The values of x- for which the 6th term in the expansion of -2log2-9x - 1 - 7 - 121-5log23x - 1 - 1-7 is 84- is equal to

The given expression reduced to [√(9^x 1 + 7) + 1(3^x 1 + 1)^1/5]^7 Given, T_6 = 84 ⇒ ^7C_5(√(9^x 1 + 7))^7 5(1(3^x 1 + 1)^1/5)^5=84 ⇒ ^7C_5(9^x 1 + 7)·1( ...

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Standard XII

Mathematics

General Term of Binomial Expansion

The values of...

Question

The value(s) of $x,$ for which the $6^{t h}$ term in the expansion of ${[2^{{log}_{2} \sqrt{(9^{x - 1} + 7)}} + \frac{1}{2^{(1 / 5) {log}_{2} (3^{x - 1} + 1)}}]}^{7}$ is $84$ , is equal to

4

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Solution

The correct option is D $1$
The given expression reduced to
${[\sqrt{9^{x - 1} + 7} + \frac{1}{(3^{x - 1} + 1)^{1 / 5}}]}^{7}$
Given, $T_{6} = 84$
$\Rightarrow^{7} C_{5} (\sqrt{9^{x - 1} + 7})^{7 - 5} {(\frac{1}{(3^{x - 1} + 1)^{1 / 5}})}^{5} = 84$
$\Rightarrow^{7} C_{5} (9^{x - 1} + 7) \cdot \frac{1}{(3^{x - 1} + 1)} = 84$
$\Rightarrow 9^{x - 1} + 7 = 4 (3^{x - 1} + 1)$
$\Rightarrow 3^{2 x} - 12 \cdot 3^{x} + 27 = 0$
$\Rightarrow (3^{x} - 3) (3^{x} - 9) = 0$
$\Rightarrow 3^{x} = 3, 9 \Rightarrow x = 1, 2$

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Similar questions

Q. In the expansion of

{⎡ ⎣ 2^{{log}_{2} \sqrt{9^{x - 1} + 7}} + \frac{1}{2^{\frac{1}{5} {log}_{2} (3^{x - 1} + 1)}} ⎤ ⎦}^{7}

, 6th term is

84

. Then

x =

Q. The value of

x

, for which the

6 t h

term in the expansion

{⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 2^{{log}_{2} \sqrt{(9^{x - 1} + 7)}} + \frac{1}{2^{⎛ ⎝ \frac{1}{5} ⎞ ⎠} {log}_{2} (3^{x - 1} + 1)} ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭}^{7}

84

, is equal to

Q. The value of

x

, for which the

6 t h

term in the expansion

{2^{{log}_{2} \sqrt{(9^{x - 1} + 7)}} + \frac{1}{2^{\frac{1}{5} {log}_{2} (3^{x - 1} + 1)}}}^{7} is 84

, is equal to

Q. Determine the value of x for which the 6th term in

(2^{l o g_{2} (\sqrt{g^{x - y} + 7})} + \frac{1}{2^{\frac{1}{5} l o g_{2} (3^{x - 1} + 1)}})

is 84.

Q. The value of x for which the sixth term in the expansion of

{[2^{{log}_{2} \sqrt{9^{x - 1} + 7}} + \frac{1}{2^{\frac{1}{5} {log}_{2} (3^{x - 1} + 1)}}]}^{7}

is equal to 84 is

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General Term of Binomial Expansion

Standard XII Mathematics

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The value(s) of x, for which the 6th term in the expansion of [2log2√(9x−1+7)+12(1/5)log2(3x−1+1)]7 is 84, is equal to

The correct option is D 1The given expression reduced to [√9x−1+7+1(3x−1+1)1/5]7 Given, T6=84 ⇒ 7C5(√9x−1+7)7−5(1(3x−1+1)1/5)5=84 ⇒ 7C5(9x−1+7)⋅1(3x−1+1)=84 ⇒9x−1+7=4(3x−1+1) ⇒32x−12⋅3x+27=0 ⇒(3x−3)(3x−9)=0 ⇒3x=3,9⇒x=1,2

The value(s) of $x,$ for which the $6^{t h}$ term in the expansion of ${[2^{{log}_{2} \sqrt{(9^{x - 1} + 7)}} + \frac{1}{2^{(1 / 5) {log}_{2} (3^{x - 1} + 1)}}]}^{7}$ is $84$ , is equal to