Albert · guest · Mathematical Methods · Units 1 & 2

01 If A and B are independent, then

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

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

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

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

Written for icaijy.com · U1–2 · Probability · answered at 2604 ms

02 The expected value of a discrete random variable is

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

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

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

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

Written for icaijy.com · U1–2 · Probability · answered at 5462 ms

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

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 8362 ms

04 Which response best completes this VCE-style item? \(\sin^2x+\cos^2x\) equals

A 1 This is the Pythagorean identity.

B 0 This does not satisfy the stated relationship.

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

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

Written for icaijy.com · U1–2 · Trigonometry · answered at 10060 ms

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

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

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

Written for icaijy.com · U1–2 · Logarithms · answered at 12264 ms

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

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

Written for icaijy.com · U1–2 · Algebra · answered at 13985 ms

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

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

B periodic This does not satisfy the stated relationship.

C strictly decreasing This does not satisfy the stated relationship.

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

Written for icaijy.com · U1–2 · Exponentials · answered at 16178 ms

08 Which response best completes this VCE-style item? At a stationary point of a differentiable function

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

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

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

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

Written for icaijy.com · U1–2 · Differentiation · answered at 20158 ms

09 A student is checking a worked solution. The average rate of change from x=a to x=b is

A \(\frac{b-a}{f(b)-f(a)}\) This does not satisfy the stated relationship.

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 \(\frac{f(b)-f(a)}{b-a}\) It is change in output divided by change in input.

Written for icaijy.com · U1–2 · Rates of change · answered at 23093 ms

10 A student is checking a worked solution. 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 26858 ms

11 During a one-minute revision round: 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 31488 ms

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

A \(\pi\) Period is 2π/2 = π.

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

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

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

Written for icaijy.com · U1–2 · Trigonometry · answered at 33534 ms

13 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 35416 ms

14 A student is checking a worked solution. The solution of \(|x|<3\) is

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

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

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

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

Written for icaijy.com · U1–2 · Algebra · answered at 37107 ms

15 During a one-minute revision round: The period of \(\sin(2x)\) is

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

B \(\pi/2\) 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 39148 ms

16 During a one-minute revision round: For \(a>1\), \(y=a^x\) is

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

B strictly decreasing This does not satisfy the stated relationship.

C periodic This does not satisfy the stated relationship.

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

Written for icaijy.com · U1–2 · Exponentials · answered at 42111 ms

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

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

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

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

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

Written for icaijy.com · U1–2 · Probability · answered at 44344 ms

18 Which response best completes this VCE-style item? The graph of \(y=f(x-2)+3\) is translated

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

B 2 right and 3 up Horizontal translation uses the opposite sign inside.

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

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

Written for icaijy.com · U1–2 · Transformations · answered at 46928 ms

19 A student is checking a worked solution. 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 \((2,5)\) This does not satisfy the stated relationship.

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

Written for icaijy.com · U1–2 · Quadratics · answered at 49624 ms

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

A 0 This does not satisfy the stated relationship.

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

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

D 1 This is the Pythagorean identity.

Written for icaijy.com · U1–2 · Trigonometry · answered at 51497 ms

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

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

B 0 This does not satisfy the stated relationship.

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

D 1 This is the Pythagorean identity.

Written for icaijy.com · U1–2 · Trigonometry · answered at 52899 ms

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

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

Written for icaijy.com · U1–2 · Differentiation · answered at 55423 ms

23 During a one-minute revision round: \(\log_a(xy)\) equals

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

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

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

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

Written for icaijy.com · U1–2 · Logarithms · answered at 57340 ms

24 During a one-minute revision round: If \(P(3)=0\), then a factor of \(P(x)\) is

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

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

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

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

Written for icaijy.com · U1–2 · Polynomials · not answered before time expired