November 22, 2019, 09:09:55 am

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Just another student

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« Reply #18195 on: October 19, 2019, 07:56:49 am »
+1
Hi,

does anyone have any tips for quickly solving the simultaneous linear equation question (multiple choice) which seems to appear in almost all exams?

for example:

2x-my=m and (1-m)x+y=2 have a unique solutions for: options A-E are given.

I see this type of question all the time, and understand the Q and the theory behind it, but it seems to take quite long to do. So I would appreciate if anyone had some valuable tips for these general category of MC questions like step by step process.
Thanks

AlphaZero

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« Reply #18196 on: October 19, 2019, 02:50:29 pm »
0
Hi,

does anyone have any tips for quickly solving the simultaneous linear equation question (multiple choice) which seems to appear in almost all exams?

for example:

2x-my=m and (1-m)x+y=2 have a unique solutions for: options A-E are given.

I see this type of question all the time, and understand the Q and the theory behind it, but it seems to take quite long to do. So I would appreciate if anyone had some valuable tips for these general category of MC questions like step by step process.
Thanks

While this is no longer in the course, you can write the simultaneous equations \begin{align*}ax+by&=\alpha\\ cx+dy&=\beta \end{align*} as the matrix equation $\begin{bmatrix}a&b\\c&d\end{bmatrix}\begin{bmatrix}x\\ y\end{bmatrix}=\begin{bmatrix}\alpha\\ \beta\end{bmatrix}.$ The linear system will have a unique solution for $x$ and $y$ provided  $\det\begin{bmatrix}a&b\\c&d\end{bmatrix}=ad-bc\neq 0$.

Note however that this is (pretty much) the same as rearranging both equations for $y$ and enforcing that the lines have distinct gradients.
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studyingg

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« Reply #18197 on: October 19, 2019, 03:09:16 pm »
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what type of transformations impact the mean and variance of a probability density function?

In a question from the insight 2019 exam, a function with E(X) = 2 and Var (X)=2 was dilated by a factor of 1/3 from the y-axis, and 3 from the x-axis. In the solutions, they found E(X) by multiplying 2*1/3 and did the same for var(x); however, I am confused as to why they didn't take the dialation from the x-axis into consideration.

So my question is, how would transformations (mainly dilations from the x and yaxis) impact the mean and variance of a pdf function -- from a general perspective?

KiNSKi01

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« Reply #18198 on: October 19, 2019, 08:07:31 pm »
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Yo anyone got general advice for confidence interval questions on exam 1 that involve long and laborious calculations

-super frustrating to drop an answer mark cos of a small mistake you make during one of the calculations involved
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jkay__

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« Reply #18199 on: October 19, 2019, 09:02:23 pm »
+3
what type of transformations impact the mean and variance of a probability density function?

In a question from the insight 2019 exam, a function with E(X) = 2 and Var (X)=2 was dilated by a factor of 1/3 from the y-axis, and 3 from the x-axis. In the solutions, they found E(X) by multiplying 2*1/3 and did the same for var(x); however, I am confused as to why they didn't take the dialation from the x-axis into consideration.

So my question is, how would transformations (mainly dilations from the x and yaxis) impact the mean and variance of a pdf function -- from a general perspective?

If you dilate from the y-axis, you are modifying the graph horizantally, so the spread (variance), and the mean are shifted. However, if you dilate from the x-axis, then the graph is modified vertically, so the spread or mean aren't affected. However, by changing the height, you're changing the probability at a certain point (so more/less people at one point or in a region if it's higher/lower).

Yo anyone got general advice for confidence interval questions on exam 1 that involve long and laborious calculations

-super frustrating to drop an answer mark cos of a small mistake you make during one of the calculations involved

I don't think I've seen a confidence interval question in Exam 1 (VCAA), and if it does pop up, it probably won't involve any long calculations. The msot I've seen is just doing it to 1 decimal place, things like 8/5 (1.6).

However, on the off chance that they do come up, you could use your remaining time to check over any working out that you don't feel 100% confident with
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Matthew_Whelan

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« Reply #18200 on: October 19, 2019, 10:04:58 pm »
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Are there any concepts or formulas that are used in specialist/other that can be applied to methods to solve more difficult problems easier? If so, are they worth learning at this point?
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DrDusk

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« Reply #18201 on: October 19, 2019, 10:19:02 pm »
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Are there any concepts or formulas that are used in specialist/other that can be applied to methods to solve more difficult problems easier? If so, are they worth learning at this point?
The way they do it in HSC and I presume VCE does the same, is that they design the papers such that anyone taking the higher level course won't have an unfair advantage because they've learnt more theory. The questions are designed so that the best method of doing it is the the ones that are taught in that course.

Best of luck for the VCE!
« Last Edit: October 19, 2019, 10:20:58 pm by DrDusk »
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AlphaZero

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« Reply #18202 on: October 19, 2019, 10:27:55 pm »
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Are there any concepts or formulas that are used in specialist/other that can be applied to methods to solve more difficult problems easier? If so, are they worth learning at this point?

The absolute value function is quite handy: $|x|=\begin{cases}x,&x\geq 0\\-x,&x<0\end{cases}$ especially in integrating functions of the form $(ax+b)^{-1}$. Instead of case-breaking, one can simply write $\int\frac{1}{ax+b}\,\text{d}x=\frac{1}{a}\log_e|ax+b|+c,\quad a\neq 0,\ \ c\in\mathbb{R}.$

Some knowledge of vectors is sometimes useful when looking at geometry in the plane. For example, to find the angle between two lines, one can form two vectors $\mathbf{a}$ and $\mathbf{b}$ that are parallel to those lines and then use the fact that $\mathbf{a}\cdot\mathbf{b}=|\mathbf{a}||\mathbf{b}|\cos(\theta).$

I seriously doubt it, but partial fraction decomposition and a few 'advanced' integration techniques could be useful in verifying your answers.

I do want to echo DrDusk's point though. The questions will not require any knowledge in higher level courses. If you're finding that you want to apply a concept that isn't thought in Methods, it's more than likely you are overthinking the question. Of course, there are exceptions to this though.
« Last Edit: October 19, 2019, 11:16:23 pm by AlphaZero »
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Tau

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« Reply #18203 on: October 19, 2019, 10:33:38 pm »
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The absolute value function is quite handy: $|x|=\begin{cases}x,&x\geq 0\\-x,&x<0\end{cases}$ especially in integrating functions of the form $(ax+b)^{-1}$. Instead of case-breaking, one can simply write $\int\frac{1}{ax+b}\,\text{d}x=\frac{1}{a}\log_e(ax+b)+c,\quad a\neq 0,\ \ c\in\mathbb{R}.$
Think you might have missed the absolute value sign in that integration.
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Tau

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« Reply #18204 on: October 19, 2019, 10:37:53 pm »
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Are there any concepts or formulas that are used in specialist/other that can be applied to methods to solve more difficult problems easier? If so, are they worth learning at this point?

The double derivative test can also be useful for determining the nature of stationary points.
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DrDusk

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« Reply #18205 on: October 19, 2019, 10:50:52 pm »
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The double derivative test can also be useful for determining the nature of stationary points.
This is only taught in Specialist? :O
How else do Methods students find the nature of Stationary points?
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Tau

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« Reply #18206 on: October 19, 2019, 10:54:09 pm »
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This is only taught in Specialist? :O
How else do Methods students find the nature of Stationary points?

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undefined

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« Reply #18207 on: October 19, 2019, 10:55:57 pm »
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Are there any concepts or formulas that are used in specialist/other that can be applied to methods to solve more difficult problems easier? If so, are they worth learning at this point?
Integration by parts is useful if you can’t figure out how to use recognition.
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redpanda83

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« Reply #18208 on: October 20, 2019, 12:19:49 am »
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What would I actually have to do to use the gradient method? That's what I was trying to do but I'm not sure how to proceed.
all you have to show that gradient is always positive, so pretty much find the derivative and state that the derivative in x1- x2 is always positive thus function is strictly increasing. (strictly increasing function m>or = 0) keep that in mind
« Last Edit: October 20, 2019, 12:21:26 am by redpanda83 »

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