Algebra
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If x = y = z and x + y + z ≠ 0, then each ratio is xa + yb + zc ya + zb + xc za + xb + yc
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x = y = z xa + yb + zc ya + zb + xc za + xb + yc = x + y + z xa + yb + zc + ya + zb + xc + za + xb + yc = x + y + z xa + ya + za + yb + ya + yc + zc + zb + zc + za = x + y + z a(x + y + z) + b(x + y + z) + c (x + y + z) = x + y + z (a + b + c)(x + y + z) = 1 a + b + c Correct Option: D
x = y = z xa + yb + zc ya + zb + xc za + xb + yc = x + y + z xa + yb + zc + ya + zb + xc + za + xb + yc = x + y + z xa + ya + za + yb + ya + yc + zc + zb + zc + za = x + y + z a(x + y + z) + b(x + y + z) + c (x + y + z) = x + y + z (a + b + c)(x + y + z) = 1 a + b + c
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If x : y = 3 : 2, then the value of x + y is x - y
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x + 3 y 2
By componendo and dividend,x + y = 3 + 2 x - y 3 - 2 ⇒ x + y = 5 5 : 1 x - y 1 Correct Option: A
x + 3 y 2
By componendo and dividend,x + y = 3 + 2 x - y 3 - 2 ⇒ x + y = 5 5 : 1 x - y 1
- If a² + b² + c² – ab – bc – ca = 0, Then a : b : c is :
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a² + b² + c² – ab – bc – ca = 0
⇒ 2a² + 2b² + 2c² – 2ab – 2bc – 2ca = 0
⇒ (a² + b² – 2ab) + (b² + c² – 2bc) + (c² + a² – 2ca) = 0
⇒ (a – b)2 + (b – c)2 + (c – a)2 = 0
[If x² + y² + z² = 0 then, x = 0, y = 0, z = 0]
∴ a – b = 0 ⇒ a = b
b – c = 0 ⇒ b = c
c – a = 0 ⇒ c = a
∴ a = b = c
∴ a : b : c = 1 : 1 : 1Correct Option: B
a² + b² + c² – ab – bc – ca = 0
⇒ 2a² + 2b² + 2c² – 2ab – 2bc – 2ca = 0
⇒ (a² + b² – 2ab) + (b² + c² – 2bc) + (c² + a² – 2ca) = 0
⇒ (a – b)2 + (b – c)2 + (c – a)2 = 0
[If x² + y² + z² = 0 then, x = 0, y = 0, z = 0]
∴ a – b = 0 ⇒ a = b
b – c = 0 ⇒ b = c
c – a = 0 ⇒ c = a
∴ a = b = c
∴ a : b : c = 1 : 1 : 1
- If a² + 13b² + c² – 4ab – 6bc = 0, then a : b : c is
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a² + 13b² + c² – 4ab – 6bc = 0
⇒ a² – 4ab + 4b² + 9b² + c² – 6bc = 0
⇒ a² – 4ab + 4b² + c² – 6bc + 9b² = 0
⇒ (a – 2b)² + (c – 3b)² = 0
⇒ a – 2b = 0 and c – 3b = 0
⇒ a = 2b and c = 3b⇒ a = 2 and b = 1 b 1 c 3
∴ a : b : c = 2 : 1 : 3Correct Option: C
a² + 13b² + c² – 4ab – 6bc = 0
⇒ a² – 4ab + 4b² + 9b² + c² – 6bc = 0
⇒ a² – 4ab + 4b² + c² – 6bc + 9b² = 0
⇒ (a – 2b)² + (c – 3b)² = 0
⇒ a – 2b = 0 and c – 3b = 0
⇒ a = 2b and c = 3b⇒ a = 2 and b = 1 b 1 c 3
∴ a : b : c = 2 : 1 : 3
- If (2x – y)² + (3y – 2z)² = 0, then the ratio x : y : z is :
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If a² + b² = 0
⇒ a = 0 and b = 0
∴ (2x – y)² + (3y – 2z)² = 0
∴ 2x – y = 0 ⇒ 2x = y
⇒ x : y = 1 : 2 and, 3y – 2z = 0
⇒ 3y = 2z
⇒ y : z = 2 : 3
∴ x : y : z = 1 : 2 : 3Correct Option: B
If a² + b² = 0
⇒ a = 0 and b = 0
∴ (2x – y)² + (3y – 2z)² = 0
∴ 2x – y = 0 ⇒ 2x = y
⇒ x : y = 1 : 2 and, 3y – 2z = 0
⇒ 3y = 2z
⇒ y : z = 2 : 3
∴ x : y : z = 1 : 2 : 3
