Math, asked by bemlasgst, 5 hours ago

please solve 13 number​

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Answers

Answered by MrImpeccable
13

ANSWER:

Given:

  • When, p(x) = x^3 - 6x^2 + 2x - k, is divided by (1 - 2x), Remainder = R1
  • When, g(x) = kx^3 + 12x^2 + 14x - 3, is divided by (2x + 1), Remainder = R2
  • R1 - R2 = 25/8

To Find:

  • Value of k

Solution:

We are given that,

\implies x^3-6x^2+2x-k\div(1-2x)=R_1

And,

\implies kx^3+12x^2+14x-3\div(2x+1)=R_2

Taking p(x) first,

We have,

\implies x^3-6x^2+2x-k\div(1-2x)=R_1

That is,

\implies p(x)\div(1-2x)=R_1

Let, 1 - 2x = 0.

So, x = 1/2

As,

\implies p(x)\div(1-2x)=R_1

So, by Factor Theorem,

\implies p\left(\dfrac{1}{2}\right)=R_1

So,

\implies\left(\dfrac{1}{2}\right)^3-6\left(\dfrac{1}{2}\right)^2+2\left(\dfrac{1}{2}\right)-k=R_1

\implies\dfrac{1}{8}-6\!\!\!/^3\left(\dfrac{1}{4\!\!\!/_2}\right)+2\!\!\!/\left(\dfrac{1}{2\!\!\!/}\right)-k=R_1

So,

\implies \dfrac{1}{8}-\dfrac{3}{2}+1-k=R_1

Taking LCM,

\implies \dfrac{1-3(4)+1(8)-k(8)}{8}=R_1

\implies \dfrac{1-12+8+8k}{8}=R_1

So,

\implies \dfrac{-3-8k}{8}=R_1 - - - -(1)

Taking g(x) now,

We have,

\implies kx^3+12x^2+14x-3\div(2x+1)=R_2

That is,

\implies g(x)\div(2x+1)=R_2

Let, 2x + 1 = 0.

So, x = -1/2

As,

\implies g(x)\div(2x+1)=R_2

So, by Factor Theorem,

\implies g\left(\dfrac{-1}{2}\right)= R_2

So,

\implies k\left(\dfrac{-1}{2}\right)^3+12\left(\dfrac{-1}{2}\right)^2+14\left(\dfrac{-1}{2}\right)-3= R_2

\implies k\dfrac{-1}{8}+12\!\!\!/^3\left(\dfrac{1}{4\!\!\!/}\right)+14\!\!\!\!\!/^7\left(\dfrac{-1}{2\!\!\!/}\right)-3= R_2

So,

\implies \dfrac{-k}{8}+3\!\!\!/-7-3\!\!\!/= R_2

\implies \dfrac{-k}{8}-7= R_2

Taking LCM,

\implies \dfrac{-k-7(8)}{8}= R_2

So,

\implies \dfrac{-k-56}{8}= R_2 - - - -(2)

Now, we are also given that,

\implies R_1-R_2=\dfrac{25}{8}

From (1) & (2),

\implies R_1-R_2=\dfrac{25}{8}

\implies \dfrac{-3-8k}{8} - \dfrac{-k-56}{8} =\dfrac{25}{8}

Combining fractions,

\implies \dfrac{-3-8k+k+56}{8\!\!\!/}=\dfrac{25}{8\!\!\!}

\implies 53-7k=25

\implies 53-25=7k

\implies 7k=28

So,

\implies k=\dfrac{28\!\!\!\!\!/^{\:\:4}}{7\!\!\!/}

Hence,

\implies\bf k=4

Therefore, the value of k is 4.

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