Let $P(x) = x^3 - ax^2 + x + b$ be a polynomial, where $a$ is a real number. When $P(x)$ is divided by $x - 3$ the remainder is 12. Find the remainder when $P(x)$ is divided by $x + 1$. The function $f(x) = ext{cos}(2 heta) - x$ has a zero near $x = rac{1}{2}$. Use one application of Newton's method to obtain another approximation to this zero. Give your answer correct to two decimal places. Find an expression for the coefficient of $x^2$ in the expansion of $\left(3x - 4 - \frac{8}{x}\right)^8$. Sketch the graph of the function $f(x) = 2 ext{cos}^{-1}(x)$. Clearly indicate the domain and range of the function. Alex's playlist consists of 40 different songs that can be arranged in any order. (i) How many arrangements are there for the 40 songs? (ii) Alex decides that she wants to play her three favourite songs first, in any order. How many arrangements of the 40 songs are now possible?

SSCE HSC Mathematics Extension 1 Velocity and acceleration as rates of change: Let $P(x) = x^3

Let $P(x) = x^3 - ax^2 + x + b$ be a polynomial, where $a$ is a real number - HSC - SSCE Mathematics Extension 1 - Question 2 - 2011 - Paper 1

Question 2

Let $P(x) = x^3 - ax^2 + x + b$ be a polynomial, where $a$ is a real number. When $P(x)$ is divided by $x - 3$ the remainder is 12. Find the remainder when $P(x)$ ... show full transcript

Worked Solution & Example Answer:Let $P(x) = x^3 - ax^2 + x + b$ be a polynomial, where $a$ is a real number - HSC - SSCE Mathematics Extension 1 - Question 2 - 2011 - Paper 1

Step 1

Let $P(x) = x^3 - ax^2 + x + b$ be a polynomial, where $a$ is a real number.

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Answer

To find the remainder when dividing by $x + 1$ , we apply the Polnomial Remainder Theorem. We know that:

$P(3) = 12$

Substituting $x = 3$ into $P(x)$ :

$3^3 - a(3^2) + 3 + b = 12$ $27 - 9a + 3 + b = 12$ $b - 9a + 30 = 0$ $b = 9a - 30$

Now to find the remainder when dividing by $x + 1$ , we compute:

$P(-1) = (-1)^3 - a(-1)^2 + (-1) + b$ $= -1 - a - 1 + (9a - 30)$ $= 8a - 32$

Thus, the remainder when $P(x)$ is divided by $x + 1$ is $8a - 32$ .

Step 2

The function $f(x) = ext{cos}(2 heta) - x$ has a zero near $x = rac{1}{2}$.

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Answer

To find another approximation using Newton's method, we first calculate:

$f'(x) = -2 ext{sin}(2 heta)$

Let $x_0 = rac{1}{2}$ and compute:

$f(x_0) = ext{cos}(2( rac{1}{2})) - rac{1}{2}$

Using Newton’s method formula:

$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$

g Compute $x_1$ . Simplifying down to two decimal places gives us the next approximate zero.

Step 3

Find an expression for the coefficient of $x^2$ in the expansion of $\left(3x - 4 - \frac{8}{x}\right)^8$.

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Answer

Using the binomial expansion, we need to focus on the term that contributes $x^2$ :

$\text{The general term} = \binom{8}{k} (3x)^{8-k}(-4 - \frac{8}{x})^k$

Setting up for $x^2$ , we find terms such that combined they yield $x^2$ .

Step 4

Sketch the graph of the function $f(x) = 2 ext{cos}^{-1}(x)$. Clearly indicate the domain and range of the function.

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Answer

The domain of $f(x) = 2 ext{cos}^{-1}(x)$ is $[-1, 1]$ since the function is defined only in that interval. The range, as $y$ , will be $[0, 2oldsymbol{ rac{ ext{ ext{Pi}}}}{2}]$ . A sketch would show the parabola intersecting the $y$ -axis at $0$ and $2 ext{Pi}$ and having the flip of symmetry.

Step 5

How many arrangements are there for the 40 songs?

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Answer

The total number of arrangements is given by $40!$ which is computed as:

$40! = 40 \times 39 \times 38 \times ... \times 1$

Step 6

Alex decides that she wants to play her three favourite songs first, in any order. How many arrangements of the 40 songs are now possible?

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Answer

First, the 3 songs can be arranged in $3!$ ways. The rest of the 37 songs can be arranged in $37!$ ways. Thus, the total arrangements are:

$3! \times 37!$

Let $P(x) = x^3 - ax^2 + x + b$ be a polynomial, where $a$ is a real number - HSC - SSCE Mathematics Extension 1 - Question 2 - 2011 - Paper 1

Worked Solution & Example Answer:Let $P(x) = x^3 - ax^2 + x + b$ be a polynomial, where $a$ is a real number - HSC - SSCE Mathematics Extension 1 - Question 2 - 2011 - Paper 1

Let $P(x) = x^3 - ax^2 + x + b$ be a polynomial, where $a$ is a real number.

The function $f(x) = ext{cos}(2 heta) - x$ has a zero near $x = rac{1}{2}$.

Find an expression for the coefficient of $x^2$ in the expansion of $\left(3x - 4 - \frac{8}{x}\right)^8$.

Sketch the graph of the function $f(x) = 2 ext{cos}^{-1}(x)$. Clearly indicate the domain and range of the function.

How many arrangements are there for the 40 songs?

Alex decides that she wants to play her three favourite songs first, in any order. How many arrangements of the 40 songs are now possible?

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