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CAT 2017 Slot 2 — QA Question 25

IndicesEasy

If 9x-12 - 22x - 2 = 4x - 32x - 3, then x is

Answer & solution

Correct answer: 3 2

  • 32

  • B

    25

  • C

    34

  • D

    49

Solution

Easy

Rewrite every term with prime bases 22 and 33. Group the powers of 33 on one side and powers of 22 on the other, factor, and compare.

1

Convert each term to base 22 or base 33. Note 9 x−12=(32)x−12=32x−19^{\,x-\frac12}=(3^2)^{x-\frac12}=3^{2x-1} and 4x=(22)x=22x4^x=(2^2)^x=2^{2x}.

3 2x−1−2 2x−2=2 2x−3 2x−3\begin{aligned} &3^{\,2x-1}-2^{\,2x-2}=2^{\,2x}-3^{\,2x-3} \end{aligned}
2

Separate the primes. Move all powers of 33 left and all powers of 22 right.

3 2x−1+3 2x−3=2 2x+2 2x−2⇒ 3 2x−3(32+1)=2 2x−2(22+1)(factor out lowest power)⇒ 3 2x−3⋅10=2 2x−2⋅5\begin{aligned} &3^{\,2x-1}+3^{\,2x-3}=2^{\,2x}+2^{\,2x-2}\\ &\Rightarrow\ 3^{\,2x-3}\big(3^{2}+1\big)=2^{\,2x-2}\big(2^{2}+1\big) \quad\text{(factor out lowest power)}\\ &\Rightarrow\ 3^{\,2x-3}\cdot 10=2^{\,2x-2}\cdot 5 \end{aligned}
3

Form a single ratio and match. Divide to isolate the prime powers.

3 2x−32 2x−2=510=12=3021\begin{aligned} &\frac{3^{\,2x-3}}{2^{\,2x-2}}=\frac{5}{10}=\frac12=\frac{3^{0}}{2^{1}} \end{aligned}

Equating the matching prime powers on both sides gives 3 2x−3=303^{\,2x-3}=3^{0} and 2 2x−2=212^{\,2x-2}=2^{1}, i.e. 2x−3=02x-3=0.

2x−3=0⇒ x=32\begin{aligned} &2x-3=0\\ &\Rightarrow\ x=\tfrac{3}{2} \end{aligned}
x=32x=\dfrac{3}{2}

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CAT 2017 Slot 2 QA Q25: If 9 x - 1 2 - 2 2x - 2 = 4 x - 3 2x - 3 , then x is — Solution | TheCATExam