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1

WB JEE 2009

MCQ (Single Correct Answer)

The coefficient of xn, where n is any positive integer, in the expansion of (1 + 2x + 3x2 + ....... $$\infty$$)1/2 is

A
1
B
$${{n + 1} \over 2}$$
C
2n + 1
D
n + 1

Explanation

$${(1 + 2x + 3{x^2} + ....\,\infty )^{1/2}} = {({(1 - x)^{ - 2}})^{1/2}}$$

$$ = {(1 - x)^{ - 1}} = 1 + {{( - 1)( - x)} \over {\left| \!{\underline {\, 1 \,}} \right. }} + {{( - 1)( - 1 - 1)} \over {\left| \!{\underline {\, 2 \,}} \right. }}{( - x)^2} + ..... + {{( - 1)( - 1 - 1)( - 1 - 2).....( - 1 - n + 1)} \over {\left| \!{\underline {\, n \,}} \right. }}{( - x)^n} + .....$$

$$\therefore$$ Coefficient of xn

$$ = {{( - 1)( - 1 - 1)( - 2 - 1)......( - n)} \over {\left| \!{\underline {\, n \,}} \right. }}{( - 1)^n}$$

$$ = {{1\,.\,2\,.\,3\,.\,.....\,.\,n} \over {\left| \!{\underline {\, n \,}} \right. }} = {{\left| \!{\underline {\, n \,}} \right. } \over {\left| \!{\underline {\, n \,}} \right. }} = 1$$

2

WB JEE 2009

MCQ (Single Correct Answer)

For what value of m, $${{{a^{m + 1}} + {b^{m + 1}}} \over {{a^m} + {b^m}}}$$ is the arithmetic mean of a and b?

A
1
B
0
C
2
D
None of these

Explanation

We know, arithmetic mean of a and b is $${{a + b} \over 2}$$

So $${{{a^{m + 1}} + {b^{m + 1}}} \over {{a^m} + {b^m}}} = {{a + b} \over 2}$$

$$ \Rightarrow 2{a^{m + 1}} + 2{b^{m + 1}} = {a^{m + 1}} + {b^{m + 1}} + a{b^m} + {a^m}b$$

$$ \Rightarrow {a^{m + 1}} + {b^{m + 1}} = {a^m}b + a{b^m} \Rightarrow {a^{m + 1}} - {a^m}b = a{b^m} - {b^{m + 1}}$$

$$ \Rightarrow {a^m}(a - b) = (a - b){b^m}$$

$$ \Rightarrow {a^m} = {b^m} \Rightarrow {\left( {{a \over b}} \right)^m} = 1 = {\left( {{a \over b}} \right)^0}$$ $$\because$$ $$a - b \ne 0$$

$$ \Rightarrow m = 0$$

3

WB JEE 2009

MCQ (Single Correct Answer)

If three positive real numbers a, b, c are in A.P. and abc = 4 then the minimum possible value of b is

A
23/2
B
22/3
C
21/3
D
25/2

Explanation

$${{a + c} \over 2} \ge \sqrt {ac} \Rightarrow b \ge \sqrt {{4 \over b}} $$

$$ \Rightarrow {b^{3/2}} \ge 2 \Rightarrow b \ge {2^{2/3}}$$

4

WB JEE 2009

MCQ (Single Correct Answer)

If a, b, c are G.P. (a > 1, b > 1, c > 1), then for any real number x (with x > 0, x $$\ne$$ 1) logax, logbx, logcx are in

A
G.P.
B
A.P.
C
H.P.
D
G.P. but not in H.P.

Explanation

$${b^2} = ac$$ ($$\because$$ a, b, c are in G.P.)

$$ \Rightarrow 2{\log _x}b = {\log _x}ac = {\log _x}a + {\log _x}c$$

(Taking log on base x)

$$ \Rightarrow {2 \over {{{\log }_b}x}} = {1 \over {{{\log }_a}x}} + {1 \over {{{\log }_c}x}}$$

$$\Rightarrow$$ logax, logbx, logcx are in H.P.

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