Algebra
- The value of the expression x4 – 17x3 + 17x2 – 17x + 17 at x = 16 is
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x4 – 17x3 + 17x2 – 17x + 17
= x4 – 16x3 + 16x2 – 16x – x3 + x2 – x + 17
When x = 16,
Expression = 164 – 164 + 163 – 162 – 163 + 162 – 16 + 17 = 1Correct Option: B
x4 – 17x3 + 17x2 – 17x + 17
= x4 – 16x3 + 16x2 – 16x – x3 + x2 – x + 17
When x = 16,
Expression = 164 – 164 + 163 – 162 – 163 + 162 – 16 + 17 = 1
- If a, b, c are real and a2 + b2 + c2 = 2 (a – b – c) – 3, then the value of 2a – 3b + 4c is
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a2 + b2 + c2 = 2a – 2b – 2c – 3
⇒ a2 – 2a + b2 + 2b + c2 + 2c + 1 + 1 + 1 = 0
⇒ (a2 – 2a + 1) + (b2 + 2b + 1) + (c2 + 2c + 1) = 0
⇒ (a – 1)2 + (b + 1)2 +(c + 1)2 = 0
⇒ a – 1 = 0 ⇒ a = 1
⇒ b + 1 = 0 ⇒ b = –1
and c + 1 = 0 ⇒ c = –1
∴ 2a – 3b + 4c = 2 + 3 – 4 = 1Correct Option: C
a2 + b2 + c2 = 2a – 2b – 2c – 3
⇒ a2 – 2a + b2 + 2b + c2 + 2c + 1 + 1 + 1 = 0
⇒ (a2 – 2a + 1) + (b2 + 2b + 1) + (c2 + 2c + 1) = 0
⇒ (a – 1)2 + (b + 1)2 +(c + 1)2 = 0
⇒ a – 1 = 0 ⇒ a = 1
⇒ b + 1 = 0 ⇒ b = –1
and c + 1 = 0 ⇒ c = –1
∴ 2a – 3b + 4c = 2 + 3 – 4 = 1
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The value of 4 + 3√3 is 7 + 4√3
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Expression = 4 + 3√3 7 + 4√3
Rationalising the denominator,= (4 + 3√3)(7 − 4√3) (7 + 4√3)(7 − 4√3) = 28 − 16√3 + 21 √3 − 12 × 3 49 − 48
= 28 + 5√3 – 36 = 5√3 – 8Correct Option: A
Expression = 4 + 3√3 7 + 4√3
Rationalising the denominator,= (4 + 3√3)(7 − 4√3) (7 + 4√3)(7 − 4√3) = 28 − 16√3 + 21 √3 − 12 × 3 49 − 48
= 28 + 5√3 – 36 = 5√3 – 8
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If for non-zero, x, x2 – 4x – 1 = 0, the value of x2 + 1 is x2
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x2 – 4x – 1 = 0
⇒ x2 –1 = 4x
Dividing by xx − 1 = 4 x
On squaring both sides,
x − 1 
2 = 16 x ⇒ x2 + 1 – 2 = 16 x2 ⇒ x2 + 1 = 16 + 2 = 18 x2
Second Method :
Using Rule 5,
Here, x2 – 4x – 1 = 0
⇒ x2 – 1 = 4 x⇒ x2 – 1 = 4 x
We know thatx2 + 1 = x − 1 2 + 2 x2 x
= 42 + 2 = 18Correct Option: D
x2 – 4x – 1 = 0
⇒ x2 –1 = 4x
Dividing by xx − 1 = 4 x
On squaring both sides,x − 1 2 = 16 x ⇒ x2 + 1 – 2 = 16 x2 ⇒ x2 + 1 = 16 + 2 = 18 x2
Second Method :
Using Rule 5,
Here, x2 – 4x – 1 = 0
⇒ x2 – 1 = 4 x⇒ x2 – 1 = 4 x
We know thatx2 + 1 = x − 1 2 + 2 x2 x
= 42 + 2 = 18
- If x = z = 225 and y = 226 then the value of x3 + y3 + z3 - 3xyz is :
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Using Rule 22,
x = z = 225, y = 226
∴ x + y + z = 225 + 226 + 225 = 676∴ x3 + y3 + z3 – 3 xyz = 1 (x + y + z) [ (x - y)2 + (y - z)2 + (z - x)2 ] 2 ⇒ x3 + y3 + z3 – 3 xyz = 1 × 676 × [ (225 – 226)2 + (226 – 225)2 + (225 – 225)2 ] 2 ⇒ x3 + y3 + z3 – 3 xyz = 1 × 676 × (1 + 1) = 676 2
Correct Option: B
Using Rule 22,
x = z = 225, y = 226
∴ x + y + z = 225 + 226 + 225 = 676∴ x3 + y3 + z3 – 3 xyz = 1 (x + y + z) [ (x - y)2 + (y - z)2 + (z - x)2 ] 2 ⇒ x3 + y3 + z3 – 3 xyz = 1 × 676 × [ (225 – 226)2 + (226 – 225)2 + (225 – 225)2 ] 2 ⇒ x3 + y3 + z3 – 3 xyz = 1 × 676 × (1 + 1) = 676 2
