Algebra
- What is the value of m in the quadratic equation x2 + mx + 24 = 0 if one of its roots is (3/2)
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Putting x = 3 in x2 + mx + 24 = 0 2 
3 
2 + m × 3 + 24 = 0 2 2 ⇒ 9 + 3m + 24 = 0 4 2 ⇒ 3m = − 24 + 9 2 4 ⇒ 3m = − 96 + 9 2 4 ⇒ 3m = − 105 2 4 ⇒ m = − 105 × 2 4 3 − 35 2 Correct Option: D
Putting x = 3 in x2 + mx + 24 = 0 2 3 2 + m × 3 + 24 = 0 2 2 ⇒ 9 + 3m + 24 = 0 4 2 ⇒ 3m = − 24 + 9 2 4 ⇒ 3m = − 96 + 9 2 4 ⇒ 3m = − 105 2 4 ⇒ m = − 105 × 2 4 3 − 35 2
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If a + 1 = 1 and b + 1 = 1 then c + 1 is equal to : b c a
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a + 1 = 1 ⇒ a = 1 ; b = 2 b 2 b + 1 = 1 ⇒ b = 2, c = –1 c ∴ c + 1 = –1 + 2 = 1 a Correct Option: C
a + 1 = 1 ⇒ a = 1 ; b = 2 b 2 b + 1 = 1 ⇒ b = 2, c = –1 c ∴ c + 1 = –1 + 2 = 1 a
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If for non-zero x, x2 – 4x –1 = 0, the value of x2 + 1 is : x2
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x2 – 4x –1 = 0
⇒ x2 – 1 = 4x⇒ x2 – 1 = 4x x x ⇒ x – 1 = 4 x
On squaring both sides,x − 1 2 = 16 x ⇒ x2 + 1 − 2 = 16 x2 ⇒ x2 + 1 = 16 + 2 = 18 x2 Correct Option: C
x2 – 4x –1 = 0
⇒ x2 – 1 = 4x⇒ x2 – 1 = 4x x x ⇒ x – 1 = 4 x
On squaring both sides,x − 1 2 = 16 x ⇒ x2 + 1 − 2 = 16 x2 ⇒ x2 + 1 = 16 + 2 = 18 x2
- If a2 + b2 + c2 = 2 (a + b + c) – 3, then the value of a + b + c is :
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a2 + b2 + c2 = 2 (a – b – c) – 3
⇒ a2 + b2 + c2 – 2a + 2b + 2c + 3 = 0
⇒ a2 – 2a + 1 + b2 + 2b + 1 + c2 + 2c + 1 = 0
⇒ (a – 1)2 + (b + 1)2 + (c + 1)2 = 0
∴ a – 1 = 0 ⇒ a = 1
b + 1 = 0 ⇒ b = –1
c + 1 = 0 ⇒ c = –1
[If x2 + y2 + z2 = 0 ⇒ x = 0, y = 0, z = 0]
∴ a + b + c = 1 – 1 – 1 = –1Correct Option: B
a2 + b2 + c2 = 2 (a – b – c) – 3
⇒ a2 + b2 + c2 – 2a + 2b + 2c + 3 = 0
⇒ a2 – 2a + 1 + b2 + 2b + 1 + c2 + 2c + 1 = 0
⇒ (a – 1)2 + (b + 1)2 + (c + 1)2 = 0
∴ a – 1 = 0 ⇒ a = 1
b + 1 = 0 ⇒ b = –1
c + 1 = 0 ⇒ c = –1
[If x2 + y2 + z2 = 0 ⇒ x = 0, y = 0, z = 0]
∴ a + b + c = 1 – 1 – 1 = –1
- If x + y = 4, x2 + y2 = 14 and x > y, then the correct value of x and y is :
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x + y = 4 --- (i)
x2 + y2 = 14 --- (ii)
∴ (x + y)2 = x2 + y2 + 2xy
⇒ 16 = 14 + 2xy
⇒ 2xy = 16 – 14 = 2
⇒ xy = 1 --- (iii)
∴ (x – y)2 = (x + y)2 – 4xy
= (4)2 – 4 = 16 – 4 = 12
⇒ x – y = √12 = 2 √3 --- (iv)
∴ On adding equations (i) and (iv)x + y = 4 x − y = 2√3 ____________ 2x = 4 + 2√3 ⇒ x = 2 + √3
From equation (i),
2 + √3 + y = 4
⇒ y = 4 – 2 – √3 = 2 – √3Correct Option: A
x + y = 4 --- (i)
x2 + y2 = 14 --- (ii)
∴ (x + y)2 = x2 + y2 + 2xy
⇒ 16 = 14 + 2xy
⇒ 2xy = 16 – 14 = 2
⇒ xy = 1 --- (iii)
∴ (x – y)2 = (x + y)2 – 4xy
= (4)2 – 4 = 16 – 4 = 12
⇒ x – y = √12 = 2 √3 --- (iv)
∴ On adding equations (i) and (iv)x + y = 4 x − y = 2√3 ____________ 2x = 4 + 2√3 ⇒ x = 2 + √3
From equation (i),
2 + √3 + y = 4
⇒ y = 4 – 2 – √3 = 2 – √3
