CAT Quantitative Aptitude Questions | CAT Algebra - Linear Equations; Quadratic Equations

The question is about finding Roots of an Equation. A complicated equation is given and we need to find out one of the roots of the equation. Framing and solving equations is an integral part of Linear Equations and Quadratic Equations. Get as much practice as you can in these two topics because the benefits of being good at framing equations will be useful in other concepts.

Question 15: (3 + 2√2)^{(x2 - 3)} + (3 - 2√2)^{(x2 - 3)} = b which of the following can be the value of b?

1. 2
2. √2
3. -√2
4. All of the above

Explanatory Answer

Method of solving this CAT Question from CAT Algebra - Quadratic Equations: Possible Value of a complicated equation!

(3 + 2√2)^{(x2 - 3)} be y.
(3 + 2√2) and (3 - 2√2) are conjugate numbers. Since they are conjugate numbers,
(3 + 2√2) * (3 - 2√2) = 1
So, (3 + 2√2) = \\frac {1}{(3-2√2)}\\)
or (3 - 2√2) = \\frac {1}{(3+2√2)}\\)
Now, (3 - 2√2)^{(x2 - 3)} = (\\frac {1}{(3+2√2)}\\))^{(x2 - 3)} = \\frac {1}{y}\\)

Equation can be written as:-
(3 + 2√2)^{(x2 - 3)} + (3 - 2√2)^{(x2 - 3)} = y + \\frac {1}{y}\\) = b
As a rule, the expression: y + \\frac {1}{y}\\) ≥ 2 or y + \\frac {1}{y}\\) ≤ -2
From the options, it is clear that y + \\frac {1}{y}\\) can take the value 2.

So, b can take the value 2

The question is "(3 + 2√2)^{(x2 - 3)} + (3 - 2√2)^{(x2 - 3)} = b which of the following can be the value of b"

Hence the answer is "2"

Choice A is the correct answer.