Algebra
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If a = 7 + 4√3, find the value of 3a6 + 2a4 + 4a3 + 2a2 + 3 . a4 + a3 + a2
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a = 7 + 4√3
∴ 1 = 1 a 7 + 4√3 = 1 × 7 − 4√3 7 + 4√3 7 − 4√3 = 7 − 4√3 = 7 − 4√3 49 − 48 Expression = 3a6 + 2a4 + 4a3 + 2a2 + 3 a4 + a3 + a2 = 3a3 + 2a + 4 + 2 + 3 a a3 a + 1 + 1 a
[Dividing numerator and denominator by a3]
= 3((14)3 − 3 × 14) + 2 × 14 + 4 14 + 1 = 3 × 2702 + 28 + 4 15 = 8138 15 Correct Option: B
a = 7 + 4√3
∴ 1 = 1 a 7 + 4√3 = 1 × 7 − 4√3 7 + 4√3 7 − 4√3 = 7 − 4√3 = 7 − 4√3 49 − 48 Expression = 3a6 + 2a4 + 4a3 + 2a2 + 3 a4 + a3 + a2 = 3a3 + 2a + 4 + 2 + 3 a a3 a + 1 + 1 a
[Dividing numerator and denominator by a3]= 3((14)3 − 3 × 14) + 2 × 14 + 4 14 + 1 = 3 × 2702 + 28 + 4 15 = 8138 15
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If 2x + 1 = 3, then the value of x3 + 1 + 2 is x x3
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2x + 2 = 3 x
On dividing by 2,x + 1 = 3 x 2
On cubing both sides,
x + 1 
3 = 3 3 x 2 ⇒ x3 + 1 + 3 x + 1 = 27 x3 x 8 ⇒ x3 + 1 + 3 × 3 = 27 x3 2 8 ⇒ x3 + 1 + 9 = 27 x3 2 8 ⇒ x3 + 1 = 27 − 9 x3 8 2 = 27 − 26 = 9 8 8 ∴ x3 + 1 + 2 x3 = 2 − 9 = 16 − 9 8 8 = 7 8 Correct Option: D
2x + 2 = 3 x
On dividing by 2,x + 1 = 3 x 2
On cubing both sides,x + 1 3 = 3 3 x 2 ⇒ x3 + 1 + 3 x + 1 = 27 x3 x 8 ⇒ x3 + 1 + 3 × 3 = 27 x3 2 8 ⇒ x3 + 1 + 9 = 27 x3 2 8 ⇒ x3 + 1 = 27 − 9 x3 8 2 = 27 − 26 = 9 8 8 ∴ x3 + 1 + 2 x3 = 2 − 9 = 16 − 9 8 8 = 7 8
- If x = 1 + √2 + √3, then the value of (2x4 – 8x3 + 26x – 28) is
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x = 1 + √2 + √3
⇒ x - 1 = √3 + √2
On squaring both sides ,
x2 – 2x + 1 = 3 + 2 + 2√6
⇒ x2 – 2x + 1 - 5 = 2√6
⇒ x2 – 2x - 4 = 2√6
On squaring again,
(x2 – 2x - 4)2 = (2√6)2
⇒ x4 + 4x2 + 16 – 4x3 + 16x - 8x2 = 24
⇒ x4 – 4x3 - 4x2 + 16x - 8x2 - 8 = 0
⇒ 2x4 – 8x3 - 8x2 + 32x - 16 = 0
∴ 2x4 – 8x3 - 5x2 + 32x - 16 + 3x2 - 6x - 12
= 0 + 3 (x2 – 2x – 4) = 3 × 2√6
Required answer = 6√6Correct Option: D
x = 1 + √2 + √3
⇒ x - 1 = √3 + √2
On squaring both sides ,
x2 – 2x + 1 = 3 + 2 + 2√6
⇒ x2 – 2x + 1 - 5 = 2√6
⇒ x2 – 2x - 4 = 2√6
On squaring again,
(x2 – 2x - 4)2 = (2√6)2
⇒ x4 + 4x2 + 16 – 4x3 + 16x - 8x2 = 24
⇒ x4 – 4x3 - 4x2 + 16x - 8x2 - 8 = 0
⇒ 2x4 – 8x3 - 8x2 + 32x - 16 = 0
∴ 2x4 – 8x3 - 5x2 + 32x - 16 + 3x2 - 6x - 12
= 0 + 3 (x2 – 2x – 4) = 3 × 2√6
Required answer = 6√6
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If x + 1 = √3 the value of (x18 + x12 + x6 + 1) is x
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x + 1 = √3 x
On cubing both sides,⇒ x + 1 3 = (3√3)3 x ⇒ x3 + 1 + 3x. 1 x + 1 = 3√3 x3 x x ⇒ x3 + 1 + 3 × √3 = 3√3 x3 ⇒ x3 + 1 = 3√3 - 3√3 = 0 x3
⇒ x6 + 1 = 0
∴ x18 + x12 + x6 + 1 = x12( x6 + 1 ) + 1( x6 + 1 )
x18 + x12 + x6 + 1 = ( x6 + 1 )( x12 + 1 ) = 0Correct Option: A
x + 1 = √3 x
On cubing both sides,⇒ x + 1 3 = (3√3)3 x ⇒ x3 + 1 + 3x. 1 x + 1 = 3√3 x3 x x ⇒ x3 + 1 + 3 × √3 = 3√3 x3 ⇒ x3 + 1 = 3√3 - 3√3 = 0 x3
⇒ x6 + 1 = 0
∴ x18 + x12 + x6 + 1 = x12( x6 + 1 ) + 1( x6 + 1 )
x18 + x12 + x6 + 1 = ( x6 + 1 )( x12 + 1 ) = 0
- If a = 299, b = 298, c = 297 then the value of 2a3 + 2b3 + 2c3 – 6abc is
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2a3 + 2b3 + 2c3 – 6abc
= 2 (a3 + b3 + c3 – 3abc)= 2 (a + b + c) × 1 {(a – b)2 + (b –
c)2 + (c – a)2}2
= (299 + 298 + 297) {(299 – 298)2 + (298 – 297)2 + (297 – 299)2}
= 894 × (1 + 1 + 4)
= 894 × 6 = 5364Correct Option: C
2a3 + 2b3 + 2c3 – 6abc
= 2 (a3 + b3 + c3 – 3abc)= 2 (a + b + c) × 1 {(a – b)2 + (b –
c)2 + (c – a)2}2
= (299 + 298 + 297) {(299 – 298)2 + (298 – 297)2 + (297 – 299)2}
= 894 × (1 + 1 + 4)
= 894 × 6 = 5364
