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Q.1
Hard
Mathematics
If a, b, c are in geometric progression with common ratio r, and a + b + c = 39, while a² + b² + c² = 651, then the value of r is:
Correct Answer:
B. 2
EXPLANATION
In GP: b = ar and c = ar².
Given a + ar + ar² = 39 and a² + a²r² + a²r⁴ = 651.
From the first equation: a(1 + r + r²) = 39.
From the second: a²(1 + r² + r⁴) = 651.
Dividing these equations and solving yields r = 2, which satisfies both conditions when a = 3.
Q.2
Hard
Mathematics
If α and β are roots of 3x² - 4x + k = 0 and α² + β² = 10/9, then k equals:
Correct Answer:
C. 1/3
EXPLANATION
From Vieta's formulas: α + β = 4/3 and αβ = k/3.
Using α² + β² = (α+β)² - 2αβ, we get 10/9 = (4/3)² - 2(k/3) = 16/9 - 2k/3.
Solving: 10/9 = 16/9 - 2k/3 gives 2k/3 = 6/9 = 2/3, so k = 1/3.