  D)
  \[\frac{n}{2}\] done clear
  View Solution play_arrow

• question_answer31)
  If G.M. = 18 and A.M. = 27, then H.M. is [RPET 1996]
  A)
    \[\frac{1}{18}\] done clear
  B)
    \[\frac{1}{12}\] done clear
  C)
  12 done clear
  D)
  \[9\sqrt{6}\] done clear
  View Solution play_arrow

• question_answer32)
  If the A.M. is twice the G.M. of the numbers \[a\] and \[b\], then \[a:b\]will be [Roorkee 1953]
  A)
  \[\frac{2-\sqrt{3}}{2+\sqrt{3}}\] done clear
  B)
  \[\frac{2+\sqrt{3}}{2-\sqrt{3}}\] done clear
  C)
  \[\frac{\sqrt{3}-2}{\sqrt{3}+2}\] done clear
  D)
  \[\frac{\sqrt{3}+2}{\sqrt{3}-2}\] done clear
  View Solution play_arrow

• question_answer33)
  \[x+y+z=15\] if \[9,\ x,\ y,\ z,\ a\] are in A.P.; while \[\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{5}{3}\] if \[9,\ x,\ y,\ z,\ a\] are in H.P., then the value of \[a\] will be [IIT 1978]
  A)
  1 done clear
  B)
  2 done clear
  C)
  3 done clear
  D)
  9 done clear
  View Solution play_arrow

• question_answer34)
  If 9 A.M.'s and H.M.'s are inserted between the 2 and 3 and if the harmonic mean \[H\]is corresponding to arithmetic mean \[A\], then \[A+\frac{6}{H}=\] [ISM Dhanbad 1987]
  A)
  1 done clear
  B)
  3 done clear
  C)
  5 done clear
  D)
  6 done clear
  View Solution play_arrow

• question_answer35)
  If the \[{{p}^{th}},\ {{q}^{th}}\] and \[{{r}^{th}}\]term of a G.P. and H.P. are \[a,\ b,\ c\], then \[a(b-c)\log a+b(c-a)\]  \[\log b+c(a-b)\log c=\] [Dhanbad Engg. 1976]
  A)
  \[-1\] done clear
  B)
  0 done clear
  C)
  1 done clear
  D)
  Does not exist done clear
  View Solution play_arrow

• question_answer36)
  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)
  \[a=b=c\] done clear
  B)
  \[2b=3a+c\] done clear
  C)
  \[{{b}^{2}}=\sqrt{(ac/8)}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer37)
  In the four numbers first three are in G.P. and last three are in A.P. whose common difference is 6. If the first and last numbers are same, then first will be [IIT 1974]
  A)
  2 done clear
  B)
  4 done clear
  C)
  6 done clear
  D)
  8 done clear
  View Solution play_arrow

• question_answer38)
  The numbers \[(\sqrt{2}+1),\ 1,\ (\sqrt{2}-1)\] will be in  [AMU 1983]
  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

• question_answer39)
  If the ratio of H.M. and G.M. of two quantities is \[12:13\], then the ratio of the numbers is  [RPET 1990]
  A)
  \[1:2\] done clear
  B)
  \[2:3\] done clear
  C)
  \[3:4\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer40)
  If  \[\frac{b+a}{b-a}=\frac{b+c}{b-c}\], then\[a,\ b,\ c\] 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

• question_answer41)
  If the ratio of two numbers be \[9:1\], then the ratio of geometric and harmonic means between them will be
  A)
  \[1:9\] done clear
  B)
  \[5:3\] done clear
  C)
  \[3:5\] done clear
  D)
  \[2:5\] done clear
  View Solution play_arrow

• question_answer42)
  If \[a,\ b,\ c\] are in H.P., then \[\frac{a}{b+c},\ \frac{b}{c+a},\ \frac{c}{a+b}\]  are in  [Roorkee 1980]
  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

• question_answer43)
  If \[\frac{x+y}{2},\ y,\ \frac{y+z}{2}\] are in H.P., then \[x,\ y,\ z\]are in  [RPET 1989; MP PET 2003]
  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

• question_answer44)
  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)
  \[a\ge b\ge c\] done clear
  B)
  \[a+c=b\] done clear
  C)
  \[ac-{{b}^{2}}=0\] done clear
  D)
  (a) and (c) both done clear
  View Solution play_arrow

• question_answer45)
  An A.P., a G.P. and a H.P. have the same first and last terms and the same odd number of terms. The middle terms of the three series 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

• question_answer46)
  If \[,a,\ b,\,c\] be in G.P. and \[a+x,\ b+x,\ c+x\] in H.P., then the value of \[x\] is (\[a,\ b,\ c\] are distinct numbers)
  A)
  \[c\] done clear
  B)
  \[b\] done clear
  C)
  \[a\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer47)
  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. done clear
  B)
  G.P. done clear
  C)
  H.P. done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer48)
  If all the terms of an A.P. are squared, then new series will be 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

• question_answer49)
  If \[2(y-a)\] is the H.M. between \[y-x\] and \[y-z\], then \[x-a,\ y-a,\ z-a\] 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

• question_answer50)
  If the ratio of A.M. between two positive real numbers \[a\] and \[b\]to their H.M. is \[m:n\], then \[a:b\] is
  A)
    \[\frac{\sqrt{m-n}+\sqrt{n}}{\sqrt{m-n}-\sqrt{n}}\] done clear
  B)
  \[\frac{\sqrt{n}+\sqrt{m-n}}{\sqrt{n}-\sqrt{m-n}}\] done clear
  C)
    \[\frac{\sqrt{m}+\sqrt{m-n}}{\sqrt{m}-\sqrt{m-n}}\] done clear
  D)
    None of these done clear
  View Solution play_arrow

• question_answer51)
  If \[{{\log }_{x}}y,\ {{\log }_{z}}x,\ {{\log }_{y}}z\] are in G.P. \[xyz=64\] and \[{{x}^{3}},\ {{y}^{3}},\ {{z}^{3}}\] are in A.P., then
  A)
  \[x=y=z\] done clear
  B)
  \[x=4\] done clear
  C)
  \[x,\ y,\,z\] are in G.P. done clear
  D)
  All the above done clear
  View Solution play_arrow

• question_answer52)
  If three unequal numbers \[p,\ q,\ r\] are in H.P. and their squares are in A.P., then the ratio \[p:q:r\] is
  A)
  \[1-\sqrt{3}:2:1+\sqrt{3}\] done clear
  B)
  \[1:\sqrt{2}:-\sqrt{3}\] done clear
  C)
  \[1:-\sqrt{2}:\sqrt{3}\] done clear
  D)
  \[1\mp \sqrt{3}:-2:1\pm \sqrt{3}\] done clear
  View Solution play_arrow

• question_answer53)
  The geometric mean of two numbers is 6 and their arithmetic mean is 6.5 . The numbers are [MP PET 1994]
  A)
  (3, 12) done clear
  B)
  (4, 9) done clear
  C)
  (2, 18) done clear
  D)
  (7, 6) done clear
  View Solution play_arrow

• question_answer54)
  If \[\frac{a+bx}{a-bx}=\frac{b+cx}{b-cx}=\frac{c+dx}{c-dx}(x\ne 0)\], then \[a,\ b,\ c,\ d\] are in [RPET 1986]
  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

• question_answer55)
  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)
  \[1:3:5\] done clear
  B)
  \[1:2:4\] done clear
  C)
  \[1:2:3\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer56)
  If \[a,\ b,\ c\] are in H.P., then for all \[n\in N\] the true statement is [RPET 1995]
  A)
  \[{{a}^{n}}+{{c}^{n}}<2{{b}^{n}}\] done clear
  B)
  \[{{a}^{n}}+{{c}^{n}}>2{{b}^{n}}\] done clear
  C)
  \[{{a}^{n}}+{{c}^{n}}=2{{b}^{n}}\] done clear
  D)
  None of the above done clear
  View Solution play_arrow

• question_answer57)
  If A.M. of two terms is 9 and H.M. is 36, then G.M. will be [RPET 1995]
  A)
  18 done clear
  B)
  12 done clear
  C)
  16 done clear
  D)
  None of the above done clear
  View Solution play_arrow

• question_answer58)
  If \[{{x}^{a}}={{x}^{b/2}}{{z}^{b/2}}={{z}^{c}}\],  then \[a,b,c\] 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

• question_answer59)
  If the product of three terms of G.P. is 512. If 8 added to first and 6 added to second term, so that number may be in A.P., then the numbers are [Roorkee 1964]
  A)
  2, 4, 8 done clear
  B)
  4, 8, 16 done clear
  C)
  3, 6, 12 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer60)
  If the ratio of H.M. and G.M. between two numbers \[a\] and \[b\] is \[4:5\], then the ratio of the two numbers will be [IIT 1992; MP PET 2000]
  A)
  \[1:2\] done clear
  B)
  \[2:1\] done clear
  C)
  \[4:1\] done clear
  D)
  \[1:4\] done clear
  View Solution play_arrow

• question_answer61)
  If the A.M., G.M. and H.M. between two positive numbers \[a\] and \[b\] are equal, then [RPET 2003]
  A)
  \[a=b\] done clear
  B)
  \[ab=1\] done clear
  C)
  \[a>b\] done clear
  D)
  \[a<b\] done clear
  View Solution play_arrow

• question_answer62)
  If \[a,\ b,\ c\] are in A.P., then \[{{10}^{ax+10}},\ {{10}^{bx+10}},\ {{10}^{cx+10}}\] will be in [Pb. CET 1989]
  A)
  A.P. done clear
  B)
  G.P. only when \[x>0\] done clear
  C)
  G.P. for all values of \[x\] done clear
  D)
  G.P. for \[x<0\] done clear
  View Solution play_arrow

• question_answer63)
  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. done clear
  B)
  G.P. done clear
  C)
  H.P. done clear
  D)
  None of the above done clear
  View Solution play_arrow

• question_answer64)
  \[{{\log }_{3}}2,\ {{\log }_{6}}2,\ {{\log }_{12}}2\]are in   [RPET 1993, 2001]
  A)
  A.P. done clear
  B)
  G.P. done clear
  C)
  H.P. done clear
  D)
  None of the above done clear
  View Solution play_arrow

• question_answer65)
  Three non-zero real numbers form an A.P. and the squares of these numbers taken in the same order form a G.P. Then the number of all possible common ratios of the G.P. is
  A)
  1 done clear
  B)
  2 done clear
  C)
  3 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer66)
  If \[{{a}^{x}}={{b}^{y}}={{c}^{z}}\,\text{and}\,\,a,b,c\] are in G.P. then \[x,y,z\] are in [Pb. CET 1993; DCE 1999; AMU 1999]
  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

• question_answer67)
  If \[{{G}_{1}}\] and \[{{G}_{2}}\] are two geometric means and A the arithmetic mean inserted between two numbers, then the value of \[\frac{G_{1}^{2}}{{{G}_{2}}}+\frac{G_{2}^{2}}{{{G}_{1}}}\]is [DCE 1999]
  A)
  \[\frac{A}{2}\] done clear
  B)
  A done clear
  C)
  2 A done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer68)
  If \[\log (x+z)+\log (x+z-2y)=2\log (x-z),\,\]  then\[x,\,y,\,z\] are in   [RPET 1999]
  A)
  H.P. done clear
  B)
  G.P. done clear
  C)
  A.P. done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer69)
  If \[\frac{a}{b+c},\frac{b}{c+a},\frac{c}{a+b}\]are in H.P., then \[a,b,c\] are in  [RPET 1999]
  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

• question_answer70)
  If a ,b, c  are in A.P., then  \[\frac{1}{\sqrt{a}+\sqrt{b}},\,\frac{1}{\sqrt{a}+\sqrt{c}},\] \[\frac{1}{\sqrt{b}+\sqrt{c}}\] are in                         [Roorkee 1999; Kerala (Engg.) 2005]
  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

• question_answer71)
  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 done clear
  B)
  15, 9, 3 done clear
  C)
  13, 9, 5 done clear
  D)
  17, 9, 1 done clear
  View Solution play_arrow

• question_answer72)
  If \[p,\ q,\ r\] are in one geometric progression and  \[a,\ b,\ c\] in another geometric progression, then \[cp,\ bq,\ ar\] are in [Roorkee 1998]
  A)
  A.P. done clear
  B)
  H.P. done clear
  C)
  G.P. done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer73)
  If first three terms of sequence \[\frac{1}{16},a,b,\frac{1}{6}\] are in geometric series and last three terms are in harmonic series, then the value of \[a\] and \[b\] will be [UPSEAT 1999]
  A)
  \[a=-\frac{1}{4},b=1\] done clear
  B)
  \[a=\frac{1}{12},b=\frac{1}{9}\] done clear
  C)
  (a) and (b) both are true done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer74)
  Given \[a+d>b+c\] where \[a,\ b,\ c,\ d\] are real numbers, then [Kurukshetra CEE 1998]
  A)
  \[a,\ b,\ c,\ d\] are in A.P. done clear
  B)
  \[\frac{1}{a},\ \frac{1}{b},\ \frac{1}{c},\ \frac{1}{d}\] are in A.P. done clear
  C)
  \[(a+b),\ (b+c),\ (c+d),\ (a+d)\]are in A.P. done clear
  D)
    \[\frac{1}{a+b},\ \frac{1}{b+c},\ \frac{1}{c+d},\ \frac{1}{a+d}\] are in A.P. done clear
  View Solution play_arrow

• question_answer75)
  If \[{{A}_{1}},\ {{A}_{2}};{{G}_{1}},\ {{G}_{2}}\] and \[{{H}_{1}},\ {{H}_{2}}\] be two A.M.s, G.M.s and  H.M.s between two numbers respectively, then\[\frac{{{G}_{1}}{{G}_{2}}}{{{H}_{1}}{{H}_{2}}}\times \frac{{{H}_{1}}+{{H}_{2}}}{{{A}_{1}}+{{A}_{2}}}\] = [RPET 1997]
  A)
  1 done clear
  B)
  0 done clear
  C)
  2 done clear
  D)
  3 done clear
  View Solution play_arrow

• question_answer76)
  The common difference of an A.P. whose first term is unity and whose second, tenth and thirty fourth terms are in G.P., is [AMU 2000]
  A)
  \[\frac{1}{5}\] done clear
  B)
  \[\frac{1}{3}\] done clear
  C)
  \[\frac{1}{6}\] done clear
  D)
  \[\frac{1}{9}\] done clear
  View Solution play_arrow

• question_answer77)
  Let the positive numbers a, b, c, d be in A.P.,  then abc, abd acd, bcd are [IIT Screening 2001]
  A)
  Not in A.P./G.P./H.P. done clear
  B)
  In A.P. done clear
  C)
  In G.P. done clear
  D)
  In H.P. done clear
  View Solution play_arrow

• question_answer78)
  If in the equation \[a{{x}^{2}}+bx+c=0,\] the sum of roots is equal to sum of square of their reciprocals, then \[\frac{c}{a},\frac{a}{b},\frac{b}{c}\] are in [RPET 2000]
  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

• question_answer79)
  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. done clear
  B)
  G.P. only when \[x>\text{0}\] done clear
  C)
  G.P. if \[x<0\] done clear
  D)
  G.P. for all \[x\ne 0\] done clear
  View Solution play_arrow

• question_answer80)
  If \[b+c,\] \[c+a,\] \[a+b\] are in H.P., then \[\frac{a}{b+c},\frac{b}{c+a},\frac{c}{a+b}\] are in [RPET 2000]
  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

• question_answer81)
  If \[\frac{a}{b},\frac{b}{c},\frac{c}{a}\] are in H.P., then [UPSEAT 2002]
  A)
  \[{{a}^{2}}b,\,{{c}^{2}}a,\,{{b}^{2}}c\] are in A.P. done clear
  B)
  \[{{a}^{2}}b,\,{{b}^{2}}c,\,{{c}^{2}}a\]are in H.P. done clear
  C)
  \[{{a}^{2}}b,\,{{b}^{2}}c,\,{{c}^{2}}a\]are in G.P. done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer82)
  If A is the A.M. of the roots of the equation \[{{x}^{2}}-2ax+b=0\] and \[G\] is the G.M. of the roots of the equation \[{{x}^{2}}-2bx+{{a}^{2}}=0,\] then [UPSEAT 2001]
  A)
  \[A>G\] done clear
  B)
  \[A\ne G\] done clear
  C)
  \[A=G\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer83)
  If A and G are arithmetic and geometric means and \[{{x}^{2}}-2Ax+{{G}^{2}}=0\], then [UPSEAT 2001]
  A)
  \[A=G\] done clear
  B)
  \[A>G\] done clear
  C)
  \[A<G\] done clear
  D)
  \[A=-\,G\] done clear
  View Solution play_arrow

• question_answer84)
  If ln \[(a+c)\], In \[(c-a)\], In \[(a-2b+c)\] are in A.P., then [IIT Screening 1994]
  A)
  \[a,\ b,\ c\]are in A.P. done clear
  B)
  \[{{a}^{2}},\ {{b}^{2}},\ {{c}^{2}}\]are in A.P. done clear
  C)
  \[a,\ b,\ c\]are in G.P. done clear
  D)
  \[a,\ b,\ c\] are in H.P done clear
  View Solution play_arrow

• question_answer85)
  If the altitudes of a triangle are in A.P., then the sides of the  triangle are in [EAMCET 2002]
  A)
  A.P. done clear
  B)
  H.P. done clear
  C)
  G.P. done clear
  D)
  Arithmetico-geometric progression done clear
  View Solution play_arrow

• question_answer86)
  If \[a,b,c\]are in G.P. then \[{{\log }_{a}}x,{{\log }_{b}}x,{{\log }_{c}}x\] are in   [RPET 2002]
  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

• question_answer87)
  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 done clear
  B)
  2: 3 : 1 done clear
  C)
  1 : 3 : 2 done clear
  D)
  3 : 2 : 1 done clear
  View Solution play_arrow

• question_answer88)
  If \[(y-x),\,\,2(y-a)\] and \[(y-z)\] are in H.P., then \[x-a,\] \[y-a,\] \[z-a\] are in [RPET 2001]
  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

• question_answer89)
  If \[a,\,b,\,c\] are in A.P. and \[{{a}^{2}},\,{{b}^{2}},{{c}^{2}}\]are in H.P.,  then  [UPSEAT 2001]
  A)
  \[a\ne b\ne c\] done clear
  B)
  \[{{a}^{2}}={{b}^{2}}=\frac{{{c}^{2}}}{2}\] done clear
  C)
  \[a,\,b,\,c\] are in G.P. done clear
  D)
  \[\frac{-a}{2},b,c\]are in G.P done clear
  View Solution play_arrow

• question_answer90)
  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)
  \[n{{(2c)}^{1/n}}\] 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

• question_answer91)
  If arithmetic mean of two positive numbers is \[A\], their geometric mean is \[G\] and harmonic mean is \[H\], then \[H\]is equal to [MP PET 2004]
  A)
  \[1.2+2.3+3.4+4.5+.........\] done clear
  B)
  \[\frac{G}{{{A}^{2}}}\] done clear
  C)
  \[\frac{{{A}^{2}}}{G}\] done clear
  D)
  \[\frac{A}{{{G}^{2}}}\] done clear
  View Solution play_arrow

• question_answer92)
  The harmonic mean between two numbers is                                                                                                                and the geometric mean 24. The greater number them is   [UPSEAT 2004]
  A)
  72 done clear
  B)
  54 done clear
  C)
  36 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer93)
  When \[\frac{1}{a}+\frac{1}{c}+\frac{1}{a-b}+\frac{1}{c-d}=0\] and \[b\ne a\ne c\], then \[a,\ b,\ c\] are [MP PET 2004]
  A)
  In H.P. done clear
  B)
  In G.P. done clear
  C)
  In A.P. done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer94)
  If \[{{a}^{2}},\,{{b}^{2}},\,{{c}^{2}}\] be in A.P., then \[\frac{a}{b+c},\,\frac{b}{c+a},\,\frac{c}{a+b}\] will be 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

• question_answer95)
  If p,q,r are in G.P and \[{{\tan }^{-1}}p\], \[{{\tan }^{-1}}q,{{\tan }^{-1}}r\]are in A.P. then p, q, r are satisfies the relation [DCE 2005]
  A)
  \[p=q=r\] done clear
  B)
  \[p\ne q\ne r\] done clear
  C)
  \[p+q=r\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer96)
  If A.M and G.M of x and y are in the ratio p : q, then x : y is [Kerala (Engg.) 2005]
  A)
  \[p-\sqrt{{{p}^{2}}+{{q}^{2}}}\]:\[p+\sqrt{{{p}^{2}}+{{q}^{2}}}\] done clear
  B)
  \[p+\sqrt{{{p}^{2}}-{{q}^{2}}}\]:\[p-\sqrt{{{p}^{2}}-{{q}^{2}}}\] done clear
  C)
  \[p:q\] done clear
  D)
  \[p+\sqrt{{{p}^{2}}+{{q}^{2}}}\]:\[p-\sqrt{{{p}^{2}}+{{q}^{2}}}\] done clear
  E)
  \[q+\sqrt{{{p}^{2}}-{{q}^{2}}}\]:\[q-\sqrt{{{p}^{2}}-{{q}^{2}}}\] done clear
  View Solution play_arrow