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1. With a kettle, Peter warms up 1,2 litres of water (20°C) till it starts to boil. The power of the kettle is 2000 W. The heating up takes four minutes. The formula for specific heat of the water is 4,2 J/g · °C. How high is the efficiency of the kettle?

2. A kettle can contain 1,4 litres of water. The device has a power of 2000 W. The formula for specific heat of the water is 4,2 J/g · °C and the density is 1,0 g/mL. Calculate how long it would take before the kettle filled to the brim with water (20°C) would be boiling.

Q = E

How many distinct ordered pairs of positive integers \((m,n)\) are there
so that the sum of the reciprocals of \(m\) and \(n\) is \(\dfrac14\)?

\(\text{From the relationship} \\ \dfrac{1}{z} = \dfrac{1}{m} + \dfrac{1}{n} \\ \text{follows immediately that $m>z$ and $n> z$ must be.}\\ \text{You can write $m=z+a$ and $n=z+b$ }\\ \text{Now the result:}\\ \dfrac{1}{z} = \dfrac{1}{z+a} + \dfrac{1}{z+b} \\ \)

\(\begin{array}{|rcll|} \hline \dfrac{1}{z} &=& \dfrac{1}{z+a} + \dfrac{1}{z+b} \\\\ \dfrac{1}{z} &=& \dfrac{2z+a+b}{z^2+za+zb+ab} \\\\ z^2+za+zb+ab &=& z(2z+a+b) \\ z^2+za+zb+ab &=& 2z^2+za+zb \\ z^2+za+zb+{\color{red}ab} &=& z^2+za+zb + {\color{red}z^2} \quad & \quad \text{by comparison follows } \boxed{z^2=ab} \\ \hline \end{array} \)

\(\text{Each pair $(a, b)=$ (divider, co-divider) of $n^2$ gives a solution }\\ \text{ from $\dfrac{1}{z} = \dfrac{1}{z+a} + \dfrac{1}{z+b} $.}\)

\(\text{if z = 4:}\\ \text{The divisors of $z^2=16$ are $1, 2, 4, 8, 16$ ($5$ divisors) }\)

\(\text{So there are $ \mathbf{5}$ distinct ordered pairs of positive integers $(m,n)$ }\)

\(\begin{array}{|c|c|c|c|c|} \hline 4^2 & divider & co-divider & \\ = 16 & a & b & ab & \dfrac{1}{4} = \dfrac{1}{4+a} + \dfrac{1}{4+b} \\ \hline & 1 & 16 & 1\cdot 16 = 16& \mathbf{\dfrac{1}{4} =} \dfrac{1}{4+1} + \dfrac{1}{4+16} = \mathbf{\dfrac{1}{5} + \dfrac{1}{20}} \\ \hline & 2 & 8 & 2\cdot 8 = 16& \mathbf{\dfrac{1}{4} =} \dfrac{1}{4+2} + \dfrac{1}{4+8}= \mathbf{\dfrac{1}{6} + \dfrac{1}{12}} \\ \hline & 4 & 4 & 4\cdot 4 = 16& \mathbf{\dfrac{1}{4} =} \dfrac{1}{4+4} + \dfrac{1}{4+4}= \mathbf{\dfrac{1}{8} + \dfrac{1}{8}} \\ \hline & 8 & 2 & 8\cdot 2 = 16& \mathbf{\dfrac{1}{4} =} \dfrac{1}{4+8} + \dfrac{1}{4+2}= \mathbf{\dfrac{1}{12} + \dfrac{1}{6}} \\ \hline & 16 & 1 & 16\cdot 1 = 16& \mathbf{\dfrac{1}{4} =} \dfrac{1}{4+16} + \dfrac{1}{4+1}= \mathbf{\dfrac{1}{20} + \dfrac{1}{5}} \\ \hline \end{array}\)

The distinct ordered pairs of positive integers \((m,n) = \sigma_0(z^2)\)

Let the plane = p miles/hour

and wind speed = w miles/hour

It takes more time to get there than to get back so on the way there the wind is causing the speed to decrease, and vise verse on the way back

So

p-w = 1620/3       and     p+w=1620/2.5

now solve simultaneously.

Ahh I understand it now :) I find it hard to put the information into the formulas

"Eric warms up 1kg milk in a pan on a gas stove from 20 °C to 80 °C. Of the energy the gas supplies, 50% is lost and 20% is absorbed by the pan. The remaining energy goes to the milk. The formula for the specific heat of the milk is 3,9 J/g · °C. The heat of combustion of the gas is 32 MJ/m3. How many cubic meters of gas has been burnt by Eric?"

Let \(\large{a = 4 + 3i}\), \(\large{b = 1 -2i}\), and \(\large{c = 8 - 5i}\).  
The complex number d is such that a, b, c, and d form the vertices of a parallelogram,
when plotted in the complex plane. Enter all possible values of d, separated by commas.

\(\begin{array}{|rclclcl|} \hline \mathbf{d} &=& -a+b+c &=& -(4+3i)+ (1 -2i) + (8 - 5i) &=& \mathbf{5-10i} \\\\ \mathbf{d} &=& a-b+c &=& (4+3i)- (1 -2i) + (8 - 5i) &=& \mathbf{11+0i} \\\\ \mathbf{d} &=& a+b-c &=& (4+3i)+ (1 -2i) - (8 - 5i) &=& \mathbf{-3+6i} \\ \hline \end{array} \)

There are four complex numbers  such that
\(\large{z \overline{z}^3 + \overline{z} z^3 = 350}\),
and both the real and imaginary parts of  are integers.
These four complex numbers are plotted in the complex plane.
Find the area of the quadrilateral formed by the four complex numbers as vertices.

\(\text{Let $z=a+bi$}\\ \text{Let $\overline{z}=a-bi$}\)

\(\begin{array}{|rcll|} \hline \mathbf{z \overline{z}^3 + \overline{z} z^3} &=& \mathbf{350} \\\\ z \overline{z}\left( \overline{z}^2+z^2 \right) &=& 350 \\\\ && \boxed{z\overline{z} = |z|^2 =a^2+b^2\\ \overline{z}^2+z^2 =a^2-2abi-b^2 +a^2+2abi-b^2 =2(a^2-b^2)} \\\\ 2(a^2-b^2)(a^2+b^2) &=& 350 \\ (a^2-b^2)(a^2+b^2) &=& 175 \quad | \quad 175 = 7\cdot 25 \\ (a^2-b^2)(a^2+b^2) &=& 7\cdot 25 \\\\ \text{if } a>b \\ a^2-b^2 &=& 7 \quad (1) \\ a^2+b^2 &=& 25 \quad (2) \\ \hline (1)+(2): \quad 2a^2 &=& 32 \\ a^2 &=& 16 \\ \mathbf{ a } &=& \mathbf{\pm4} \\ \hline (2)-(1): \quad 2b^2 &=& 18 \\ b^2 &=& 9 \\ \mathbf{ b } &=& \mathbf{\pm3} \\ \hline \hline \end{array}\\ \begin{array}{|lcll|} \hline \text{The four complex numbers are: $\mathbf{(4+3i), (4-3i), (-4-3i), (-4+3i)} $} \\ \text{The four vertices are: $\mathbf{(4,3), (4,-3), (-4,-3), (-4,3)} $} \\ \text{The area of this rectangle is $8\cdot 6 = \mathbf{48} $} \\ \hline \end{array}\)

Evaluate

\(i^{11} + i^{16} + i^{21} + i^{26} + i^{31}\)

Formula: \(i^{4n+a} = i^a\)

\(\begin{array}{|rcll|} \hline &&\mathbf{ i^{11} + i^{16} + i^{21} + i^{26} + i^{31} } \\ &=& i^{4\cdot 2+3} + i^{4\cdot 4+0} + i^{4\cdot 5+1} + i^{4\cdot 6+2} + i^{4\cdot 7+3} \\ &=& i^{3} + i^{0} + i^{1} + i^{2} + i^{3} \\ &=& i^{2}i + 1 + i + i^{2} + i^{2}i \quad | \quad i^2 = -1 \\ &=& -i + 1 + i -1 + -i \\ &=& \mathbf{ -i } \\ \hline \end{array}\)