Algebra
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If x + 1 = 66 , then the value of x2 + x + 2 is x x2( 1 - x )
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⇒ x + 2 = 1 x Expression = x2 + x + 2 = [ x + 1 + ( 2 / x ) ] x2( 1 - x ) x( 1 - x )
(Dividing numerator and denominator by x)Expression = x + 2 = [ x + 1 + ( 2 / x ) ] x2( 1 - x ) x( 1 - x ) Expression = x + 2 + 1 x x( 1 - x ) Expression = 1 + 1 2 × 2 x Expression = 2 = 1 2 Correct Option: A
⇒ x + 2 = 1 x Expression = x2 + x + 2 = [ x + 1 + ( 2 / x ) ] x2( 1 - x ) x( 1 - x )
(Dividing numerator and denominator by x)Expression = x + 2 = [ x + 1 + ( 2 / x ) ] x2( 1 - x ) x( 1 - x ) Expression = x + 2 + 1 x x( 1 - x ) Expression = 1 + 1 2 × 2 x Expression = 2 = 1 2
- If p = 99, then the value of p(p2 + 3p + 3) is
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Using Rule 8,
p = 99 (Given)
∴ p(p2 + 3p + 3) = p3 + 3p2 + 3p
= p3 + 3p2 + 3p + 1 - 1
= (p + 1)3 - 1 = (99 + 1)3 - 1
∴ p(p2 + 3p + 3) = 1003 – 1 = 999999Correct Option: C
Using Rule 8,
p = 99 (Given)
∴ p(p2 + 3p + 3) = p3 + 3p2 + 3p
= p3 + 3p2 + 3p + 1 - 1
= (p + 1)3 - 1 = (99 + 1)3 - 1
∴ p(p2 + 3p + 3) = 1003 – 1 = 999999
- Find the value of √(x² + y² + z)(x + y - 3z) ÷ ³√xy³z² when x = + 1, y = –3, z = – 1.
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Expression = √(x² + y² + z)(x + y - 3z) = 1 ³√xy³z²
Putting x = 1, y = –3, z = –1Expression = √(1 + 9 - 1)(1 - 3 + 3) = 1 ³√ 1 × (-27) × 1 Expression = 3 = -1 -3
Note : Original question is :
√(x² + y² + z)(x + y - 3z) ÷ ³√xy³z²
which gives answer = – √7 which is not in options.Correct Option: C
Expression = √(x² + y² + z)(x + y - 3z) = 1 ³√xy³z²
Putting x = 1, y = –3, z = –1Expression = √(1 + 9 - 1)(1 - 3 + 3) = 1 ³√ 1 × (-27) × 1 Expression = 3 = -1 -3
Note : Original question is :
√(x² + y² + z)(x + y - 3z) ÷ ³√xy³z²
which gives answer = – √7 which is not in options.
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The simplest form of the expression ( p² - p ) ÷ ( p² - 1 ) ÷ p² is 2p³ + 6p² p² + 3p p + 1
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Expression = ( p² - p ) ÷ ( p² - 1 ) ÷ p² 2p³ + 6p² p² + 3p p + 1 Expression = p( p - 1 ) ÷ ( p + 1 )( p - 1 ) ÷ p² 2p²( p + 3 ) p( p + 3 ) ( p + 1 ) Expression = p( p - 1 ) × p( p + 3 ) × ( p + 1 ) 2p²( p + 3 ) ( p + 1 )( p - 1 ) p² Expression = 1 2p²
Correct Option: B
Expression = ( p² - p ) ÷ ( p² - 1 ) ÷ p² 2p³ + 6p² p² + 3p p + 1 Expression = p( p - 1 ) ÷ ( p + 1 )( p - 1 ) ÷ p² 2p²( p + 3 ) p( p + 3 ) ( p + 1 ) Expression = p( p - 1 ) × p( p + 3 ) × ( p + 1 ) 2p²( p + 3 ) ( p + 1 )( p - 1 ) p² Expression = 1 2p²
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If x + 1 = 2 , then the value of 
x2 + 1 
x3 + 1 is x x2 x3
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⇒ x + 1 = 2 x
On squaring both sides,⇒ x2 + 1 + 2 = 4 x2 ⇒ x2 + 1 = 4 – 2 = 2 x2 Again , x + 1 = 2 x
On cubing both sides,⇒ x + 1 3 = 8 x ⇒ x3 + 1 + 3 x + 1 = 8 x3 x ⇒ x3 + 1 = 8 - 3 × 2 = 2 x3 ∴ x2 + 1 x3 + 1 = 2 × 2 = 4 x2 x3
Second Method :
Using Rule 14,Here, x + 1 = 2 x x2 + 1 = 2 and x3 + 1 = 2 x2 x3 ∴ x2 + 1 x3 + 1 = 2 × 2 = 4 x2 x3 Correct Option: B
⇒ x + 1 = 2 x
On squaring both sides,⇒ x2 + 1 + 2 = 4 x2 ⇒ x2 + 1 = 4 – 2 = 2 x2 Again , x + 1 = 2 x
On cubing both sides,⇒ x + 1 3 = 8 x ⇒ x3 + 1 + 3 x + 1 = 8 x3 x ⇒ x3 + 1 = 8 - 3 × 2 = 2 x3 ∴ x2 + 1 x3 + 1 = 2 × 2 = 4 x2 x3
Second Method :
Using Rule 14,Here, x + 1 = 2 x x2 + 1 = 2 and x3 + 1 = 2 x2 x3 ∴ x2 + 1 x3 + 1 = 2 × 2 = 4 x2 x3
