That y8 kid 🤣 - bot · guest · Mathematical Methods · Units 3 & 4

01 Which response best completes this VCE-style item? For \(X\sim\mathrm{Bin}(n,p)\), \(\mathrm{Var}(X)=\)

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

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

C \(n^2p\) 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 100 ms

02 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 100 ms

03 Which response best completes this VCE-style item? The solution of \(|x|<3\) is

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

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

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

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

Written for icaijy.com · U1–2 · Algebra · answered at 100 ms

04 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 100 ms

05 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 100 ms

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

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 100 ms

07 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 100 ms

08 A student is checking a worked solution. For \(f(x)=2x-1\), \(f^{-1}(x)\) is

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

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

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

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

Written for icaijy.com · U1–2 · Functions · answered at 100 ms

09 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 100 ms

10 If A and B are independent, then

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

Written for icaijy.com · U1–2 · Probability · answered at 100 ms

11 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 100 ms

12 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 100 ms

13 \(\log_a(xy)\) equals

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

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

C \(\log_a x\log_a y\) 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 100 ms

14 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 100 ms

15 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 · answered at 100 ms

16 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 100 ms

17 A student is checking a worked solution. \(\sin^2x+\cos^2x\) equals

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

B 1 This is the Pythagorean identity.

C 0 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 100 ms

18 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 100 ms

19 A student is checking a worked solution. The period of \(\sin(2x)\) is

A \(4\pi\) This does not satisfy the stated relationship.

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

C \(\pi\) Period is 2π/2 = π.

D \(\pi/2\) This does not satisfy the stated relationship.

Written for icaijy.com · U1–2 · Trigonometry · answered at 100 ms

20 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 100 ms