CAT 2019 Quant Questions with Solutions

Question:
If m and n are integers such that $(\sqrt{2})^{19} 3^{4} 4^{2} 9^{m} 8^{n}=3^{n} 16^{m}(\sqrt[4]{64})$ then m is

$-20$
$-12$
$-24$
$-16$
Show Answer

Correct Answer: Option: 2

$(\sqrt{2})^{19} 3^{4} 4^{2} 9^{m} 8^{n}=3^{n} 16^{m}(\sqrt[4]{64})$

$\Rightarrow 2^{19 / 2} \times 3^{4} \times 2^{4} \times 3^{2 m} \times 2^{3 n}=3^{n} \times 2^{4 m} \times 2^{3 / 2}$

$\Rightarrow {{2}^{(19/2+4+3n)}}\times {{3}^{(4+2m)}}={{2}^{(4m+3)}}\times {{3}^{n}}$

Comparing the powers of same bases we get

$\frac{19}{2}+4+3 n=4 m+\frac{3}{2} \cdots(1)$

$4+2 m=n \cdots(2)$

Substitute the value of n from (2) in (1) and solving for m, we get m = -12


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