A-Level Edexcel Maths Pure Compound & Double Angle Formulae: A curve C has equation $y = f(x)

A curve C has equation $y = f(x)$ Given that - $f'(x) = 6x^2 + ax - 23$ where $a$ is a constant - the $y$ intercept of C is $-12$ - $(x + 4)$ is a factor of $f(x)$ find, in simplest form, $f(x)$ - Edexcel - A-Level Maths Pure - Question 10 - 2020 - Paper 2

Question 10

A-curve-C-has-equation-$y-=-f(x)$----Given-that------$f'(x)-=-6x^2-+-ax---23$-where-$a$-is-a-constant-----the-$y$-intercept-of-C-is-$-12$-----$(x-+-4)$-is-a-factor-of-$f(x)$----find,-in-simplest-form,-$f(x)$-Edexcel-A-Level Maths Pure-Question 10-2020-Paper 2.png

A curve C has equation $y = f(x)$ Given that - $f'(x) = 6x^2 + ax - 23$ where $a$ is a constant - the $y$ intercept of C is $-12$ - $(x + 4)$ is a factor o... show full transcript

Worked Solution & Example Answer:A curve C has equation $y = f(x)$ Given that - $f'(x) = 6x^2 + ax - 23$ where $a$ is a constant - the $y$ intercept of C is $-12$ - $(x + 4)$ is a factor of $f(x)$ find, in simplest form, $f(x)$ - Edexcel - A-Level Maths Pure - Question 10 - 2020 - Paper 2

Step 1

Find the value of the constant $a$

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Answer

To find the constant $a$ , we need to use the information given about the $y$ -intercept of the curve, which is $-12$ . The $y$ -intercept occurs when $x = 0$ .

First, we calculate $f'(0)$ :
$f'(0) = 6(0)^2 + a(0) - 23 = -23.$
Since the curve is continuous, $f(0) = f(0) = -12$ . Now we can integrate $f'(x)$ :

$f(x) = rac{6}{3}x^3 + rac{a}{2}x^2 - 23x + C = 2x^3 + rac{a}{2}x^2 - 23x + C.$
Next, we substitute $x = 0$ :

ightarrow C = -12.$$ Thus, $$f(x) = 2x^3 + rac{a}{2}x^2 - 23x - 12.$$

Step 2

Use that $(x + 4)$ is a factor of $f(x)$

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Answer

Since $(x + 4)$ is a factor of $f(x)$ , we know that $f(-4) = 0$ .
We can substitute $x = -4$ into $f(x)$ :

$f(-4) = 2(-4)^3 + rac{a}{2}(-4)^2 - 23(-4) - 12 = 0.$
Calculating $f(-4)$ yields:
$f(-4) = 2(-64) + rac{a}{2}(16) + 92 - 12 = 0$
$-128 + 8a + 80 - 12 = 0.$
This simplifies to:

ightarrow 8a = 60 ightarrow a = rac{60}{8} = 7.5.$$

Step 3

Substituting value of $a$ back into $f(x)$

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Answer

Now that we have determined $a$ , we substitute it back into the equation for $f(x)$ :
$f(x) = 2x^3 + rac{7.5}{2}x^2 - 23x - 12.$
Thus, simplifying gives:
$f(x) = 2x^3 + 3.75x^2 - 23x - 12.$
This is the final form of $f(x)$ .

Therefore, in simplest form,
$f(x) = 2x^3 + 3.75x^2 - 23x - 12.$

A curve C has equation $y = f(x)$ Given that - $f'(x) = 6x^2 + ax - 23$ where $a$ is a constant - the $y$ intercept of C is $-12$ - $(x + 4)$ is a factor of $f(x)$ find, in simplest form, $f(x)$ - Edexcel - A-Level Maths Pure - Question 10 - 2020 - Paper 2

Worked Solution & Example Answer:A curve C has equation $y = f(x)$ Given that - $f'(x) = 6x^2 + ax - 23$ where $a$ is a constant - the $y$ intercept of C is $-12$ - $(x + 4)$ is a factor of $f(x)$ find, in simplest form, $f(x)$ - Edexcel - A-Level Maths Pure - Question 10 - 2020 - Paper 2

Find the value of the constant $a$

Use that $(x + 4)$ is a factor of $f(x)$

Substituting value of $a$ back into $f(x)$

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