Project ForgeGeneral Aptitude
Quantitative Aptitude
GATE 2015
Hard
If a^2 + b^2 + c^2 = 1, then ab + bc + ac lies in the interval
- A[1, 2/3]
- [-1/2, 1]
- C[-1, 1/2]
- D[2, -4]
Explanation
Lower bound: (a+b+c)^2 >= 0 expands to a^2+b^2+c^2 + 2(ab+bc+ac) >= 0, i.e. 1 + 2S >= 0, so S >= -1/2. Upper bound: (a-b)^2+(b-c)^2+(c-a)^2 >= 0 expands to 2(a^2+b^2+c^2) - 2(ab+bc+ac) >= 0, i.e. 2 - 2S >= 0, so S <= 1. Combined: S is in [-1/2, 1].
Source: GATE ME 2015 (Session 1) (2015)
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