Progressions Test 5

Total Questions:50 Total Time: 75 Min

Remaining:

 

Questions 1 of 50

Question:If \(a,\;b,\;c\) are in A.P., then \(\frac{{{{(a - c)}^2}}}{{({b^2} - ac)}} = \)

Answers Choices:

1

2

3

4

Questions 2 of 50

Question:If \({\log _3}2,\;{\log _3}({2^x} - 5)\) and \({\log _3}\left( {{2^x} - \frac{7}{2}} \right)\) are in A.P., then \(x\) is equal to

Answers Choices:

\(1,\;\frac{1}{2}\)

\(1,\;\frac{1}{3}\)

\(1,\;\frac{3}{2}\)

None of these

Questions 3 of 50

Question:If the \({p^{th}},\;{q^{th}}\) and \({r^{th}}\) term of an arithmetic sequence are a , b and \(c\) respectively, then the value of \([a(q - r)\) + \(b(r - p)\) \( + c(p - q)] = \)

Answers Choices:

1

\( - 1\)

0

2-Jan

Questions 4 of 50

Question:If the \({p^{th}}\) term of an A.P. be \(\frac{1}{q}\) and \({q^{th}}\) term be\(\frac{1}{p}\), then the sum of its \(p{q^{th}}\)terms will be

Answers Choices:

\(\frac{{pq - 1}}{2}\)

\(\frac{{1 - pq}}{2}\)

\(\frac{{pq + 1}}{2}\)

\( - \frac{{pq + 1}}{2}\)

Questions 5 of 50

Question:The sum of first \(n\) natural numbers is

Answers Choices:

\(n\,(n - 1)\)

\(\frac{{n\,(n - 1)}}{2}\)

\(n\,(n + 1)\)

\(\frac{{n\,(n + 1)}}{2}\)

Questions 6 of 50

Question:The first term of an A.P. is 2 and common difference is 4. The sum of its 40 terms will be

Answers Choices:

3200

1600

200

2800

Questions 7 of 50

Question:The sum of the numbers between 100 and 1000 which is divisible by 9 will be

Answers Choices:

55350

57228

97015

62140

Questions 8 of 50

Question:The ratio of sum of \(m\) and \(n\) terms of an A.P. is \({m^2}:{n^2}\), then the ratio of \({m^{th}}\)and \({n^{th}}\) term will be

Answers Choices:

\(\frac{{m - 1}}{{n - 1}}\)

\(\frac{{n - 1}}{{m - 1}}\)

\(\frac{{2m - 1}}{{2n - 1}}\)

\(\frac{{2n - 1}}{{2m - 1}}\)

Questions 9 of 50

Question:The value of \(\sum\limits_{r = 1}^n {\log \left( {\frac{{{a^r}}}{{{b^{r - 1}}}}} \right)} \) is

Answers Choices:

\(\frac{n}{2}\log \left( {\frac{{{a^n}}}{{{b^n}}}} \right)\)

\(\frac{n}{2}\log \left( {\frac{{{a^{n + 1}}}}{{{b^n}}}} \right)\)

\(\frac{n}{2}\log \left( {\frac{{{a^{n + 1}}}}{{{b^{n - 1}}}}} \right)\)

\(\frac{n}{2}\log \left( {\frac{{{a^{n + 1}}}}{{{b^{n + 1}}}}} \right)\)

Questions 10 of 50

Question:Let the sequence \({a_1},{a_2},{a_3},.............{a_{2n}}\) form an A.P. Then \(a_1^2 - a_2^2 + a_3^3 - ......... + a_{2n - 1}^2 - a_{2n}^2 = \)

Answers Choices:

\(\frac{n}{{2n - 1}}(a_1^2 - a_{2n}^2)\)

\(\frac{{2n}}{{n - 1}}(a_{2n}^2 - a_1^2)\)

\(\frac{n}{{n + 1}}(a_1^2 + a_{2n}^2)\)

None of these

Questions 11 of 50

Question:If sum of \(n\) terms of an A.P. is \(3{n^2} + 5n\) and \({T_m} = 164\) then \(m = \)

Answers Choices:

26

27

28

None of these

Questions 12 of 50

Question:If \({S_n} = nP + \frac{1}{2}n(n - 1)Q\), where \({S_n}\) denotes the sum of the first \(n\) terms of an A.P., then the common difference is

Answers Choices:

\(P + Q\)

\(2P + 3Q\)

\(2Q\)

\(Q\)

Questions 13 of 50

Question:The sum of \(n\)arithmetic means between \(a\) and \(b\), is

Answers Choices:

\(\frac{{n(a + b)}}{2}\)

\(n(a + b)\)

\(\frac{{(n + 1)(a + b)}}{2}\)

\((n + 1)(a + b)\)

Questions 14 of 50

Question:After inserting \(n\) A.M.'s between 2 and 38, the sum of the resulting progression is 200. The value of \(n\) is

Answers Choices:

10

8

9

None of these

Questions 15 of 50

Question:The mean of the series \(a,a + nd,\,\,a + 2nd\) is

Answers Choices:

\(a + (n - 1)\,d\)

\(a + nd\)

\(a + (n + 1)\,d\)

None of these

Questions 16 of 50

Question:Four numbers are in arithmetic progression. The sum of first and last term is 8 and the product of both middle terms is 15. The least number of the series is

Answers Choices:

4

3

2

1

Questions 17 of 50

Question:If twice the 11th term of an A.P. is equal to 7 times of its 21st term, then its 25th term is equal to

Answers Choices:

24

120

0

None of these

Questions 18 of 50

Question:If \(x,y,z\) are in A.P. and \({\tan ^{ - 1}}x,{\tan ^{ - 1}}y\)and \({\tan ^{ - 1}}z\) are also in A.P., then

Answers Choices:

\(x = y = z\)

\(x = y = - z\)

\(x = 1;y = 2;z = 3\)

\(x = 2;y = 4;z = 6\)

\(x = 2y = 3z\)

Questions 19 of 50

Question:The \({20^{th}}\) term of the series \(2 \times 4 + 4 \times 6 + 6 \times 8 + .......\)will be

Answers Choices:

1600

1680

420

840

Questions 20 of 50

Question:If \(a,\;b,\;c\) are \({p^{th}},\;{q^{th}}\) and \({r^{th}}\)terms of a G.P., then \({\left( {\frac{c}{b}} \right)^p}{\left( {\frac{b}{a}} \right)^r}{\left( {\frac{a}{c}} \right)^q}\) is equal to

Answers Choices:

1

\({a^P}{b^q}{c^r}\)

\({a^q}{b^r}{c^p}\)

\({a^r}{b^p}{c^q}\)

Questions 21 of 50

Question:If every term of a G.P. with positive terms is the sum of its two previous terms, then the common ratio of the series is

Answers Choices:

1

\(\frac{2}{{\sqrt 5 }}\)

\(\frac{{\sqrt 5 - 1}}{2}\)

\(\frac{{\sqrt 5 + 1}}{2}\)

Questions 22 of 50

Question:The sum of first two terms of a G.P. is 1 and every term of this series is twice of its previous term, then the first term will be

Answers Choices:

4-Jan

3-Jan

3-Feb

4-Mar

Questions 23 of 50

Question:If the geometric mean between \(a\) and \(b\) is \(\frac{{{a^{n + 1}} + {b^{n + 1}}}}{{{a^n} + {b^n}}}\), then the value of n is

Answers Choices:

1

1/2

2-Jan

2

Questions 24 of 50

Question:If \(G\) be the geometric mean of \(x\) and \(y\), then \(\frac{1}{{{G^2} - {x^2}}} + \frac{1}{{{G^2} - {y^2}}} = \)

Answers Choices:

\({G^2}\)

\(\frac{1}{{{G^2}}}\)

\(\frac{2}{{{G^2}}}\)

\(3{G^2}\)

Questions 25 of 50

Question:\(x = 1 + a + {a^2} + ....\infty \,(a < 1)\), \(y = 1 + b + {b^2}.......\infty \,(b < 1)\). Then the value of \(1 + ab + {a^2}{b^2} + ..........\infty \) is

Answers Choices:

\(\frac{{xy}}{{x + y - 1}}\)

\(\frac{{xy}}{{x + y + 1}}\)

\(\frac{{xy}}{{x - y - 1}}\)

\(\frac{{xy}}{{x - y + 1}}\)

Questions 26 of 50

Question:The first term of a G.P. whose second term is 2 and sum to infinity is 8, will be [MNR 1979; RPET 1992, 95]

Answers Choices:

6

3

4

1

Questions 27 of 50

Question:The sum of infinite terms of the geometric progression \(\frac{{\sqrt 2 + 1}}{{\sqrt 2 - 1}},\frac{1}{{2 - \sqrt 2 }},\frac{1}{2}.....\) is

Answers Choices:

\(\sqrt 2 {(\sqrt 2 + 1)^2}\)

\({(\sqrt 2 + 1)^2}\)

\(5\sqrt 2 \)

\(3\sqrt 2 + \sqrt 5 \)

Questions 28 of 50

Question:Sum of infinite number of terms in G.P. is 20 and sum of their square is 100. The common ratio of G.P. is

Answers Choices:

5

5-Mar

5-Aug

5-Jan

Questions 29 of 50

Question:If \({5^{th}}\) term of a H.P. is \(\frac{1}{{45}}\)and \({11^{th}}\) term is \(\frac{1}{{69}}\), then its \({16^{th}}\) term will be

Answers Choices:

Jan-89

Jan-85

Jan-80

Jan-79

Questions 30 of 50

Question:The first term of a harmonic progression is 1/7 and the second term is 1/9. The \({12^{th}}\) term is

Answers Choices:

19-Jan

29-Jan

17-Jan

27-Jan

Questions 31 of 50

Question:If \(a,\;b,\;c\) be in H.P., then

Answers Choices:

\({a^2} + {c^2} > {b^2}\)

\({a^2} + {b^2} > 2{c^2}\)

\({a^2} + {c^2} > 2{b^2}\)

\({a^2} + {b^2} > {c^2}\)

Questions 32 of 50

Question:If \(a,\;b,\;c,\;d\) are in H.P., then

Answers Choices:

\(a + d > b + c\)

\(ad > bc\)

Both (1) and (2)

None of these

Questions 33 of 50

Question:If \({b^2},\,{a^2},\,{c^2}\) are in A.P., then \(a + b,\,b + c,\,c + a\) will be in

Answers Choices:

A.P.

G.P.

H.P.

None of these

Questions 34 of 50

Question:If \(a,\;b,\;c\) are in A.P. as well as in G.P., then

Answers Choices:

\(a = b \ne c\)

\(a \ne b = c\)

\(a \ne b \ne c\)

\(a = b = c\)

Questions 35 of 50

Question:If G.M. = 18 and A.M. = 27, then H.M. is

Answers Choices:

\(\frac{1}{{18}}\)

\(\frac{1}{{12}}\)

12

\(9\sqrt 6 \)

Questions 36 of 50

Question:If the A.M. is twice the G.M. of the numbers \(a\) and \(b\), then \(a:b\)will be

Answers Choices:

\(\frac{{2 - \sqrt 3 }}{{2 + \sqrt 3 }}\)

\(\frac{{2 + \sqrt 3 }}{{2 - \sqrt 3 }}\)

\(\frac{{\sqrt 3 - 2}}{{\sqrt 3 + 2}}\)

\(\frac{{\sqrt 3 + 2}}{{\sqrt 3 - 2}}\)

Questions 37 of 50

Question:If \(\frac{{a + b}}{{1 - ab}},\;b,\;\frac{{b + c}}{{1 - bc}}\) are in A.P., then \(a,\;\frac{1}{b},\;c\) are in

Answers Choices:

A.P.

G.P.

H.P.

None of these

Questions 38 of 50

Question:If all the terms of an A.P. are squared, then new series will be in

Answers Choices:

A.P.

G.P.

H.P.

None of these

Questions 39 of 50

Question: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

Answers Choices:

A.P.

G.P.

H.P.

None of the above

Questions 40 of 50

Question:\({\log _3}2,\;{\log _6}2,\;{\log _{12}}2\)are in

Answers Choices:

A.P.

G.P.

H.P.

None of the above

Questions 41 of 50

Question:If a,b,c are in A.P., then \({2^{ax + 1}},{2^{bx + 1}},\,{2^{cx + 1}},x \ne 0\) are in

Answers Choices:

A.P.

G.P. only when \(x > {\rm{0}}\)

G.P. if \(x < 0\)

G.P. for all \(x \ne 0\)

Questions 42 of 50

Question: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

Answers Choices:

A.P.

G.P.

H.P.

None of these

Questions 43 of 50

Question: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

Answers Choices:

\(p = q = r\)

\(p \ne q \ne r\)

\(p + q = r\)

None of these

Questions 44 of 50

Question:If A.M and G.M of x and y are in the ratio p : q, then x : y is

Answers Choices:

\(p - \sqrt {{p^2} + {q^2}} \):\(p + \sqrt {{p^2} + {q^2}} \)

\(p + \sqrt {{p^2} - {q^2}} \):\(p - \sqrt {{p^2} - {q^2}} \)

\(p:q\)

\(p + \sqrt {{p^2} + {q^2}} \):\(p - \sqrt {{p^2} + {q^2}} \)

\(q + \sqrt {{p^2} - {q^2}} \):\(q - \sqrt {{p^2} - {q^2}} \)

Questions 45 of 50

Question:The sum of \(i - 2 - 3i + 4 + .........\)upto 100 terms, where \(i = \sqrt { - 1} \) is

Answers Choices:

\(50(1 - i)\)

\(25i\)

\(25(1 + i)\)

\(100(1 - i)\)

Questions 46 of 50

Question:\({99^{th}}\) term of the series \(2 + 7 + 14 + 23 + 34 + .....\) is

Answers Choices:

9998

9999

10000

100000

Questions 47 of 50

Question:Sum of the squares of first \(n\) natural numbers exceeds their sum by 330, then \(n = \)

Answers Choices:

8

10

15

20

Questions 48 of 50

Question:Sum of first \(n\) terms in the following series \({\cot ^{ - 1}}3 + {\cot ^{ - 1}}7 + {\cot ^{ - 1}}13 + {\cot ^{ - 1}}21 + .............\) is given by

Answers Choices:

\({\tan ^{ - 1}}\left( {\frac{n}{{n + 2}}} \right)\)

\({\cot ^{ - 1}}\left( {\frac{{n + 2}}{n}} \right)\)

\({\tan ^{ - 1}}(n + 1) - {\tan ^{ - 1}}1\)

All of these

Questions 49 of 50

Question:First term of the \({11^{th}}\) group in the following groups (1),(2, 3, 4), (5, 6, 7, 8, 9), .................is

Answers Choices:

89

97

101

123

Questions 50 of 50

Question:\({11^2} + {12^2} + {13^2} + {.......20^2} = \)

Answers Choices:

2481

2483

2485

2487