CAT 2017 Question Paper | Quants Slot 1

CAT Questions

This question is about finding the value of a given expression. A simple quadratic equation is given but how you're going to use this quadratic equation will determine whether you arrive at the correct answer or not? Making effective use of an equation is an art that you have to master to ace CAT. Get as much practice as you can on this topic because the benefits of being good at solving equations will be useful in other topics.

Question 22 : If x + 1 = x² and x > 0, then 2x⁴ is :

1. 6 + 4√5
2. 3 + 5√5
3. 5 + 3√5
4. 7 + 3√5

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Video Explanation

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Explanatory Answer

Method of solving this CAT Question from Algebra

Given that x + 1 = x² and x > 0 then we have to find the value of 2x⁴.
This 2x⁴ can be found by either finding x first so let us take,
⟹ x + 1 = x²
⟹ x² - x – 1 = 0
⟹ x² - x + \\frac{1}{4}) = 1 + \\frac{1}{4})
⟹ (x - \\frac{1}{2}))² = \\frac{5}{4})
⟹ x - \\frac{1}{2}) = ± \\frac{√5}{2})
⟹ x = \\frac{1}{2}) ± \\frac{√5}{2})
⟹ x + 1 = x²
⟹ (x + 1)² = (x²)²
⟹ x² + 2x + 1 = x⁴
⟹ x + 1 + 2x + 1 = x⁴
⟹ 3x + 2 = x⁴
⟹ 6x + 4 = 2x⁴
We have already found the value of x = \\frac{1}{2}) ± \\frac{√5}{2}), so we can substitute that in place of x. We will get,
⟹ 6(\\frac{1 + √5}{2})) + 4 = 2x⁴
⟹ 3 + 3√5 + 4 = 2x⁴
⟹ 7 + 3√5 = 2x⁴
Hence if x + 1 = x² and x > 0, then 2x⁴ is equal to 7 + 3√5.
Key thing here is to somehow find x, and then write x⁴ in terms of x and then find the value of 2x⁴

The question is "If x + 1 = x² and x > 0, then 2x⁴ is :"

Hence, the answer is 7 + 3√5

Choice D is the correct answer