Algebra
- If ³√a + ³√b = ³√c , then the simplest value of (a + b – c)3 + 27abc is
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Given , ³√a + ³√b = ³√c
⇒ ³√a + ³√b - ³√c = 0
∴ a + b – c = -3(abc)1 / 3
On cubing both sides,
(a + b – c)3 = -27abc
∴ (a + b – c)3 + 27abc = 0Correct Option: D
Given , ³√a + ³√b = ³√c
⇒ ³√a + ³√b - ³√c = 0
∴ a + b – c = -3(abc)1 / 3
On cubing both sides,
(a + b – c)3 = -27abc
∴ (a + b – c)3 + 27abc = 0
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If t2 – 4t + 1 = 0, then the value of t3 + 1 is t3
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t2 – 4t + 1 = 0
⇒ t2 + 1 = 4t⇒ t2 + 1 = 4 t ⇒ t + 1 = 4 t
On cubing both sides,⇒ 
t + 1 
3 = 43 t ⇒ t3 + 1 + 3 t + 1 = 64 t3 t ⇒ t3 + 1 + 3 × 4 = 64 t3 ⇒ t3 + 1 = 64 – 12 = 52 t3 Correct Option: C
t2 – 4t + 1 = 0
⇒ t2 + 1 = 4t⇒ t2 + 1 = 4 t ⇒ t + 1 = 4 t
On cubing both sides,⇒ t + 1 3 = 43 t ⇒ t3 + 1 + 3 t + 1 = 64 t3 t ⇒ t3 + 1 + 3 × 4 = 64 t3 ⇒ t3 + 1 = 64 – 12 = 52 t3
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If a3 + b3 = 9 and a + b = 3, then the value of 1 + 1 is : a b
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a3 + b3 = (a + b)(a2 - ab + b2)
⇒ 9 = 3(a2 - ab + b2) { ∴ a + b = 3 }
⇒ a2 - ab + b2 = 9 ÷ 3 = 3
⇒ (a - b)2 – 2ab – ab = 3
⇒ 9 – 3ab = 3
⇒ 3ab = 9 – 3 = 6
⇒ ab = 2∴ 1 + 1 = a + b = 3 a b ab 2
Correct Option: B
a3 + b3 = (a + b)(a2 - ab + b2)
⇒ 9 = 3(a2 - ab + b2) { ∴ a + b = 3 }
⇒ a2 - ab + b2 = 9 ÷ 3 = 3
⇒ (a - b)2 – 2ab – ab = 3
⇒ 9 – 3ab = 3
⇒ 3ab = 9 – 3 = 6
⇒ ab = 2∴ 1 + 1 = a + b = 3 a b ab 2
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The value of 4x3 - x when x = 9999 is (2x + 1)(6x - 3)
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Expression = 4x3 - x (2x + 1)(6x - 3) Expression = x(4x2 - 1) (2x + 1) × 3(2x - 1) Expression = x(2x + 1)(2x - 1) 3(2x + 1)(2x - 1) Expression = x = 9999 = 3333 3 3
Correct Option: C
Expression = 4x3 - x (2x + 1)(6x - 3) Expression = x(4x2 - 1) (2x + 1) × 3(2x - 1) Expression = x(2x + 1)(2x - 1) 3(2x + 1)(2x - 1) Expression = x = 9999 = 3333 3 3
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If x > 1 and x + 1 = 2 1 , then the value of x4 + 25 is x 12 x4
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x + 1 = 2 1 = 25 x 12 12
On squaring both sides,⇒ x + 1 2 = 25 2 x 12 ⇒ x2 + 1 + 2 = 625 x2 144 ⇒ x2 + 1 = 625 - 2 x2 144 ⇒ x2 + 1 = 625 - 228 = 337 x2 144 144 ⇒ x - 1 2 + 2 = 337 x 144 ⇒ x - 1 2 = 337 - 2 x 144 = 337 - 288 = 49 144 144 ⇒ x - 1 = √ 49 = 7 x 144 12 
Required answer = 337 × 25 × 7 = 58975 144 12 12 20736 Correct Option: A
x + 1 = 2 1 = 25 x 12 12
On squaring both sides,⇒ x + 1 2 = 25 2 x 12 ⇒ x2 + 1 + 2 = 625 x2 144 ⇒ x2 + 1 = 625 - 2 x2 144 ⇒ x2 + 1 = 625 - 228 = 337 x2 144 144 ⇒ x - 1 2 + 2 = 337 x 144 ⇒ x - 1 2 = 337 - 2 x 144 = 337 - 288 = 49 144 144 ⇒ x - 1 = √ 49 = 7 x 144 12 Required answer = 337 × 25 × 7 = 58975 144 12 12 20736
