Progressions Test 1

Total Questions:50 Total Time: 75 Min

Remaining:

 

Questions 1 of 50

Question:If \(\tan \,n\theta = \tan m\theta \), then the different values of \(\theta \)  will be in

Answers Choices:

A.P.

G.P.

H.P.

None of these

Questions 2 of 50

Question:\({n^{th}}\) term of the series \(3.8 + 6.11 + \) \(9.14 + 12.17 + .....\)will be

Answers Choices:

\(3n(3n + 5)\)

\(3n(n + 5)\)

\(n(3n + 5)\)

\(n(n + 5)\)

Questions 3 of 50

Question:The sum of integers from 1 to 100 that are divisible by 2 or 5 is

Answers Choices:

3000

3050

4050

None of these

Questions 4 of 50

Question:If \(p\)times the \({p^{th}}\) term of an A.P. is equal to \(q\) times the \({q^{th}}\) term of an A.P., then \({(p + q)^{th}}\) term is

Answers Choices:

0

1

2

3

Questions 5 of 50

Question:The sums of \(n\) terms of two arithmatic series are in the ratio \(2n + 3:6n + 5\), then the ratio of their \({13^{th}}\) terms is

Answers Choices:

53 : 155

27 : 77

29 : 83

31 : 89

Questions 6 of 50

Question:If \({a_{\rm{m}}}\) denotes the \({m^{{\rm{th}}}}\) term of an A.P. then \({a_m}\) =

Answers Choices:

\(\frac{2}{{{a_{m + k}} + {a_{m - k}}}}\)

\(\frac{{{a_{m + k}} - {a_{m - k}}}}{2}\)

\(\frac{{{a_{m + k}} + {a_{m - k}}}}{2}\)

None of these

Questions 7 of 50

Question:If \({a_1},\;{a_2},............,{a_n}\) are in A.P. with common difference , \(d\), then the sum of the following series is \(\sin d\left( {\cos ec{a_1}.\cos ec{a_2} + \cos ec{a_2}.\cos ec{a_3}} \right) + ...... + \cos ec\,{a_{n - 1}}\cos ec\,{a_n}\)

Answers Choices:

\(\sec {a_1} - \sec {a_n}\)

\(\cot {a_1} - \cot {a_n}\)

\(\tan {a_1} - \tan {a_n}\)

\(c{\rm{osec}}\;{a_1} - {\rm{cosec}}\;{a_n}\)

Questions 8 of 50

Question:If the sum of the series \(2 + 5 + 8 + 11............\)is 60100, then the number of terms are

Answers Choices:

100

200

150

250

Questions 9 of 50

Question:The sum of all natural numbers between 1 and 100 which are multiples of 3 is

Answers Choices:

1680

1683

1681

1682

Questions 10 of 50

Question:If \({S_k}\)denotes the sum of first \(k\)terms of an arithmetic progression whose first term and common difference are \(a\)and \(d\) respectively, then \({S_{kn}}/{S_n}\)be independent of \(n\)if

Answers Choices:

\(2a - d = 0\)

\(a - d = 0\)

\(a - 2d = 0\)

None of these

Questions 11 of 50

Question:A series whose nth term is \(\left( {\frac{n}{x}} \right) + y,\)the sum of r terms will be

Answers Choices:

\(\left\{ {\frac{{r(r + 1)}}{{2x}}} \right\} + ry\)

\(\left\{ {\frac{{r(r - 1)}}{{2x}}} \right\}\)

\(\left\{ {\frac{{r(r - 1)}}{{2x}}} \right\} - ry\)

\(\left\{ {\frac{{r(r + 1)}}{{2y}}} \right\} - rx\)

Questions 12 of 50

Question:The sum of the integers from 1 to 100 which are not divisible by 3 or 5 is

Answers Choices:

2489

4735

2317

2632

Questions 13 of 50

Question:There are 15 terms in an arithmetic progression. Its first term is 5 and their sum is 390. The middle term is

Answers Choices:

23

26

29

32

Questions 14 of 50

Question:If the sum of the 10 terms of an A.P. is 4 times to the sum of its 5 terms, then the ratio of first term and common difference is

Answers Choices:

\(1:2\)

\(2:1\)

\(2:3\)

\(3:2\)

Questions 15 of 50

Question:Three numbers are in A.P. whose sum is 33 and product is 792, then the smallest number from these numbers is

Answers Choices:

4

8

11

14

Questions 16 of 50

Question:If \(a,\;b,\;c,\;d,\;e,\;f\) are in A.P., then the value of \(e - c\) will be

Answers Choices:

\(2(c - a)\)

\(2(f - d)\)

\(2(d - c)\)

\(d - c\)

Questions 17 of 50

Question:The number which should be added to the numbers 2, 14, 62 so that the resulting numbers may be in G.P., is

Answers Choices:

1

2

3

4

Questions 18 of 50

Question:If \({(p + q)^{th}}\) term of a G.P. be \(m\) and \({(p - q)^{th}}\)term be \(n\), then the \({p^{th}}\) term will be [RPET 1997; MP PET 1985, 99]

Answers Choices:

\(m/n\)

\(\sqrt {mn} \)

\(mn\)

0

Questions 19 of 50

Question:If the nth term of geometric progression \(5, - \frac{5}{2},\frac{5}{4}, - \frac{5}{8},...\) is \(\frac{5}{{1024}}\), then the value of n is

Answers Choices:

11

10

9

4

Questions 20 of 50

Question:The third term of a G.P. is the square of first term. If the second term is 8, then the \({6^{th}}\) term is

Answers Choices:

120

124

128

132

Questions 21 of 50

Question:The solution of the equation \(1 + a + {a^2} + {a^3} + ....... + {a^x}\) \( = (1 + a)(1 + {a^2})(1 + {a^4})\) is given by \(x\) is equal to

Answers Choices:

3

5

7

None of these

Questions 22 of 50

Question:If in a geometric progression \(\left\{ {{a_n}} \right\},\;{a_1} = 3,\;{a_n} = 96\) and \({S_n} = 189\) then the value of \(n\) is

Answers Choices:

5

6

7

8

Questions 23 of 50

Question:The two geometric means between the number 1 and 64 are

Answers Choices:

1 and 64

4 and 16

2 and 16

8 and 16

Questions 24 of 50

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

Answers Choices:

\({a^2},\;{b^2},\;{c^2}\)are in G.P.

\({a^2}(b + c),\;{c^2}(a + b),\;{b^2}(a + c)\) are in G.P.

\(\frac{a}{{b + c}},\;\frac{b}{{c + a}},\;\frac{c}{{a + b}}\) are in G.P.

None of the above

Questions 25 of 50

Question:If \(S\) is the sum to infinity of a G.P., whose first term is \(a\), then the sum of the first \(n\) terms is

Answers Choices:

\(S{\left( {1 - \frac{a}{S}} \right)^n}\)

\(S\left[ {1 - {{\left( {1 - \frac{a}{S}} \right)}^n}} \right]\)

\(a\left[ {1 - {{\left( {1 - \frac{a}{S}} \right)}^n}} \right]\)

None of these

Questions 26 of 50

Question:0.14189189189,....................... can be expressed as a rational number

Answers Choices:

\(\frac{7}{{3700}}\)

\(\frac{7}{{50}}\)

\(\frac{{525}}{{111}}\)

\(\frac{{21}}{{148}}\)

Questions 27 of 50

Question:If s is the sum of an infinite G.P., the first term a then the common ratio r given by

Answers Choices:

\(\frac{{a - s}}{s}\)

\(\frac{{s - a}}{s}\)

\(\frac{a}{{1 - s}}\)

\(\frac{{s - a}}{a}\)

Questions 28 of 50

Question:The sum to infinity of the progression \(9 - 3 + 1 - \frac{1}{3} + .....\) is

Answers Choices:

9

2-Sep

27/4

15/2

Questions 29 of 50

Question:n a H.P., pth term is q and the qth term is p. Then pqth term is

Answers Choices:

0

1

pq

\(pq(p + q)\)

Questions 30 of 50

Question:The 4th term of a H.P. is \(\frac{3}{5}\) and 8th term is \(\frac{1}{3},\) then its 6th term is

Answers Choices:

\(\frac{1}{6}\)

\(\frac{3}{7}\)

\(\frac{1}{7}\)

\(\frac{3}{5}\)

Questions 31 of 50

Question:If \({\log _a}x,\;{\log _b}x,\;{\log _c}x\) be in H.P., then \(a,\;b,\;c\) are in

Answers Choices:

A.P.

H.P.

G.P.

None of these

Questions 32 of 50

Question:If three numbers be in G.P., then their logarithms will be in

Answers Choices:

A.P.

G.P.

H.P.

None of these

Questions 33 of 50

Question:If \(a,\;b,\;c\) are in A.P. and \(|a|,\;|b|,\;|c|\; < 1\) and \(x = 1 + a + {a^2} + ........\infty \)\(y = 1 + b + {b^2} + .......\infty \)\(z = 1 + c + {c^2}........\infty \). Then\(x,\;y,\;z\) shall be in

Answers Choices:

A.P.

G.P.

H.P.

None of these

Questions 34 of 50

Question:If three unequal non-zero real numbers \(a,\;b,\;c\)are in G.P. and \(b - c,\;c - a,\;a - b\)are in H.P., then the value of \(a + b + c\) is independent of

Answers Choices:

\(a\)

\(b\)

\(c\)

None of these

Questions 35 of 50

Question:The numbers \((\sqrt 2 + 1),\;1,\;(\sqrt 2 - 1)\) will be in

Answers Choices:

A.P.

G.P.

H.P.

None of these

Questions 36 of 50

Question:If the ratio of H.M. and G.M. of two quantities is \(12:13\), then the ratio of the numbers is

Answers Choices:

\(1:2\)

\(2:3\)

\(3:4\)

None of these

Questions 37 of 50

Question:If \(\frac{{a + bx}}{{a - bx}} = \frac{{b + cx}}{{b - cx}} = \frac{{c + dx}}{{c - dx}}(x \ne 0)\), then \(a,\;b,\;c,\;d\) are in

Answers Choices:

A.P.

G.P.

H.P.

None of these

Questions 38 of 50

Question:If \(a,\;b,\;c\) are in A.P. and \(a,\;c - b,\;b - a\) are in G.P. \((a \ne b \ne c)\), then \(a:b:c\) is

Answers Choices:

\(1:3:5\)

\(1:2:4\)

\(1:2:3\)

None of these

Questions 39 of 50

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

Answers Choices:

A.P.

G.P.

H.P.

None of these

Questions 40 of 50

Question:The sum of three decreasing numbers in A.P. is 27. If \( - 1,\, - 1,\,3\) are added to them respectively, the resulting series is in G.P. The numbers are

Answers Choices:

5, 9, 13

15, 9, 3

13, 9, 5

17, 9, 1

Questions 41 of 50

Question:If \(a,b,c\)are in G.P. then \({\log _a}x,{\log _b}x,{\log _c}x\) are in

Answers Choices:

A.P.

G.P.

H.P.

None of these

Questions 42 of 50

Question:If \(a,\,b,\,c\) are three unequal numbers such that \(a,\,b,\,c\) are in A.P. and b - a, c - b, a are in G.P., then a : b : c is

Answers Choices:

1:2:3

2:3:1

1:3:2

3:2:1

Questions 43 of 50

Question:\(2 + 4 + 7 + 11 + 16 + ......\)to \(n\) terms =

Answers Choices:

\(\frac{1}{6}({n^2} + 3n + 8)\)

\(\frac{n}{6}({n^2} + 3n + 8)\)

\(\frac{1}{6}({n^2} - 3n + 8)\)

\(\frac{n}{6}({n^2} - 3n + 8)\)

Questions 44 of 50

Question:Sum of n terms of series \(12 + 16 + 24 + 40 + .....\) will be

Answers Choices:

\(2\,({2^n} - 1) + 8n\)

\(2({2^n} - 1) + 6n\)

\(3({2^n} - 1) + 8n\)

\(4({2^n} - 1) + 8n\)

Questions 45 of 50

Question:The sum of the series \(1 + (1 + 2) + (1 + 2 + 3) + ............\)upto \(n\) terms, will be

Answers Choices:

\({n^2} - 2n + 6\)

\(\frac{{n(n + 1)(2n - 1)}}{6}\)

\({n^2} + 2n + 6\)

\(\frac{{n(n + 1)(n + 2)}}{6}\)

Questions 46 of 50

Question:The sum of \(n\) terms of the series whose \({n^{hth}}\) term is \(n(n + 1)\) is equal to

Answers Choices:

\(\frac{{n(n + 1)(n + 2)}}{3}\)

\(\frac{{(n + 1)(n + 2)(n + 3)}}{{12}}\)

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

\(n(n + 1)(n + 2)\)

Questions 47 of 50

Question:Sum of the \(n\) terms of the series \(\frac{3}{{{1^2}}} + \frac{5}{{{1^2} + {2^2}}} + \frac{7}{{{1^2} + {2^2} + {3^2}}}\, + ...\,\,{\rm{is}}\)

Answers Choices:

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

\(\frac{{4n}}{{n + 1}}\)

\(\frac{{6n}}{{n + 1}}\)

\(\frac{{9n}}{{n + 1}}\)

Questions 48 of 50

Question:The sum of the series \(1.3.5 + .2.5.8 + 3.7.11 + .........\)upto \('n'\) terms is

Answers Choices:

\(\frac{{n\,(n + 1)(9{n^2} + 23n + 13)}}{6}\)

\(\frac{{n\,(n - 1)(9{n^2} + 23n + 12)}}{6}\)

\(\frac{{(n + 1)(9{n^2} + 23n + 13)}}{6}\)

\(\frac{{n\,(9{n^2} + 23n + 13)}}{6}\)

Questions 49 of 50

Question:If \(\frac{1}{{{1^4}}} + \frac{1}{{{2^4}}} + \frac{1}{{{3^4}}} + ..... + \infty = \frac{{{\pi ^4}}}{{90}}\), then the value of \(\frac{1}{{{1^4}}} + \frac{1}{{{3^4}}} + \frac{1}{{{5^4}}} + .....\infty \)is

Answers Choices:

\(\frac{{{\pi ^4}}}{{96}}\)

\(\frac{{{\pi ^4}}}{{45}}\)

\(\frac{{89}}{{90}}{\pi ^4}\)

None of these

Questions 50 of 50

Question:The value of \(\frac{1}{{(1 + a)(2 + a)}} + \frac{1}{{(2 + a)(3 + a)}}\) \(\frac{1}{{(3 + a)(4 + a)}}\)+ ..... +\(\infty \) is, (where a is a constant)

Answers Choices:

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

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

\(\infty \)

None of these