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CAT 2017 Slot 1 — QA Question 24

IndicesEasy

If 92x–1 – 81x-1 = 1944, then x is

Answer & solution

Correct answer: 9/4

  • A

    3

  • 9/4

  • C

    4/9

  • D

    1/3

Solution

Easy

Write everything in base 33 (or base 99). Pull out the common power, simplify the constant, and match exponents to read off xx.

1

Rewrite with a common power of 99. Note 81=9281 = 9^2, so 81x−1=92x−281^{x-1} = 9^{2x-2}.

92x−1−81x−1=1944⇒ 92x−2⋅9−92x−2=1944(since 92x−1=92x−2⋅9)⇒ 92x−2(9−1)=1944\begin{aligned} &9^{2x-1} - 81^{x-1} = 1944\\ &\Rightarrow\ 9^{2x-2}\cdot 9 - 9^{2x-2} = 1944 \quad\text{(since }9^{2x-1}=9^{2x-2}\cdot9\text{)}\\ &\Rightarrow\ 9^{2x-2}(9-1) = 1944 \end{aligned}
2

Isolate the power and factor 19441944. Divide by 88 and write the result as a power of 33.

92x−2⋅8=1944⇒ 92x−2=243=35(1944÷8=243)⇒ 34x−4=35(92x−2=34x−4)\begin{aligned} &9^{2x-2}\cdot 8 = 1944\\ &\Rightarrow\ 9^{2x-2} = 243 = 3^5 \quad\text{(}1944\div 8 = 243\text{)}\\ &\Rightarrow\ 3^{4x-4} = 3^5 \quad\text{(}9^{2x-2}=3^{4x-4}\text{)} \end{aligned}
3

Equate exponents. The bases match, so the exponents are equal.

4x−4=5⇒ x=94\begin{aligned} &4x - 4 = 5\\ &\Rightarrow\ x = \tfrac{9}{4} \end{aligned}
x=94x = \dfrac{9}{4}

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CAT 2017 Slot 1 QA Q24 — Indices solution | TheCATExam