# 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 $$\frac{{{{(a - c)}^2}}}{{({b^2} - ac)}} =$$

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

$$1,\;\frac{1}{2}$$

$$1,\;\frac{1}{3}$$

$$1,\;\frac{3}{2}$$

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 $$[a(q - r)$$ + $$b(r - p)$$ $$+ c(p - q)] =$$

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

$$\frac{{pq - 1}}{2}$$

$$\frac{{1 - pq}}{2}$$

$$\frac{{pq + 1}}{2}$$

$$- \frac{{pq + 1}}{2}$$

## Questions 5 of 50

Question:The sum of first $$n$$ natural numbers is

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

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

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

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

## 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

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

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

$$\frac{{m - 1}}{{n - 1}}$$

$$\frac{{n - 1}}{{m - 1}}$$

$$\frac{{2m - 1}}{{2n - 1}}$$

$$\frac{{2n - 1}}{{2m - 1}}$$

## Questions 9 of 50

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

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

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

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

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

## Questions 10 of 50

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

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

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

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

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 $$m =$$

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

$$P + Q$$

$$2P + 3Q$$

$$2Q$$

$$Q$$

## Questions 13 of 50

Question:The sum of $$n$$arithmetic means between $$a$$ and $$b$$, is

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

$$n(a + b)$$

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

$$(n + 1)(a + b)$$

## 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

10

8

9

None of these

## Questions 15 of 50

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

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

$$a + nd$$

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

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

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

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

$$x = y = z$$

$$x = y = - z$$

$$x = 1;y = 2;z = 3$$

$$x = 2;y = 4;z = 6$$

$$x = 2y = 3z$$

## Questions 19 of 50

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

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

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

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

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

1

1/2

2-Jan

2

## Questions 24 of 50

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

$${G^2}$$

$$\frac{1}{{{G^2}}}$$

$$\frac{2}{{{G^2}}}$$

$$3{G^2}$$

## 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

$$\frac{{xy}}{{x + y - 1}}$$

$$\frac{{xy}}{{x + y + 1}}$$

$$\frac{{xy}}{{x - y - 1}}$$

$$\frac{{xy}}{{x - y + 1}}$$

## 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]

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

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

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

$$5\sqrt 2$$

$$3\sqrt 2 + \sqrt 5$$

## 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

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

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

19-Jan

29-Jan

17-Jan

27-Jan

## Questions 31 of 50

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

$${a^2} + {c^2} > {b^2}$$

$${a^2} + {b^2} > 2{c^2}$$

$${a^2} + {c^2} > 2{b^2}$$

$${a^2} + {b^2} > {c^2}$$

## Questions 32 of 50

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

$$a + d > b + c$$

$$ad > bc$$

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

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

$$a = b \ne c$$

$$a \ne b = c$$

$$a \ne b \ne c$$

$$a = b = c$$

## Questions 35 of 50

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

$$\frac{1}{{18}}$$

$$\frac{1}{{12}}$$

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

$$\frac{{2 - \sqrt 3 }}{{2 + \sqrt 3 }}$$

$$\frac{{2 + \sqrt 3 }}{{2 - \sqrt 3 }}$$

$$\frac{{\sqrt 3 - 2}}{{\sqrt 3 + 2}}$$

$$\frac{{\sqrt 3 + 2}}{{\sqrt 3 - 2}}$$

## 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

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

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

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

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

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

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

$$p = q = r$$

$$p \ne q \ne r$$

$$p + q = r$$

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

$$p - \sqrt {{p^2} + {q^2}}$$:$$p + \sqrt {{p^2} + {q^2}}$$

$$p + \sqrt {{p^2} - {q^2}}$$:$$p - \sqrt {{p^2} - {q^2}}$$

$$p:q$$

$$p + \sqrt {{p^2} + {q^2}}$$:$$p - \sqrt {{p^2} + {q^2}}$$

$$q + \sqrt {{p^2} - {q^2}}$$:$$q - \sqrt {{p^2} - {q^2}}$$

## Questions 45 of 50

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

$$50(1 - i)$$

$$25i$$

$$25(1 + i)$$

$$100(1 - i)$$

## Questions 46 of 50

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

9998

9999

10000

100000

## Questions 47 of 50

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

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

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

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

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

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

89

97

101

123

## Questions 50 of 50

Question:$${11^2} + {12^2} + {13^2} + {.......20^2} =$$