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Mathematical Methods VCE practice questions

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Question 1Algebra, number and structure4 marks
Solve the quadratic equation $x^2 + 5x + 6 = 0$. Find the sum and product of the roots.
What a full answer covers
The answer should show the steps to factorise the quadratic equation, and find the roots. The sum and product of the roots should be clearly calculated.
Question 2Functions, relations and graphs9 marks
Sketch the graph of the function $f(x) = x^3 - 2x^2 - 5x + 6$. Find the x-intercepts and the local maximum and minimum points.
What a full answer covers
The answer should show the sketch of the graph, and identify the x-intercepts and the local maximum and minimum points. The answer should be well presented.
Question 3Calculus3 marks
Find the derivative of the function $f(x) = 3x^2 sin x$. Use the product rule.
What a full answer covers
The answer should show the steps to apply the product rule, and simplify the derivative. The final answer should be clearly stated.
Question 4Functions, relations and graphs8 marks
Consider the function $f(x) = 2x + 1$. Find the inverse function $f^{-1}(x)$, and sketch the graphs of $f(x)$ and $f^{-1}(x)$ on the same set of axes.
What a full answer covers
The answer should show the steps to find the inverse function, and sketch the graphs of $f(x)$ and $f^{-1}(x)$. The graphs should be clearly labelled, and the answer should be well presented.
Question 5Calculus10 marks
Evaluate the definite integral $int_{0}^{2} (x^2 + 1) dx$. Use the fundamental theorem of calculus to find the area under the curve.
What a full answer covers
The answer should show the steps to evaluate the definite integral, and apply the fundamental theorem of calculus. The area under the curve should be clearly calculated, and the answer should be well presented.
Question 6Data analysis, probability and statistics12 marks
A set of exam scores has a mean of 75 and a standard deviation of 10. Find the z-scores of the scores 60, 70, 80, and 90. Determine which scores are within one standard deviation of the mean.
What a full answer covers
The answer should show the steps to calculate the z-scores, and determine which scores are within one standard deviation of the mean. The answer should be well presented.
Question 7Functions, relations and graphs3 marks
Sketch the graph of the function $f(x) = x^2 - 4x + 3$. Label the x-intercepts and the vertex.
What a full answer covers
The answer should include a sketch of the parabola, with the x-intercepts and vertex correctly labelled. The graph should be drawn in a standard cartesian plane.
Question 8Algebra, number and structure2 marks
Solve the system of equations: $2x + 3y = 7$, $x - 2y = -3$. Find the value of $x$.
What a full answer covers
The answer should show the steps to solve the system of equations, using either substitution or elimination. The final value of $x$ should be clearly stated.
Question 9Data analysis, probability and statistics2 marks
A coin is tossed three times. Find the probability of getting exactly two heads.
What a full answer covers
The answer should show the sample space, and calculate the probability of getting exactly two heads. The final answer should be clearly stated.
Question 10Algebra, number and structure6 marks
Solve the system of equations: $x + 2y = 4$, $3x - 2y = 5$. Find the values of $x$ and $y$.
What a full answer covers
The answer should show the steps to solve the system of equations, using either substitution or elimination. The final values of $x$ and $y$ should be clearly stated, and the answer should be well presented.

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Mathematical Methods VCE practice questions

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