2021 QCE Maths Methods Paper 2 Mini Test 6
External Assessment Paper 2 — Technology-active
Number of marks: 9
Perusal time: 1 minute
Writing time: 15 minutes
Section 1
Instructions
• This section has 10 questions and is worth 10 marks.
• Use a 2B pencil to fill in the A, B, C or D answer bubble completely.
• Choose the best answer for Questions 1 10.
• If you change your mind or make a mistake, use an eraser to remove your response and fill in the new answer bubble completely.
The graphs of the functions \(f(x) = 2e^x + 5\) and \(g(x) = \frac{3}{e^x}\) intersect at point A. Determine the coordinates of point A.
- (A) (1.609, 15)
- (B) (1.099, 1)
- (C) (0.4065, 2)
- (D) (-0.693, 6)
An object travels in a straight line so that its velocity at time \(t\) seconds is given by \(v(t) = 2t + \sin(2t)\). Determine the expression for acceleration as a function of time.
- (A) \(a(t) = 2 + 2\cos(2t)\)
- (B) \(a(t) = 2 - \frac{1}{2}\cos(2t)\)
- (C) \(a(t) = t^2 + 2\cos(2t)\)
- (D) \(a(t) = t^2 - \frac{1}{2}\cos(2t)\)
Section 2
Instructions
• Write using black or blue pen.
• Questions worth more than one mark require mathematical reasoning and/or working to be shown to support answers.
• If you need more space for a response, use the additional pages at the back of this book.
– On the additional pages, write the question number you are responding to.
– Cancel any incorrect response by ruling a single diagonal line through your work.
– Write the page number of your alternative/additional response, i.e. See page …
– If you do not do this, your original response will be marked.
• This section has nine questions and is worth 45 marks.
A random variable \(X\), defined over the interval \(a \le x \le b\), is uniformly distributed if its probability density function is defined by: \[ f(x) = \begin{cases} \frac{1}{b-a}, & a \le x \le b \\ 0, & \text{otherwise} \end{cases} \]
The expected value and variance of a uniform random variable \(X\) are \[ E(X) = \frac{(a+b)}{2} \quad \text{Var}(X) = \frac{(b-a)^2}{12} \]
A manufacturer has observed that the time that elapses between placing an order with a supplier and the delivery of the order is uniformly distributed between 100 and 180 minutes.
Determine the probability that the time between placing an order and delivery of the order will be within one standard deviation of the expected time.
The random variable \(B\) is normally distributed with a mean of 0 and a standard deviation of 1.
Determine the probability that the quadratic equation \( x^2 + 3x + 2B = 0 \) has real roots.
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