Question

(ix) If 3 ^ (x - 1) - 3 ^ (x - 3) = 8 then the value of (x ^ 2 - x + 4) * i_{5} (D) 15 (C) 12 (B) 8 (A) 10​

Answer 1

$\large \underline{ \underline{ \text{Question:}}}$

If 3^(x - 1) - 3^(x - 3) = 8 then the value of: x² - x + 4.

• (A) 10

• (B) 8

• (C) 12

• (D) 15

$\large \underline{ \underline{ \text{Solution:}}}$

We have,

• ${3}^{(x - 1)} - {3}^{(x - 3)} = 8$

Solving for 'x',

$\implies {3}^{(x - 1)} - {3}^{(x - 3)} = 8 \\ \\ \bigg [ {x}^{(a - b)} = \frac{ {x}^{a} }{ {x}^{b} } \bigg] \\ \\ \implies \frac{ {3}^{x} }{ {3}^{1} } - \frac{ {3}^{x} }{ {3}^{3} } = 8 \\ \\ \implies \frac{ {3}^{x} }{3} - \frac{ {3}^{x} }{27} = 8 \\ \\ \implies \frac{9( {3}^{x} ) - {3}^{x} }{27} = 8 \\ \\ \implies 9( {3}^{x} ) - {3}^{x} = 8 \times 27 \\ \\ \implies {3}^{x} (9 - 1) = 216 \\ \\ \implies {3}^{x} (8) = 216 \\ \\ \implies {3}^{x} = \frac{216}{8} \\ \\ \implies {3}^{x} = 27 \\ \\ [27 = 3 \times 3 \times 3 = {3}^{3} ] \\ \\ \implies {3}^{x} = {3}^{3}$

Comparing both sides. We get,

• $x = 3$

Hence,

• The value of 'x' is 3.

We have,

• $p(x) = {x}^{2} - x + 4$

Substituting this value of 'x' is given Polynomial,

$\implies p(x) = {x}^{2} - x + 4 \\ \\ \implies p(3) = {(3)}^{2} - (3) + 4 \\ \\ \implies p(3) = 9 - 3 + 4 \\ \\ \implies p(3) = 13 - 3 \\ \\ \implies p(3) = 10$

Therefore,

• The value of 'x² - x + 4' is 10.

$\\ \large \underline{ \underline{ \text{Required Answer:}}}$

• Option (A) 10 is correct.