Question

Values of x for which the sixth term of the expansion of

E=$\left\lbrack 3^{\log_{3}\sqrt{9^{|x - 2|}}} + 7^{\frac{1}{5}\log_{7}\lbrack(4).3^{|x - 2|} - 9\rbrack} \right\rbrack$is 567, is/are :

• A. 3
• B. 2
• C. 4
• D. None

Answer 1

Answer: (A)

3

Answer 2

Let y = $3{}{\log_{3}{}_{\sqrt{9^{|x - 2|}}}}$ z = $7^{\frac{1}{5}\log_{7}\lbrack 4.3^{|x - 2|} - 9\rbrack}$

Žlog₃y =$\log_{3}\sqrt{9^{|x - 2|}}$ log₇ z =$\frac{1}{5}$ log₇[4.3^|x–2| –9]

Ž $y = \sqrt{9^{|x - 2|}}$ z = (4.3^|x–2| –9)^1/5

Ž y = 3^{|x – 2|} z = (4.3^{|x – 2|} – 9)^1/5

Now E = (y + z)⁷ and the sixth term is given by

t₆ = ⁷C₅ y²z⁵ Ž 21.(3^{|x – 2|})².[(4.3^{|x – 2|} – 9)^1/5]⁵

= 567(given)

Ž 21.3^{2|x – 2|}.[4.3^{|x – 2|} – 9] = 567 [put 3^{|x – 2|} = k]

Ž 21.k². [4.k – 9] = 567

Ž 4k³ – 9k² – 27 = 0

Ž (k – 3)(4k² + 3k + 9) = 0

Ž k = 3