TheCATExam
Sign in

CAT 2002 — QA Question 6

Number TheoryEasy

If x2 + 5y2 + z2 = 2y(2x + z), then which of the following statements are necessarily true?

I. x = 2y
II. x = 2z
III. 2x = z

Answer & solution

Correct answer: Only I and II

  • A

    Only I

  • B

    Only II

  • C

    Only III

  • Only I and II

Solution

x² + 5y² + z² = 4xy + 2yz

∴ x² + 4y² + y² + z² – 4xy – 2yz = 0

∴ (x² + 4y² – 4xy) + (y² – 2yz + z²) = 0

∴ (x – 2y)² + (y – z)² = 0            …(i)

As per equation (i),

∴ (x – 2y)² = 0 and (y – z)² = 0

∴ x = 2y and y = z

Hence, option (d).

Related Number Theory questions

See all Number Theory questions →
CAT 2002 QA Q6: If x 2 + 5y 2 + z 2 = 2y(2x + z), then which of the following statements are necessarily true? I. x = 2y II. x — Solution | TheCATExam