elephant!!!!! · guest · Mathematical Methods · Units 3 & 4

01 A student is checking a worked solution. \(\frac d{dx}[xe^x]=\)

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

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

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

D \(e^x(1+x)\) Use uv′+u′v.

Written for icaijy.com · U3–4 · Product rule · answered at 1225 ms

02 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 2637 ms

03 If \(y=(3x+1)^5\), then \(dy/dx=\)

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

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

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

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

Written for icaijy.com · U3–4 · Chain rule · answered at 4452 ms

04 A student is checking a worked solution. For events A and B, \(P(A\cup B)=\)

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

B \(P(A)+P(B)-P(A\cap B)\) Subtract the intersection counted twice.

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

D \(P(A)+P(B)\) This does not satisfy the stated relationship.

Written for icaijy.com · U1–2 · Probability · answered at 6386 ms

05 A student is checking a worked solution. The vertex of \(y=(x-2)^2-5\) is

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

B \((-5,2)\) This does not satisfy the stated relationship.

C \((-2,-5)\) This does not satisfy the stated relationship.

D \((2,5)\) This does not satisfy the stated relationship.

Written for icaijy.com · U1–2 · Quadratics · answered at 9740 ms

06 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 11751 ms

07 Which response best completes this VCE-style item? 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 \(x^2\) This does not satisfy the stated relationship.

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

Written for icaijy.com · U1–2 · Differentiation · answered at 13575 ms

08 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 15484 ms

09 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 17018 ms

10 During a one-minute revision round: \(\int 3x^2\,dx=\)

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

B \(3x^3+C\) 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 19085 ms

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

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

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

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

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

Written for icaijy.com · U1–2 · Rates of change · answered at 20402 ms

12 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 21995 ms

13 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 23535 ms

14 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 25136 ms

15 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 27024 ms

16 During a one-minute revision round: The domain of \(f(x)=\sqrt{x-3}\) is

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

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

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

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

Written for icaijy.com · U1–2 · Functions · answered at 29279 ms

17 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 30734 ms

18 Which response best completes this VCE-style item? \(P(A\mid B)=\)

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

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

C \(\frac{P(A\cap B)}{P(B)}\) Restrict the sample space to B.

D \(\frac{P(A)}{P(B)}\) This does not satisfy the stated relationship.

Written for icaijy.com · U3–4 · Conditional probability · answered at 32836 ms

19 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 34787 ms

20 Which response best completes this VCE-style item? For \(a>1\), \(y=a^x\) is

A strictly decreasing This does not satisfy the stated relationship.

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

C periodic This does not satisfy the stated relationship.

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

Written for icaijy.com · U1–2 · Exponentials · answered at 36743 ms

21 Which response best completes this VCE-style item? If \(y=(3x+1)^5\), then \(dy/dx=\)

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

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

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

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

Written for icaijy.com · U3–4 · Chain rule · answered at 38851 ms

22 A student is checking a worked solution. If \(P(3)=0\), then a factor of \(P(x)\) is

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

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

C \(x+3\) 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 40375 ms

23 \(\sin^2x+\cos^2x\) 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 41792 ms

24 Which response best completes this VCE-style item? 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 43575 ms

25 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 45411 ms

26 Which response best completes this VCE-style item? \(\int_a^b f(x)\,dx\) 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 the gradient at b This does not satisfy the stated relationship.

D total unsigned area always This does not satisfy the stated relationship.

Written for icaijy.com · U3–4 · Fundamental theorem · answered at 47539 ms

27 Which response best completes this VCE-style item? \(\frac d{dx}[xe^x]=\)

A \(e^x(1+x)\) Use uv′+u′v.

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

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

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

Written for icaijy.com · U3–4 · Product rule · answered at 48935 ms

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

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

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

Written for icaijy.com · U1–2 · Differentiation · answered at 51003 ms

29 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 52547 ms

30 During a one-minute revision round: A solution to \(dy/dt=ky\) has the form

A \(y=A+k/t\) This does not satisfy the stated relationship.

B \(y=A\sin kt\) This does not satisfy the stated relationship.

C \(y=Ae^{kt}\) A quantity with rate proportional to itself changes exponentially.

D \(y=Akt\) This does not satisfy the stated relationship.

Written for icaijy.com · U3–4 · Exponentials · answered at 54475 ms

31 Which response best completes this VCE-style item? The vertex of \(y=(x-2)^2-5\) is

A \((-5,2)\) This does not satisfy the stated relationship.

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

C \((2,5)\) This does not satisfy the stated relationship.

D \((-2,-5)\) This does not satisfy the stated relationship.

Written for icaijy.com · U1–2 · Quadratics · answered at 56411 ms

32 \(\int 3x^2\,dx=\)

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

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

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

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

Written for icaijy.com · U3–4 · Integration · answered at 58095 ms

33 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 59516 ms

34 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 · not answered before time expired