Algebra
- If x (x + y + z) = 20, y (x + y + z) = 30, and z (x + y + z) = 50, then the value of 2 (x + y + z) is :
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x (x + y + z) = 20
⇒ x2 + xy + xz = 20 --- (i)
Again, y (x + y + z) = 30
⇒ xy + y2 + yz = 30 --- (ii)
and, z (x + y + z) = 50
⇒ xz + yz + z2 = 50 --- (iii)
On adding all three equations,
x2 + y2 + z2 + 2xy + 2yz + 2zx = 20 + 30 + 50
⇒ (x + y + z)2 = 100
⇒ x + y + z = 10
⇒ 2 (x + y + z) = 20Correct Option: B
x (x + y + z) = 20
⇒ x2 + xy + xz = 20 --- (i)
Again, y (x + y + z) = 30
⇒ xy + y2 + yz = 30 --- (ii)
and, z (x + y + z) = 50
⇒ xz + yz + z2 = 50 --- (iii)
On adding all three equations,
x2 + y2 + z2 + 2xy + 2yz + 2zx = 20 + 30 + 50
⇒ (x + y + z)2 = 100
⇒ x + y + z = 10
⇒ 2 (x + y + z) = 20
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If x = a + 2 , then the value of x2 − y2 is : y a − 2 x2 + y2
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x = a + 2 y a − 2
On squaring both sides,x2 = (a + 2)2 y2 (a − 2)2
By componendo and dividendo,x2 − y2 = (a + 2)2 − (a − 2)2 x2 + y2 (a + 2)2 + (a − 2)2 ⇒ x2 − y2 = 4 × a × 2 x2 + y2 4(a2 + 4) = 4a a2 + 4
[∵ (a + b)2 + (a – b)2 = 2(a2 + b2);
(a + b)2 – (a – b)2 = 4ab]
Correct Option: C
x = a + 2 y a − 2
On squaring both sides,x2 = (a + 2)2 y2 (a − 2)2
By componendo and dividendo,x2 − y2 = (a + 2)2 − (a − 2)2 x2 + y2 (a + 2)2 + (a − 2)2 ⇒ x2 − y2 = 4 × a × 2 x2 + y2 4(a2 + 4) = 4a a2 + 4
[∵ (a + b)2 + (a – b)2 = 2(a2 + b2);
(a + b)2 – (a – b)2 = 4ab]
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If √y = 4x, then x2 is : y
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√y = 4x ⇒ y = (4x)2 = 16x2
∴ x2 = x2 = 1 y 16x2 16 Correct Option: D
√y = 4x ⇒ y = (4x)2 = 16x2
∴ x2 = x2 = 1 y 16x2 16
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If a + b = 2, then the value of (a – b) is : b a
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a + b = 2 b a ⇒ a2 + b2 = 2 ab
⇒ a2 + b2 = 2ab
⇒ a2 + b2 – 2ab = 0
⇒ (a – b)2 = 0 ⇒ a – b = 0Correct Option: D
a + b = 2 b a ⇒ a2 + b2 = 2 ab
⇒ a2 + b2 = 2ab
⇒ a2 + b2 – 2ab = 0
⇒ (a – b)2 = 0 ⇒ a – b = 0
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If x + 1 = a and 1 − y = b , then the value of x − y is equal to x − 1 b 1 + y a 1 + xy
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x + 1 = a x − 1 b
By componendo and dividendo,x + 1 + x − 1 = a + b x + 1 − x + 1 a − b ⇒ 2x = a + b 2 a − b ⇒ x = a + b a − b
Again,1 − y = b 1 + y a ⇒ 1 + y = a 1 − y b ⇒ 1 + y + 1 − y = a + b 1 + y − 1 + y a − b ⇒ 2 = a + b 2y a − b ⇒ y = a − b a + b ∴ x − y = a + b − a − b a − b a + b = (a + b) 2 − (a − b)2 = 4ab (a + b)(a − b) a2 − b2 xy = a + b × a − b = 1 a − b a + b
∴ Expression= x − y = 4ab 1 + xy a2 − b2 1 + 1 = 4ab = 2ab 2(a2 − b2) a2 − b2 Correct Option: A
x + 1 = a x − 1 b
By componendo and dividendo,x + 1 + x − 1 = a + b x + 1 − x + 1 a − b ⇒ 2x = a + b 2 a − b ⇒ x = a + b a − b
Again,1 − y = b 1 + y a ⇒ 1 + y = a 1 − y b ⇒ 1 + y + 1 − y = a + b 1 + y − 1 + y a − b ⇒ 2 = a + b 2y a − b ⇒ y = a − b a + b ∴ x − y = a + b − a − b a − b a + b = (a + b) 2 − (a − b)2 = 4ab (a + b)(a − b) a2 − b2 xy = a + b × a − b = 1 a − b a + b
∴ Expression= x − y = 4ab 1 + xy a2 − b2 1 + 1 = 4ab = 2ab 2(a2 − b2) a2 − b2
