Question : If a( a + b + c) = 45, b( a + b + c) = 75, and c( a + b + c) =105, then what is the value of a2 + b2 + c2?
Option 1: 75
Option 2: 83
Option 3: 217
Option 4: 225
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Correct Answer: 83
Solution : a(a + b + c) = 45 ------------------------- (1) b(a + b + c) = 75 ------------------------- (2) c(a + b + c) = 105 ------------------------ (3) Adding (1), (2), and (3), we get (a + b + c)2 = 225 ⇒ (a + b + c) = 15 or (a + b + c) = –15 Case 1: (a + b + c) = 15 -------------------(4) Dividing (4) by (1), we get a = 3 Dividing (4) by (2), we get b = 5 Dividing (4) by (1), we get c = 7 (a2 + b2 + c2) = 32 + 52 + 72 = 83 Case 2: (a + b+ c) = –15 -------------(5) Dividing (5) by (1), we get a = –3 Dividing (5) by (2), we get b = –5 Dividing (5) by (1), we get c = –7 (a2 + b2 + c2) = (–3)2 + (–5)2 + (–7)2 = 83 Hence, the correct answer is 83.
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Question : What is the possible value of (a + b + c) – 3, if a2 + b2 + c2 = 9 and ab + bc + ca = 8?
Option 1: 5
Option 2: 3
Option 3: 9
Option 4: 2
Question : If a + b + c = 0, the value of (a3 + b3 + c3) is:
Option 1: abc
Option 2: 2abc
Option 3: 3abc
Option 4: 0
Question : a3 + b3 + c3 = 3abc if _________.
Option 1: a = b = c
Option 2: a + b + c = 0
Option 3: a + b + c = 1
Option 4: a + b + c = –1
Question : If a + b + c = 1, ab + bc + ca = –22 and abc = –40, then what is the value of a3 + b3 + c3 ?
Option 1: 67
Option 2: –53
Option 3: –51
Option 4: 27
Question : Directions: (2)* * 4 = 2, and (4)* * 4 = 16, then what is the value of A in (6)* * A = 18?
Option 1: 12
Option 2: 14
Option 3: 16
Option 4: 20
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