1 + 11x - 6x^2 \ \ (x - 3)(1 - 2x) \= A + \frac{B}{(x - 3)} + \frac{C}{(1 - 2x)}. \ (a) Find the values of the constants A, B and C. \ \ f(x) = \frac{1 + 11x - 6x^2}{(x - 3)(1 - 2x)} \ x > 3. \ (b) Prove that f(x) is a decreasing function.

A-Level Edexcel Maths Pure Trigonometric Proof: 1 + 11x - 6x^2 \ \ (x - 3)(1

1 + 11x - 6x^2 \ \ (x - 3)(1 - 2x) \= A + \frac{B}{(x - 3)} + \frac{C}{(1 - 2x)} - Edexcel - A-Level Maths Pure - Question 12 - 2018 - Paper 2

Question 12

$1-+-11x---6x^2-\-\-(x---3)(1---2x)-\=-A-+-\frac{B}{(x---3)}-+-\frac{C}{(1---2x)}-Edexcel-A-Level Maths Pure-Question 12-2018-Paper 2.png$

1 + 11x - 6x^2 \ \ (x - 3)(1 - 2x) \= A + \frac{B}{(x - 3)} + \frac{C}{(1 - 2x)}. \ (a) Find the values of the constants A, B and C. \ \ f(x) = \frac{1 + 11x - 6x^2}... show full transcript

Worked Solution & Example Answer:1 + 11x - 6x^2 \ \ (x - 3)(1 - 2x) \= A + \frac{B}{(x - 3)} + \frac{C}{(1 - 2x)} - Edexcel - A-Level Maths Pure - Question 12 - 2018 - Paper 2

Step 1

Find the values of the constants A, B and C.

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Answer

To find the values of A, B, and C, we'll start by setting up the equation:

$1 + 11x - 6x^2 = A(1 - 2x)(x - 3) + B(1 - 2x) + C(x - 3)$

Step 1: Expand and Collect Terms

Expand the right side and arrange terms. Distributing A:

$A(1 - 2x)(x - 3) = A(1 - 2x^2 - 3 + 6x) = A(-2x^2 + 6x - 2)$

Step 2: Substitute Specific Values

By substituting specific values for x, such as x = 3 and x = 1, we will find A, B, and C. Alternatively, we can equate coefficients directly:

For $x^2$ : Set coefficients equal:

$-6 = -2A \Rightarrow A = 3$
For $x^1$ :

$11 = 6A + B - 3C$

Substitute A = 3 to find B & C.
For constant terms:

$1 = -2A + 3B + C$

Then substitute A and solve for B and C. After solving these equations, you will find:
- $A = 3$
- $B = 4$
- $C = -2$

Step 2

Prove that f(x) is a decreasing function.

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Answer

To prove that f(x) is decreasing for x > 3, we need to analyze the derivative of f(x).

Step 1: Differentiate f(x)

Using the quotient rule to find f'(x):

$f'(x) = \frac{(0)(g(x)) - (g'(x))(f(x))}{(g(x))^2}$

Where g(x) = (x - 3)(1 - 2x)

Step 2: Analyze the Sign of f'(x)

Calculate the derivative and check the sign. If f'(x) < 0 for x > 3, then f(x) is decreasing.

Let’s simplify f'(x):

Find the critical points and calculate f'(x) to understand where it may change signs.
Use test points to confirm the I need for any intervals.