VCE Maths Methods Polynomials Mini Test
Number of marks: 11
Reading time: 2 minutes
Writing time: 16 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.
Let \(f: [-3, 0] \to R, f(x) = (x+2)^2(x-1)\).
a. Show that \((x+2)^2(x-1) = x^3 + 3x^2 - 4\). 1 mark
b. Sketch the graph of \(f\) on the axes below. Label the axis intercepts with their coordinates. 3 marks
The function \(f: \mathbb{R} \to \mathbb{R}\), \(f(x)\) is a polynomial function of degree 4. Part of the graph of \(f\) is shown below.
The graph of \(f\) touches the \(x\)-axis at the origin.
End of examination questions
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