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

We start with the function $f(x) = ext{cos}(2x) - x$, with a known zero near x = rac{1}{2}.

1. Calculate the derivative: $f'(x) = -2 ext{sin}(2x) - 1$

2. Use Newton's method formula: $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$

3. Substitute $x_0 = 0.5$ into $f(x)$:

   $f(0.5) = ext{cos}(1) - 0.5$

Compute $ext{cos}(1)$ using a calculator, approximately $0.5403$:

$f(0.5) = 0.5403 - 0.5 \approx 0.0403$

4. Now find $f'(0.5)$:

   $f'(0.5) = -2 ext{sin}(1) - 1\ \approx -2(0.8415) - 1 \approx -2.683$

5. Plug values into Newton's method: $x_{1} = 0.5 - \frac{0.0403}{-2.683} \approx 0.515$

Finally, rounded to two decimal places, we find: $x_1 \approx 0.52$.

To find the coefficient of $x^2$ in the expansion of $\left(3x - 4 - \frac{8}{x}\right)^8$, we can use the binomial theorem:

$(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$

Setting $a = 3x - 4$ and $b = -\frac{8}{x}$, we need to find the term involving $x^2$:

1. Let $k$ be such that the total power of $x$ is 2:

   $(3x - 4)^{8-k} \left(-\frac{8}{x}\right)^k$

   This gives us:

   $$$$

ight)^k = 3^{8-k} (-4)^{8-k} (-8)^k x^{(8 - k) - k} = 3^{8-k} (-4)^{8-k} (-8)^k x^{8 - 2k} $$

2. To find $k$, set $8 - 2k = 2 \Rightarrow k = 3$:

3. Substitute $k$ back to find the coefficient:

   $T_k = 3^{5} (-4)^{5} (-8)^{3} \binom{8}{3}$

4. Now, calculate:

   $\binom{8}{3} = 56 \Rightarrow 3^{5} = 243, (-4)^{5} = -1024, (-8)^{3} = -512$

Thus, coefficient = $56 * 243 * -1024 * -512$.

To sketch the graph of the function $f(x) = 2 \text{cos}^{-1}(x)$:

1. Domain: The inverse cosine function is defined for $[-1, 1]$, hence the domain of $f(x)$ is $[-1, 1]$.

2. Range: The values of $ext{cos}^{-1}(x)$ range from $0$ to rac{\pi}{2}, so the range of $f(x)$ is $[0, \pi]$.

3. Plotting points:

   • At $x = -1$, $f(-1) = 2\text{cos}^{-1}(-1) = 2\pi$.
   • At $x = 0$, $f(0) = 2\text{cos}^{-1}(0) = \pi$.
   • At $x = 1$, $f(1) = 0$.

4. Shape: The graph will be decreasing, starting from $2\pi$ and ending at $0$.

5. Draw the curve above the $x$-axis, and ensure that the endpoints align with the domain's boundaries.

The total number of arrangements for 40 different songs can be calculated using the factorial function:

$40! = 40 \times 39 \times 38 \times ... \times 2 \times 1$

Thus, the answer is $40!$ arrangements.

If Alex wants to play her three favorite songs first in any order, we first arrange those three songs and the remaining 37 songs.

1. The three favorite songs can be arranged among themselves in $3! = 6$ ways.
2. Then, we have $37$ songs left, which can be arranged in $37!$ ways.
3. Therefore, the total number of arrangements becomes: $3! \times 37! = 6 \times 37!$ Hence, there are $6 \times 37!$ arrangements.