Question
If, 6x + y = 20, and 2xy = 32, and 6x > y, then find
the value of 216x³ – y³.Solution
2xy = 32 So, xy = (32/2) = 16 ......(i) 6x + y = 20 On squaring the above equation, we get, (6x + y)² = 20² Or, 36x² + y² + 2 × 6x × y = 400 Or, 36x² + y² + 12 × 16 = 400 (since, xy = 16) Or, 36x² + y² = 400 - 192 So, 36x² + y² = 208 ....(ii) (6x - y)² = 36x² + y² - 2 × 6x × y = 208 - 12 × 16 = 208 - 192 = 16 {using values from equations (i) and (ii)} Since, 6x > y, so, 6x - y, cannot be negative. So, 6x - y = 4 ....(iii) We know that, a³ - b³ = (a - b) × (a² + b² + ab). So, (6x)³ - (y)³ = 216x³ - y³ = (6x - y) × (36x² + y² + 6x × y) Putting the value of equations (i), (ii) and (iii) in the above equation, we get, = 4 × (208 + 6 × 16) = 4 × (208 + 96) = 4 × 304 = 1,216 Hence, option b.
Statements: P > R = S; T > S > U; Q < U = V
Conclusions:
I. Q < P
II. T > V
III. R ≥ V
In the question, assuming the given statements to be true, find which of the conclusion (s) among given three conclusions is /are definitely true and t...
In the question, assuming the given statements to be true, find which of the conclusion (s) among given four conclusions is/are definitely true and the...
Statements:
A < W ≤ T < Y = O; V < S > K ≥ F > O
Conclusions:
I). W < S
II). Y ≥ K
...Statements: J > O > B < N = T ≥ S < C < I ≤ P
Conclusion
I: P > B
II: N > S
Which of the following symbols should replace the sign (*) and ($) respectively in the expression ' B ≥ T ≥ D = Q = S * V ≤ F $ H ' in order to ma...
Statements: I % C, C & D, D $ K, K # Z
Conclusions: I. I & DÂ Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â II. D # Z
...Statement: F ≥ G > I > E ≤ P, E = S ≥ PÂ
Conclusion: I. F ≥ P         II. G > P
Statements: K > L ≥ M > N, N < O < P = J
Conclusion:
 I. K ≥ O
II. P > L
III. L > J
Statements: U < P = I ≤ V < T ≤ R = W > H = Z > O
Conclusions:
I. U < W
II. R > O
