JEE Main & Advanced Mathematics Sequence & Series Question Bank

done Critical Thinking Total Questions - 45

• question_answer1)
  If the angles of a quadrilateral are in A.P. whose common difference is\[{{10}^{o}}\], then the angles of the quadrilateral are
  A)
  \[{{65}^{o}},\,{{85}^{o}},\,{{95}^{o}},\,{{105}^{o}}\] done clear
  B)
  \[{{75}^{o}},\,{{85}^{o}},\,{{95}^{o}},\,{{105}^{o}}\] done clear
  C)
  \[{{65}^{o}},\,{{75}^{o}},\,{{85}^{o}},\,{{95}^{o}}\] done clear
  D)
  \[{{65}^{o}},\,{{95}^{o}},\,{{105}^{o}},\,{{115}^{o}}\] done clear
  View Solution play_arrow

• question_answer2)
  If the sum of first \[n\] terms of an A.P. be equal to the sum of its first \[m\] terms, \[(m\ne n)\], then the sum of its first \[(m+n)\] terms will be [MP PET 1984]
  A)
    0 done clear
  B)
  \[n\] done clear
  C)
  \[m\] done clear
  D)
  \[m+n\] done clear
  View Solution play_arrow

• question_answer3)
  If \[p,\ q,\ r\] are in A.P. and are positive, the roots of the quadratic equation \[p{{x}^{2}}+qx+r=0\] are all real for [IIT 1995]
  A)
  \[\left| \,\frac{r}{p}-7\  \right|\ \ge 4\sqrt{3}\] done clear
  B)
  \[\left| \ \frac{p}{r}-7\  \right|\ <4\sqrt{3}\] done clear
  C)
  All \[p\]and \[r\] done clear
  D)
  No \[p\] and \[r\] done clear
  View Solution play_arrow

• question_answer4)
  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)
  \[{{S}_{1}}+{{S}_{3}}={{S}_{2}}\] done clear
  B)
  \[{{S}_{1}}+{{S}_{3}}=2{{S}_{2}}\] done clear
  C)
  \[{{S}_{1}}+{{S}_{2}}=2{{S}_{3}}\] done clear
  D)
  \[{{S}_{1}}+{{S}_{2}}={{S}_{3}}\] done clear
  View Solution play_arrow

• question_answer5)
  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)
  \[x=a\] done clear
  B)
  \[x={{a}^{a}}\] done clear
  C)
  \[x={{a}^{-1/a}}\] done clear
  D)
  \[x={{a}^{1/a}}\] done clear
  View Solution play_arrow

• question_answer6)
  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 done clear
  B)
  Rs. 20475 done clear
  C)
  Rs. 20500 done clear
  D)
  Rs. 20700 done clear
  View Solution play_arrow

• question_answer7)
  Let \[{{S}_{1}},\ {{S}_{2}},.......\]be squares such that for each \[n\ge 1\], the length of a side of \[{{S}_{n}}\] equals the length of a diagonal of \[{{S}_{n+1}}\]. If the length of a side of \[{{S}_{1}}\]is\[10cm\], then for which of the following values of \[n\] is the area of \[{{S}_{n}}\] less then \[1\ sq\ cm\] [IIT 1999]
  A)
  7 done clear
  B)
  8 done clear
  C)
  9 done clear
  D)
  10 done clear
  View Solution play_arrow

• question_answer8)
  If \[{{S}_{1}},\ {{S}_{2}},\ {{S}_{3}},...........{{S}_{m}}\] are the sums of \[n\] terms of \[m\] A.P.'s whose first terms are \[1,\ 2,\ 3,\ ...............,m\] and common differences are \[1,\ 3,\ 5,\ ...........2m-1\] respectively, then \[{{S}_{1}}+{{S}_{2}}+{{S}_{3}}+.......{{S}_{m}}=\]
  A)
  \[\frac{1}{2}mn(mn+1)\] done clear
  B)
  \[mn(m+1)\] done clear
  C)
  \[\frac{1}{4}mn(mn-1)\] done clear
  D)
  None of the above done clear
  View Solution play_arrow

• question_answer9)
  If \[{{a}_{1}},\ {{a}_{2}},\,{{a}_{3}},......{{a}_{24}}\] are in arithmetic progression and \[{{a}_{1}}+{{a}_{5}}+{{a}_{10}}+{{a}_{15}}+{{a}_{20}}+{{a}_{24}}=225\], then \[{{a}_{1}}+{{a}_{2}}+{{a}_{3}}+........+{{a}_{23}}+{{a}_{24}}=\] [MP PET 1999; AMU 1997]
  A)
  909 done clear
  B)
  75 done clear
  C)
  750 done clear
  D)
  900 done clear
  View Solution play_arrow

• question_answer10)
  If the roots of the equation \[{{x}^{3}}-12{{x}^{2}}+39x-28=0\] are in A.P., then their common difference will be [UPSEAT 1994, 99, 2001; RPET 2001]
  A)
  \[\pm 1\] done clear
  B)
  \[\pm 2\] done clear
  C)
  \[\pm 3\] done clear
  D)
  \[\pm 4\] done clear
  View Solution play_arrow

• question_answer11)
  If the first term of a G.P. \[{{a}_{1}},\ {{a}_{2}},\ {{a}_{3}},..........\]is unity such that \[4{{a}_{2}}+5{{a}_{3}}\] is least, then the common ratio of G.P. is
  A)
  \[-\frac{2}{5}\] done clear
  B)
  \[-\frac{3}{5}\] done clear
  C)
  \[\frac{2}{5}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer12)
  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)
  \[\frac{R}{S}\] done clear
  B)
  \[\frac{S}{R}\] done clear
  C)
  \[{{\left( \frac{R}{S} \right)}^{n}}\] done clear
  D)
  \[{{\left( \frac{S}{R} \right)}^{n}}\] done clear
  View Solution play_arrow

• question_answer13)
  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 done clear
  B)
  63 done clear
  C)
  64 done clear
  D)
  127 done clear
  View Solution play_arrow

• question_answer14)
  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 done clear
  B)
  3 done clear
  C)
  4 done clear
  D)
  5 done clear
  View Solution play_arrow

• question_answer15)
  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 done clear
  B)
  5 done clear
  C)
  6 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer16)
  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)
  \[{{G}_{1}}.{{G}_{2}}........{{G}_{n}}=G\] done clear
  B)
  \[{{G}_{1}}.{{G}_{2}}........{{G}_{n}}={{G}^{1/n}}\] done clear
  C)
  \[{{G}_{1}}.{{G}_{2}}........{{G}_{n}}={{G}^{n}}\] done clear
  D)
  \[{{G}_{1}}.{{G}_{2}}........{{G}_{n}}={{G}^{2/n}}\] done clear
  View Solution play_arrow

• question_answer17)
  \[\alpha ,\ \beta \] 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) done clear
  B)
  (12, 3) done clear
  C)
  (2, 32) done clear
  D)
  (4, 16) done clear
  View Solution play_arrow

• question_answer18)
  \[2.\overset{\bullet \,\,\bullet \,\,\bullet }{\mathop{357}}\,=\] [IIT 1983; RPET 1995]
  A)
  \[\frac{2355}{1001}\] done clear
  B)
  \[\frac{2370}{997}\] done clear
  C)
  \[\frac{2355}{999}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer19)
  If \[1+\cos \alpha +{{\cos }^{2}}\alpha +.......\,\infty =2-\sqrt{2,}\] then \[\alpha ,\] \[(0<\alpha <\pi )\] is    [Roorkee 2000; AMU 2005]
  A)
  \[\pi /8\] done clear
  B)
  \[\pi /6\] done clear
  C)
  \[\pi /4\] done clear
  D)
  \[3\pi /4\] done clear
  View Solution play_arrow

• question_answer20)
  The first term of an infinite geometric progression is \[x\] and its sum is 5. Then [IIT Screening 2004]
  A)
  \[0\le x\le 10\] done clear
  B)
  \[0<x<10\] done clear
  C)
  \[-10<x<0\] done clear
  D)
  \[x>10\] done clear
  View Solution play_arrow

• question_answer21)
  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)
  \[\frac{2}{bc}+\frac{1}{{{b}^{2}}}\] 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

• question_answer22)
  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)
  \[b-a\] done clear
  B)
  \[ab(b-a)\] done clear
  C)
  \[a(b-a)\] done clear
  D)
  \[ab(a-b)\] done clear
  View Solution play_arrow

• question_answer23)
  A boy goes to school from his home at a speed of x km/hour and comes back at a speed of y km/hour, then the average speed is given by  [DCE 2002]
  A)
  A.M. done clear
  B)
  G.M. done clear
  C)
  H.M. done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer24)
  If  \[a,\ b,\ c,\ d\] be in H.P., then
  A)
  \[{{a}^{2}}+{{c}^{2}}>{{b}^{2}}+{{d}^{2}}\] done clear
  B)
  \[{{a}^{2}}+{{d}^{2}}>{{b}^{2}}+{{c}^{2}}\] done clear
  C)
  \[ac+bd>{{b}^{2}}+{{c}^{2}}\] done clear
  D)
  \[ac+bd>{{b}^{2}}+{{d}^{2}}\] done clear
  View Solution play_arrow

• question_answer25)
  If \[a,\ b,\ c\] are the positive integers, then \[(a+b)(b+c)(c+a)\]is [DCE 2000]
  A)
  \[<8abc\] done clear
  B)
  \[>8abc\] done clear
  C)
  \[=8abc\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer26)
  In a G.P. the sum of three numbers is 14, if 1 is added to first two numbers and subtracted from third number, the series becomes A.P., then the greatest number is  [Roorkee 1973]
  A)
  8 done clear
  B)
  4 done clear
  C)
  24 done clear
  D)
  16 done clear
  View Solution play_arrow

• question_answer27)
  If \[a,\ b,\ c\] are in G.P. and \[\log a-\log 2b,\ \log 2b-\log 3c\]and \[\log 3c-\log a\] are in A.P., then \[a,\ b,\ c\] are the length of the sides of a triangle which is
  A)
  Acute angled done clear
  B)
  Obtuse angled done clear
  C)
  Right angled done clear
  D)
  Equilateral done clear
  View Solution play_arrow

• question_answer28)
  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)
  \[\frac{{{A}_{1}}+{{A}_{2}}}{{{H}_{1}}+{{H}_{2}}}\] done clear
  B)
  \[\frac{{{A}_{1}}-{{A}_{2}}}{{{H}_{1}}+{{H}_{2}}}\] done clear
  C)
  \[\frac{{{A}_{1}}+{{A}_{2}}}{{{H}_{1}}-{{H}_{2}}}\] done clear
  D)
  \[\frac{{{A}_{1}}-{{A}_{2}}}{{{H}_{1}}-{{H}_{2}}}\] done clear
  View Solution play_arrow

• question_answer29)
  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)
  \[6,\,3\] done clear
  B)
  5, 4 done clear
  C)
  \[5,\ -2.5\] done clear
  D)
  \[-3,\ 1\] done clear
  View Solution play_arrow

• question_answer30)
  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 done clear
  B)
  12, 3 done clear
  C)
  16, 4 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer31)
  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)
  \[{{x}^{2}}-16x-25=0\] done clear
  B)
  \[{{x}^{2}}-8x+5=0\] done clear
  C)
  \[{{x}^{2}}-16x+25=0\] done clear
  D)
  \[{{x}^{2}}+16x-25=0\] done clear
  View Solution play_arrow

• question_answer32)
  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)
  \[15,\ 12,\ \frac{144}{15}\] done clear
  B)
  \[\frac{144}{15},\ 12,\ 15\] done clear
  C)
  \[12,\ 15,\ \frac{144}{15}\] done clear
  D)
  \[\frac{144}{15},\ 15,\ 12\] done clear
  View Solution play_arrow

• question_answer33)
  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)
  \[{{G}_{1}}{{G}_{2}}a\] done clear
  B)
  \[2{{G}_{1}}{{G}_{2}}a\] done clear
  C)
  \[3{{G}_{1}}{{G}_{2}}a\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer34)
  Three numbers form a G.P. If the \[{{3}^{rd}}\] term is decreased by 64, then the three numbers thus obtained will constitute an A.P. If the second term of this A.P. is decreased by 8, a G.P. will be formed again, then the numbers will be
  A)
  4, 20, 36 done clear
  B)
  4, 12, 36 done clear
  C)
  4, 20, 100 done clear
  D)
  None of the above done clear
  View Solution play_arrow

• question_answer35)
  If \[x>1,\ y>1,z>1\] are in G.P., then \[\frac{1}{1+\text{In}\,x},\ \frac{1}{1+\text{In}\,y},\] \[\ \frac{1}{1+\text{In}\,z}\]  are in       [IIT 1998; UPSEAT 2001]
  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_answer36)
  \[a,\,g,\,h\] 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 done clear
  B)
  No such relation exists between a, g and h done clear
  C)
  g  is the geometric mean between a and h done clear
  D)
  A  is the arithmetic mean between g  and h done clear
  View Solution play_arrow

• question_answer37)
  \[{{2}^{\sin \theta }}+{{2}^{\cos \theta }}\] is greater than [AMU 2000]
  A)
  \[\frac{1}{2}\] done clear
  B)
  \[\sqrt{2}\] done clear
  C)
  \[{{2}^{\frac{1}{\sqrt{2}}}}\] done clear
  D)
  \[{{2}^{\left( 1-\,\frac{1}{\sqrt{2}} \right)}}\] done clear
  View Solution play_arrow

• question_answer38)
  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)
  \[0<M\le 1\] done clear
  B)
  \[1\le M\le 2\] done clear
  C)
  \[2\le M\le 3\] done clear
  D)
  \[3\le M\le 4\] done clear
  View Solution play_arrow

• question_answer39)
   Suppose \[a,\,b,\,c\] are in A.P. and \[{{a}^{2}},{{b}^{2}},{{c}^{2}}\] are in G.P. If       a < b < c  and \[a+b+c=\frac{3}{2}\], then the value of a is [IIT Screening 2002]
  A)
  \[\frac{1}{2\sqrt{2}}\] done clear
  B)
  \[\frac{1}{2\sqrt{3}}\] done clear
  C)
  \[\frac{1}{2}-\frac{1}{\sqrt{3}}\] done clear
  D)
  \[\frac{1}{2}-\frac{1}{\sqrt{2}}\] done clear
  View Solution play_arrow

• question_answer40)
  \[{{n}^{th}}\] term of the series\[1+\frac{4}{5}+\frac{7}{{{5}^{2}}}+\frac{10}{{{5}^{3}}}+........\]will be
  A)
  \[\frac{3n+1}{{{5}^{n-1}}}\] done clear
  B)
  \[\frac{3n-1}{{{5}^{n}}}\] done clear
  C)
  \[\frac{3n-2}{{{5}^{n-1}}}\] done clear
  D)
  \[\frac{3n+2}{{{5}^{n-1}}}\] done clear
  View Solution play_arrow

• question_answer41)
  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)
  \[\frac{n({{n}^{2}}+1)}{{{n}^{2}}+n+1}\] done clear
  B)
  \[\frac{n(n+1)}{2({{n}^{2}}+n+1)}\] done clear
  C)
  \[\frac{n({{n}^{2}}-1)}{2({{n}^{2}}+n+1)}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer42)
  For any odd integer \[n\ge 1\],\[{{n}^{3}}-{{(n-1)}^{3}}+...........+{{(-1)}^{n-1}}{{1}^{3}}=\] [IIT 1996]
  A)
  \[\frac{1}{2}{{(n-1)}^{2}}(2n-1)\] done clear
  B)
  \[\frac{1}{4}{{(n-1)}^{2}}(2n-1)\] done clear
  C)
  \[\frac{1}{2}{{(n+1)}^{2}}(2n-1)\] done clear
  D)
  \[\frac{1}{4}{{(n+1)}^{2}}(2n-1)\] done clear
  View Solution play_arrow

• question_answer43)
  The sum of n terms of the series \[\frac{1}{1+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{5}}+\frac{1}{\sqrt{5}+\sqrt{7}}+.........\] is   [UPSEAT 2002]
  A)
  \[\sqrt{2n+1}\] done clear
  B)
  \[\frac{1}{2}\sqrt{2n+1}\] done clear
  C)
  \[\sqrt{2n+1}-1\] done clear
  D)
  \[\frac{1}{2}(\sqrt{2n+1}-1)\] done clear
  View Solution play_arrow

• question_answer44)
  \[{{n}^{th}}\] term of the series\[\frac{{{1}^{3}}}{1}+\frac{{{1}^{3}}+{{2}^{3}}}{1+3}+\frac{{{1}^{3}}+{{2}^{3}}+{{3}^{3}}}{1+3+5}+......\] will be [Pb. CET 2000]
  A)
  \[{{n}^{2}}+2n+1\] done clear
  B)
  \[\frac{{{n}^{2}}+2n+1}{8}\] done clear
  C)
  \[\frac{{{n}^{2}}+2n+1}{4}\] done clear
  D)
  \[\frac{{{n}^{2}}-2n+1}{4}\] done clear
  View Solution play_arrow

• question_answer45)
  The sum of the series \[\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+...+\frac{1}{\sqrt{{{n}^{2}}-1}+\sqrt{{{n}^{2}}}}\]equals                                          [AMU 2002]
  A)
  \[\frac{(2n+1)}{\sqrt{n}}\] done clear
  B)
  \[\frac{\sqrt{n}+1}{\sqrt{n}+\sqrt{n-1}}\] done clear
  C)
  \[\frac{(n+\sqrt{{{n}^{2}}-1})}{2\sqrt{n}}\] done clear
  D)
  \[n-1\] done clear
  View Solution play_arrow