， · guest · Mathematical Methods · Units 3 & 4

01 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 3893 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 6823 ms

03 The average rate of change from x=a to x=b is

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

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

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

D \(f(a)+f(b)\) This does not satisfy the stated relationship.

Written for icaijy.com · U1–2 · Rates of change · answered at 13205 ms

04 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 16796 ms

05 The domain of \(f(x)=\sqrt{x-3}\) is

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

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

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

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

Written for icaijy.com · U1–2 · Functions · answered at 20387 ms

06 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 26071 ms

07 Which response best completes this VCE-style item? \(\int 3x^2\,dx=\)

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

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

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

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

Written for icaijy.com · U3–4 · Integration · answered at 32792 ms

08 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 35613 ms

09 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 40808 ms

10 Which response best completes this VCE-style item? The period of \(\sin(2x)\) is

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

B \(4\pi\) 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 46772 ms

11 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 55770 ms

12 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 59559 ms

13 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 · not answered before time expired