Algebra
- For what value of k, the system of equations
5x + 2y = k
10x + 4y = 3 has infinite solutions ?
-
View Hint View Answer Discuss in Forum
a1x + b1y + c1 = 0 and
a2x + b2y + c2 = 0 will have infinite solutions ifa1 = b1 = c1 a2 b2 c2 ⇒ 5 = 2 = −k 10 4 −3 ⇒ 1 = k ⇒ k = 3 2 3 2 Correct Option: A
a1x + b1y + c1 = 0 and
a2x + b2y + c2 = 0 will have infinite solutions ifa1 = b1 = c1 a2 b2 c2 ⇒ 5 = 2 = −k 10 4 −3 ⇒ 1 = k ⇒ k = 3 2 3 2
- If 4x2 + 4y2 + 4z2 = 12x + 12y – 18 then x + y + z = ?
-
View Hint View Answer Discuss in Forum
4x2 + 4y2 + 4z2 – 12x – 12y + 18 = 0
⇒ (2x)2 – 2 × 2x × 3 + 9 + (2y)2 – 2 × 2y × 3 + 9 + 4z2 = 0
⇒ (2x – 3)2 + (2y – 3)2 + 4z2 = 0
⇒ 2x – 3 = 0⇒ x = 3 ; 2
2y – 3 = 0⇒ x = 3 , z = 0 2 = 6 = 3 2
[If x2 + y2 + z2 = 0 ⇒ x = 0, y = 0, z = 0]Correct Option: A
4x2 + 4y2 + 4z2 – 12x – 12y + 18 = 0
⇒ (2x)2 – 2 × 2x × 3 + 9 + (2y)2 – 2 × 2y × 3 + 9 + 4z2 = 0
⇒ (2x – 3)2 + (2y – 3)2 + 4z2 = 0
⇒ 2x – 3 = 0⇒ x = 3 ; 2
2y – 3 = 0⇒ x = 3 , z = 0 2 = 6 = 3 2
[If x2 + y2 + z2 = 0 ⇒ x = 0, y = 0, z = 0]
- What will be the value of (x – a)3 + (x – b)3 + (x – c)3 – 3 (x – a) (x – b) (x – c) if a + b+c = 3x ?
-
View Hint View Answer Discuss in Forum
x – a + x – b + x – c
= 3x – (a + b + c) = 0
∴ (x – a)3 + (x – b)3 + (x – c)3 – 3
(x – a) (x – b) (x – c) = 0
[∵ a3 + b3 + c3– 3abc = 0 when a + b + c = 0]Correct Option: C
x – a + x – b + x – c
= 3x – (a + b + c) = 0
∴ (x – a)3 + (x – b)3 + (x – c)3 – 3
(x – a) (x – b) (x – c) = 0
[∵ a3 + b3 + c3– 3abc = 0 when a + b + c = 0]
- If a + b + c = 15 and a2 + b2 + c2 = 83, then a3 + b3 + c3 – 3abc = ?
-
View Hint View Answer Discuss in Forum
a3 + b3 + c3 – 3abc = (a + b + c)
(a2 + b2 + c2 – ab – bc – ca)
Now, (a + b + c)2 = a2 + b2 + c2 + 2 (ab + bc + ca)
⇒ 152 = 83 + 2 (ab + bc + ca)
⇒ 225 = 83 + 2 (ab + bc + ca)
⇒ 142 = 2 (ab + bc + ca)⇒ ab + bc + ca = 142 = 71 2
∴ a3 + b3 + c3 – 3abc = 15 × (83 – 71) = 15 × 12 = 180Correct Option: C
a3 + b3 + c3 – 3abc = (a + b + c)
(a2 + b2 + c2 – ab – bc – ca)
Now, (a + b + c)2 = a2 + b2 + c2 + 2 (ab + bc + ca)
⇒ 152 = 83 + 2 (ab + bc + ca)
⇒ 225 = 83 + 2 (ab + bc + ca)
⇒ 142 = 2 (ab + bc + ca)⇒ ab + bc + ca = 142 = 71 2
∴ a3 + b3 + c3 – 3abc = 15 × (83 – 71) = 15 × 12 = 180
- For what value of ‘a’, the polynomial 2x3 + ax2 + 11x + a + 3, is exactly divisible by (2x – 1) ?
-
View Hint View Answer Discuss in Forum
Let, P(x) = 2x3 + ax2 + 11x + a + 3
(2x – 1) is its factor.∴ p 
1 
= 0 2 ⇒ 2 × 1 3 + a × 1 2 + 11 × 1 + a + 3 = 0 2 2 2 ⇒ 1 + a + 11 + a + 3 = 0 4 4 2 ⇒ 1 + a + 22 + 4a + 12 = 0 4 ⇒ 5a + 35 = 0 4
⇒ 5a + 35 = 0 ⇒ 5a = – 35
⇒ a = –7Correct Option: B
Let, P(x) = 2x3 + ax2 + 11x + a + 3
(2x – 1) is its factor.∴ p 1 = 0 2 ⇒ 2 × 1 3 + a × 1 2 + 11 × 1 + a + 3 = 0 2 2 2 ⇒ 1 + a + 11 + a + 3 = 0 4 4 2 ⇒ 1 + a + 22 + 4a + 12 = 0 4 ⇒ 5a + 35 = 0 4
⇒ 5a + 35 = 0 ⇒ 5a = – 35
⇒ a = –7
