Algebra
-
If x = √5 − √3 and y = √5 + √3 then the value of x2 + xy + y2 = ? √5 + √3 √5 − √3 x2 − xy + y2
-
View Hint View Answer Discuss in Forum
x = √5 − √3 , y = √5 + √3 √5 + √3 √5 − √3 ∴ x + y = √5 − √3 + √5 + √3 √5 + √3 √5 − √3 = (√5 − √3)2 + (√5 + √3)2 (√5 + √3) × (√5 − √3) = 2[(√5)2 + (√3)2] 5 − 3
= 5 + 3 = 8xy = √5 − √3 × √5 + √3 = 1 √5 + √3 √5 − √3 = x2 + xy + y2 x2 − xy + y2 ∴ (x + y)2 − xy (x + y)2 − 3xy = (8)2 − 1 (8)2 − 3 = 64 − 1 64 − 3 = 63 61 Correct Option: A
x = √5 − √3 , y = √5 + √3 √5 + √3 √5 − √3 ∴ x + y = √5 − √3 + √5 + √3 √5 + √3 √5 − √3 = (√5 − √3)2 + (√5 + √3)2 (√5 + √3) × (√5 − √3) = 2[(√5)2 + (√3)2] 5 − 3
= 5 + 3 = 8xy = √5 − √3 × √5 + √3 = 1 √5 + √3 √5 − √3 = x2 + xy + y2 x2 − xy + y2 ∴ (x + y)2 − xy (x + y)2 − 3xy = (8)2 − 1 (8)2 − 3 = 64 − 1 64 − 3 = 63 61
- If p = 99 then, the value of p(p2 + 3p + 3) is :
-
View Hint View Answer Discuss in Forum
p = 99 (Given)
Expression = p(p2 + 3p + 3)
= p3 + 3p2 + 3p
= p3 + 3p2 + 3p + 1 – 1
= (p + 1)3 – 1
= (99 + 1)3 – 1 = (100)3 – 1
= 1000000 – 1 = 999999Correct Option: C
p = 99 (Given)
Expression = p(p2 + 3p + 3)
= p3 + 3p2 + 3p
= p3 + 3p2 + 3p + 1 – 1
= (p + 1)3 – 1
= (99 + 1)3 – 1 = (100)3 – 1
= 1000000 – 1 = 999999
-
If x + 1 = 1 then the value of x2 + 3x + 1 is x x2 + 7x + 1
-
View Hint View Answer Discuss in Forum
x + 1 = 1 (Given) x Expression = x2 + 3x + 1 x2 + 7x + 1 = x + 1 + 3 x x + 1 + 7 x
(Dividing numerator and denominator by x)= 1 + 3 = 4 = 1 1 + 7 8 2 Correct Option: C
x + 1 = 1 (Given) x Expression = x2 + 3x + 1 x2 + 7x + 1 = x + 1 + 3 x x + 1 + 7 x
(Dividing numerator and denominator by x)= 1 + 3 = 4 = 1 1 + 7 8 2
-
If m − a2 + m − b2 + m − c2 = 3 , then the value of m is b2 + c2 c2 + a2 a2 + b2
-
View Hint View Answer Discuss in Forum
m − a2 + m − b2 + m − c2 − 3 = 0 b2 + c2 c2 + a2 a2 + b2 ⇒ m − a2 − 1 + m − b2 − 1 + m − c2 − 1 = 0 b2 + c2 c2 + a2 a2 + b2 ⇒ m − a2 − b2 − c2 + m − b2 − c2 − a2 + m − c2 − a2 − b2 = 0 b2 + c2 c2 + a2 a2 + b2 ⇒ m − (a2 + b2 + c2) + m − (a2 + b2 + c2) + m −(a2 + b2 + c2) = 0 b2 + c2 c2 + a2 a2 + b2
∴ Each term = 0∴ m − (a2 + b2 + c2) = 0 b2 + c2
⇒ m − (a2 + b2 + c2) = 0
⇒ m = (a2 + b2 + c2)Correct Option: C
m − a2 + m − b2 + m − c2 − 3 = 0 b2 + c2 c2 + a2 a2 + b2 ⇒ m − a2 − 1 + m − b2 − 1 + m − c2 − 1 = 0 b2 + c2 c2 + a2 a2 + b2 ⇒ m − a2 − b2 − c2 + m − b2 − c2 − a2 + m − c2 − a2 − b2 = 0 b2 + c2 c2 + a2 a2 + b2 ⇒ m − (a2 + b2 + c2) + m − (a2 + b2 + c2) + m −(a2 + b2 + c2) = 0 b2 + c2 c2 + a2 a2 + b2
∴ Each term = 0∴ m − (a2 + b2 + c2) = 0 b2 + c2
⇒ m − (a2 + b2 + c2) = 0
⇒ m = (a2 + b2 + c2)
-
If x = 1 , y = 1 , then the value of 8xy (x2 + y2) is 2 + √3 2 − √3
-
View Hint View Answer Discuss in Forum
x = 1 2 + √3 = 1 × 1 = 2 − √3 2 + √3 2 − √ 3 4 − 3
= 2 − √ 3∴ y = 1 = 2 + √ 3 2 − √ 3
∴ x + y = 2 – √ 3 + 2 + √ 3 = 4
xy = (2 – √ 3 ) (2 + √ 3 )
= 4 – 3 = 1
∴ 8xy (x2 + y2)
= 8xy [(x + y)2 – 2 xy]
= 8 × 1 (42 – 2 × 1)
= 8 (16 – 2) = 8 × 14 = 112Correct Option: C
x = 1 2 + √3 = 1 × 1 = 2 − √3 2 + √3 2 − √ 3 4 − 3
= 2 − √ 3∴ y = 1 = 2 + √ 3 2 − √ 3
∴ x + y = 2 – √ 3 + 2 + √ 3 = 4
xy = (2 – √ 3 ) (2 + √ 3 )
= 4 – 3 = 1
∴ 8xy (x2 + y2)
= 8xy [(x + y)2 – 2 xy]
= 8 × 1 (42 – 2 × 1)
= 8 (16 – 2) = 8 × 14 = 112