CAT 2006 — QA Question 15

Solving Quadratic EquationsEasy

Passage / Data

Answer the next 2 questions based on the information given below:

A punching machine is used to punch a circular hole of diameter two units from a square sheet of aluminium of width 2 units, as shown below. The hole is punched such that the circular hole touches one corner P of the square sheet and the diameter of the hole originating at P is in line with a diagonal of the square.

What values of x satisfy $x^{\frac{2}{3}} + x^{\frac{1}{3}} - 2 \leq 0$ ?

Answer & solution

–8 ≤ x ≤ 1
B
–1 ≤ x ≤ 8
C
1 < x < 8
D
1 ≤ x ≤ 8
E
–8 ≤ x ≤ 8

Solution

$x^{\frac{2}{3}} + x^{\frac{1}{3}} - 2 \leq 0$ ...(1)

Put y = x^1/3

Then equation (1) becomes y² + y – 2 ≤ 0

(y + 2)(y – 1) ≤ 0

–2 ≤ y ≤ 1

$- 2 \leq x^{\frac{1}{3}} \leq 1$

–8 ≤ x ≤ 1

Hence, option (a).

CAT 2006 QA Q15: What values of x satisfy x 2 3 + x 1 3 - 2 ≤ 0 ? — Solution | TheCATExam