Aussie Maths & Science Teachers: Save your time with SmarterEd
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| i. |  |
ii. `text(By inspection, intersection when)\ x = -1`
`text(Test:)`
| `|-1-1|` | `= -2 + 4` |
| `2` | `= 2` |
`:.\ text(Intersection at)\ (-1, 2)`
The network diagram shows the tracks connecting 8 picnic sites in a nature park. The vertices `A` to `H` represents the picnic sites. The weights on the edges represent the distance along the tracks between the picnic sites, in kilometres.
Draw a minimum spanning tree and calculate the minimum length of water pipes required to supply water to all the sites if the water pipes can only be laid along the tracks. (2 marks)
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a. `text(One strategy – using Prim’s algorithm:)`
`text(Starting at)\ A`
`text(1st edge -)\ AB,\ \ text(2nd edge -)\ BC`
`text(3rd edge -)\ CH,\ \ text(4th edge -)\ HG`
`text(5th edge -)\ GF,\ \ text(6th edge -)\ HD`
`text(7th edge -)\ DE\ text(or)\ HE`
`text(Maximum length = 4 + 5 + 3 + 2 + 1 + 5 + 5 = 25 km)`
b. `text(Shortest Path is)\ CGHE`
The diagram shows the region bounded by the curve `y = x - x^3`, and the `x`-axis between `x = 0` and `x = 1`. The region is rotated about the `x`-axis to form a solid.
Find the exact value of the volume of the solid formed. (3 marks)
The diagram shows a circle with centre `O` and radius 20 cm.
The points `A` and `B` lie on the circle such that `∠AOB = 70^@`.
Find the perimeter of the shaded segment, giving your answer correct to one decimal place. (3 marks)
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The diagram shows the graph of `y = (3x)/(x^2 + 1)`.
The region enclosed by the graph, the `x`-axis and the line `x = 3` is shaded.
Calculate the exact value of the area of the shaded region. (3 marks)
The number of leaves, `L(t)`, on a tree `t` days after the start of autumn can be modelled by
`L(t) = 200\ 000e^(-0.14t)`
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The parabola `y = x^2` meets the line `y = x + 2` at the points `(-1, 1)` and `(2, 4)`. Do NOT prove this.
By first sketching the graphs of `y = x^2` and `y = x + 2`, shade the region which simultaneously satisfies the two inequalities `y >= x^2` and `y >= x + 2`. (2 marks)
The formula below is used to calculate an estimate for blood alcohol content `(BAC)` for females.
`BAC_text(female) = (10N - 7.5H)/(5.5M)`
The number of hours required for a person to reach zero `BAC` after they stop consuming alcohol is given by the following formula.
`text(Time) = (BAC)/0.015`
The number of standard drinks in a glass of wine and a glass of spirits is shown.
Hannah weighs 60 kg. She consumed 3 glasses of wine and 4 glasses of spirits between 6:15 pm and 12:30 am the following day. She then stopped drinking alcohol.
Using the given formulae, calculate the time in the morning when Hannah's `BAC` should reach zero. (4 marks)
A project requires activities `A` to `F` to be completed. The activity chart shows the immediate prerequisite(s) and duration for each activity.
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| a. |  |
`text(Scanning forwards:)`
`text(Minimum time = 2 + 6 + 2 + 4 + 1 = 15 hours)`
`text{(Scanning forwards and backwards is highly recommended but not}`
`text{required in the network diagram.)}`
b. `text(Critical Path is)\ ABDEF.`♦♦ Mean mark 30%.
`text(Non-critical activity is)\ C.`
| `text(Float time)` | `= 5 – 2` |
| `= 3\ text(hours)` |
Evaluate `int_0^1 1/(3x + 2)^2\ dx`. (2 marks)
Last Saturday, Luke had 165 followers on social media. Rhys had 537 followers. On average, Luke gains another 3 followers per day and Rhys loses 2 followers per day.
If `x` represents the number of days since last Saturday and `y` represents the number of followers, which pair of equations model this situation?
| A. | `text(Luke:)\ \ y = 165x + 3`
`text(Rhys:)\ \ y = 537x - 2` |
| B. | `text(Luke:)\ \ y = 165 + 3x`
`text(Rhys:)\ \ y = 537 - 2x` |
| C. | `text(Luke:)\ \ y = 3x + 165`
`text(Rhys:)\ \ y = 2x - 537` |
| D. | `text(Luke:)\ \ y = 3 + 165x`
`text(Rhys:)\ \ y = 2 - 537x` |
The graph show the future values over time of `$P`, invested at three different rates of compound interest.
Which of the following correctly identifies each graph?
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B. |  |
| C. |  |
D. |  |
The Coordinated Universal Time (UTC) of Auckland is +12 hours and the UTC of Chicago is −5 hours.
When the time in Chicago is 2 pm, Thursday, what is the time in Auckland?
Chris opens a bank account and deposits $1000 into it. Interest is paid at 3.5% per annum, compounding annually.
Assuming no further deposits or withdrawals are made, what will be the balance in the account at the end of two years?
A set of bivariate data is collected by measuring the height and arm span of seven children. The graph shows a scatterplot of these measurements.
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Calculate the predicted height for this child using the equation of the least-squares regression line. (1 mark)
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Two right-angled triangles, `ABC` and `ADC`, are shown.
Calculate the size of angle `theta`, correct to the nearest minute. (3 marks)
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A person owns 1526 shares with a market value of $8.75 per share. The total dividend received for these shares is $1068.20.
Calculate the percentage dividend yield. (2 marks)
A roulette wheel has the numbers 0, 1, 2, …, 36 where each of the 37 numbers is equally likely to be spun.
If the wheel is spun 18 500 times, calculate the expected frequency of spinning the number 8. (2 marks)
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The heights, in centimetres, of 10 players on a basketball team are shown.
170, 180, 185, 188, 192, 193, 193, 194, 196, 202
Is the height of the shortest player on the team considered an outlier? Justify your answer with calculations. (3 marks)
Andrew, Brandon and Cosmo are the first three batters in the school cricket team. In a recent match, Andrew scored 30 runs, Brandon scored 25 runs and Cosmo scored 40 runs.
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The diagram shows a triangle with sides of length `x` cm, 11 cm and 13 cm and an angle of 80°.
Use the cosine rule to calculate the value of `x`, correct to two significant figures. (3 marks)
A bowl is in the shape of a hemisphere with a diameter of 16 cm.
What is the volume of the bowl, correct to the nearest cubic centimetre? (2 marks)
A particle is moving along a straight line. The graph shows the acceleration of the particle.
For what value of `t` is the velocity `v` a maximum?
The diagram shows part of the graph of `y = a sin(bx) + 4`.
What are the values of `a` and `b`?
A game is played by tossing an ordinary 6-sided die and an ordinary coin at the same time. The game is won if the uppermost face of the die shows an even number or the uppermost face of the coin shows a tail (or both).
What is the probability of winning this game?
Which of the following is equal to `(log_2 9)/(log_2 3)`?
What is the value of `p` so that `(a^2a^(-3))/sqrt a = a^p`?
A projectile is fired horizontally off a cliff at an initial speed of `V` metres per second.
The projectile strikes the water, `l` metres from the base of the cliff.
Let `g` be the acceleration due to gravity and assume air resistance is negligible.
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| i. | `underset~v` | `= Vcosthetaunderset~i + (Vsintheta – g t)underset~j` |
| `= Vcos0 underset~i + (Vsin0 – g t)underset~j` | ||
| `= Vunderset~i – g tunderset~j` |
| `underset~s` | `= intunderset~v\ dt` |
| `= Vtunderset~i – 1/2g t^2 underset~j + c` |
`text(When)\ t = 0, underset~s = 0, c = 0`
`text(Time of flight:)`
`underset~j\ text(component of)\ underset~s = −d`
| `−1/2 g t^2` | `= −d` |
| `t^2` | `= (2d)/g` |
| `t` | `= sqrt((2d)/g)` |
ii. `l = 2d\ \ (text(given))`
`text(Projectile hits water at)\ theta:`
| `overset.y` | `= underset~j\ text(component of)\ underset~v` |
| `= −g t` | |
| `= −g · sqrt((2d)/g)` | |
| `= −sqrt(2dg)` | |
| `overset.x` | `= underset~i\ text(component of)\ underset~v` |
| `= V` |
`text(When)\ \ t = sqrt((2d)/g),`
`underset~i\ text(component of)\ underset~s = 2d`
| `2d` | `= V · sqrt((2d)/g)` |
| `V` | `= (2dsqrtg)/sqrt(2d) = sqrt(2dg)` |
| `tantheta` | `= (|overset.y|)/(|overset.x|)= sqrt(2dg)/sqrt(2dg)=1` |
| `:. theta` | `= 45°` |
iii. `text(Speed = magnitude of velocity)`
| `|underset~v|` | `= sqrt((sqrt(2dg))^2 + (sqrt(2dg))^2)` |
| `= sqrt(4dg)` | |
| `= 2sqrt(dg)` |
A basketball player aims to throw a basketball through a ring, the centre of which is at a horizontal distance of 4.5 m from the point of release of the ball and 3 m above floor level. The ball is released at a height of 1.75 m above floor level, at an angle of projection `alpha` to the horizontal and at a speed of `V\ text(ms)^(-1)`. Air resistance is assumed to be negligible.
The position vector of the centre of the ball at any time, `t` seconds, for `t >= 0`, relative to the point of release is given by
`qquad underset ~s(t) = Vt cos (alpha) underset ~i + (Vt sin(alpha) - 4.9t^2) underset ~j`,
where `underset ~i` is a unit vector in the horizontal direction of motion of the ball and `underset ~j` is a unit vector vertically up. Displacement components are measured in metres.
For the player’s first shot at goal, `V = 7\ text(ms)^(-1)` and `alpha = 45^@`
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A cricketer hits a ball at time `t = 0` seconds from an origin `O` at ground level across a level playing field.
The position vector `underset ~s(t)`, from `O`, of the ball after `t` seconds is given by
`qquad underset ~s(t) = 15t underset ~i + (15 sqrt 3 t - 4.9t^2)underset ~j`,
where, `underset ~i` is a unit vector in the forward direction, `underset ~j` is a unit vector vertically up and displacement components are measured in metres.
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How far horizontally from `O` is the fielder when the ball is caught? Give your answer in metres, correct to one decimal place. (2 marks)
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A manufacturer makes torches that have a probability of 0.03 of being defective.
Let `overset^p` be the random variable that represents the sample proportion of torches for samples of size `n` drawn from production.
Find the smallest integer value of `n` such that the standard deviation of `overset^p` is less than `1/50`. (2 marks)
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When a standard 6-sided die is thrown, the probability that it shows a prime number is `2/3`.
If 10 standard dice are thrown, the number, `N`, of times a prime number is showing has a binomial distribution.
What is the standard deviation of `N`, correct to 3 decimal places?
In a large population of fish, the proportion of angel fish is `1/4`.
Let `hat p` be the random variable that represents the sample proportion of angel fish for samples of size `n` drawn from the population.
Find the smallest integer value of `n` such that the standard deviation of `hat p` is less than or equal to `1/100`. (2 marks)
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Steve and Jess are two students who have agreed to take part in a psychology experiment. Each has to answer several sets of multiple-choice questions. Each set has the same number of questions, `n`, where `n` is a number greater than 20. For each question there are four possible options A, B, C or D, of which only one is correct.
Let the random variable `X` be the number of questions that Steve answers correctly in a particular set.
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If `P(Y > 23) = 6 xx P(Y = 25)`, show that the value of `p=5/6`. (2 marks)
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In a chocolate factory the material for making each chocolate is sent to a machines.
The time, `X` seconds, taken to produce a chocolate by machine is a binomial distribution where it can be shown that `P(X <= 3) = 9/32`.
A random sample of 10 chocolates is chosen. Find the probability, correct to two decimal places, that exactly 4 of these 10 chocolates took 3 or less seconds to produce. (2 marks)
A school has a class set of 22 new laptops kept in a recharging trolley. Provided each laptop is correctly plugged into the trolley after use, its battery recharges.
On a particular day, a class of 22 students uses the laptops. All laptop batteries are fully charged at the start of the lesson. Each student uses and returns exactly one laptop. The probability that a student does not correctly plug their laptop into the trolley at the end of the lesson is 10%. The correctness of any student’s plugging-in is independent of any other student’s correctness.
Determine the probability that at least one of the laptops is not correctly plugged into the trolley at the end of the lesson. Give your answer correct to three decimal places. (2 marks)
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It is known that 50% of the customers who enter a restaurant order a cup of coffee. If four customers enter the restaurant, what is the probability that more than two of these customers order coffee? (Assume that what any customer orders is independent of what any other customer orders.) (2 marks)
A biased coin tossed three times. The probability of a head from a toss of this coin is `p.`
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A paddock contains 10 tagged sheep and 20 untagged sheep. Four times each day, one sheep is selected at random from the paddock, placed in an observation area and studied, and then returned to the paddock.
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Express your answer in the form `(a/c)^c`, where `a`, `b` and `c` are positive integers. (1 mark)
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The binomial random variable, `X`, has `E(X) = 2` and `text(Var)( X ) = 4/3.`
Calculate `P(X = 1)`. (3 marks)
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Let `X` be a discrete random variable with a binomial distribution. The mean of `X` is 1.2 and the variance of `X` is 0.72
The values of `n` (the number of independent trials) and `p` (the probability of success in each trial) are
A. `n = 3,\ \ \ p = 0.6`
B. `n = 2,\ \ \ p = 0.6`
C. `n = 2,\ \ \ p = 0.4`
D. `n = 3,\ \ \ p = 0.4`
The minimum number of times that a fair coin can be tossed so that the probability of obtaining a head on each trial is less than 0.0005 is
A. `9`
B. `10`
C. `11`
D. `12`
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i. `f(x) = (1-e^x)/(1 + e^x)`
| `f(−x)` | `= (1-e^(−x))/(1 + e^(−x)) xx (e^x)/(e^x)` |
| `= (e^x-1)/(e^x + 1)` | |
| `= −(1-e^x)/(1 + e^x)` | |
| `= −f(x)` |
`:. f(x)\ text(is ODD.)`
ii. `y = (1-e^x)/(1 + e^x) xx (e^(−x))/(e^(−x)) = (e^(−x)-1)/(e^(−x) + 1) = 1-2/(e^(−x) + 1)`
`text(As)\ x -> ∞, \ 2/(e^(−x) + 1) -> 2, \ y -> −1`
`text(As)\ x ->-∞, \ 2/(e^(−x) + 1) -> 0, \ y -> 1`
`text(When)\ x = 0, \ y = 0`
Consider the function `f(x) = 1/(4x - 1)`.
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| i. | `4x – 1` | `!= 0` |
| `x` | `!= 1/4` |
`:.\ text(Domain:)\ {text(all real)\ x, x!=1/4}`
ii. `text(When)\ \ x = 0, \ y = −1`
`text(As)\ \ x -> ∞, \ y -> 0^+`
`text(As)\ \ x -> −∞, \ y -> 0^−`
Sketch the graph of `f(x) = (2x+1)/(x-1)`. Label the axis intercepts with their coordinates and label any asymptotes with the appropriate equation. (4 marks)
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| `(2x+1)/(x-1)` | `=(2x-2+3)/(x-1)` | |
| `=(2(x-1)+3)/(x-1)` | ||
| `=2 + 3/(x-1)` |
`text(Asymptotes:)\ \ x = 1,\ \ y = 2`
`text(As)\ \ x->oo,\ \ y->2(+)`
`text(As)\ \ x->-oo,\ \ y->2(-)`
`text(As)\ \ x->-1 (-),\ \ y->-oo`
`text(As)\ \ x->-1 (+),\ \ y->oo`
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On the axes below, sketch the graph of `f(x) = (2x-2)/(x + 1)`.
Label all axis intercepts. Label each asymptote with its equation. (4 marks)
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| `(2x-2)/(x + 1)` | `=(2(x+1) – 4)/(x+1)` |
| `=2- 4/((x+1)` |
`text(Asymptotes:)\ \ x = −1,\ \ y = 2`
`text(As)\ \ x->oo,\ \ y->2(-)`
`text(As)\ \ x->-oo,\ \ y->2(+)`
`text(As)\ \ x->-1 (-),\ \ y->oo`
`text(As)\ \ x->-1 (+),\ \ y->-oo`
The graph of the function `f(x) = (x - 3)/(2 - x)` has asymptotes
Consider the function `f(x) = x/(4 - x^2)`.
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i. `4-x^2 !=0`
`=> x!= 2 or -2`
`:.\ text(Domain:)\ \ text(all)\ x,\ x != +- 2`
ii. `text(Asymptotes at)\ \ x = +- 2`
`text(Passes through)\ (0,0)`
| `text(As)` | `\ \ x -> 2^(-),` | `\ y -> oo` |
| `\ \ x -> 2 ^(+),` | `\ y -> – oo` |
| `text(As)` | `\ \ x -> -2^ (-),` | `\ y -> oo` |
| `\ \ x -> -2^ (+),` | `\ y -> -oo` |
| `text(As)` | `\ \ x -> oo,` | `\ y -> 0` |
| `\ \ x -> – oo,` | `\ y -> 0` |
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`qquad y = (x − 2)/(x − 4)` and hence sketch the graph of `y = (x − 2)/(x − 4)`. (3 marks)
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i. `y = (x − 2)/(x − 4)`
`text(Vertical asymptote at)\ \ x = 4`
| `lim_(x → ∞) (x − 2)/(x − 4)` | `= lim_(x → ∞) (1 − 2/x)/(1 − 4/x)=1` |
`ytext(–intercept)\ = 1/2`
`xtext(–intercept)\ = 2`
ii. `text(Find)\ \ x\ \ text(so that)\ \ (x − 2)/(x − 4) ≤ 3`
| `(x − 2)/(x − 4)` | `= 3` |
| `x − 2` | `= 3x − 12` |
| `2x` | `= 10` |
| `x` | `= 5` |
`=>(5, 3)\ \ text(is the intersection of)\ \ y = 3\ and\ y = (x − 2)/(x − 4)`
`:. (x − 2)/(x − 4) ≤ 3\ \ text(when)\ \ x < 4\ \ text(and)\ \ x ≥ 5.`
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| i. | `y` | `= (2x^2)/(x^2 +9)` |
| `= 2/(1 + 9/(x^2))` |
`text(As)\ \ x -> oo,\ y ->2`
`text(As)\ \ x -> – oo,\ y -> 2`
`:.\ text(Horizontal asymptote at)\ y = 2`
| ii. | `text(At)\ \ x = 0,\ y = 0` |
`f(x) = (2x^2)/(x^2 + 9) >= 0\ text(for all)\ x`
`f(–x) = (2(–x)^2)/((–x)^2 + 9) = (2x^2)/(x^2 + 9) = f(x)`
`text(S)text(ince)\ \ f(x) = f(–x) \ \ =>\ text(EVEN function)`
What are the asymptotes of `y = (3x)/((x + 1)(x + 2))`
| A. | `y = 0,` | `x = −1,` | `x = −2` |
| B. | `y = 0,` | `x = 1,` | `x = 2` |
| C. | `y = 3,` | `x = −1,` | `x = −2` |
| D. | `y = 3,` | `x = 1,` | `x = 2` |
Which graph best represents the function `y = (2x^2)/(1 - x^2)`?
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B. |  |
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A slope field representing the differential equation `dy/dx = −x/(1 + y^2)` is shown below.
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| a. | |
♦♦ Mean mark part (a) 32%.
MARKER’S COMMENT: Solution curve should follow slope ticks and not cross them.
| b. | `(1 + y^2)(dy)/(dx)` | `= −x` |
| `int 1 + y^2 dy` | `= −int x\ dx` | |
| `y + (y^3)/3` | `= −(x^2)/2 + C, C ∈ R` |
`text(Substituting)\ (-1,1):`
| `1 + (1^3)/3` | `= −((−1)^2)/2 + C` |
| `1 + 1/3` | `= −1/2 + C` |
| `:. C` | `= 11/6` |
| `y + 1/3y^3` | `= −1/2x^2 + 11/6` |
| `6y + 2y^3` | `= −3x^2 + 11` |
`:. 2y^3 + 6y + 3x^2 – 11 = 0`
Bacteria are spreading over a Petri dish at a rate modelled by the differential equation
`(dP)/(dt) = P/2 (1 - P),\ 0 < P < 1`
where `P` is the proportion of the dish covered after `t` hours.
Given `2/(P(1 - P)) = 2/P + 2/(1 - P),`
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The differential equation that best represents the direction field above is
A. `(dy)/(dx) = x - y^2`
B. `(dy)/(dx) = y - x`
C. `(dy)/(dx) = y^2 - x^2`
D. `(dy)/(dx) = y^2 - x`
The differential equation that best represents the direction field above is
A. `(dy)/(dx) = (2x + y)/(y - 2x)`
B. `(dy)/(dx) = (x + 2y)/(2x - y)`
C. `(dy)/(dx) = (2x - y)/(x + 2y)`
D. `(dy)/(dx) = (x - 2y)/(y - 2x)`
The direction field for the differential equation `(dy)/(dx) + x + y = 0` is shown above.
A solution to this differential equation that includes `(0, -1)` could also include
A. `(3, –1)`
B. `(3.5, –2.5)`
C. `(–1.5, –2)`
D. `(2.5, –1)`
`=> B`
The differential equation that is best represented by the above direction field is
A. `(dy)/(dx) = 1/(x - y)`
B. `(dy)/(dx) = y - x`
C. `(dy)/(dx) = 1/(y - x)`
D. `(dy)/(dx) = x - y`
The differential equation that best represents the above direction field is
A. `(dy)/(dx) = x^2 - y^2`
B. `(dy)/(dx) = y^2 - x^2`
C. `(dy)/(dx) = −x/y`
D. `(dy)/(dx) = x/y`
The diagram that best represents the direction field of the differential equation `(dy)/(dx) = xy` is
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B. |  |
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D. |  |