Algebra
- If x = 2 then the value of x3 + 27x2 + 243x + 631 is
-
View Hint View Answer Discuss in Forum
Using Rule 8,
x3 + 27x2 + 243x + 631 = x3 + 3 . x2 × 9 + 3x.92 + 93 - 93 + 631
x3 + 27x2 + 243x + 631 = (x + 9)3 - 729 + 631
x3 + 27x2 + 243x + 631 = (2 + 9)3 – 98 { x = 2( Given ) }
x3 + 27x2 + 243x + 631 = 113 – 98 = 1331 – 98 = 1233Correct Option: A
Using Rule 8,
x3 + 27x2 + 243x + 631 = x3 + 3 . x2 × 9 + 3x.92 + 93 - 93 + 631
x3 + 27x2 + 243x + 631 = (x + 9)3 - 729 + 631
x3 + 27x2 + 243x + 631 = (2 + 9)3 – 98 { x = 2( Given ) }
x3 + 27x2 + 243x + 631 = 113 – 98 = 1331 – 98 = 1233
- Given that x3 + y3 = 72 and xy = 6 with x > y. Then the value of (x – y) is
-
View Hint View Answer Discuss in Forum
x3 + y3 = 72
x3 + y3 = 64 + 8 = 43 + 23
∴ x = 4, y = 2 ⇒ xy = 8
∴ x – y = 4 – 2 = 2Correct Option: C
x3 + y3 = 72
x3 + y3 = 64 + 8 = 43 + 23
∴ x = 4, y = 2 ⇒ xy = 8
∴ x – y = 4 – 2 = 2
-
If x + 1 = 2 , then the value of x12 - 1 is x x12
-
View Hint View Answer Discuss in Forum
x + 1 = 2 x ⇒ x2 + 1 = 2 x
⇒ x2 + 1 = 2x
⇒ x2 - 2x + 1 = 0
⇒ (x - 1)2 = 0 ⇒ x = 1∴ x12 + 1 = 1 + 1 = 2 x12
Second Method :
Using Rule 16,Here, x + 1 = 2 x ∴ x12 + 1 = 2 x12
Correct Option: A
x + 1 = 2 x ⇒ x2 + 1 = 2 x
⇒ x2 + 1 = 2x
⇒ x2 - 2x + 1 = 0
⇒ (x - 1)2 = 0 ⇒ x = 1∴ x12 + 1 = 1 + 1 = 2 x12
Second Method :
Using Rule 16,Here, x + 1 = 2 x ∴ x12 + 1 = 2 x12
- If m = – 4, n = – 2, then the value of m3 - 3m2 + 3m + 3n + 3n2 + n3 is
-
View Hint View Answer Discuss in Forum
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 (m + 1) = √n + 3, the value of 1 
m3 - 6m2 + 12m - 8 - n 
is 2 √n
-
View Hint View Answer Discuss in Forum
m + 1 = √n + 3 (Given)
⇒ m + 1 - 3 = √n
⇒ m - 2 = √n
On cubing both sides,
(m - 2)3 = (√n)3
⇒ m3 - 3m2 × 2 + 3 m(2)2 - 23 = n√n
[ ∴ (a - b)3 = a3 - 3a2b + 3ab2 - b3 ]
⇒ m3 - 6m2 + 12m - 8 = n√n⇒ m3 - 6m2 + 12m - 8 = n √n ⇒ m3 - 6m2 + 12m - 8 - n = 0 √n ⇒ 1 m3 - 6m2 + 12m - 8 - n = 0 2 √n Correct Option: A
m + 1 = √n + 3 (Given)
⇒ m + 1 - 3 = √n
⇒ m - 2 = √n
On cubing both sides,
(m - 2)3 = (√n)3
⇒ m3 - 3m2 × 2 + 3 m(2)2 - 23 = n√n
[ ∴ (a - b)3 = a3 - 3a2b + 3ab2 - b3 ]
⇒ m3 - 6m2 + 12m - 8 = n√n⇒ m3 - 6m2 + 12m - 8 = n √n ⇒ m3 - 6m2 + 12m - 8 - n = 0 √n ⇒ 1 m3 - 6m2 + 12m - 8 - n = 0 2 √n
