Algebra
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If x = 3 + 2√2 , then the value of x6 + x4 + x2 + 1 is equal to x3
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Expression = x6 + x4 + x2 + 1 x3 Expression = x6 + x4 + x2 + 1 x3 x3 x3 x3 Expression = x3 + x + 1 + 1 x x3 Expression = 
x3 + 1 
+ x + 1 x3 x Expression = x + 1 3 - 3 x + 1 + x + 1 x x x Expression = x + 1 3 - 2 x + 1 ---(i) x x
Now, x = 3 + 2√2∴ 1 = 1 x 3 + 2√2 = 1 × 3 - 2√2 3 + 2√2 3 - 2√2 1 = 3 - 2√2 x 9 - 8 ∴ x + 1 = 3 + 2√2 + 3 - 2√2 = 6 x
∴ Expression = (6)3 - 2 × 6 = 216 –12 = 204Correct Option: D
Expression = x6 + x4 + x2 + 1 x3 Expression = x6 + x4 + x2 + 1 x3 x3 x3 x3 Expression = x3 + x + 1 + 1 x x3 Expression = x3 + 1 + x + 1 x3 x Expression = x + 1 3 - 3 x + 1 + x + 1 x x x Expression = x + 1 3 - 2 x + 1 ---(i) x x
Now, x = 3 + 2√2∴ 1 = 1 x 3 + 2√2 = 1 × 3 - 2√2 3 + 2√2 3 - 2√2 1 = 3 - 2√2 x 9 - 8 ∴ x + 1 = 3 + 2√2 + 3 - 2√2 = 6 x
∴ Expression = (6)3 - 2 × 6 = 216 –12 = 204
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If x + 1 = 3 , then the value of 3x2 - 4x + 3 is x x2 - x + 1
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Given , x + 1 = 3 x Expression = 3x2 - 4x + 3 x2 - x + 1 Expression = (3x2 - 3x + 3) - x x2 - x + 1 Expression = 3(x2 - x + 1) - x x2 - x + 1 x2 - x + 1 Expression = 3 - 1 x - 1 + 1 x Expression = 3 - 1 x + 1 + 1 x Expression = 3 - 1 = 3 - 1 3 - 1 2 Expression = 6 - 1 = 5 2 2 Correct Option: C
Given , x + 1 = 3 x Expression = 3x2 - 4x + 3 x2 - x + 1 Expression = (3x2 - 3x + 3) - x x2 - x + 1 Expression = 3(x2 - x + 1) - x x2 - x + 1 x2 - x + 1 Expression = 3 - 1 x - 1 + 1 x Expression = 3 - 1 x + 1 + 1 x Expression = 3 - 1 = 3 - 1 3 - 1 2 Expression = 6 - 1 = 5 2 2
- If x = 997, y = 998 and z = 999, then the value of x2 + y2 + z2 – xy – yz – zx is
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x = 997 , y = 998 , z = 999
∴ x – y = 997 – 998 = –1
y – z = 998 – 999 = –1
z – x = 999 – 997 = 2∴ x2 + y2 + z2 – xy – yz – zx = 1 [ 2x2 + 2y2 + 2z2 – 2xy – 2yz – 2zx ] 2 ⇒ x2 + y2 + z2 – xy – yz – zx = 1 [ x2 + y2 – 2xy + y2 + z2 – 2yz + z2 + x2 – 2zx ] 2 ⇒ x2 + y2 + z2 – xy – yz – zx = 1 [ (x - y)2 + (y - z)2 + (z - x)2 ] 2 ⇒ x2 + y2 + z2 – xy – yz – zx = 1 [ (-1)2 + (-1)2 + (2)2 ] 2 ⇒ x2 + y2 + z2 – xy – yz – zx = 1 [ 1 + 1 + 4 ] 2 ⇒ x2 + y2 + z2 – xy – yz – zx = 1 × 6 = 3 2
Correct Option: D
x = 997 , y = 998 , z = 999
∴ x – y = 997 – 998 = –1
y – z = 998 – 999 = –1
z – x = 999 – 997 = 2∴ x2 + y2 + z2 – xy – yz – zx = 1 [ 2x2 + 2y2 + 2z2 – 2xy – 2yz – 2zx ] 2 ⇒ x2 + y2 + z2 – xy – yz – zx = 1 [ x2 + y2 – 2xy + y2 + z2 – 2yz + z2 + x2 – 2zx ] 2 ⇒ x2 + y2 + z2 – xy – yz – zx = 1 [ (x - y)2 + (y - z)2 + (z - x)2 ] 2 ⇒ x2 + y2 + z2 – xy – yz – zx = 1 [ (-1)2 + (-1)2 + (2)2 ] 2 ⇒ x2 + y2 + z2 – xy – yz – zx = 1 [ 1 + 1 + 4 ] 2 ⇒ x2 + y2 + z2 – xy – yz – zx = 1 × 6 = 3 2
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If a + 1 = √3 ,then the value
of a18 + a12 + a6 + 1 isa
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Using Rule 1 and 8,
a + 1 = √3 a
On squaring both sides ,⇒ a2 + 1 + 2 = 3 a2 ⇒ a2 + 1 = 3 – 2 = 1 a2
On cubing both sides,⇒ a2 + 1 3 = (1)3 a2 ⇒ a6 + 1 + 3 a2 + 1 = 1 a6 a2 ⇒ a6 + 1 = 1 - 3 = -2 a6 ⇒ a12 + 1 = -2 a6
⇒ a12 + 2a6 + 1 = 0
⇒ ( a6 + 1 )2 = 0
⇒ a6 + 1 = 0
∴ Expression = a18 + a12 + a6 + 1
Expression = a12(a6 + 1) + 1(a6 + 1) = 0
Correct Option: A
Using Rule 1 and 8,
a + 1 = √3 a
On squaring both sides ,⇒ a2 + 1 + 2 = 3 a2 ⇒ a2 + 1 = 3 – 2 = 1 a2
On cubing both sides,⇒ a2 + 1 3 = (1)3 a2 ⇒ a6 + 1 + 3 a2 + 1 = 1 a6 a2 ⇒ a6 + 1 = 1 - 3 = -2 a6 ⇒ a12 + 1 = -2 a6
⇒ a12 + 2a6 + 1 = 0
⇒ ( a6 + 1 )2 = 0
⇒ a6 + 1 = 0
∴ Expression = a18 + a12 + a6 + 1
Expression = a12(a6 + 1) + 1(a6 + 1) = 0
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If a + 1 2 = 3 , then the value of a3 + 1 is a a3
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Using Rule 8,
⇒ a + 1 2 = 3 a ⇒ a + 1 = √3 a
On cubing both sides,⇒ a + 1 3 = (√3)3 a ⇒ a3 + 1 + 3 a + 1 = 3√3 a3 a ⇒ a3 + 1 + 3 × √3 = 3√3 a3 ⇒ a3 + 1 = 3√3 - 3√3 = 0 a3
Correct Option: A
Using Rule 8,
⇒ a + 1 2 = 3 a ⇒ a + 1 = √3 a
On cubing both sides,⇒ a + 1 3 = (√3)3 a ⇒ a3 + 1 + 3 a + 1 = 3√3 a3 a ⇒ a3 + 1 + 3 × √3 = 3√3 a3 ⇒ a3 + 1 = 3√3 - 3√3 = 0 a3
