CAT 2017 Question Paper | Quants Slot 1

CAT Previous Year Paper | CAT Algebra Questions | Question 22

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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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

 

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