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Oxford University Math Professor takes German High School
Maths Exam
Tom Rocks Maths

Hello Math Fans!
Dr
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Today, I'm tackling the German high school
math exam
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However, I'll be using Google Translate to help me navigate
through the equations and figures
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To find the stationary points, I'll calculate the
derivative:
f'(x) = 2e^(2x) - (e^(2x) / x^2)
I'll set this derivative equal to zero to find the stationary points:
2e^(2x) - (e^(2x) / x^2) = 0
Simplifying further, I find:
2x^2 = x
From this, I can determine that x = 1/2
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Therefore, the extreme point occurs when x = 1/2
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Therefore, the extreme
point is a minimum
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The question asks me to show that one
of the points where the line g intersects the graph of f has an x-coordinate of 1/2
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So, one of the intersection points has an
x-coordinate of 1/2
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To do
this, I'll calculate the area of the rectangle and the area under the curve separately
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Now, I'll calculate the area under the curve using
integration
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Question 1
Calculate the total area of the shaded region in the graph below
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Question 2
Based on the graph of the derivative, determine which function it represents
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Question 3
Sketch the graph of f and determine its behavior
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Question 4
Find the largest possible value of k such that the function hk is reversible
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To understand the graph of the inverse function, we need to pay attention to the intersection of
the graphs of the function and its inverse
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The original function is y = cos(x),
and we are going from 0 to pi
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To
find the inverse function, we reflect the graph in the line y = x
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When we plot the inverse function, it starts at zero when x = 1, and it
goes up to pi when x = -1
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We can't determine the exact point of intersection,
but it should be somewhere on the graph
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One example is the exponential function y = e^x and its inverse y = ln(x)
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It seems that I am only supposed to answer one group of questions from the analysis section
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Task Group Two: Stochastic

For this task group, I have chosen Group Two, which includes pictures
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Let's start with Question 1
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One sector is labeled 0, one is labeled
1, and the other two sectors are labeled 9
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Calculate the
probability of getting the numbers 2, 0, 1, and 9 in the given order
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Find the probability that the sum of the numbers obtained is
at least 11
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The probabilities of getting these outcomes are:
Probability(9-2) = (2/5) * (1/5) = 2/25

Probability(9-9) = (4/5) * (4/5) = 16/25
Probability(2-9) = (1/5) * (2/5) = 2/25
Probability(9-9 in different order) = (2/5) * (4/5) = 8/25
The total probability is: (2/25) + (16/25) + (2/25) + (8/25) = 12/25
Question 3
A binomial distributed random variable, x, is given with n=5
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From the diagram, the probability of x=5 is 1
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43
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15
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43 - 0
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28
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Determine the probability of event
B
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Therefore, P(B) = 1 - P(A') = 1 - 1/3
= 2/3
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Geometry Problem: Intersecting Spheres
Given:
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Center of sphere K1: (1, 2, 3)
Radius of sphere K1: 5
Center of sphere K2: (3, -2, -1)
Radius of sphere K2: 5

To show that K1 and K2 intersect, we need to show that the distance between their centers is
less than the sum of their radii
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Equations of Intersection:
To find the coordinates of the center and the radius of the intersecting circle, we can subtract
the equations of the spheres
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Substituting z = -2x - 2y into one of the sphere equations, we can solve for x and y to find the
coordinates of the center of the intersecting circle
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However, there are alternative methods to solve this problem using
computational tools like Maple Learn
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To access the full version of Maple Learn, you can sign up for a discounted subscription using
the code provided in the video description
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Remember, the goal is to speed run through these questions, so let's keep going
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To solve this, let's assume that x1, x2, and x3 are equal
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Part B: Prove that there are infinitely many planes that do not contain a point with three equal
coordinates
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Finally, let's move on to the last question, which involves the intersection of two spaces
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It seems that I need to rotate and create a new
coordinate system, but my brain is not cooperating at the moment
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However, I am confident in my performance on the other questions
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That proof at the end possibly creeping into further maths territory but circle question circles
again anyway
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I can't actually mark this
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I can't find a mark scheme for this online
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So I can't market, but I'm pretty confident that everything else I did was okay
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Maybe
I'd pick up one mark of the three, but I'm confident that everything else I did felt okay, felt right
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But no, it was good fun as always
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So all that's left to say is thank you to all of you for watching
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I'd love to do exams from other countries having now
covered the UK quite a few from the UK obviously I'm biased, the US and now Germany
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So add your suggestions to the
comments or you can get in touch using the contact form on my website go to tomrocksmaths
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Thank you for watching please
like and subscribe and I'll see you all very soon take care

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