algo rank 1 · guest · Mathematical Methods · Units 3 & 4

01 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 1909 ms

02 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 4009 ms

03 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 6360 ms

04 During a one-minute revision round: A standard normal random variable has mean and variance

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

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

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

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

Written for icaijy.com · U3–4 · Normal distribution · answered at 7827 ms

05 The solution of \(|x|<3\) is

A \((-3,3)\) Distance from zero must be less than three.

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

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

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

Written for icaijy.com · U1–2 · Algebra · answered at 9253 ms

06 The graph of \(y=f(x-2)+3\) is translated

A 2 left and 3 up This does not satisfy the stated relationship.

B 3 right and 2 down This does not satisfy the stated relationship.

C 2 right and 3 down 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 12265 ms

07 \(\sin^2x+\cos^2x\) 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 13653 ms

08 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 16389 ms

09 A student is checking a worked solution. If A and B are independent, then

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

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

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

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

Written for icaijy.com · U1–2 · Probability · answered at 18506 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 19912 ms

11 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 21325 ms

12 A student is checking a worked solution. If \(y=(3x+1)^5\), then \(dy/dx=\)

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

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

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

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

Written for icaijy.com · U3–4 · Chain rule · answered at 23958 ms

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

A \(n^2p\) This does not satisfy the stated relationship.

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

C \(\sqrt{np(1-p)}\) This does not satisfy the stated relationship.

D \(np(1-p)\) The binomial variance is np(1−p).

Written for icaijy.com · U3–4 · Binomial distribution · answered at 26873 ms

14 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 28528 ms

15 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 30386 ms

16 For \(f(x)=2x-1\), \(f^{-1}(x)\) is

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

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

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

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

Written for icaijy.com · U1–2 · Functions · answered at 32816 ms

17 Which response best completes this VCE-style item? 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 35270 ms

18 Which response best completes this VCE-style item? 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 \((3,\infty)\) This does not satisfy the stated relationship.

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

Written for icaijy.com · U1–2 · Functions · answered at 37378 ms

19 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 39169 ms

20 A student is checking a worked solution. Increasing sample size while all else is fixed makes a confidence interval generally

A wider This does not satisfy the stated relationship.

B narrower Standard error decreases as sample size grows.

C unchanged This does not satisfy the stated relationship.

D centred at zero This does not satisfy the stated relationship.

Written for icaijy.com · U3–4 · Confidence intervals · answered at 41044 ms

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

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 42854 ms

22 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 44838 ms

23 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 46229 ms

24 During a one-minute revision round: \(\int_a^b f(x)\,dx\) represents

A signed area between the graph and x-axis A definite integral is signed area.

B the mean of a and b This does not satisfy the stated relationship.

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 48021 ms

25 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 50273 ms

26 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 51878 ms

27 A student is checking a worked solution. \(\int 3x^2\,dx=\)

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

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

C \(6x+C\) This does not satisfy the stated relationship.

D \(x^3+C\) Reverse the power rule and include C.

Written for icaijy.com · U3–4 · Integration · answered at 54484 ms

28 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 56590 ms

29 During a one-minute revision round: For \(f(x)=2x-1\), \(f^{-1}(x)\) is

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

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

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

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

Written for icaijy.com · U1–2 · Functions · answered at 58629 ms

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