elephant!!!! · guest · Mathematical Methods · Units 3 & 4

01 Which response best completes this VCE-style item? 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 1394 ms

02 \(\sin^2x+\cos^2x\) equals

A \(\sin 2x\) This does not satisfy the stated relationship.

B 1 This is the Pythagorean identity.

C \(\tan x\) This does not satisfy the stated relationship.

D 0 This does not satisfy the stated relationship.

Written for icaijy.com · U1–2 · Trigonometry · answered at 3181 ms

03 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 4525 ms

04 During a one-minute revision round: 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 \(f(b)-f(a)\) This does not satisfy the stated relationship.

C \(f(a)+f(b)\) 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 6013 ms

05 Which response best completes this VCE-style item? If \(f(x)=x^3\), then \(f\prime(x)=\)

A \(3x^2\) Apply the power rule.

B \(3x\) This does not satisfy the stated relationship.

C \(x^2\) This does not satisfy the stated relationship.

D \(x^4/4\) This does not satisfy the stated relationship.

Written for icaijy.com · U1–2 · Differentiation · answered at 7590 ms

06 Which response best completes this VCE-style item? \(\log_a(xy)\) equals

A \(\log_a(x+y)\) This does not satisfy the stated relationship.

B \(\log_a x\log_a y\) This does not satisfy the stated relationship.

C \(x+y\) This does not satisfy the stated relationship.

D \(\log_a x+\log_a y\) The logarithm of a product is a sum.

Written for icaijy.com · U1–2 · Logarithms · answered at 9498 ms

07 During a one-minute revision round: For \(X\sim\mathrm{Bin}(n,p)\), \(E(X)=\)

A \(np(1-p)\) This does not satisfy the stated relationship.

B \(np\) The binomial mean is np.

C \(p/n\) This does not satisfy the stated relationship.

D \(n+p\) This does not satisfy the stated relationship.

Written for icaijy.com · U3–4 · Binomial distribution · answered at 11025 ms

08 During a one-minute revision round: If \(y=(3x+1)^5\), then \(dy/dx=\)

A \(15(3x+1)^5\) This does not satisfy the stated relationship.

B \(15(3x+1)^4\) Differentiate the outside and multiply by the inner derivative.

C \(5(3x+1)^4\) This does not satisfy the stated relationship.

D \((3x+1)^4\) This does not satisfy the stated relationship.

Written for icaijy.com · U3–4 · Chain rule · answered at 13257 ms

09 \(P(A\mid B)=\)

A \(\frac{P(A\cap B)}{P(B)}\) Restrict the sample space to B.

B \(P(A)P(B)\) always This does not satisfy the stated relationship.

C \(P(A\cup B)\) This does not satisfy the stated relationship.

D \(\frac{P(A)}{P(B)}\) This does not satisfy the stated relationship.

Written for icaijy.com · U3–4 · Conditional probability · answered at 14897 ms

10 A student is checking a worked solution. For \(a>1\), \(y=a^x\) is

A strictly decreasing This does not satisfy the stated relationship.

B undefined at x=0 This does not satisfy the stated relationship.

C strictly increasing An exponential with base greater than one increases.

D periodic This does not satisfy the stated relationship.

Written for icaijy.com · U1–2 · Exponentials · answered at 16781 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 18493 ms

12 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 20144 ms

13 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 22003 ms

14 A student is checking a worked solution. A solution to \(dy/dt=ky\) has the form

A \(y=A+k/t\) This does not satisfy the stated relationship.

B \(y=Ae^{kt}\) A quantity with rate proportional to itself changes exponentially.

C \(y=A\sin kt\) This does not satisfy the stated relationship.

D \(y=Akt\) This does not satisfy the stated relationship.

Written for icaijy.com · U3–4 · Exponentials · answered at 24541 ms

15 For \(a>1\), \(y=a^x\) is

A periodic This does not satisfy the stated relationship.

B strictly decreasing This does not satisfy the stated relationship.

C strictly increasing An exponential with base greater than one increases.

D undefined at x=0 This does not satisfy the stated relationship.

Written for icaijy.com · U1–2 · Exponentials · answered at 26094 ms

16 A student is checking a worked solution. If \(P(3)=0\), then a factor of \(P(x)\) is

A \(3x-1\) This does not satisfy the stated relationship.

B \(x-3\) The factor theorem gives x−3.

C \(x+3\) 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 27637 ms

17 Which response best completes this VCE-style item? A solution to \(dy/dt=ky\) has the form

A \(y=Ae^{kt}\) A quantity with rate proportional to itself changes exponentially.

B \(y=Akt\) This does not satisfy the stated relationship.

C \(y=A+k/t\) This does not satisfy the stated relationship.

D \(y=A\sin kt\) This does not satisfy the stated relationship.

Written for icaijy.com · U3–4 · Exponentials · answered at 29686 ms

18 A student is checking a worked solution. For events A and B, \(P(A\cup B)=\)

A \(1-P(A)\) This does not satisfy the stated relationship.

B \(P(A)+P(B)-P(A\cap B)\) Subtract the intersection counted twice.

C \(P(A)P(B)\) 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 · answered at 31224 ms

19 Which response best completes this VCE-style item? The vertex of \(y=(x-2)^2-5\) is

A \((-5,2)\) This does not satisfy the stated relationship.

B \((2,-5)\) Vertex form y=(x−h)²+k has vertex (h,k).

C \((2,5)\) This does not satisfy the stated relationship.

D \((-2,-5)\) This does not satisfy the stated relationship.

Written for icaijy.com · U1–2 · Quadratics · answered at 34537 ms

20 Which response best completes this VCE-style item? For events A and B, \(P(A\cup B)=\)

A \(P(A)P(B)\) This does not satisfy the stated relationship.

B \(1-P(A)\) This does not satisfy the stated relationship.

C \(P(A)+P(B)-P(A\cap B)\) Subtract the intersection counted twice.

D \(P(A)+P(B)\) This does not satisfy the stated relationship.

Written for icaijy.com · U1–2 · Probability · answered at 36304 ms

21 Which response best completes this VCE-style item? If A and B are independent, then

A \(P(A)=P(B)\) This does not satisfy the stated relationship.

B \(P(A\mid B)=0\) This does not satisfy the stated relationship.

C \(P(A\cap B)=P(A)P(B)\) Independence gives the product rule.

D \(P(A\cap B)=0\) This does not satisfy the stated relationship.

Written for icaijy.com · U1–2 · Probability · answered at 38577 ms

22 During a one-minute revision round: 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 40480 ms

23 If \(f(x)=x^3\), then \(f\prime(x)=\)

A \(3x\) This does not satisfy the stated relationship.

B \(x^4/4\) This does not satisfy the stated relationship.

C \(3x^2\) Apply the power rule.

D \(x^2\) This does not satisfy the stated relationship.

Written for icaijy.com · U1–2 · Differentiation · answered at 43171 ms

24 A student is checking a worked solution. The domain of \(f(x)=\sqrt{x-3}\) is

A \(\mathbb R\) This does not satisfy the stated relationship.

B \(({-\infty},3]\) This does not satisfy the stated relationship.

C \([3,\infty)\) The radicand requires x−3 ≥ 0.

D \((3,\infty)\) This does not satisfy the stated relationship.

Written for icaijy.com · U1–2 · Functions · answered at 45086 ms

25 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 47181 ms

26 During a one-minute revision round: \(\sin^2x+\cos^2x\) equals

A 0 This does not satisfy the stated relationship.

B 1 This is the Pythagorean identity.

C \(\tan x\) This does not satisfy the stated relationship.

D \(\sin 2x\) This does not satisfy the stated relationship.

Written for icaijy.com · U1–2 · Trigonometry · answered at 48828 ms

27 A student is checking a worked solution. At a stationary point of a differentiable function

A \(f(x)=0\) This does not satisfy the stated relationship.

B \(f\prime\prime(x)=0\) always This does not satisfy the stated relationship.

C \(f(x)=1\) This does not satisfy the stated relationship.

D \(f\prime(x)=0\) The tangent gradient is zero.

Written for icaijy.com · U1–2 · Differentiation · answered at 51986 ms

28 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 53653 ms

29 A student is checking a worked solution. The expected value of a discrete random variable is

A \(\max x\) This does not satisfy the stated relationship.

B \(\sum xP(X=x)\) Expectation is the probability-weighted mean.

C \(\sum P(X=x)\) This does not satisfy the stated relationship.

D \(\sqrt{\mathrm{Var}(X)}\) This does not satisfy the stated relationship.

Written for icaijy.com · U1–2 · Probability · answered at 55733 ms

30 At a stationary point of a differentiable function

A \(f\prime(x)=0\) The tangent gradient is zero.

B \(f(x)=0\) This does not satisfy the stated relationship.

C \(f(x)=1\) This does not satisfy the stated relationship.

D \(f\prime\prime(x)=0\) always This does not satisfy the stated relationship.

Written for icaijy.com · U1–2 · Differentiation · answered at 58217 ms

31 A student is checking a worked solution. \(\frac d{dx}[xe^x]=\)

A \(x^2e^x\) This does not satisfy the stated relationship.

B \(e^x\) This does not satisfy the stated relationship.

C \(xe^x\) This does not satisfy the stated relationship.

D \(e^x(1+x)\) Use uv′+u′v.

Written for icaijy.com · U3–4 · Product rule · answered at 59885 ms

32 Which response best completes this VCE-style item? 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 · not answered before time expired