A test consists of five multiple-choice questions - HSC - SSCE Mathematics Extension 1 - Question 4 - 2009 - Paper 1

To find this probability, we need to consider the cases where Huong selects 3, 4, or 5 correct answers.

1. Three correct: As calculated above, the probability for this case is: $P(3) = \frac{45}{512}.$

2. Four correct: Using the same method: $C(5, 4) = 5$ $P(4) = C(5, 4) \cdot \left(\frac{1}{4}\right)^4 \cdot \left(\frac{3}{4}\right)^1 = 5 \cdot \frac{1}{256} \cdot \frac{3}{4} = \frac{15}{1024}.$

3. Five correct: All answers correct: $P(5) = 1 \cdot \left(\frac{1}{4}\right)^5 = \frac{1}{1024}.$

Thus the total probability is:

$P(3 \text{ or more}) = P(3) + P(4) + P(5) = \frac{45}{512} + \frac{15}{1024} + \frac{1}{1024}.$

Converting \frac{45}{512} to a common denominator:

$\frac{45 \cdot 2}{1024} = \frac{90}{1024}.$

Adding these:

$P(3 \text{ or more}) = \frac{90 + 15 + 1}{1024} = \frac{106}{1024} = \frac{53}{512}.$

To find this probability, we can first calculate the probability of selecting no incorrect answers (i.e., all correct) and then subtract from 1.

The probability of selecting all correct answers (5 correct):

$P(5) = \left(\frac{1}{4}\right)^5 = \frac{1}{1024}.$

Thus, the probability of selecting at least one incorrect answer is:

$P(\text{at least one incorrect}) = 1 - P(5) = 1 - \frac{1}{1024} = \frac{1023}{1024}.$

To verify that $f(x)$ is an even function, we need to show that $f(-x) = f(x)$. Let's calculate:

$f(-x) = \frac{(-x)^4 + 3(-x)^2}{(-x)^4 + 3} = \frac{x^4 + 3x^2}{x^4 + 3} = f(x).$

Since $f(-x) = f(x)$, we conclude that $f(x)$ is indeed an even function.

To find the horizontal asymptote, we examine the behavior of $f(x)$ as $x \to \infty$:

$f(x) = \frac{x^4 + 3x^2}{x^4 + 3}.$

As $x$ grows large, the lower order terms become negligible:

$f(x) \approx \frac{x^4}{x^4} = 1.$

Thus, the horizontal asymptote is given by:

$y = 1.$

To find the stationary points, we need to differentiate $f(x)$ and set the derivative equal to zero:

1. First, apply the quotient rule: $f'(x) = \frac{(4x^3)(x^4 + 3) - (x^4 + 3x^2)(4x^3)}{(x^4 + 3)^2}.$

2. Simplifying the numerator, we set it to zero to find the stationary points. After some algebra, solving will yield the x-coordinates: $x = 0.$

   Focusing on the conditions when $f'(x) = 0$ may yield other roots depending on the leftover terms in the numerator.

To sketch the graph of $y = f(x)$, observe the following:

• The function is even, implying symmetry about the y-axis.
• As $x \to \infty$, $y \to 1$ (as established, it approaches the horizontal asymptote).
• The stationary point calculated allows us to examine local maxima or minima.

Plotting these points and using behaviors near the asymptote will assist in sketching a smooth curve respecting these aspects.