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RUN REVIEW · Mathematical Methods · Units 3 & 4 · Tech-free

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12 Oct 2026, 1:52 p.m.

23Serious score
01 Which response best completes this VCE-style item? \(\frac d{dx}[xe^x]=\)
A \(e^x(1+x)\) Use uv′+u′v.
B \(e^x\) This does not satisfy the stated relationship.
C \(x^2e^x\) This does not satisfy the stated relationship.
D \(xe^x\) This does not satisfy the stated relationship.

Written for icaijy.com · U3–4 · Product rule · answered at 1948 ms

02 If \(P(3)=0\), then a factor of \(P(x)\) is
A \(x-3\) The factor theorem gives x−3.
B \(x+3\) This does not satisfy the stated relationship.
C \(3x-1\) This does not satisfy the stated relationship.
D \(x^3\) This does not satisfy the stated relationship.

Written for icaijy.com · U1–2 · Polynomials · answered at 4380 ms

03 During a one-minute revision round: The vertex of \(y=(x-2)^2-5\) is
A \((-5,2)\) This does not satisfy the stated relationship.
B \((-2,-5)\) This does not satisfy the stated relationship.
C \((2,-5)\) Vertex form y=(x−h)²+k has vertex (h,k).
D \((2,5)\) This does not satisfy the stated relationship.

Written for icaijy.com · U1–2 · Quadratics · answered at 7176 ms

04 Which response best completes this VCE-style item? If \(y=(3x+1)^5\), then \(dy/dx=\)
A \((3x+1)^4\) This does not satisfy the stated relationship.
B \(15(3x+1)^5\) This does not satisfy the stated relationship.
C \(5(3x+1)^4\) This does not satisfy the stated relationship.
D \(15(3x+1)^4\) Differentiate the outside and multiply by the inner derivative.

Written for icaijy.com · U3–4 · Chain rule · answered at 10044 ms

05 The expected value of a discrete random variable is
A \(\sum P(X=x)\) This does not satisfy the stated relationship.
B \(\max x\) This does not satisfy the stated relationship.
C \(\sum xP(X=x)\) Expectation is the probability-weighted mean.
D \(\sqrt{\mathrm{Var}(X)}\) This does not satisfy the stated relationship.

Written for icaijy.com · U1–2 · Probability · answered at 12289 ms

06 During a one-minute revision round: The period of \(\sin(2x)\) is
A \(\pi/2\) This does not satisfy the stated relationship.
B \(2\pi\) This does not satisfy the stated relationship.
C \(4\pi\) This does not satisfy the stated relationship.
D \(\pi\) Period is 2π/2 = π.

Written for icaijy.com · U1–2 · Trigonometry · answered at 14963 ms

07 A solution to \(dy/dt=ky\) has the form
A \(y=A+k/t\) This does not satisfy the stated relationship.
B \(y=A\sin kt\) This does not satisfy the stated relationship.
C \(y=Ae^{kt}\) A quantity with rate proportional to itself changes exponentially.
D \(y=Akt\) This does not satisfy the stated relationship.

Written for icaijy.com · U3–4 · Exponentials · answered at 18600 ms

08 A student is checking a worked solution. The average rate of change from x=a to x=b is
A \(\frac{f(b)-f(a)}{b-a}\) It is change in output divided by change in input.
B \(f(a)+f(b)\) This does not satisfy the stated relationship.
C \(f(b)-f(a)\) This does not satisfy the stated relationship.
D \(\frac{b-a}{f(b)-f(a)}\) This does not satisfy the stated relationship.

Written for icaijy.com · U1–2 · Rates of change · answered at 20323 ms

09 Which response best completes this VCE-style item? \(\int 3x^2\,dx=\)
A \(x^3\) This does not satisfy the stated relationship.
B \(x^3+C\) Reverse the power rule and include C.
C \(3x^3+C\) This does not satisfy the stated relationship.
D \(6x+C\) This does not satisfy the stated relationship.

Written for icaijy.com · U3–4 · Integration · answered at 23652 ms

10 Which response best completes this VCE-style item? \(\sin^2x+\cos^2x\) equals
A \(\tan x\) This does not satisfy the stated relationship.
B 0 This does not satisfy the stated relationship.
C 1 This is the Pythagorean identity.
D \(\sin 2x\) This does not satisfy the stated relationship.

Written for icaijy.com · U1–2 · Trigonometry · answered at 25456 ms

11 During a one-minute revision round: At a stationary point of a differentiable function
A \(f\prime\prime(x)=0\) always This does not satisfy the stated relationship.
B \(f(x)=0\) This does not satisfy the stated relationship.
C \(f\prime(x)=0\) The tangent gradient is zero.
D \(f(x)=1\) This does not satisfy the stated relationship.

Written for icaijy.com · U1–2 · Differentiation · answered at 28312 ms

12 A student is checking a worked solution. The graph of \(y=f(x-2)+3\) is translated
A 2 right and 3 down This does not satisfy the stated relationship.
B 3 right and 2 down This does not satisfy the stated relationship.
C 2 left and 3 up This does not satisfy the stated relationship.
D 2 right and 3 up Horizontal translation uses the opposite sign inside.

Written for icaijy.com · U1–2 · Transformations · answered at 31653 ms

13 Increasing sample size while all else is fixed makes a confidence interval generally
A unchanged This does not satisfy the stated relationship.
B wider This does not satisfy the stated relationship.
C narrower Standard error decreases as sample size grows.
D centred at zero This does not satisfy the stated relationship.

Written for icaijy.com · U3–4 · Confidence intervals · answered at 34696 ms

14 \(\int_a^b f(x)\,dx\) represents
A the mean of a and b This does not satisfy the stated relationship.
B signed area between the graph and x-axis A definite integral is signed area.
C total unsigned area always This does not satisfy the stated relationship.
D the gradient at b This does not satisfy the stated relationship.

Written for icaijy.com · U3–4 · Fundamental theorem · answered at 37089 ms

15 The period of \(\sin(2x)\) is
A \(\pi\) Period is 2π/2 = π.
B \(\pi/2\) This does not satisfy the stated relationship.
C \(2\pi\) This does not satisfy the stated relationship.
D \(4\pi\) This does not satisfy the stated relationship.

Written for icaijy.com · U1–2 · Trigonometry · answered at 39583 ms

16 During a one-minute revision round: \(\frac d{dx}[xe^x]=\)
A \(e^x\) This does not satisfy the stated relationship.
B \(x^2e^x\) This does not satisfy the stated relationship.
C \(e^x(1+x)\) Use uv′+u′v.
D \(xe^x\) This does not satisfy the stated relationship.

Written for icaijy.com · U3–4 · Product rule · answered at 41348 ms

17 Which response best completes this VCE-style item? The period of \(\sin(2x)\) is
A \(\pi/2\) This does not satisfy the stated relationship.
B \(4\pi\) This does not satisfy the stated relationship.
C \(2\pi\) This does not satisfy the stated relationship.
D \(\pi\) Period is 2π/2 = π.

Written for icaijy.com · U1–2 · Trigonometry · answered at 43095 ms

18 The average rate of change from x=a to x=b is
A \(f(b)-f(a)\) This does not satisfy the stated relationship.
B \(\frac{f(b)-f(a)}{b-a}\) It is change in output divided by change in input.
C \(\frac{b-a}{f(b)-f(a)}\) This does not satisfy the stated relationship.
D \(f(a)+f(b)\) This does not satisfy the stated relationship.

Written for icaijy.com · U1–2 · Rates of change · answered at 44532 ms

19 Which response best completes this VCE-style item? \(P(A\mid B)=\)
A \(P(A\cup B)\) This does not satisfy the stated relationship.
B \(P(A)P(B)\) always This does not satisfy the stated relationship.
C \(\frac{P(A\cap B)}{P(B)}\) Restrict the sample space to B.
D \(\frac{P(A)}{P(B)}\) This does not satisfy the stated relationship.

Written for icaijy.com · U3–4 · Conditional probability · answered at 47331 ms

20 For \(X\sim\mathrm{Bin}(n,p)\), \(\mathrm{Var}(X)=\)
A \(\sqrt{np(1-p)}\) This does not satisfy the stated relationship.
B \(np(1-p)\) The binomial variance is np(1−p).
C \(np\) This does not satisfy the stated relationship.
D \(n^2p\) This does not satisfy the stated relationship.

Written for icaijy.com · U3–4 · Binomial distribution · answered at 50368 ms

21 A student is checking a worked solution. For \(X\sim\mathrm{Bin}(n,p)\), \(E(X)=\)
A \(n+p\) This does not satisfy the stated relationship.
B \(np\) The binomial mean is np.
C \(np(1-p)\) This does not satisfy the stated relationship.
D \(p/n\) This does not satisfy the stated relationship.

Written for icaijy.com · U3–4 · Binomial distribution · answered at 52200 ms

22 Which response best completes this VCE-style item? The expected value of a discrete random variable is
A \(\sqrt{\mathrm{Var}(X)}\) This does not satisfy the stated relationship.
B \(\max x\) This does not satisfy the stated relationship.
C \(\sum P(X=x)\) This does not satisfy the stated relationship.
D \(\sum xP(X=x)\) Expectation is the probability-weighted mean.

Written for icaijy.com · U1–2 · Probability · answered at 56627 ms

23 During a one-minute revision round: \(P(A\mid B)=\)
A \(\frac{P(A)}{P(B)}\) This does not satisfy the stated relationship.
B \(P(A\cup B)\) This does not satisfy the stated relationship.
C \(P(A)P(B)\) always This does not satisfy the stated relationship.
D \(\frac{P(A\cap B)}{P(B)}\) Restrict the sample space to B.

Written for icaijy.com · U3–4 · Conditional probability · answered at 58857 ms

24 During a one-minute revision round: For events A and B, \(P(A\cup B)=\)
A \(P(A)+P(B)-P(A\cap B)\) Subtract the intersection counted twice.
B \(P(A)+P(B)\) This does not satisfy the stated relationship.
C \(1-P(A)\) This does not satisfy the stated relationship.
D \(P(A)P(B)\) This does not satisfy the stated relationship.

Written for icaijy.com · U1–2 · Probability · not answered before time expired

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