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Answered by ashwinsraj007
1

Answer:

a = 1, when R1 = R2.  a = -2.354 when R1 + R2 = 0. a = 0.1417, when 2R1 - R2 = 0.

Step-by-step explanation:

f(x) = ax^{3} + 3x^{2} - 3

f(x) when divided by (x - 4) leaves remainder R_{1}

i.e. when f(x) - R_{1} is divided by (x -4), remainder will be zero.

⇒ (x-4) = 0, is a factor of f(x) - R_{1}

⇒ x = 4, is a factor of f(x) - R_{1}

f(x) - R_{1} = 0, for x = 4

a x^{3} + 3x^{2} - 3 - R_{1} = 0, for x = 4

⇒ a × 4^{3} + 3 × 4^{2} - 3 - R_{1} = 0

64a + 3 × 16 - 3 - R_{1} = 0

R_{1} = 64a + 48 - 3 = 64a + 45

similarly,

g(x) - R_{2} = 0, for x = 4

2x^{3}  - 5x + a - R_{2} = 0, for x = 4

2 × 4^{3}  - 5 × 4 + a - R_{2} = 0 = 0

2 × 64  - 20 + a - R_{2} = 0

R_{2} = 128 - 20 + a = 108 + a

when R_{1} = R_{2},

64a + 45 = 108 + a

64a - a = 108 - 45

63a = 63, ⇒ a = 1

when R_{1} + R_{2} = 0,

(i.e. when R_{1} = - R_{2})

64a + 45 = -(108 + a)

64a + 45 = -108 - a

64a + a = -108 -45

65a = -(108 + 45)

65a = -153

a = \frac{-153}{65}

a = - ( 2\frac{23}{65} )

a = - ( 2 + \frac{23}{65} )

a = - 2.354

(153 when divided by 65 leaves a remainder 23, quotient = 2)

(answer = quotient + (remainder/divisor), 153 = 2 + (23/65)

when 2R_{1} - R_{2} = 0,

(i.e. R_{2} = 2R_{1})

108 + a = 2 × (64a + 45)

108 + a = 2 × 64a + 2 × 45

108 + a = 128a + 90

128a - a = 108 - 90

127a = 18

a = \frac{18}{127} ≅ 0.1417

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