six · guest · Mathematical Methods · Units 3 & 4

01 A student is checking a worked solution. \(\log_a(xy)\) equals

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

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

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

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

Written for icaijy.com · U1–2 · Logarithms · answered at 7357 ms

02 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 10057 ms

03 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 \(n+p\) 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 13873 ms

04 A student is checking a worked solution. \(P(A\mid B)=\)

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

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

Written for icaijy.com · U3–4 · Conditional probability · answered at 17940 ms

05 A student is checking a worked solution. If \(f(x)=x^3\), then \(f\prime(x)=\)

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

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

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

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

Written for icaijy.com · U1–2 · Differentiation · answered at 20190 ms

06 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 24123 ms

07 During a one-minute revision round: The domain of \(f(x)=\sqrt{x-3}\) is

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

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

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

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

Written for icaijy.com · U1–2 · Functions · answered at 29040 ms

08 A standard normal random variable has mean and variance

A 0 and 1 Standardisation produces mean 0 and variance 1.

B 1 and 1 This does not satisfy the stated relationship.

C 1 and 0 This does not satisfy the stated relationship.

D 0 and 0 This does not satisfy the stated relationship.

Written for icaijy.com · U3–4 · Normal distribution · answered at 31557 ms

09 Which response best completes this VCE-style item? For \(a>1\), \(y=a^x\) is

A strictly decreasing This does not satisfy the stated relationship.

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

C periodic This does not satisfy the stated relationship.

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

Written for icaijy.com · U1–2 · Exponentials · answered at 37090 ms

10 Which response best completes this VCE-style item? For \(f(x)=2x-1\), \(f^{-1}(x)\) is

A \(2x+1\) This does not satisfy the stated relationship.

B \((x-1)/2\) This does not satisfy the stated relationship.

C \((x+1)/2\) Swap x and y, then solve for y.

D \(1/(2x-1)\) This does not satisfy the stated relationship.

Written for icaijy.com · U1–2 · Functions · answered at 44956 ms

11 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 · not answered before time expired