2022 VCE Maths Methods Mini Test 11

The function \( f(x) = \frac{1}{3}x^3 + mx^2 + nx + p \), for \( m, n, p \in \mathbb{R} \), has turning points at \( x = -3 \) and \( x = 1 \) and passes through the point (3, 4).
The values of \( m, n \) and \( p \) respectively are

• A. \( m = 0, n = -\frac{7}{3}, p = 2 \)
• B. \( m = 1, n = -3, p = -5 \)
• C. \( m = -1, n = -3, p = 13 \)
• D. \( m = \frac{5}{4}, n = \frac{3}{2}, p = -\frac{83}{4} \)
• E. \( m = \frac{5}{2}, n = 6, p = -\frac{91}{2} \)

A function \( g \) is continuous on the domain \( x \in [a, b] \) and has the following properties:
• The average rate of change of \( g \) between \( x = a \) and \( x = b \) is positive.
• The instantaneous rate of change of \( g \) at \( x = \frac{a + b}{2} \) is negative.
Therefore, on the interval \( x \in [a, b] \), the function must be

• A. many-to-one
• B. one-to-many
• C. one-to-one
• D. strictly decreasing
• E. strictly increasing

If \( X \) is a binomial random variable where \( n = 20, p = 0.88 \), and \( \Pr(X \geq 16 \mid X \geq a) = 0.9175 \), correct to four decimal places, then \( a \) is equal to

• A. 11
• B. 12
• C. 13
• D. 14
• E. 15

A box is formed from a rectangular sheet of cardboard, which has a width of \( a \) units and a length of \( b \) units, by first cutting out squares of side length \( x \) units from each corner and then folding up to form an open-top container.
The maximum volume of the box occurs when \( x \) is equal to

• A. \( \frac{a - b + \sqrt{a^2 - ab + b^2}}{6} \)
• B. \( \frac{a + b + \sqrt{a^2 - ab + b^2}}{6} \)
• C. \( \frac{a - b - \sqrt{a^2 - ab + b^2}}{6} \)
• D. \( \frac{a + b - \sqrt{a^2 - ab + b^2}}{6} \)
• E. \( \frac{a + b - \sqrt{a^2 - 2ab + b^2}}{6} \)

A soccer player kicks a ball with an angle of elevation \( \theta^\circ \), where \( \theta \) is a normally distributed random variable with a mean of 42° and a standard deviation of 8°.
The horizontal distance that the ball travels before landing is given by the function \( d = 50 \sin(2\theta) \).
The probability that the ball travels more than 40 m horizontally before landing is closest to

• A. 0.969
• B. 0.937
• C. 0.226
• D. 0.149
• E. 0.027

Part of the graph of \(y = f(x)\) is shown below. The rule \(A(k) = k \sin(k)\) gives the area bounded by the graph of \(f\), the horizontal axis and the line \(x = k\).

a. State the value of \( A\left(\frac{\pi}{3}\right) \). 1 mark

b. Evaluate \( f\left(\frac{\pi}{3}\right) \). 2 marks

c. Consider the average value of the function \(f\) over the interval \(x \in [0, k]\), where \(k \in [0, 2]\). Find the value of \(k\) that results in the maximum average value. 2 marks