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

01 For \(f(x)=e^{2x}\), \(f^{-1}(x)=\)

A \(\tfrac12\ln x\) Take ln and divide by 2.

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

C \(e^{x/2}\) This does not satisfy the stated relationship.

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

Written for icaijy.com · U3–4 · Inverse functions · answered at 100 ms

02 A student is checking a worked solution. If \(P(A)=0.6\), \(P(B)=0.5\), \(P(A\cap B)=0.3\), A and B are

A mutually exclusive This does not satisfy the stated relationship.

B complements This does not satisfy the stated relationship.

C independent 0.6×0.5=0.3.

D impossible This does not satisfy the stated relationship.

Written for icaijy.com · U3–4 · Probability · answered at 100 ms

03 A student is checking a worked solution. For X taking 0,1,2 with probabilities 0.2,0.5,0.3, \(E(X)=\)

A 1.1 0×0.2+1×0.5+2×0.3=1.1.

B 1.0 This does not satisfy the stated relationship.

C 1.5 This does not satisfy the stated relationship.

D 0.8 This does not satisfy the stated relationship.

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

04 \(\int_0^2 (3x^2+1)\,dx=\)

A 9 This does not satisfy the stated relationship.

B 10 \([x^3+x]_0^2=10\).

C 8 This does not satisfy the stated relationship.

D 12 This does not satisfy the stated relationship.

Written for icaijy.com · U3–4 · Definite integrals · answered at 100 ms

05 A student is checking a worked solution. For \(f(x)=\log(x-4)\), the domain is

A \(({-\infty},4)\) This does not satisfy the stated relationship.

B \((4,\infty)\) The logarithm argument must be positive.

C \([4,\infty)\) This does not satisfy the stated relationship.

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

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

06 A residual is defined as

A observed times predicted This does not satisfy the stated relationship.

B observed value minus predicted value Residual = y−ŷ.

C gradient minus intercept This does not satisfy the stated relationship.

D predicted minus observed only This does not satisfy the stated relationship.

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

07 A student is checking a worked solution. For \(X\sim\mathrm{Bin}(10,0.5)\), \(P(X=0)=\)

A \(10/1024\) This does not satisfy the stated relationship.

B \(1/1024\) \((0.5)^{10}=1/1024\).

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

D \(1/10\) This does not satisfy the stated relationship.

Written for icaijy.com · U3–4 · Binomial probability · answered at 100 ms

08 During a one-minute revision round: A correlation coefficient near −1 indicates

A strong negative linear association Values near −1 show strong negative linear association.

B weak positive association This does not satisfy the stated relationship.

C a horizontal residual plot only This does not satisfy the stated relationship.

D causation This does not satisfy the stated relationship.

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

09 Solve \(2^x=10\), correct to two decimal places.

A 2.30 This does not satisfy the stated relationship.

B 10.00 This does not satisfy the stated relationship.

C 5.00 This does not satisfy the stated relationship.

D 3.32 \(x=\log_2 10\approx3.32\).

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

10 During a one-minute revision round: If \(P(A)=0.6\), \(P(B)=0.5\), \(P(A\cap B)=0.3\), A and B are

A independent 0.6×0.5=0.3.

B complements This does not satisfy the stated relationship.

C impossible This does not satisfy the stated relationship.

D mutually exclusive This does not satisfy the stated relationship.

Written for icaijy.com · U3–4 · Probability · answered at 100 ms

11 A student is checking a worked solution. Solve \(2^x=10\), correct to two decimal places.

A 3.32 \(x=\log_2 10\approx3.32\).

B 10.00 This does not satisfy the stated relationship.

C 2.30 This does not satisfy the stated relationship.

D 5.00 This does not satisfy the stated relationship.

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

12 A student is checking a worked solution. For X with mean 50 and standard deviation 5, x=60 has z-score

A 5 This does not satisfy the stated relationship.

B 12 This does not satisfy the stated relationship.

C −2 This does not satisfy the stated relationship.

D 2 \(z=(60-50)/5=2\).

Written for icaijy.com · U3–4 · Normal standardisation · answered at 100 ms

13 During a one-minute revision round: For \(f(x)=\log(x-4)\), the domain is

A \(({-\infty},4)\) This does not satisfy the stated relationship.

B \((4,\infty)\) The logarithm argument must be positive.

C \([4,\infty)\) This does not satisfy the stated relationship.

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

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

14 The maximum of \(f(x)=-x^2+6x+1\) is

A 19 This does not satisfy the stated relationship.

B 10 The vertex occurs at x=3, giving f(3)=10.

C 1 This does not satisfy the stated relationship.

D 7 This does not satisfy the stated relationship.

Written for icaijy.com · U3–4 · Optimisation · answered at 100 ms

15 A student is checking a worked solution. The graphs \(y=x^2\) and \(y=2x+3\) intersect at x-values

A \(x=-1,3\) Solve x²−2x−3=0.

B \(x=0,2\) This does not satisfy the stated relationship.

C \(x=-3,1\) This does not satisfy the stated relationship.

D \(x=1,3\) This does not satisfy the stated relationship.

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

16 Which response best completes this VCE-style item? For \(f(x)=x^2+1\), the tangent gradient at x=3 is

A 7 This does not satisfy the stated relationship.

B 10 This does not satisfy the stated relationship.

C 3 This does not satisfy the stated relationship.

D 6 \(f\prime(x)=2x\).

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

17 During a one-minute revision round: For \(f(x)=x^2+1\), the tangent gradient at x=3 is

A 10 This does not satisfy the stated relationship.

B 7 This does not satisfy the stated relationship.

C 6 \(f\prime(x)=2x\).

D 3 This does not satisfy the stated relationship.

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

18 A student is checking a worked solution. An arithmetic sequence has \(u_1=4\), common difference 3. Then \(u_{10}=\)

A 34 This does not satisfy the stated relationship.

B 30 This does not satisfy the stated relationship.

C 27 This does not satisfy the stated relationship.

D 31 u10=4+9×3=31.

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

19 How many x-intercepts does \(y=x^4-5x^2+4\) have?

A 5 This does not satisfy the stated relationship.

B 0 This does not satisfy the stated relationship.

C 4 Let u=x²: (u−1)(u−4)=0 gives x=±1,±2.

D 2 This does not satisfy the stated relationship.

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

20 If \(f(x)=x+2\) and \(g(x)=x^2\), then \((g\circ f)(3)=\)

A 25 f(3)=5, then g(5)=25.

B 11 This does not satisfy the stated relationship.

C 13 This does not satisfy the stated relationship.

D 7 This does not satisfy the stated relationship.

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