# Progressions Test 6

Total Questions:50 Total Time: 75 Min

Remaining:

## Questions 1 of 50

Question:Which term of the sequence (-8, +18i), (-6 + 15i); (-4+12i)............... is purely imaginary.

5th

7th

8th

6th

## Questions 2 of 50

Question:If $${n^{th}}$$ terms of two A.P.'s are $$3n + 8$$ and $$7n + 15$$, then the ratio of their $${12^{th}}$$ terms will be

9-Apr

16-Jul

7-Mar

15-Aug

## Questions 3 of 50

Question:If $${a_1} = {a_2} = 2,\;{a_n} = {a_{n - 1}} - 1\;(n > 2)$$, then $${a_5}$$is

1

$$- 1$$

0

$$- 2$$

## Questions 4 of 50

Question:If $$n$$ be odd or even, then the sum of $$n$$ terms of the series $$1 - 2 +$$ $$3 -$$$$4 + 5 - 6 + ......$$ will be

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

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

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

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

1 and 3 are correct

## Questions 5 of 50

Question:If the first, second and last terms of an A.P. be $$a,\;b,\;2a$$ respectively, then its sum will be

$$\frac{{ab}}{{b - a}}$$

$$\frac{{ab}}{{2(b - a)}}$$

$$\frac{{3ab}}{{2(b - a)}}$$

$$\frac{{3ab}}{{4(b - a)}}$$

## Questions 6 of 50

Question:The solution of the equation$$(x + 1) + (x + 4) + (x + 7) + ......... + (x + 28) = 155$$ is

1

2

3

4

## Questions 7 of 50

Question:The sum of all two digit numbers which, when divided by 4, yield unity as a remainder is

1190

1197

1210

None of these

## Questions 8 of 50

Question:Let $${S_n}$$denotes the sum of $$n$$ terms of an A.P. If $${S_{2n}} = 3{S_n}$$, then ratio $$\frac{{{S_{3n}}}}{{{S_n}}} =$$

4

6

8

10

## Questions 9 of 50

Question:The first term of an A.P. of consecutive integers is $${p^2} + 1$$ The sum of $$(2p + 1)$$ terms of this series can be expressed as

$${(p + 1)^2}$$

$${(p + 1)^3}$$

$$(2p + 1){(p + 1)^2}$$

$${p^3} + {(p + 1)^3}$$

## Questions 10 of 50

Question:If $$f(x + y,x - y) = xy\,,$$ then the arithmetic mean of $$f(x,y)$$ and $$f(y,x)$$ is

$$x$$

$$y$$

0

1

## Questions 11 of 50

Question:If $$\log 2,\;\log ({2^n} - 1)$$ and $$\log ({2^n} + 3)$$ are in A.P., then n =

2-May

$${\log _2}5$$

$${\log _3}5$$

2-Mar

## Questions 12 of 50

Question:If $$a,\,b,\,c$$ are in G.P., then

$$a({b^2} + {a^2}) = c({b^2} + {c^2})$$

$$a({b^2} + {c^2}) = c({a^2} + {b^2})$$

$${a^2}(b + c) = {c^2}(a + b)$$

None of these

## Questions 13 of 50

Question:$${7^{th}}$$ term of the sequence $$\sqrt 2 ,\;\sqrt {10} ,\;5\sqrt 2 ,\;.......$$is

$$125\sqrt {10}$$

$$25\sqrt 2$$

125

$$125\sqrt 2$$

## Questions 14 of 50

Question:The first and last terms of a G.P. are $$a$$ and $$l$$ respectively; $$r$$ being its common ratio; then the number of terms in this G.P. is

$$\frac{{\log l - \log a}}{{\log r}}$$

$$1 - \frac{{\log l - \log a}}{{\log r}}$$

$$\frac{{\log a - \log l}}{{\log r}}$$

$$1 + \frac{{\log l - \log a}}{{\log r}}$$

## Questions 15 of 50

Question:If $${\log _x}a,\;{a^{x/2}}$$ and $${\log _b}x$$ are in G.P., then $$x =$$

$$- \log ({\log _b}a)$$

$$- {\log _a}({\log _a}b)$$

$${\log _a}({\log _e}a) - {\log _a}({\log _e}b)$$

$${\log _a}({\log _e}b) - {\log _a}({\log _e}a)$$

## Questions 16 of 50

Question:If the roots of the cubic equation $$a{x^3} + b{x^2} + cx + d = 0$$ are in G.P., then

$${c^3}a = {b^3}d$$

$$c{a^3} = b{d^3}$$

$${a^3}b = {c^3}d$$

$$a{b^3} = c{d^3}$$

## Questions 17 of 50

Question:If the sum of $$n$$ terms of a G.P. is 255 and $${n^{th}}$$terms is 128 and common ratio is 2, then first term will be

1

3

7

None of these

## Questions 18 of 50

Question:The sum of $$n$$ terms of the following series $$1 + (1 + x) + (1 + x + {x^2}) + ..........$$will be

$$\frac{{1 - {x^n}}}{{1 - x}}$$

$$\frac{{x(1 - {x^n})}}{{1 - x}}$$

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

None of these

## Questions 19 of 50

Question:If the sum of first 6 term is 9 times to the sum of first 3 terms of the same G.P., then the common ratio of the series will be

$$- 2$$

2

1

2-Jan

## Questions 20 of 50

Question:If three geometric means be inserted between 2 and 32, then the third geometric mean will be

8

4

16

12

## Questions 21 of 50

Question:If five G.M.'s are inserted between 486 and 2/3 then fourth G.M. will be

4

6

12

6

## Questions 22 of 50

Question:The G.M. of roots of the equation $${x^2} - 18x + 9 = 0$$ is

3

4

2

1

## Questions 23 of 50

Question:$$0.\mathop {423}\limits^{\,\,\,\, \bullet \,\,\, \bullet \,} =$$

$$\frac{{419}}{{990}}$$

$$\frac{{419}}{{999}}$$

$$\frac{{417}}{{990}}$$

$$\frac{{417}}{{999}}$$

## Questions 24 of 50

Question:If $$y = x - {x^2} + {x^3} - {x^4} + ......\infty$$, then value of x will be

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

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

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

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

## Questions 25 of 50

Question:If $$x = \sum\limits_{n = 0}^\infty {{a^n}} ,\;y = \sum\limits_{n = 0}^\infty {{b^n},\;z = \sum\limits_{n = 0}^\infty {{{(ab)}^n}} }$$, where$$a,\;b < 1$$, then

$$xyz = x + y + z$$

$$xz + yz = xy + z$$

$$xy + yz = xz + y$$

$$xy + xz = yz + x$$

## Questions 26 of 50

Question:If in an infinite G.P. first term is equal to the twice of the sum of the remaining terms, then its common ratio is

1

2

3-Jan

1/3

## Questions 27 of 50

Question:If the sum of the series $$1 + \frac{2}{x} + \frac{4}{{{x^2}}} + \frac{8}{{{x^3}}} + ....\infty$$ is a finite number, then

$$x > 2$$

$$x > - 2$$

$$x > \frac{1}{2}$$

None of these

## Questions 28 of 50

Question:$$0.5737373...... =$$

$$\frac{{284}}{{497}}$$

$$\frac{{284}}{{495}}$$

$$\frac{{568}}{{990}}$$

$$\frac{{567}}{{990}}$$

## Questions 29 of 50

Question:If $$a,\;b,\;c$$ are three distinct positive real numbers which are in H.P., then $$\frac{{3a + 2b}}{{2a - b}} + \frac{{3c + 2b}}{{2c - b}}$$ is

Greater than or equal to 10

Less than or equal to 10

Only equal to 10

None of these

## Questions 30 of 50

Question:If $$a,\;b,\;c,\;d$$ are in H.P., then $$ab + bc + cd$$ is equal to

$$3ad$$

$$(a + b)(c + d)$$

$$3ac$$

None of these

## Questions 31 of 50

Question: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

$$A > G > H$$

$$A > G < H$$

$$H > G > A$$

$$G > A > H$$

## Questions 32 of 50

Question:If the arithmetic, geometric and harmonic means between two positive real numbers be $$A,\;G$$ and $$H$$, then

$${A^2} = GH$$

$${H^2} = AG$$

$$G = AH$$

$${G^2} = AH$$

## Questions 33 of 50

Question: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

0

1

2

$$\frac{1}{2}$$

## Questions 34 of 50

Question:If $$a,\;b,\;c$$ are in A.P. and $$a,\;b,\;d$$ in G.P., then $$a,\;a - b,\;d - c$$ will be in

A.P.

G.P.

H.P.

None of these

## Questions 35 of 50

Question:$$x + y + z = 15$$ 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

1

2

3

9

## Questions 36 of 50

Question: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 $$A + \frac{6}{H} =$$

1

3

5

6

## Questions 37 of 50

Question:If $$2(y - a)$$ is the H.M. between $$y - x$$ and $$y - z$$, then $$x - a,\;y - a,\;z - a$$ are in

A.P.

G.P.

H.P.

None of these

## Questions 38 of 50

Question: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

$$\frac{{\sqrt {m - n} + \sqrt n }}{{\sqrt {m - n} - \sqrt n }}$$

$$\frac{{\sqrt n + \sqrt {m - n} }}{{\sqrt n - \sqrt {m - n} }}$$

$$\frac{{\sqrt m + \sqrt {m - n} }}{{\sqrt m - \sqrt {m - n} }}$$

None of these

## Questions 39 of 50

Question: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

1

2

3

None of these

## Questions 40 of 50

Question:If $${a^x} = {b^y} = {c^z}\,{\rm{and}}\,\,a,b,c$$ are in G.P. then $$x,y,z$$ are in

A. P.

G. P.

H. P.

None of these

## Questions 41 of 50

Question:If $$\frac{a}{b},\frac{b}{c},\frac{c}{a}$$ are in H.P., then

$${a^2}b,\,{c^2}a,\,{b^2}c$$ are in A.P.

$${a^2}b,\,{b^2}c,\,{c^2}a$$are in H.P.

$${a^2}b,\,{b^2}c,\,{c^2}a$$are in G.P.

None of these

## Questions 42 of 50

Question: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

$$A > G$$

$$A \ne G$$

$$A = G$$

None of these

## Questions 43 of 50

Question:If $$|x|\, < 1$$, then the sum of the series $$1 + 2x + 3{x^2} + 4{x^3} + ...........\infty$$ will be

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

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

$$\frac{1}{{{{(1 + x)}^2}}}$$

$$\frac{1}{{{{(1 - x)}^2}}}$$

## Questions 44 of 50

Question:$$1 + \frac{3}{2} + \frac{5}{{{2^2}}} + \frac{7}{{{2^3}}} + ......\,\infty \,$$is equal to

3

6

9

12

## Questions 45 of 50

Question:If the set of natural numbers is partitioned into subsets $${S_1} = \left\{ 1 \right\},\;{S_2} = \left\{ {2,\;3} \right\},\;{S_3} = \left\{ {4,\;5,\;6} \right\}$$ and so on. Then the sum of the terms in $${S_{50}}$$ is

62525

25625

62500

None of these

## Questions 46 of 50

Question:The sum of $$(n + 1)$$ terms of $$\frac{1}{1} + \frac{1}{{1 + 2}} + \frac{1}{{1 + 2 + 3}} + ......\,\,{\rm{is }}$$

$$\frac{n}{{n + 1}}$$

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

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

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

## Questions 47 of 50

Question:The natural numbers are written as follows $$1$$$$\begin{array}{*{20}{c}} 2 & 3 \\\end{array}$$$$\begin{array}{*{20}{c}} 4 & 5 & 6 \\\end{array}$$$$\begin{array}{*{20}{c}} 7 & 8 & 9 & {10} \\ . & . & . & . \\ . & . & . & . \\ . & . & . & . \\\end{array}$$.The sum of the numbers in the $${n^{th}}$$ row is

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

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

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

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

## Questions 48 of 50

Question:The $${n^{th}}$$ term of series $$\frac{1}{1} + \frac{{1 + 2}}{2} + \frac{{1 + 2 + 3}}{3} + .......$$ will be

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

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

$$\frac{{{n^2} + 1}}{2}$$

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

## Questions 49 of 50

Question:The sum to infinity of the following series $$\frac{1}{{1.2}} + \frac{1}{{2.3}} + \frac{1}{{3.4}} + ...........$$ shall be

$$\infty$$

1

0

None of these

## Questions 50 of 50

Question:$$\frac{{\frac{1}{2}.\frac{2}{2}}}{{{1^3}}} + \frac{{\frac{2}{2}.\frac{3}{2}}}{{{1^3} + {2^3}}} + \frac{{\frac{3}{2}.\frac{4}{2}}}{{{1^3} + {2^3} + {3^3}}} + .....n$$ terms =

$${\left( {\frac{n}{{n + 1}}} \right)^2}$$
$${\left( {\frac{n}{{n + 1}}} \right)^3}$$
$$\left( {\frac{n}{{n + 1}}} \right)$$
$$\left( {\frac{1}{{n + 1}}} \right)$$