Algebra
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If x + 1 = 2 , then the value of x7 + 
1 
is x x5
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x + 1 = 2 x
⇒ x2 + 1 = 2x
⇒ x2 - 2x + 1 = 0
⇒ (x - 1)2 = 0 ⇒ x = 1∴ x7 + 1 = 1 + 1 = 2 x5
Second Method :
Using Rule 16,Here, x + 1 = 2 x ∴ x7 + 1 = 2 x5
Correct Option: B
x + 1 = 2 x
⇒ x2 + 1 = 2x
⇒ x2 - 2x + 1 = 0
⇒ (x - 1)2 = 0 ⇒ x = 1∴ x7 + 1 = 1 + 1 = 2 x5
Second Method :
Using Rule 16,Here, x + 1 = 2 x ∴ x7 + 1 = 2 x5
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If p4 = 119 - 1 , then the value of p3 - 1 is p4 p3
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p4 = 119 - 1 p4 p4 + 1 = 119 p4 ⇒ p2 + 1 2 - 2 = 119 p2 ⇒ p2 + 1 2 = 119 + 2 = 121 p2 ⇒ p2 + 1 2 = (11)2 p2 ⇒ p2 + 1 = 11 p2
Again ,⇒ p - 1 2 + 2 = 11 p ⇒ p - 1 2 = 11 - 2 = 9 p ⇒ p2 + 1 = √9 = ±3 p2
On cubing both sides ,⇒ p - 1 3 = ±27 p ⇒ p3 - 1 - 3 × p × 1 p - 1 = ±27 p3 p p ⇒ p3 - 1 - 3 × (±3) = ±27 p3 ⇒ p3 - 1 = ±27 ± 9 = ±36 p3
Correct Option: C
p4 = 119 - 1 p4 p4 + 1 = 119 p4 ⇒ p2 + 1 2 - 2 = 119 p2 ⇒ p2 + 1 2 = 119 + 2 = 121 p2 ⇒ p2 + 1 2 = (11)2 p2 ⇒ p2 + 1 = 11 p2
Again ,⇒ p - 1 2 + 2 = 11 p ⇒ p - 1 2 = 11 - 2 = 9 p ⇒ p2 + 1 = √9 = ±3 p2
On cubing both sides ,⇒ p - 1 3 = ±27 p ⇒ p3 - 1 - 3 × p × 1 p - 1 = ±27 p3 p p ⇒ p3 - 1 - 3 × (±3) = ±27 p3 ⇒ p3 - 1 = ±27 ± 9 = ±36 p3
- If x + y + z = 6, then the value of (x - 1)3 + (y - 2)3 + (z - 3)3 is
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Using Rule 21,
If. a + b + c = 0, then a3 + b3 + c3 = 3abc
Here, x – 1 + y – 2 + z – 3 = x + y + z – 6
= 6 – 6 = 0
∴ (x - 1)3 + (y - 2)3 + (z - 3)3 = 3(x – 1 )(y – 2 )(z – 3 )Correct Option: A
Using Rule 21,
If. a + b + c = 0, then a3 + b3 + c3 = 3abc
Here, x – 1 + y – 2 + z – 3 = x + y + z – 6
= 6 – 6 = 0
∴ (x - 1)3 + (y - 2)3 + (z - 3)3 = 3(x – 1 )(y – 2 )(z – 3 )
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If a = 2 + √3 , then the value of a6 + a4 + a2 + 1 is a3
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a = 2 + √3
⇒ 1 = 1 a 2 + √3 ⇒ 1 = ( 2 - √3 ) a ( 2 + √3 )( 2 - √3 ) ⇒ 1 = 2 - √3 = 2 - √3 a 4 - 3 ∴ a + 1 = 2 + √3 + 2 - √3 = 4 a ∴ Expression = a6 + a4 + a2 + 1 a3 Expression = a3 + a + 1 + 1 a a3 Expression = a3 + 1 + a + 1 a3 a Expression = a + 1 3 - 3 a + 1 + a + 1 a a a Expression = a + 1 3 - 2 a + 1 a a
Expression = (4)3 – 2 × 4 = 64 – 8 = 56
Correct Option: D
a = 2 + √3
⇒ 1 = 1 a 2 + √3 ⇒ 1 = ( 2 - √3 ) a ( 2 + √3 )( 2 - √3 ) ⇒ 1 = 2 - √3 = 2 - √3 a 4 - 3 ∴ a + 1 = 2 + √3 + 2 - √3 = 4 a ∴ Expression = a6 + a4 + a2 + 1 a3 Expression = a3 + a + 1 + 1 a a3 Expression = a3 + 1 + a + 1 a3 a Expression = a + 1 3 - 3 a + 1 + a + 1 a a a Expression = a + 1 3 - 2 a + 1 a a
Expression = (4)3 – 2 × 4 = 64 – 8 = 56
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If a = 2 + √3 , then the value of a6 + a4 + a2 + 1 is a3
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a = 2 + √3
⇒ 1 = 1 a 2 + √3 ⇒ 1 = ( 2 - √3 ) a ( 2 + √3 )( 2 - √3 ) ⇒ 1 = 2 - √3 = 2 - √3 a 4 - 3 ∴ a + 1 = 2 + √3 + 2 - √3 = 4 a ∴ Expression = a6 + a4 + a2 + 1 a3 Expression = a3 + a + 1 + 1 a a3 Expression = a3 + 1 + a + 1 a3 a Expression = a + 1 3 - 3 a + 1 + a + 1 a a a Expression = a + 1 3 - 2 a + 1 a a
Expression = (4)3 – 2 × 4 = 64 – 8 = 56
Correct Option: D
a = 2 + √3
⇒ 1 = 1 a 2 + √3 ⇒ 1 = ( 2 - √3 ) a ( 2 + √3 )( 2 - √3 ) ⇒ 1 = 2 - √3 = 2 - √3 a 4 - 3 ∴ a + 1 = 2 + √3 + 2 - √3 = 4 a ∴ Expression = a6 + a4 + a2 + 1 a3 Expression = a3 + a + 1 + 1 a a3 Expression = a3 + 1 + a + 1 a3 a Expression = a + 1 3 - 3 a + 1 + a + 1 a a a Expression = a + 1 3 - 2 a + 1 a a
Expression = (4)3 – 2 × 4 = 64 – 8 = 56
