Algebra
- If a + b + c = 0, then
2a2 + 2b2 + 2c2 + 3 = ? b2 + c2 − a2 c2 + a2 − b2 a2 + b2 − c2
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If a + b + c = 0
a + b = – c
On squaring both sides,
⇒ a2 + b2 + 2ab = c2
Similarly,
a2 = b2 + c2 + 2ac
b2 = a2 + c2 + 2ac∴ Expression = 2a2 + 2b2 + 2c2 + 3 b2 + c2 − a2 c2 + a2 − b2 a2 + b2 − c2 L.H.S. = 2a2 + 1 + 2b2 + 1 + 2c2 + 1 b2 + c2 − a2 c2 + a2 − b2 a2 + b2 − c2 L.H.S. = 2a2 + 1 + 2b2 + 1 + 2c2 + 1 b2 + c2 − a2 c2 + a2 − b2 a2 + b2 − c2 = a2 + b2 + c2 + a2 + b2 + c2 + a2 + b2 + c2 = (a2 + b2 + c2) b2 + c2 − a2 c2 + a2 − b2 a2 + b2 − c2 
1 + 1 + 1 
b2 + c2 − b2 − c2 − 2bc c2 + a2 − c2 − a2 − 2ac a2 + b2 − a2 − b2 − 2ab = (a2 + b2 + c2) 
1 − 1 − 1 
−2bc 2ac 2ab = (a2 + b2 + c2) − a − b − c = 0 −2abc Correct Option: C
If a + b + c = 0
a + b = – c
On squaring both sides,
⇒ a2 + b2 + 2ab = c2
Similarly,
a2 = b2 + c2 + 2ac
b2 = a2 + c2 + 2ac∴ Expression = 2a2 + 2b2 + 2c2 + 3 b2 + c2 − a2 c2 + a2 − b2 a2 + b2 − c2 L.H.S. = 2a2 + 1 + 2b2 + 1 + 2c2 + 1 b2 + c2 − a2 c2 + a2 − b2 a2 + b2 − c2 L.H.S. = 2a2 + 1 + 2b2 + 1 + 2c2 + 1 b2 + c2 − a2 c2 + a2 − b2 a2 + b2 − c2 = a2 + b2 + c2 + a2 + b2 + c2 + a2 + b2 + c2 = (a2 + b2 + c2) b2 + c2 − a2 c2 + a2 − b2 a2 + b2 − c2 1 + 1 + 1 b2 + c2 − b2 − c2 − 2bc c2 + a2 − c2 − a2 − 2ac a2 + b2 − a2 − b2 − 2ab = (a2 + b2 + c2) 1 − 1 − 1 −2bc 2ac 2ab = (a2 + b2 + c2) − a − b − c = 0 −2abc
- If a2 + b2 + c2 = 2(2a – 3b – 5c) – 38, find the value of (a – b – c).
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a2 + b2 + c2 = 4a – 6b – 10c – 38
⇒ a2 + b2 + c2 –4a + 6b + 10c + 38 = 0
⇒ a2 – 4a + 4 + b2 + 6b + 9 + c2 + 10c + 25 = 0
⇒ (a – 2)2 + (b + 3)2 + (c + 5)2 = 0
∴ a – 2 = 0 ⇒ a = 2
b + 3 = 0 ⇒ b = –3
c + 5 = 0 ⇒ c = –5
∴ a – b – c = 2 + 3 + 5 = 10Correct Option: B
a2 + b2 + c2 = 4a – 6b – 10c – 38
⇒ a2 + b2 + c2 –4a + 6b + 10c + 38 = 0
⇒ a2 – 4a + 4 + b2 + 6b + 9 + c2 + 10c + 25 = 0
⇒ (a – 2)2 + (b + 3)2 + (c + 5)2 = 0
∴ a – 2 = 0 ⇒ a = 2
b + 3 = 0 ⇒ b = –3
c + 5 = 0 ⇒ c = –5
∴ a – b – c = 2 + 3 + 5 = 10
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If x2 – 3x + 1 = 0, find the value of x3 + 1 . x3
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x2 – 3x + 1 = 0
⇒ x2 + 1 = 3x⇒ x2 + 1 = 3 x ⇒ x + 1 = 3 x
On cubing both sides,x + 1 3 = 27 x ⇒ x3 + 1 + 3 x + 1 = 27 x3 x ⇒ x3 + 1 + 3 × 3 = 27 x3 ⇒ x3 + 1 = 27 − 9 = 18 x3 Correct Option: A
x2 – 3x + 1 = 0
⇒ x2 + 1 = 3x⇒ x2 + 1 = 3 x ⇒ x + 1 = 3 x
On cubing both sides,x + 1 3 = 27 x ⇒ x3 + 1 + 3 x + 1 = 27 x3 x ⇒ x3 + 1 + 3 × 3 = 27 x3 ⇒ x3 + 1 = 27 − 9 = 18 x3
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If ax + by = 1 and bx + ay = 2ab then (x2 + y2)(a2 + b2)is equal to a2 + b2
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ax + by – 1 = 0
bx + ay – 2ab = 0 a2 + b2
By cross-multiplication.Correct Option: A
ax + by – 1 = 0
bx + ay – 2ab = 0 a2 + b2
By cross-multiplication.
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If a + 1 = b + 1 = c + 1 (where a ≠ b ≠ c), then abc is equal to b c a
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a + 1 = b + 1 = c + 1 = ±1 (let) b c a ⇒ a + 1 = 1 b
⇒ ab + 1 = b ⇒ ab = b – 1b + 1 = 1 ⇒ 1 = 1 – b c c c = 1 1 – b ∴ abc = b – 1 = – 1 1 – b Again, a + 1 = – 1 b
⇒ ab + 1 = –b ⇒ ab = –b – 1b + 1 = –1 ⇒ 1 = – 1 – b c c c = 1 – 1 – b
∴ abc = 1
∴ abc = ±1Correct Option: C
a + 1 = b + 1 = c + 1 = ±1 (let) b c a ⇒ a + 1 = 1 b
⇒ ab + 1 = b ⇒ ab = b – 1b + 1 = 1 ⇒ 1 = 1 – b c c c = 1 1 – b ∴ abc = b – 1 = – 1 1 – b Again, a + 1 = – 1 b
⇒ ab + 1 = –b ⇒ ab = –b – 1b + 1 = –1 ⇒ 1 = – 1 – b c c c = 1 – 1 – b
∴ abc = 1
∴ abc = ±1
