Year 12 VCE Maths - Mathematical Methods sample questions
Year 12 VCE Maths - Mathematical Methods sample questions
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Year 12 VCE Mathematical Methods practice questions
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Practice functions, relations, algebra, calculus, probability, transformations, graphs, and mathematical modelling.
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Sample VCE Mathematical Methods questions
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VCE Year 12Mathematical MethodsDifferentiation and integrationhardtangent and area graph
1. The graph shows y = x(4 - x). Tangent T is drawn at P where x = 1, and R is the region between T and the curve from x = 1 to x = 3. What is the exact area of R?
Choices
8/3
4/3
2
10/3
Explanation:
f'(x) = 4 - 2x, so at x = 1 the tangent is y = 2x + 1. On 1 ≤ x ≤ 3, T - f(x) = 2x + 1 - x(4 - x) = (x - 1)2. The area is the integral from 1 to 3 of (x - 1)2, which is 8/3.
VCE Year 12Mathematical MethodsFunctions and transformationshardtransformation graph
2. The graph of y = f(x) has stationary point A(-2, 5). For g(x) = 3 - 2f(x + 1), which point is the corresponding stationary point on y = g(x)?
Choices
(-3, -7)
(-1, -7)
(-3, 13)
(1, -7)
Explanation:
The input x + 1 must equal -2, so x = -3. The output becomes 3 - 2(5) = -7, so the corresponding stationary point is (-3, -7).
VCE Year 12Mathematical MethodsContinuous probabilityharddensity graph
3. For the continuous random variable X with density f(x) = cx(4 - x), 0 ≤ x ≤ 4, what is P(X > 3 | X > 2)?
Choices
5/16
5/32
11/16
3/8
Explanation:
The constant c cancels in the conditional probability. The integral of x(4 - x) from 3 to 4 is 5/3, and from 2 to 4 is 16/3. Hence P(X > 3 | X > 2) = (5/3)/(16/3) = 5/16.
VCE Year 12Mathematical MethodsOptimisationhardoptimisation diagram
4. A rectangle has top corners on y = 12 - x2 and base on the x-axis, symmetric about the y-axis. If the right top corner is P(x, 12 - x2), x > 0, which value of x gives the maximum rectangle area?
Choices
2
√2
√(6)
3
Explanation:
The rectangle has width 2x and height 12 - x2, so A(x) = 2x(12 - x2). Then A'(x) = 24 - 6x2, giving x = 2. Also A''(x) = -12x is negative at x = 2, so the area is maximal.
VCE Year 12Mathematical MethodsExponential and logarithmic modellinghardmodel graph
5. A revision model is N(t) = 80/(1 + 7e-0.4t), where t is measured in weeks. At what time is N'(t) greatest?
Choices
2.5 ln 7 weeks
ln 7 / 2.5 weeks
0.4 ln 7 weeks
7 / 2.5 weeks
Explanation:
For a logistic model, growth is greatest at half the limiting value, so N = 40. Solve 80/(1 + 7e-0.4t) = 40 to get 7e-0.4t = 1, so t = ln 7 / 0.4 = 2.5 ln 7 weeks.
VCE Year 12Mathematical MethodsNormal distributionhardnormal curve
6. For X ~ N(μ, σ2), P(X < 62) = 0.1587 and P(X < 74) = 0.8413. Using Φ(-1) = 0.1587 and Φ(1) = 0.8413, what are μ and σ?
Choices
μ = 68, σ = 6
μ = 68, σ = 12
μ = 62, σ = 6
μ = 74, σ = 6
Explanation:
The probabilities place 62 one standard deviation below the mean and 74 one standard deviation above it. The mean is the midpoint (62 + 74)/2 = 68, and the standard deviation is 74 - 68 = 6.
For parents comparing VCE Mathematical Methods support
VCE Mathematical Methods practice should make functions, calculus, probability, and modelling feel testable without hiding the method. These examples preview graph-backed, single-best-answer questions before the no-login Mathematical Methods demo.
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Practice Summary
Year 12 · VCE Maths - Mathematical Methods · Test mode · 3 questions · 10 min
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