Algebra
- If a = 0.1039, then the value of √4a2 − 4a + 1 + 3a is
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a = 0.1039 (Given)
Now, √4a2 − 4a + 1 + 3a
= √(1 − 2a)2 + 3a
= 1 – 2a + 3a
= 1 + a = 1 + 0.1039
= 1.1039Correct Option: C
a = 0.1039 (Given)
Now, √4a2 − 4a + 1 + 3a
= √(1 − 2a)2 + 3a
= 1 – 2a + 3a
= 1 + a = 1 + 0.1039
= 1.1039
- If a + b + c = 9 (where a, b, c are real numbers), then the minimum value of a2 + b2 + c2 is
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a + b + c = 9
a2 + b2 + c2
= (a + b + c)2 – 2 (ab + bc + ca)
[ab + bc + ca will be maximum if a = b = c]
a2 + b2 + c2 = 92 – 2 × 27
= 81 – 54 = 27Correct Option: C
a + b + c = 9
a2 + b2 + c2
= (a + b + c)2 – 2 (ab + bc + ca)
[ab + bc + ca will be maximum if a = b = c]
a2 + b2 + c2 = 92 – 2 × 27
= 81 – 54 = 27
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If xy + yz + zx = 0, then 
1 + 1 + 1 
(x, y, z ≠ 0) is equal to: x2 − yz y2 − zx z2 − xy
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x2 − yz = x2 − + xy + zx = x (x + y + z)
[∵ xy + yz + zx = 0
⇒ yz = −xy −zx]
Similarly,
y2 − zx = y (x + y + z)
z2 − xy = x (x + y + z)∴ Expression = 1 + 1 + 1 x(x + y + z) y(x + y + z) z(x + y + z) = yz + zx + xy = 0 xyz(x + y + z) Correct Option: D
x2 − yz = x2 − + xy + zx = x (x + y + z)
[∵ xy + yz + zx = 0
⇒ yz = −xy −zx]
Similarly,
y2 − zx = y (x + y + z)
z2 − xy = x (x + y + z)∴ Expression = 1 + 1 + 1 x(x + y + z) y(x + y + z) z(x + y + z) = yz + zx + xy = 0 xyz(x + y + z)
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If a2 + b2 = 5ab, then the value of a2 + b2 is : b2 a2
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a2 + b2 = 5ab
⇒ a2 + b2 = 5 ab ⇒ a + b = 5 b a
On squaring both sides,∴ a + b 2 = 25 b a ⇒ a2 + b2 + 2 = 25 b2 a2 ⇒ a2 + b2 = 25 − 2 = 23 b2 a2 Correct Option: C
a2 + b2 = 5ab
⇒ a2 + b2 = 5 ab ⇒ a + b = 5 b a
On squaring both sides,∴ a + b 2 = 25 b a ⇒ a2 + b2 + 2 = 25 b2 a2 ⇒ a2 + b2 = 25 − 2 = 23 b2 a2
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If x2 − 3x + 1 = 0, then the value of x2 + x + 1 + 1 is x x2
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x2 − 3x + 1 = 0
⇒ x2 + 1 = 3x
Dividing both sides by x,⇒ x + 1 = 3 x ∴ x2 + x + 1 + 1 x x2 = x2 + 1 + x + 1 x2 x = x + 1 2 − 2 + x + 1 x x
= 9 – 2 + 3 = 10Correct Option: A
x2 − 3x + 1 = 0
⇒ x2 + 1 = 3x
Dividing both sides by x,⇒ x + 1 = 3 x ∴ x2 + x + 1 + 1 x x2 = x2 + 1 + x + 1 x2 x = x + 1 2 − 2 + x + 1 x x
= 9 – 2 + 3 = 10
