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1

### WB JEE 2009

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

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

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

2

### WB JEE 2009

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.

3

### WB JEE 2008

The sum of the infinite series $${\left( {{1 \over 3}} \right)^2} + {1 \over 3}{\left( {{1 \over 3}} \right)^4} + {1 \over 5}{\left( {{1 \over 3}} \right)^6} + ...$$ is

A
$${1 \over 4}{\log _e}2$$
B
$${1 \over 2}{\log _e}2$$
C
$${1 \over 6}{\log _e}2$$
D
$${1 \over 4}{\log _e}{3 \over 2}$$

## Explanation

$$\because$$ $${\log _e}(1 + x) = x - {{{x^2}} \over 2} + {{{x^3}} \over 3} - {{{x^4}} \over 4} + ...\,|x| < 1$$

$${\log _e}(1 - x) = - x - {{{x^2}} \over 2} - {{{x^3}} \over 3} - {{{x^4}} \over 4} - ...$$

On subtracting

$$lo{g_e}(1 + x) - {\log _e}(1 - x) = 2\left( {x + {{{x^3}} \over 3} + {{{x^5}} \over 5} + ...} \right)$$ ..... (i)

$$\therefore$$ $${\left( {{1 \over 3}} \right)^2} + {1 \over 3}{\left( {{1 \over 3}} \right)^4} + {1 \over 5}{\left( {{1 \over 3}} \right)^6} + ...$$

$$= {1 \over 3}\left[ {{1 \over 3} + {1 \over 3}{{\left( {{1 \over 3}} \right)}^3} + {1 \over 5}{{\left( {{1 \over 3}} \right)}^5} + ....} \right]$$

$$= {1 \over {2 \times 3}}\left[ {{{\log }_e}\left( {1 + {1 \over 3}} \right) - \log \left( {1 - {1 \over 3}} \right)} \right] = {1 \over 6}{\log _e}\left[ {{4 \over 3} \times {3 \over 2}} \right]$$

$$= {1 \over 6}{\log _e}2$$.

4

### WB JEE 2008

If three real numbers a, b, c are in Harmonic Progression, then which of the following is true?

A
$${1 \over a},b,{1 \over c}$$ are in A.P.
B
$${1 \over {bc}},{1 \over {ca}},{1 \over {ab}}$$ are in H.P.
C
ab, bc, ca are in H.P.
D
$${a \over b},{b \over c},{c \over a}$$ are in H.P.

## Explanation

a, b, c are in H.P.

or, $${1 \over a},{1 \over b},{1 \over c}$$ are in A.P. (reciprocal of H.P. is A.P.)

or, $${{abc} \over a},{{abc} \over b},{{abc} \over c}$$ are in A.P. (multiplying by abc)

or, bc, ac, ab are in A.P.

or, $${1 \over {bc}},{1 \over {ac}},{1 \over {ab}}$$ are in H.P. (reciprocal of A.P. is H.P.)

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