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VCE Maths Methods Basic Probability Mini Test 1
Adapted for Year 11. Unit 1 & 2.
Number of marks: 10
Reading time: 2 minutes
Writing time: 15 minutes
Section A – Calculator Allowed
Instructions
• Answer all questions in pencil on your Multiple-Choice Answer Sheet.
• Choose the response that is correct for the question.
• A correct answer scores 1; an incorrect answer scores 0.
• Marks will not be deducted for incorrect answers.
• No marks will be given if more than one answer is completed for any question.
• Unless otherwise indicated, the diagrams in this book are not drawn to scale.
For the parabola with equation \( y = ax^2 + 2bx + c \), where \( a, b, c \in \mathbb{R} \), the equation of the axis of symmetry is
- A. \( x = -\frac{b}{a} \)
- B. \( x = -\frac{b}{2a} \)
- C. \( y = c \)
- D. \( x = \frac{b}{a} \)
- E. \( x = \frac{b}{2a} \)
A polynomial has the equation \( y = x(3x - 1)(x + 3)(x + 1) \).
The number of tangents to this curve that pass through the positive x-intercept is
- A. 0
- B. 1
- C. 2
- D. 3
- E. 4
Find all values of \( k \), such that the equation
\( x^2 + (4k + 3)x + 4k^2- \frac{9}{4} = 0 \)
has two real solutions for \( x \), one positive and one negative.
- A. \( k > -\frac{3}{4} \)
- B. \( k \geq -\frac{3}{4} \)
- C. \( k > \frac{3}{4} \)
- D. \( -\frac{3}{4} < k < \frac{3}{4} \)
- E. \( k < -\frac{3}{4} \text{ or } k > \frac{3}{4} \)
Let \(p(x) = x^3 - 2ax^2 + x - 1\), where \(a \in R\). When \(p\) is divided by \(x + 2\), the remainder is 5.
The value of \(a\) is
- A. 2
- B. \(-\frac{7}{4}\)
- C. \(-\frac{1}{2}\)
- D. \(\frac{3}{2}\)
- E. -2
The set of values of \(k\) for which \(x^2 + 2x - k = 0\) has two real solutions is
- A. \(\{-1, 1\}\)
- B. \((-1, \infty)\)
- C. \((-\infty, -1)\)
- D. \(\{-1\}\)
- E. \([-1, \infty)\)
The equation \((p - 1)x^2 + 4x = 5 - p\) has no real roots when
- A. \(p^2 - 6p + 6 < 0\)
- B. \(p^2 - 6p + 1 > 0\)
- C. \(p^2 - 6p - 6 < 0\)
- D. \(p^2 - 6p + 1 < 0\)
- E. \(p^2 - 6p + 6 > 0\)
End of Section A
Section B – No Calculator
Instructions
• Answer all questions in the spaces provided.
• Write your responses in English.
• In questions where a numerical answer is required, an exact value must be given unless otherwise specified.
• In questions where more than one mark is available, appropriate working must be shown.
• Unless otherwise indicated, the diagrams in this book are not drawn to scale.
A car manufacturer is reviewing the performance of its car model X. It is known that at any given six-month service, the probability of model X requiring an oil change is \(\frac{17}{20}\), the probability of model X requiring an air filter change is \(\frac{3}{20}\) and the probability of model X requiring both is \(\frac{1}{20}\).
a. State the probability that at any given six-month service model X will require an air filter change without an oil change. 1 mark
b. The car manufacturer is developing a new model, Y. The production goals are that the probability of model Y requiring an oil change at any given six-month service will be \(\frac{m}{m+n}\), the probability of model Y requiring an air filter change will be \(\frac{n}{m+n}\) and the probability of model Y requiring both will be \(\frac{1}{m+n}\), where \(m, n \in Z^+\).
Determine \(m\) in terms of \(n\) if the probability of model Y requiring an air filter change without an oil change at any given six-month service is 0.05. 2 marks
The only possible outcomes when a coin is tossed are a head or a tail. When an unbiased coin is tossed, the probability of tossing a head is the same as the probability of tossing a tail.
Jo has three coins in her pocket; two are unbiased and one is biased. When the biased coin is tossed, the probability of tossing a head is \(\frac{1}{3}\).
Jo randomly selects a coin from her pocket and tosses it.
a. Find the probability that she tosses a head. 2 marks
b. Find the probability that she selected an unbiased coin, given that she tossed a head. 1 mark
End of examination questions
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