Algebra
- If x (x – 3) = – 1, then the value of x3(x3 - 18) is
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x (x – 3) = –1
⇒ x2 – 3x = –1
⇒ x2 – 3x + 1 = 0
Expression = x3( x3 - 18 ) = x6 - 18x3
On dividing ( x6 - 18x3 ) by ( x2 – 3x + 1 )
∴ x6 - 18x3 = (x4 + 3x3 + 8x2 + 3x)
( x2 – 3x + 1 ) + x2 – 3x
= 0 + x (x – 3) = –1Correct Option: A
x (x – 3) = –1
⇒ x2 – 3x = –1
⇒ x2 – 3x + 1 = 0
Expression = x3( x3 - 18 ) = x6 - 18x3
On dividing ( x6 - 18x3 ) by ( x2 – 3x + 1 )
∴ x6 - 18x3 = (x4 + 3x3 + 8x2 + 3x)
( x2 – 3x + 1 ) + x2 – 3x
= 0 + x (x – 3) = –1
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If a2 + b2 + c2 = ab + bc + ca ,then the value of a + c is b
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a2 + b2 + c2 = ab + bc + ca
⇒ a2 + b2 + c2 - ab - bc - ca = 0
On multiplying by 2,
2a2 + 2b2 + 2c2 - 2ab - 2bc - 2ca = 0
⇒ a2 + b2 - 2ab + b2 + c2 - 2bc + c2 + a2 - 2ca = 0
⇒ (a - b)2 + (b - c)2 + (c - a)2 = 0
⇒ a – b = 0 ⇒ a = b
b – c = 0 ⇒ b = c
c – a = 0 ⇒ c = a∴ a + c = 2a = 2 b a
Correct Option: B
a2 + b2 + c2 = ab + bc + ca
⇒ a2 + b2 + c2 - ab - bc - ca = 0
On multiplying by 2,
2a2 + 2b2 + 2c2 - 2ab - 2bc - 2ca = 0
⇒ a2 + b2 - 2ab + b2 + c2 - 2bc + c2 + a2 - 2ca = 0
⇒ (a - b)2 + (b - c)2 + (c - a)2 = 0
⇒ a – b = 0 ⇒ a = b
b – c = 0 ⇒ b = c
c – a = 0 ⇒ c = a∴ a + c = 2a = 2 b a
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If ab + bc + ca = 0 then the value of 1 + 1 + 1 is ( a² - bc ) ( b² - ac ) ( c² - ab )
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ab + bc + ca = 0
⇒ ab + ca = – bc
∴ a2 – bc = a2 + ab + ac = a(a + b + c)
Similarly,
b2 – ca = b(a + b + c)
c2 – ab = c(a + b + c)∴ Expression = 1 + 1 + 1 ( a² - bc ) ( b² - ac ) ( c² - ab ) Expression = 1 + 1 + 1 a(a + b + c) b(a + b + c) c(a + b + c) Expression = 1 
1 + 1 + 1 
(a + b + c) a b c Expression = 1 bc + ca + ab (a + b + c) abc Expression = 1 × 0 = 0 (a + b + c) abc
Correct Option: A
ab + bc + ca = 0
⇒ ab + ca = – bc
∴ a2 – bc = a2 + ab + ac = a(a + b + c)
Similarly,
b2 – ca = b(a + b + c)
c2 – ab = c(a + b + c)∴ Expression = 1 + 1 + 1 ( a² - bc ) ( b² - ac ) ( c² - ab ) Expression = 1 + 1 + 1 a(a + b + c) b(a + b + c) c(a + b + c) Expression = 1 1 + 1 + 1 (a + b + c) a b c Expression = 1 bc + ca + ab (a + b + c) abc Expression = 1 × 0 = 0 (a + b + c) abc
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If 3x + 3 = 1, then x3 + 1 + 1 is : x x3
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3x + 3 = 1 x ⇒ x + 1 = 1 x 3
On cubing both sides,⇒ x3 + 1 + 3 x + 1 = 1 x3 x 27 ⇒ x3 + 1 + 3 × 1 = 1 x3 3 27 ⇒ x3 + 1 + 1 = 1 x3 27
Correct Option: B
3x + 3 = 1 x ⇒ x + 1 = 1 x 3
On cubing both sides,⇒ x3 + 1 + 3 x + 1 = 1 x3 x 27 ⇒ x3 + 1 + 3 × 1 = 1 x3 3 27 ⇒ x3 + 1 + 1 = 1 x3 27
- The factors of (a2 + 4b2 + 4b – 4ab – 2a – 8) are
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a2 + 4b2 + 4b – 4ab – 2a – 8 = a2 + 4b2 – 4ab – 2a + 4b – 8
= (a – 2b)2 - 2(a – 2b) – 8
Let (a – 2b) = x
∴ Expression = x2 – 2x – 8
Expression = x2 – 4x + 2x – 8
Expression = x (x – 4) + 2(x – 4)
Expression = (x – 4)(x + 2)
Expression = (a – 2b – 4)(a – 2b + 2)Correct Option: A
a2 + 4b2 + 4b – 4ab – 2a – 8 = a2 + 4b2 – 4ab – 2a + 4b – 8
= (a – 2b)2 - 2(a – 2b) – 8
Let (a – 2b) = x
∴ Expression = x2 – 2x – 8
Expression = x2 – 4x + 2x – 8
Expression = x (x – 4) + 2(x – 4)
Expression = (x – 4)(x + 2)
Expression = (a – 2b – 4)(a – 2b + 2)
