Algebra
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If x + y + z = 1, 1 + 1 + 1 = 1 and xyz = – 1, then x³ + y³ + z³ is equal to x y z
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x + y + z = 1 ..... (i)
Again,1 + 1 + 1 = yz + zx + xy = 1 x y z xyz
⇒ xy + yz + zx = xyz = –1 .. (ii)
∴ (x + y + z)² = x² + y² + z² + 2 (xy + yz + zx)
⇒ 1 = x² + y² + z² – 2
⇒ x² + y² + z² = 2 + 1 = 3 .(iii)
∴ x³ + y³ + z³ – 3xyz
= (x + y + z) (x² + y² + z² – xy – yz – zx)
= 1 (3 + 1) = 4
⇒ x³ + y³ + z³ + 3 = 4
⇒ x³ + y³ + z³ = 4 – 3 = 1Correct Option: B
x + y + z = 1 ..... (i)
Again,1 + 1 + 1 = yz + zx + xy = 1 x y z xyz
⇒ xy + yz + zx = xyz = –1 .. (ii)
∴ (x + y + z)² = x² + y² + z² + 2 (xy + yz + zx)
⇒ 1 = x² + y² + z² – 2
⇒ x² + y² + z² = 2 + 1 = 3 .(iii)
∴ x³ + y³ + z³ – 3xyz
= (x + y + z) (x² + y² + z² – xy – yz – zx)
= 1 (3 + 1) = 4
⇒ x³ + y³ + z³ + 3 = 4
⇒ x³ + y³ + z³ = 4 – 3 = 1
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If 1 (a² + 1) = 3 the value of 
a6 + 1 
is : a a³
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a² + 1 = 3 a ⇒ a + 1 = 3 a
On cubing both sides,a + 1 ³ = 3³ a ⇒ a³ + 1 + 3 a + 1 = 27 a³ a ⇒ a³ + 1 + 3 × 3 = 27 a³ ⇒ a6 + 1 = 27 - 9 = 18 a³ Correct Option: B
a² + 1 = 3 a ⇒ a + 1 = 3 a
On cubing both sides,a + 1 ³ = 3³ a ⇒ a³ + 1 + 3 a + 1 = 27 a³ a ⇒ a³ + 1 + 3 × 3 = 27 a³ ⇒ a6 + 1 = 27 - 9 = 18 a³
- The third proportional of the following numbers (x – y)², (x² – y²)² is :
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If c be the third proportional between a and b, then
a = b b c ⇒ a³ + 1 + 3 × 3 = 27 a³ ⇒ c = b² = {(x² - y²)²}² a (x - y)² = {(x + y)}4 (x - y)²
= (x + y)4 (x – y)²Correct Option: B
If c be the third proportional between a and b, then
a = b b c ⇒ a³ + 1 + 3 × 3 = 27 a³ ⇒ c = b² = {(x² - y²)²}² a (x - y)² = {(x + y)}4 (x - y)²
= (x + y)4 (x – y)²
- If (x – 5)² + (y – 2)² + (z – 9)² = 0, then value of (x + y – z) is :
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If a² + b² + c² = 0
⇒ a = 0, b = 0, c = 0
∴ (x – 5)² + (y – 2)² + (z – 9)² = 0
∴ x – 5 = 0 ⇒ x = 5
y – 2 = 0 ⇒ y = 2
z – 9 = 0 ⇒ z = 9
∴ x + y – z = 5 + 2 – 9 = –2Correct Option: C
If a² + b² + c² = 0
⇒ a = 0, b = 0, c = 0
∴ (x – 5)² + (y – 2)² + (z – 9)² = 0
∴ x – 5 = 0 ⇒ x = 5
y – 2 = 0 ⇒ y = 2
z – 9 = 0 ⇒ z = 9
∴ x + y – z = 5 + 2 – 9 = –2
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If x + 1 = 3 then x8 + 1 is equal to x x8
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x = 1 = 3 x
On squaring both sides,x + 1 ² = 9 x ⇒ x² + 1 + 2 = 9 x² ⇒ x² + 1 = 9 - 2 = 7 x²
On squaring again,x² + 1 = 49 x² ⇒ x4 + 1 + 2 = 49 a4 ⇒ x4 + 1 = 49 - 2 = 47 a4
On squaring again,(x4)² + 1 ² + 2 = 47² = 2209 x4 ⇒ x8 + 1 = 2209 - 2 = 2207 x8 Correct Option: C
x = 1 = 3 x
On squaring both sides,x + 1 ² = 9 x ⇒ x² + 1 + 2 = 9 x² ⇒ x² + 1 = 9 - 2 = 7 x²
On squaring again,x² + 1 = 49 x² ⇒ x4 + 1 + 2 = 49 a4 ⇒ x4 + 1 = 49 - 2 = 47 a4
On squaring again,(x4)² + 1 ² + 2 = 47² = 2209 x4 ⇒ x8 + 1 = 2209 - 2 = 2207 x8
