CAT 2017 Slot 1 — QA Question 22
Number TheoryEasy
If x + 1 = x2 and x > 0, then 2x4 is:
Answer & solution
- A
6 + 4√5
- B
3 + 5√5
- C
5 + 3√5
7 + 3√5
Solution
x + 1 = x2
⇒ x2 - x – 1 = 0
⇒ x = (1+√5)/2 (âµ x > 0)
Now, squaring the original equation, we get
(x +1)2 = x4
⇒ x2 + 1 + 2x = x4
⇒ x4 = x + 1 + 1 + 2x (Since, x2 = x + 1)
⇒ x4 = 3x + 2
⇒ 2x4 = 6x + 4
⇒ 2x4 = 6((1+√5)/2) + 4
âââââââ⇒ 2x4 = 7 + 3√5
Alternately,
We know, x = (1+√5)/2
⇒ x ≈ 1.62
∴ 2x4 ≈ 13.8
option I ≈ 14.8,
option II ≈ 11.6,
option IV ≈13.6
Hence, option (d).