CAT 2023 Slot 1 QA Question 1

Multiple choice (+3 / −1) · Modern Math · Logarithms · Try it, then check the answer and solution below.

x

and

y

are positive real numbers such that

\log_{x}\left(x^{2} + 12\right)=4

and $3 \log _{y} x=1 , then

x+y

equals

⌨️ Shortcuts:[1-4 / A-D] Select[Enter] Check[B] Star[S] Solution

Official Convenor Answer Key: D

Answer and solution

Answer: D) $10$

📌 Core Concept

The governing mathematical theorems are:

Logarithm Definition:

\log_{b} a = c \implies b^c = a

Quadratic Equation: For

az^2 + bz + c = 0 , solutions are

z = \frac{-b ± \

\sqrt{b^2 - 4ac}

}{2a}$.

🔢 Step-by-Step Solution

First Equation:

\log_{x}(x^2 + 12) = 4 \implies x^4 = x^2 + 12

Rearranging:

x^4 - x^2 - 12 = 0 Let z = x^2: z^2 - z - 12 = 0

Solving using quadratic formula: $

z = \frac{1 \pm \sqrt{1 + 48}}{2} = \frac{1 \pm 7}{2}

Thus,

z = 4

(since

z = -3

is discarded asz = x^2 > 0). Hence,

x^2 = 4 \implies x = 2

Second Equation:

&#36;3 \log_{y} x = 1 \implies \log_{y} x = \frac{1}{3} \implies y^{1/3} = x \implies y = x^3

Substituting

x = 2

y = 2^3 = 8

Sum Calculation:

x + y = 2 + 8 = 10

⚡ 30-Second Shortcut

Recognize the logarithmic equations can be converted to exponential forms.

Solve the quadratic in terms of

x^2

for the first equation.

Use the relationship

y = x^3

from the second equation.

Sum

x

and

y

directly.

🎯 Final Answer

Correct Answer: D

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