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.

Answers Choices:

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

Answers Choices:

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

Answers Choices:

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

Answers Choices:

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

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

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

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

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

Answers Choices:

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

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

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

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

Questions 6 of 50

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

Answers Choices:

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

Answers Choices:

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 \(\frac{{{S_{3n}}}}{{{S_n}}} = \)

Answers Choices:

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

Answers Choices:

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

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

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

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

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

Answers Choices:

\(x\)

\(y\)

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 =

Answers Choices:

2-May

\({\log _2}5\)

\({\log _3}5\)

2-Mar

Questions 12 of 50

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

Answers Choices:

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

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

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

None of these

Questions 13 of 50

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

Answers Choices:

\(125\sqrt {10} \)

\(25\sqrt 2 \)

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

Answers Choices:

\(\frac{{\log l - \log a}}{{\log r}}\)

\(1 - \frac{{\log l - \log a}}{{\log r}}\)

\(\frac{{\log a - \log l}}{{\log r}}\)

\(1 + \frac{{\log l - \log a}}{{\log r}}\)

Questions 15 of 50

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

Answers Choices:

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

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

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

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

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

Answers Choices:

\({c^3}a = {b^3}d\)

\(c{a^3} = b{d^3}\)

\({a^3}b = {c^3}d\)

\(a{b^3} = c{d^3}\)

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

Answers Choices:

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

Answers Choices:

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

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

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

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

Answers Choices:

\( - 2\)

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

Answers Choices:

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

Answers Choices:

4

6

12

6

Questions 22 of 50

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

Answers Choices:

3

4

2

1

Questions 23 of 50

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

Answers Choices:

\(\frac{{419}}{{990}}\)

\(\frac{{419}}{{999}}\)

\(\frac{{417}}{{990}}\)

\(\frac{{417}}{{999}}\)

Questions 24 of 50

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

Answers Choices:

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

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

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

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

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

Answers Choices:

\(xyz = x + y + z\)

\(xz + yz = xy + z\)

\(xy + yz = xz + y\)

\(xy + xz = yz + x\)

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

Answers Choices:

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

Answers Choices:

\(x > 2\)

\(x > - 2\)

\(x > \frac{1}{2}\)

None of these

Questions 28 of 50

Question:\(0.5737373...... = \)

Answers Choices:

\(\frac{{284}}{{497}}\)

\(\frac{{284}}{{495}}\)

\(\frac{{568}}{{990}}\)

\(\frac{{567}}{{990}}\)

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

Answers Choices:

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

Answers Choices:

\(3ad\)

\((a + b)(c + d)\)

\(3ac\)

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

Answers Choices:

\(A > G > H\)

\(A > G < H\)

\(H > G > A\)

\(G > A > H\)

Questions 32 of 50

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

Answers Choices:

\({A^2} = GH\)

\({H^2} = AG\)

\(G = AH\)

\({G^2} = AH\)

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

Answers Choices:

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

Answers Choices:

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

Answers Choices:

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 \(A + \frac{6}{H} = \)

Answers Choices:

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

Answers Choices:

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

Answers Choices:

\(\frac{{\sqrt {m - n} + \sqrt n }}{{\sqrt {m - n} - \sqrt n }}\)

\(\frac{{\sqrt n + \sqrt {m - n} }}{{\sqrt n - \sqrt {m - n} }}\)

\(\frac{{\sqrt m + \sqrt {m - n} }}{{\sqrt m - \sqrt {m - n} }}\)

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

Answers Choices:

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

Answers Choices:

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

Answers Choices:

\({a^2}b,\,{c^2}a,\,{b^2}c\) 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

Answers Choices:

\(A > G\)

\(A \ne G\)

\(A = G\)

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

Answers Choices:

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

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

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

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

Questions 44 of 50

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

Answers Choices:

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

Answers Choices:

62525

25625

62500

None of these

Questions 46 of 50

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

Answers Choices:

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

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

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

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

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

Answers Choices:

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

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

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

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

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

Answers Choices:

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

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

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

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

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

Answers Choices:

\(\infty \)

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 =

Answers Choices:

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

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

\(\left( {\frac{n}{{n + 1}}} \right)\)

\(\left( {\frac{1}{{n + 1}}} \right)\)