2021 VCE Maths Methods Mini Test 9

A value of \(k\) for which the average value of \( y = \cos\left(kx - \frac{\pi}{2}\right) \) over the interval [0, π] is equal to the average value of \( y= \sin(x)\) over the same interval is

• A. \( \frac{1}{6} \)
• B. \( \frac{1}{5} \)
• C. \( \frac{1}{4} \)
• D. \( \frac{1}{3} \)
• E. \( \frac{1}{2} \)

Four fair coins are tossed at the same time.
The outcome for each coin is independent of the outcome for any other coin.
The probability that there is an equal number of heads and tails, given that there is at least one head, is

• A. \( \frac{1}{2} \)
• B. \( \frac{1}{3} \)
• C. \( \frac{3}{4} \)
• D. \( \frac{2}{5} \)
• E. \( \frac{4}{7} \)

Let \( \cos(x) = \frac{3}{5} \) and \( \sin^2(y) = \frac{25}{169} \), where \( x \in \left[\frac{3\pi}{2}, 2\pi\right] \) and \( y \in \left[\frac{3\pi}{2}, 2\pi\right] \).
The value of \(\sin(x) + \cos(y)\) is

• A. \( \frac{8}{65} \)
• B. \( -\frac{112}{65} \)
• C. \( \frac{112}{65} \)
• D. \( -\frac{8}{65} \)
• E. \( \frac{64}{65} \)

A discrete random variable \(X\) has a binomial distribution with a probability of success of \(p = 0.1\) for \(n\) trials, where \(n > 2\).
If the probability of obtaining at least two successes after \(n\) trials is at least 0.5, then the smallest possible value of \(n\) is

• A. 15
• B. 16
• C. 17
• D. 18
• E. 19

Let \( f: R \to R, f(x) = (2x-1)(2x+1)(3x-1) \) and \( g: (-\infty, 0) \to R, g(x) = x \log_e(-x) \).
The maximum number of solutions for the equation \( f(x-k) = g(x) \), where \( k \in R \), is

• A. 0
• B. 1
• C. 2
• D. 3
• E. 4

The gradient of a function is given by \( \frac{dy}{dx} = \sqrt{x+6} - \frac{x}{2} - \frac{3}{2} \).
The graph of the function has a single stationary point at \( \left(3, \frac{29}{4}\right) \).

a. Find the rule of the function. 3 marks

b. Determine the nature of the stationary point. 2 marks