Aussie Maths & Science Teachers: Save your time with SmarterEd
Two unbiased coins are tossed.
What is the financial expectation of this game? (2 marks)
A company makes large marshmallows. They are in the shape of a cylinder with diameter 5 cm and height 3 cm, as shown in the diagram.
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A cake is to be made by stacking 24 of these large marshmallows and filling the gaps between them with chocolate. The diagrams show the cake and its top view. The shading shows the gaps to be filled with chocolate.
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| i. | `V` | `= pir^2h` |
| `= pi xx 2.5^2 xx 3` | ||
| `= 58.904…` | ||
| `= 58.9\ text{cm³ (1 d.p.)}` |
| ii. |  |
`text(Volume of rectangle)`
`= 15 xx 10 xx 6`
`= 900\ text(cm)^3`
`text(Volume of marshmallows in rectangle)`
`= 6 xx 2 xx 58.9`
`= 706.8\ text(cm)^3`
`:.\ text(Volume of chocolate)`
`= 900-706.8`
`= 193.2`
`= 193\ text{cm}^3 \ text{(nearest cm}^3 text{)}`
Jacob has a large jar of silver coins. He adds 20 gold coins into the jar. He then seals the jar and shakes it to ensure that the gold coins are mixed in thoroughly with the silver coins. Jacob then opens the jar and takes a handful of coins. In his hand he has 33 silver coins and 4 gold coins.
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Melbourne time is 6 hours ahead of Dubai time.
A plane leaves Melbourne on Friday at 11.30 pm. The flight time to Dubai is 15 hours.
What will be the time and the day in Dubai when the plane is due to land? (2 marks)
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Marge borrowed $19 000 to buy a used car. Interest on the loan was charged at 4.8% pa at the end of each month. She made a repayment of $436 at the end of every month. The table below sets out her monthly repayment schedule for the first four months of the loan.
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What is the total amount that Marge repaid? (1 mark)
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Theo is completing his tax return. He has a gross salary of $82 521 and income from a rental property totalling $10 920. He is claiming $13 420 in allowable deductions.
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\begin{array} {|l|l|}
\hline
\rule{0pt}{2.5ex}\textit{ Taxable income}\rule[-1ex]{0pt}{0pt} & \textit{ Tax payable}\\
\hline
\rule{0pt}{2.5ex}\text{\$0 – \$18 200}\rule[-1ex]{0pt}{0pt} & \text{Nil}\\
\hline
\rule{0pt}{2.5ex}\text{\$18 201 – \$37 000}\rule[-1ex]{0pt}{0pt} & \text{19 cents for each \$1 over \$18 200}\\
\hline
\rule{0pt}{2.5ex}\text{\$37 001 – \$80 000}\rule[-1ex]{0pt}{0pt} & \text{\$3572 plus 32.5 cents for each \$1 over \$37 000}\\
\hline
\rule{0pt}{2.5ex}\text{\$80 001 – \$180 000}\rule[-1ex]{0pt}{0pt} & \text{\$17 547 plus 37 cents for each \$1 over \$80 000}\\
\hline
\rule{0pt}{2.5ex}\text{\$180 001 and over}\rule[-1ex]{0pt}{0pt} & \text{\$54 547 plus 45 cents for each \$1 over \$180 000}\\
\hline
\end{array}
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Jenny earns a yearly salary of $63 752. Her annual leave loading is 17.5% of four weeks pay.
Calculate her total pay for her four weeks of annual leave. (3 marks)
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The diagram shows a block of land `ABCD` that has been surveyed. All measurements are in metres.
Calculate the length of `AB`, correct to the nearest metre. (2 marks)
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`text(Using Pythagoras,)`
| `AB` | `= sqrt(32^2 + 33^2)` |
| `= 45.967…` | |
| `= 46\ text{m (nearest m)}` |
Simplify `(8x^4y)/(24x^3y^5)`. (2 marks)
Which of the following correctly expresses `Q` as the subject of `e = iR + Q/C`?
The box-and-whisker plots show the results of a History test and a Geography test.
In History, 112 students completed the test. The number of students who scored above 30 marks was the same for the History test and the Geography test.
How many students completed the Geography test?
Isabella works a 35-hour week and is paid at an hourly rate of $18. Any overtime hours worked are paid at time-and-a-half. In a particular week, she earned $1008.
How many hours in total did Isabella work in this week to earn this amount?
The width (`W`) of a river can be calculated using two similar triangles, as shown in the diagram.
What is the approximate width of the river?
The graph shows a line which has an equation in the form `y = mx + c`.
Which of the following statements is true?
The speed limit outside a school is 40 km/h. Year 11 students measured the speed of passing vehicles over a period of time. They found the set of data to be normally distributed with a mean speed of 36 km/h and a standard deviation of 2 km/h.
What percentage of the vehicles passed the school at a speed greater than 40 km/h?
| `z` | `= (x – mu)/sigma` |
| `= (40 – 36)/2` | |
| `= 2` |
`=> A`
Caroline drinks two small bottles of wine over a three-hour period. Each of these bottles contains 2.3 standard drinks. Caroline weighs 53 kg.
Using the formula below, what is Caroline's approximate blood alcohol content (BAC) at the end of this period?
`BAC_text(Female) = (10N - 7.5H)/(5.5M)`
where `N` is the number of standard drinks consumed
`H` is the number of hours drinking
`M` is the person's mass in kilograms
The table shows the future value of an investment of $1000, compounding yearly, at varying interest rates for different periods of time.
Based on the information provided, what is the future value of an investment of $2500 over 3 years at 4% pa?
Which expression is equivalent to `2(3x-4) + 2`?
The graph shows a scatterplot for a set of data.
Which of the following is the best approximation for the correlation coefficient of this set of data?
Solve `sin (x/2) = 1/2` for `0 <= x <= 2pi.` (2 marks)
Find the gradient of the tangent to the curve `y = tan x` at the point where `x = pi/8.`
Give your answer correct to 3 significant figures. (2 marks)
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Which expression is equivalent to `4 + log_2 x?`
(A) `log_2 (2x)`
(B) `log_2 (16 + x)`
(C) `4 log_2 (2x)`
(D) `log_2 (16x)`
The circle centred at `O` has radius 5. Arc `AB` has length 7 as shown in the diagram.
What is the area of the shaded sector `OAB?`
Let `p(x) = 1 + x + x^2 + x^3 + … + x^12.`
What is the coefficient of `x^8` in the expansion of `p (x + 1)?`
Multiplying a non-zero complex number by `(1 - i)/(1 + i)` results in a rotation about the origin on an Argand diagram.
What is the rotation?
The sum of the eccentricities of two different conics is `3/4.`
Which pair of conics could this be?
Which polynomial has a multiple root at `x = 1?`
Jo has either tea or coffee at morning break. If she has tea one morning, the probability she has tea the next morning is 0.4. If she has coffee one morning, the probability she has coffee the next morning is 0.3. Suppose she has coffee on a Monday morning. What is the probability that she has tea on the following Wednesday morning? (3 marks)
`text(Pr)(C\ T\ T) + text(Pr)(C\ C\ T)`
`= 0.7 xx 0.4 + 0.3 xx 0.7`
`= 0.28 + 0.21`
`= 0.49`
The probability density function of a continuous random variable `X` is given by `f(x) = {(x/12,\ \ 1 <= x <= 5), (\ 0,\ \ text(otherwise)):}` --- 5 WORK AREA LINES (style=lined) --- --- 5 WORK AREA LINES (style=lined) ---
Let `X` be a normally distributed random variable with a mean of 72 and a standard deviation of 8. Let `Z` be the standard normal random variable. Use the result that `text(Pr) (Z < 1) = 0.84`, correct to two decimal places, to find --- 4 WORK AREA LINES (style=lined) --- --- 4 WORK AREA LINES (style=lined) --- --- 4 WORK AREA LINES (style=lined) ---
`text(Pr) (X > 80)` `= text(Pr) (Z > (80 – 72)/8)` `= text(Pr) (Z > 1)` `= 0.16` b. `text(Pr) (64 < X < 72)` `= text(Pr) (72 < X < 80)\ \ \ text(due to symmetry)` `= 0.5 – 0.16` `= 0.34` c. `text(Conditional probability)`Show Worked Solution
a.
♦ Mean mark 45%.♦ Mean mark 40%.
MARKER’S COMMENT: Notation was poor, showing a lack of understanding in this area.
`text(Pr) (X < 64 | X < 72)`
`= (text{Pr} (X < 64))/(text{Pr} (X < 72))`
`= 0.16/0.50`
`= 8/25 or 0.32`
Let `g(x) = log_e(tan(x))`. Evaluate `g^{prime}(pi/4)`. (2 marks)
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There is a daily flight from Paradise Island to Melbourne. The probability of the flight departing on time, given that there is fine weather on the island, is 0.8, and the probability of the flight departing on time, given that the weather on the island is not fine, is 0.6. In March the probability of a day being fine is 0.4. Find the probability that on a particular day in March --- 6 WORK AREA LINES (style=lined) --- --- 5 WORK AREA LINES (style=lined) ---
`= 0.4 xx 0.8 + 0.6 xx 0.6` `= 0.32 + 0.36` `= 0.68` b. `text(Conditional probability:)` `:. text(Pr)(F\ text(|)\ T) = 8/17`Show Worked Solution
a.
`text{Pr(FT)}\ +\ text{Pr(F′T)}`♦♦ Mean mark 29%.
MARKER’S COMMENT: Students continue to struggle with conditional probability. Attention required here.
`text(Pr)(F\ text(|)\ T)`
`= (text(Pr)(F ∩ T))/(text(Pr)(T))`
`= 0.32/0.68`
The area of the region bounded by the curve with equation `y = kx^(1/2)`, where `k` is a positive constant, the `x`-axis and the line with equation `x = 9` is 27. Find `k`. (3 marks)
`text(Sketch:)`
| `text(Area)` | `= 27` |
| `27` | `= k int_0^9(x ^(1/2))\ dx` |
| `27` | `= 2/3k [x^(3/2)]_0^9` |
| `27` | `= 2/3k [27 – 0]` |
| `:. k` | `= 3/2` |
The graph of `f: R -> R`, `f(x) = e^(x/2) + 1` is shown. The normal to the graph of `f` where it crosses the `y`-axis is also shown.
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Let `f: R -> R`, `f(x) = sin((2pix)/3)`.
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Solve the equation `log_e(3x + 5) + log_e(2) = 2`, for `x`. (2 marks)
Let `f: R -> R,\ \ f(x) = e^(2x)-1`.
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a. `text(Let)\ \ y = f(x)`
`text(For Inverse, swap)\ x ↔ y`
| `x` | `= e^(2y)-1` |
| `x + 1` | `= e^(2y)` |
| `2y` | `= log_e(x + 1)` |
| `y` | `= 1/2 log_e(x + 1)` |
| `text(Domain)(f^(-1))` | `=\ text(Range)\ (f)` |
| `= (-1,∞)` |
`:. f^(-1)(x) = 1/2log_e(x + 1),quadx ∈ (-1,∞)`
b. `f(f^(-1)(x)) = x`
`text(Domain is)\ \ (-1, oo)`
| c. | `-f^(-1)(2x)` | `= -1/2 ln(2x + 1)` |
| `:. f(-f^(-1)(2x))` | `= e^(-log_e(2x + 1))-1` | |
| `=(2x+1)^-1-1` | ||
| `= 1/(2x + 1)-1` | ||
| `= (-2x)/(2x + 1)` |
A plastic brick is made in the shape of a right triangular prism. The triangular end is an equilateral triangle with side length `x` cm and the length of the brick is `y` cm.
The volume of the brick is 1000 cm³.
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Every Friday Jean-Paul goes to see a movie. He always goes to one of two local cinemas – the Dandy or the Cino.
If he goes to the Dandy one Friday, the probability that he goes to the Cino the next Friday is 0.5. If he goes to the Cino one Friday, then the probability that he goes to the Dandy the next Friday is 0.6.
On any given Friday the cinema he goes to depends only on the cinema he went to on the previous Friday.
If he goes to the Cino one Friday, what is the probability that he goes to the Cino on exactly two of the next three Fridays? (3 marks)
Jane drives to work each morning and passes through three intersections with traffic lights. The number `X` of traffic lights that are red when Jane is driving to work is a random variable with probability distribution given b --- 1 WORK AREA LINES (style=lined) --- --- 3 WORK AREA LINES (style=lined) ---
The area of the region bounded by the `y`-axis, the `x`-axis, the curve `y = e^(2x)` and the line `x = C`, where `C` is a positive real constant, is `5/2`. Find `C`. (3 marks)
| `text(Area)` | `= 5/2` |
| `int_0^C e^(2x) dx` | `= 5/2` |
| `[1/2 e^(2x)]_0^C` | `= 5/2` |
| `e^(2C) – e^0` | `= 5` |
| `e^(2C)` | `= 6` |
| `2C` | `= ln6` |
| `:. C` | `= 1/2 ln6` |
The function `f(x) = {{:(k),(0):}{:(sin(pix)qquadtext(if)qquadx ∈ [0,1]),(qquadqquadqquadqquadquadtext(otherwise)):}` is a probability density function for the continuous random variable `X`. --- 5 WORK AREA LINES (style=lined) --- --- 5 WORK AREA LINES (style=lined) ---
Solve the equation `cos((3x)/2) = 1/2` for `x ∈ [-pi/2,pi/2]`. (2 marks)
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Solve the equation `2 log_e (x) - log_e (x + 3) = log_e (1/2)` for `x.` (4 marks)
The random variable `X` has this probability distribution Find --- 5 WORK AREA LINES (style=lined) --- --- 6 WORK AREA LINES (style=lined) ---
Four identical balls are numbered 1, 2, 3 and 4 and put into a box. A ball is randomly drawn from the box, and not returned to the box. A second ball is then randomly drawn from the box. --- 3 WORK AREA LINES (style=lined) --- --- 3 WORK AREA LINES (style=lined) --- --- 4 WORK AREA LINES (style=lined) ---
Solve the equation `tan (2x) = sqrt 3` for `x in (-pi/4, pi/4) uu (pi/4, (3 pi)/4).` (3 marks)
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Let `f: R\ text(\)\ {0} -> R` where `f(x) = 3/x-4.` Find `f^-1`, the inverse function of `f.` (3 marks)
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Find an anti-derivative of `1/(1-2x)` with respect to `x`. (2 marks)
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The line `y = ax-1` is a tangent to the curve `y = x^(1/2) + d` at the point `(9, c)` where `a, c` and `d` are real constants.
Find the values of `a, c` and `d`. (4 marks)
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The discrete random variable `X` has the probability distribution
Find the value of `p.` (3 marks)
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The continuous random variable `X` has a distribution with probability density function given by `f(x) = {(ax(5 - x), \ text(if)\ \ 0 <= x <= 5), (0,\ text (if)\ \ x < 0\ \ text(or if)\ \ x > 5):}` where `a` is a positive constant. --- 5 WORK AREA LINES (style=lined) --- --- 1 WORK AREA LINES (style=lined) ---
Let `X` be a normally distributed random variable with mean 5 and variance 9 and let `Z` be the random variable with the standard normal distribution. --- 2 WORK AREA LINES (style=lined) --- --- 4 WORK AREA LINES (style=lined) ---
Let `f: R^+ -> R` where `f(x) = 1/x^2.`
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Differentiate `x^3 e^(2x)` with respect to `x`. (2 marks)
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The figure shown represents a wire frame where `ABCE` is a convex quadrilateral. The point `D` is on line segment `EC` with `AB = ED = 2\ text(cm)` and `BC = a\ text(cm)`, where `a` is a positive constant.
`/_ BAE = /_ CEA = pi/2`
Let `/_ CBD = theta` where `0 < theta < pi/2.`
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a. `text(In)\ \ Delta BCD,`
| `cos theta` | `= (BD)/a` |
| `:. BD` | `= a cos theta` |
| `sin theta` | `= (CD)/a` |
| `:. CD` | `= a sin theta` |
| b. | `L` | `= 4 + 2 BD + CD + a` |
| `= 4 + 2a cos theta + a sin theta + a` | ||
| `= 4 + a + 2a cos theta + a sin theta` |
c. `text(Noting that)\ a\ text(is a constant:)`
`(dL)/(d theta)= – 2 a sin theta + a cos theta`
`text(When)\ \ (dL)/(d theta) = 0`,
| `- 2 a sin theta+ a cos theta` | `= 0` |
| `a cos theta` | `= 2 a sin theta` |
| `:. BD` | `= 2CD\ \ text{(using part (a))}` |
d. `text(SP’s when)\ \ (dL)/(d theta)=0,`
| `- 2 a sin theta+ a cos theta` | `= 0` |
| `sin theta` | `=1/2 cos theta` |
| `tan theta` | `=1/2` |
`text(If)\ \ tan theta=1/2,\ \ cos theta = 2/sqrt5,\ \ sin theta = 1/sqrt5`
| `L_(max)` | `= 4 + a + 2a cos theta + a sin theta` |
| `= 4 + (3 sqrt 5) + 2 (3 sqrt 5) (2/sqrt 5) + (3 sqrt 5) (1/sqrt 5)` | |
| `= 4 + 3 sqrt 5 + 12 + 3` | |
| `= 19 + 3 sqrt 5\ text(cm)` |
Two events, `A` and `B`, are such that `text(Pr) (A) = 3/5` and `text(Pr) (B) = 1/4.` If `A^{′}` denotes the compliment of `A`, calculate `text(Pr) (A^{′} nn B)` when --- 5 WORK AREA LINES (style=lined) --- --- 4 WORK AREA LINES (style=lined) ---
a. `text(Sketch Venn Diagram)` `:.\ text(Pr) (A^{′} nn B) = 1/4-1/10 = 3/20` `text(Pr) (A∩ B)=0\ \ text{(mutually exclusive)},` `:.\ text(Pr) (A^{′} nn B) = text(Pr) (B) = 1/4`Show Worked Solution
`text(Pr) (A uu B)`
`= text(Pr) (A) + text(Pr) (B)-text(Pr) (A nn B)`
`3/4`
`= 3/5 + 1/4-text(Pr) (A nn B)`
`text(Pr) (A nn B)`
`= 1/10`
b.
A biased coin tossed three times. The probability of a head from a toss of this coin is `p.` --- 2 WORK AREA LINES (style=lined) --- --- 2 WORK AREA LINES (style=lined) --- --- 5 WORK AREA LINES (style=lined) ---
State the range and period of the function
`h: R -> R,\ \ h(x) = 4 + 3 cos ((pi x)/2)`. (2 marks)
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Differentiate `sqrt (4 - x)` with respect to `x.` (1 mark)