Algebra and Applications Test 1

Question : 4 of 40

Marks: +1, -0

Consider the following all real numbers t, u and v:
1. If

t^2 + u^2 + v^2 = tu + uv + vt

, then

t = u = v

2. If

t + u + v = 0

, then

t^3 = u^3 = v^3 = 3tuv

3. If

t + u = v

, then

t^3 = u^3 = v^3 + 3tuv = 0

Which of the above statement(s) is/arecorrect?

Solution:

(1)

t^ 2 + u^2 + v^ 2 = tu + uv + vt

⇒ t^ 2 + u^2 + v^ 2 - tu - uv - vt = 0

⇒ t(t - u) (+u) (u - v) + v(v - t) = 0

This is possible only if

t = u = v

(2) Since

t + u + v = 0

t^3 + u^3 + v^ 3 = 3tuv

(3)

t - u - v = 0

∴ t^ 3 + u^3 - v^ 3 = 3tuv = 0

Thus only Statements (1) and (2) are correct

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