Algebra
-
The numerical value of (a - b)2 + (b - c)2 + (c - a)2 is (a ≠ b ≠ c) (b - c)(c - a) (c - a)(a - b) (a - b)(b - c)
-
View Hint View Answer Discuss in Forum
Expression = (a - b)2 + (b - c)2 + (c - a)2 (b - c)(c - a) (c - a)(a - b) (a - b)(b - c) Expression = (a - b)3 + (b - c)3 + (c - a)3 (a - b)(b - c)(c - a) Expression = 3(a - b)(b - c)(c - a) = 3 (a - b)(b - c)(c - a)
[Here, a – b + b – c + c – a = 0.
If x + y + z = 0, x3 + y3 + z3 = 3xyz ]Correct Option: D
Expression = (a - b)2 + (b - c)2 + (c - a)2 (b - c)(c - a) (c - a)(a - b) (a - b)(b - c) Expression = (a - b)3 + (b - c)3 + (c - a)3 (a - b)(b - c)(c - a) Expression = 3(a - b)(b - c)(c - a) = 3 (a - b)(b - c)(c - a)
[Here, a – b + b – c + c – a = 0.
If x + y + z = 0, x3 + y3 + z3 = 3xyz ]
- An example of an equality relation of two expressions in x, which is not an identity is
-
View Hint View Answer Discuss in Forum
According to equality relation (x + 2)2 = x2 + 4x + 4 is not an identity
Correct Option: C
According to equality relation (x + 2)2 = x2 + 4x + 4 is not an identity
- If x = 332, y = 333, z = 335, then the value of x3 + y3 + z3 – 3xyz is
-
View Hint View Answer Discuss in Forum
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
-
If p2 + q2 = 1 , then the value of (p6 + q6) is q2 p2
-
View Hint View Answer Discuss in Forum
p2 + q2 = 1 q2 p2 ⇒ p4 + q4 = 1 p2q2
⇒ p4 + q4 = p2q2
⇒ p4 + q4 - p2q2 = 0 ...... (i)
∴ p6 + q6 = (p2)3 + (q2)3
p6 + q6 = (p2 + q2)(p4 + q4 - p2q2)
[ ∴ a3 + b3 = (a + b)(a2 + b2 - ab) ]
p6 + q6 = (p2 + q2) × 0 = 0Correct Option: A
p2 + q2 = 1 q2 p2 ⇒ p4 + q4 = 1 p2q2
⇒ p4 + q4 = p2q2
⇒ p4 + q4 - p2q2 = 0 ...... (i)
∴ p6 + q6 = (p2)3 + (q2)3
p6 + q6 = (p2 + q2)(p4 + q4 - p2q2)
[ ∴ a3 + b3 = (a + b)(a2 + b2 - ab) ]
p6 + q6 = (p2 + q2) × 0 = 0
- If a + b – c = 0 then the value of 2b2c2 + 2c2a2 + 2a2b2 – a4 – b4 – c4
-
View Hint View Answer Discuss in Forum
Expression = 2b2c2 + 2c2a2 + 2a2b2 - a4 - b4 - c4
Expression = 4b2c2 - (2b2c2 - 2c2a2 - 2a2b2 + a4 + b4 + c4)
Expression = (2bc)2 - (a2 - b2 - c2)2
Expression = (2bc + a2 - b2 - c2)(2bc - a2 + b2 + c2)
Expression = {a2 - (b2 + c2 - 2bc){(b2 + c2 + 2bc) - a2}
Expression = { a2 - (b - c)2 }{ (b + c)2 - a2 }
Expression = (a – b + c) (a + b – c)(a + b + c) (b + c – a)
If a + b – c = 0,
∴ Expression = 0Correct Option: B
Expression = 2b2c2 + 2c2a2 + 2a2b2 - a4 - b4 - c4
Expression = 4b2c2 - (2b2c2 - 2c2a2 - 2a2b2 + a4 + b4 + c4)
Expression = (2bc)2 - (a2 - b2 - c2)2
Expression = (2bc + a2 - b2 - c2)(2bc - a2 + b2 + c2)
Expression = {a2 - (b2 + c2 - 2bc){(b2 + c2 + 2bc) - a2}
Expression = { a2 - (b - c)2 }{ (b + c)2 - a2 }
Expression = (a – b + c) (a + b – c)(a + b + c) (b + c – a)
If a + b – c = 0,
∴ Expression = 0
