If a, 8, b are in AP, a, 4, b are in GP, and if a, x, b are in HP then x is

(a)

2

(b)

1

(c)

4

(d)

16

The sum up to n terms of the series \(\frac { 1 }{ \sqrt { 1 } +\sqrt { 3 } } +\frac { 1 }{ \sqrt { 3 } +\sqrt { 5 } } +\frac { 1 }{ \sqrt { 5 } +\sqrt { 7 } } +\)....is 

(a)

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

(b)

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

(c)

\(\sqrt { 2n+1 } -1\)

(d)

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

The value of the series\(\frac { 1 }{ 2 } +\frac { 7 }{ 4 } +\frac { 13 }{ 8 } +\frac { 19 }{ 16 } +\).....is

(a)

14

(b)

7

(c)

4

(d)

6

If \(\frac { { T }_{ 2 } }{ { T }_{ 3 } } \)is the expansion of (a+b)ⁿ and \(\frac { { T }_{ 3 } }{ { T }_{ 4 } } \) is the expansion of (a+b)ⁿ⁺³ are equal, then n = ______________

(a)

3

(b)

4

(c)

5

(d)

6

If in an infinite G. P. first term is equal to 10 times the sum of all successive terms, then its common ratio is ______________

(a)

\(\frac { 1 }{ 10 } \)

(b)

\(\frac { 1 }{ 11 } \)

(c)

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

(d)

\(\frac { 1 }{ 20 } \)

If \(\Sigma n=210\) then \(\Sigma { n }^{ 2 }\)= ______________

(a)

2870

(b)

2160

(c)

2970

(d)

none of these

The series for log \(\left( \frac { 1+x }{ 1-x } \right) is\) ______________

(a)

\(x+\frac { { x }^{ 3 } }{ 3 } +\frac { { x }^{ 5 } }{ 5 } +...+\infty \)

(b)

\(2\left[ x+\frac { { x }^{ 3 } }{ 3 } +\frac { { x }^{ 5 } }{ 5 } +...+\infty \right] \)

(c)

\(\frac { { x }^{ 2 } }{ 2 } +\frac { { x }^{ 4 } }{ 4 } +\frac { { x }^{ 6 } }{ 6 } +...+\infty \)

(d)

\(2\left[ \frac { { x }^{ 2 } }{ 2 } +\frac { { x }^{ 4 } }{ 4 } +\frac { { x }^{ 6 } }{ 6 } +...+\infty \right] \)

The coefficient of x⁶ in (2 + 2x)¹⁰ is

(a)

¹⁰C₆

(b)

2⁶

(c)

¹⁰C₆2⁶

(d)

¹⁰C₆2¹⁰

The ratio of the coefficient of x ¹⁵ to the term independent of x in \([x^2+(\frac{2}{x})]^{15}\) is ______________

(a)

1:16

(b)

1:8

(c)

1:32

(d)

1:64

Sum of the binomial coefficients is ______________

(a)

2n

(b)

n²

(c)

2ⁿ

(d)

n+17

Expand \(\left( { 2x }^{ 2 }-\frac { 3 }{ x } \right) ^{ 3 }\)

 If n is a positive integer and R is a nonnegative integer. prove that the co-efficients of x^r and x^n-r Expansion of (1+x)ⁿ are equal

Find the 5^th term in the sequence whose first three terms are 3, 3, 6 and each term after the second is the sum of the two terms preceding it.

Find the middle terms in the expansion of (x + y)⁷.

Find the \(\sqrt [ 3 ]{ 126 } \) approximately to two decimal places.

If the roots of the equation (q - r) x² + (r - p)x + p - q = 0 are equal, then show that p, q and r are in A.P.

A man repays an amount of Rs. 3250 by paying Rs. 20 in the first month and then increases the payment by Rs.15 per month. How long will it take him to clear the amount?

if n is an odd positive integer, prove that the Co-efficients of the middle terms in the expansion equal

The first three terms in the expansion of (1 + ax)ⁿ are 1 + 12x + 64x². Find n and a

Prove that in the expansion of (1+x)ⁿ, the Co-efficient of terms equidistant from the beginning and from the end are equal

Find the general terms and sum to n terms of the sequence 1, \(\frac{4}{3},\frac{7}{9},\frac{10}{27},....\)

If S _n denotes that Sum of n terms of a G. P., prove that (s₁₀-s₂₀ )² = s₁₀ (s₃₀ - s₂₀)

If S₁, S₂, S₃ be respectively the sums of n, 2n, 3n, terms of a G.P. , then prove that S₁ (S₃ - S₂) = (S₂ - S₁)².