Algebra
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The value of 1 + 1 + 1 is ( a² + ax + x² ) ( a² - ax + x² ) ( a4 + a2x2 + x4 )
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Expression = 1 + 1 + 2ax a² + ax + x² a² - ax + x² a4 + a2x2 + x4 Expression = ( a² - ax + x² - a² - ax - x² ) + 2ax ( a² + ax + x² )( a² - ax + x² ) ( a4 + a2x2 + x4 ) Expression = -2ax + 2ax = 0 ( a4 + a2x2 + x4 ) ( a4 + a2x2 + x4 ) Correct Option: D
Expression = 1 + 1 + 2ax a² + ax + x² a² - ax + x² a4 + a2x2 + x4 Expression = ( a² - ax + x² - a² - ax - x² ) + 2ax ( a² + ax + x² )( a² - ax + x² ) ( a4 + a2x2 + x4 ) Expression = -2ax + 2ax = 0 ( a4 + a2x2 + x4 ) ( a4 + a2x2 + x4 )
- If x = 11, then the value of x5 – 12x4 + 12x3 – 12x2 + 12x – 1 is
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x = 11 (Given)
∴ x5 – 12x4 + 12x3 – 12x2 + 12x – 1
= x5 – (11 + 1)x4 + (11 + 1)x3 – (11 + 1)x2 + (11 + 1)x – 1
= x5 – 11x4 - x4 + 11x3 + x3 – 11x2 - x2 + 11x + x – 1
When x = 11,
= 115 – 115 - 114 + 114 + 113 – 113 - 112 + 112 + 11 – 1
= 0 - 0 + 0 - 0 + 10 = 10Correct Option: B
x = 11 (Given)
∴ x5 – 12x4 + 12x3 – 12x2 + 12x – 1
= x5 – (11 + 1)x4 + (11 + 1)x3 – (11 + 1)x2 + (11 + 1)x – 1
= x5 – 11x4 - x4 + 11x3 + x3 – 11x2 - x2 + 11x + x – 1
When x = 11,
= 115 – 115 - 114 + 114 + 113 – 113 - 112 + 112 + 11 – 1
= 0 - 0 + 0 - 0 + 10 = 10
- If x = k3 – 3k2 and y = 1 – 3k then for what value of k, will be x = y ?
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Using Rule 9
x = k3 – 3k2
y = 1 – 3k
When x = y, then
k3 – 3k2 = 1 – 3k
⇒ k3 – 3k2 + 3k – 1 = 0
⇒ (k – 1)3 = 0 ⇒ k – 1 = 0
⇒ k = 1Correct Option: B
Using Rule 9
x = k3 – 3k2
y = 1 – 3k
When x = y, then
k3 – 3k2 = 1 – 3k
⇒ k3 – 3k2 + 3k – 1 = 0
⇒ (k – 1)3 = 0 ⇒ k – 1 = 0
⇒ k = 1
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If 1 = a ³√4 + b ³√2 + c and a, b, c are rational numbers, then a + b + c is equal to [ ³√4 + ³√2 + 1 ]
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1 = a ³√4 + b ³√2 + c [ ³√4 + ³√2 + 1 ] ⇒ 1 = a.22 / 3 + b.21 / 3 + c ( 22 / 3 + 21 / 3 + 1 ) ⇒ ( 21 / 3 - 1 ) = a.22 / 3 + b.21 / 3 + c ( 21 / 3 - 1 )( 22 / 3 + 21 / 3 + 1 ) ⇒ ( 21 / 3 - 1 ) = a.22 / 3 + b.21 / 3 + c ( 2 - 1 )
[ ∵ ( a + b )(a2 + b2 - ab) = a3 - b3 ]
⇒ a = 0, b = 1, c = – 1
∴ a + b + c = 0 + 1 – 1 = 0Correct Option: A
1 = a ³√4 + b ³√2 + c [ ³√4 + ³√2 + 1 ] ⇒ 1 = a.22 / 3 + b.21 / 3 + c ( 22 / 3 + 21 / 3 + 1 ) ⇒ ( 21 / 3 - 1 ) = a.22 / 3 + b.21 / 3 + c ( 21 / 3 - 1 )( 22 / 3 + 21 / 3 + 1 ) ⇒ ( 21 / 3 - 1 ) = a.22 / 3 + b.21 / 3 + c ( 2 - 1 )
[ ∵ ( a + b )(a2 + b2 - ab) = a3 - b3 ]
⇒ a = 0, b = 1, c = – 1
∴ a + b + c = 0 + 1 – 1 = 0
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If x = ³√2 + √3 , then the value of x3 + 1 is x3
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x = ³√2 + √3
Cubing on both sides ,
⇒ x3 = 2 + √31 = 1 x3 2 + √3 1 = 1 × ( 2 - √3 ) x3 ( 2 + √3 ) × ( 2 - √3 ) = 2 - √3 = 2 - √3 4 - 3 ∴ x3 + 1 = 2 + √3 + 2 - √3 = 4 x3 Correct Option: D
x = ³√2 + √3
Cubing on both sides ,
⇒ x3 = 2 + √31 = 1 x3 2 + √3 1 = 1 × ( 2 - √3 ) x3 ( 2 + √3 ) × ( 2 - √3 ) = 2 - √3 = 2 - √3 4 - 3 ∴ x3 + 1 = 2 + √3 + 2 - √3 = 4 x3
