Algebra
- If m = – 4, n = – 2, then the value of m3 - 3m2 + 3m + 3n + 3n2 + n3 is
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Using Rule 8 and 9,
Expression = m3 - 3m2 + 3m + 3n + 3n2 + n3
Expression = m3 - 3m2 + 3m - 1 + n3 + 3n2 + 3n + 1
Expression = (m - 1)3 + (n + 1)3
Expression = (-4 - 1)3 + (-2 + 1)3
Expression = (-5)3 + (-1)3 = – 125 – 1 = – 126Correct Option: A
Using Rule 8 and 9,
Expression = m3 - 3m2 + 3m + 3n + 3n2 + n3
Expression = m3 - 3m2 + 3m - 1 + n3 + 3n2 + 3n + 1
Expression = (m - 1)3 + (n + 1)3
Expression = (-4 - 1)3 + (-2 + 1)3
Expression = (-5)3 + (-1)3 = – 125 – 1 = – 126
- If x = 332, y = 333, z = 335, then the value of x3 + y3 + z3 – 3xyz is
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Using Rule 22,
x = 332, y = 333, z = 335
∴ x + y + z = 332 + 333 + 335 = 1000∴ x3 + y3 + z3 – 3xyz = 1 (x + y + z)[ (x - y)2 + (y - z)2 + (z - x)2 ] 2 ⇒ x3 + y3 + z3 – 3xyz = 1000 [ (332 – 333)2 + (333 – 335)2 + (335 - 332)2 ] 2
⇒ x3 + y3 + z3 – 3xyz = 500 (1 + 4 + 9) = 500 × 14 = 7000
Correct Option: B
Using Rule 22,
x = 332, y = 333, z = 335
∴ x + y + z = 332 + 333 + 335 = 1000∴ x3 + y3 + z3 – 3xyz = 1 (x + y + z)[ (x - y)2 + (y - z)2 + (z - x)2 ] 2 ⇒ x3 + y3 + z3 – 3xyz = 1000 [ (332 – 333)2 + (333 – 335)2 + (335 - 332)2 ] 2
⇒ x3 + y3 + z3 – 3xyz = 500 (1 + 4 + 9) = 500 × 14 = 7000
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If x + 1 = 2 , then the value of x7 + 
1 
is x x5
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x + 1 = 2 x
⇒ x2 + 1 = 2x
⇒ x2 - 2x + 1 = 0
⇒ (x - 1)2 = 0 ⇒ x = 1∴ x7 + 1 = 1 + 1 = 2 x5
Second Method :
Using Rule 16,Here, x + 1 = 2 x ∴ x7 + 1 = 2 x5
Correct Option: B
x + 1 = 2 x
⇒ x2 + 1 = 2x
⇒ x2 - 2x + 1 = 0
⇒ (x - 1)2 = 0 ⇒ x = 1∴ x7 + 1 = 1 + 1 = 2 x5
Second Method :
Using Rule 16,Here, x + 1 = 2 x ∴ x7 + 1 = 2 x5
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If p4 = 119 - 1 , then the value of p3 - 1 is p4 p3
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p4 = 119 - 1 p4 p4 + 1 = 119 p4 ⇒ p2 + 1 2 - 2 = 119 p2 ⇒ p2 + 1 2 = 119 + 2 = 121 p2 ⇒ p2 + 1 2 = (11)2 p2 ⇒ p2 + 1 = 11 p2
Again ,⇒ p - 1 2 + 2 = 11 p ⇒ p - 1 2 = 11 - 2 = 9 p ⇒ p2 + 1 = √9 = ±3 p2
On cubing both sides ,⇒ p - 1 3 = ±27 p ⇒ p3 - 1 - 3 × p × 1 p - 1 = ±27 p3 p p ⇒ p3 - 1 - 3 × (±3) = ±27 p3 ⇒ p3 - 1 = ±27 ± 9 = ±36 p3
Correct Option: C
p4 = 119 - 1 p4 p4 + 1 = 119 p4 ⇒ p2 + 1 2 - 2 = 119 p2 ⇒ p2 + 1 2 = 119 + 2 = 121 p2 ⇒ p2 + 1 2 = (11)2 p2 ⇒ p2 + 1 = 11 p2
Again ,⇒ p - 1 2 + 2 = 11 p ⇒ p - 1 2 = 11 - 2 = 9 p ⇒ p2 + 1 = √9 = ±3 p2
On cubing both sides ,⇒ p - 1 3 = ±27 p ⇒ p3 - 1 - 3 × p × 1 p - 1 = ±27 p3 p p ⇒ p3 - 1 - 3 × (±3) = ±27 p3 ⇒ p3 - 1 = ±27 ± 9 = ±36 p3
- If x + y + z = 6, then the value of (x - 1)3 + (y - 2)3 + (z - 3)3 is
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Using Rule 21,
If. a + b + c = 0, then a3 + b3 + c3 = 3abc
Here, x – 1 + y – 2 + z – 3 = x + y + z – 6
= 6 – 6 = 0
∴ (x - 1)3 + (y - 2)3 + (z - 3)3 = 3(x – 1 )(y – 2 )(z – 3 )Correct Option: A
Using Rule 21,
If. a + b + c = 0, then a3 + b3 + c3 = 3abc
Here, x – 1 + y – 2 + z – 3 = x + y + z – 6
= 6 – 6 = 0
∴ (x - 1)3 + (y - 2)3 + (z - 3)3 = 3(x – 1 )(y – 2 )(z – 3 )
