The purpose of this test is to revise the concepts that you have learned throughout your schooling and get you to think critically about some of the concepts that we will encounter in the next few weeks.

Do send me the solutions once you are done! Have fun, take as much time as you need. Note that you are allowed to use calculators and any other resources you wish (just don’t look up the answers! If you do that, you are only harming your progress and understanding of the subject)

Algebra

1. Suppose you are given two equations, $a_1 x + b_1y = c_1$ and $a_2x+b_2y = c_2$. Here $x,y$ are unknowns and $a_2,a_2, b_1, b_2, c_1 , c_2$ are constants. Can you find the values of $x$ and $y$ in terms of these constants using the process of elimination?

2. Can you extend the process of elimination to the following system of three equations? Again, $x,y,z$ are variables all others are constants.

   $$ \begin{align*}

   a_1x + b_1 y + c_1 z &= d_1 \\ a_2x + b_2 y + c_2 z &= d_2 \\ a_3x + b_3 y + c_3 z &= d_3 \\

   \end{align*} $$

3. Can you find the solution to the following equation $x^4 + ax^2 + b = 0$ in terms of $a$ and $b$?

4. Suppose you are given two finite sets $A$ and $B$. What is the condition for these two sets to be equal?

Geometry and Trigonometry

1. Prove the basic trigonometric identities :

   $$ \begin{align*}

   &\sin^2 x + \cos^2 x = 1 \\ &\cot^2 x + 1 = \csc^2 x \\ &\tan^2x + 1 = \sec^2x

   \end{align*} $$

2. How would you prove that two triangles are congruent without using the congruence test? What does it mean for triangles to be congruent?

3. We will now see how you can calculate the circumference of earth (at the equator) using just basic trigonometry. The following method is the simplified method that Eratosthenes (~240 BC in Ancient Greece).

   The assumption of this method are as following,

   1. The Earth is perfectly spherical.
   2. The sun rays come from so far away that they are parallel for all practical purposes.
   3. We know the time when the sun is exactly overhead at point A.
   4. The measurements at point A and B are made simultaneously.
   5. The points A and B lie on the same meridian. (How would this affect the measurement if they didn’t lie on the same meridian? Think about it)

   Here’s the process (and a bit of history) — Eratosthenes travelled between two cities, the city of Alexandria (which we call point B) and the city of Syene (which we call point A for simplicity). He knew exactly at what time of the day the sun was directly overhead in Syene, thus casting no shadow, this was noon. He then went to Alexandria and calculated the angle at of sun’s elevation.

   He did this by keeping a stick vertically upright (assume the length of the stick was 1 meter) and then measuring the shadow cast by the sun rays of the stick. This is depicted in the following figure,

   

   1. Here, AC is the stick, and AB is the shadow cast by the stick. Assuming the length of the stick to be one meter, and the angle BCA to be $7^\circ$ calculate the length of the shadow.
   2. If 360 degrees gives you the entire circumference, then what fraction of the circumference correspond to an angle of $7^\circ$ ?

   Now, suppose the distance between Syene and Alexandria was 800 km. In other words, going from Alexandria to Syene causes a change of about $7^\circ$ in the angle of incidence of the sun rays. So how much fraction of the earth’s circumference would you have travelled to achieve this difference of $7^\circ$ ?

   Based on this can you calculate the circumference of the earth? As a corollary, also find the diameter of earth.

Physics

1. Suppose that a person travels with constant acceleration $a$ with initial velocity 0. Can you find the position of this person as a function of time? (that is, given a particular time, you should be able to find the position of the person.)
2. Is it possible for a particle to have finite velocity, zero mass and yet non-zero momentum?
3. If an object has constant momentum, what can you say about the force acting on it? If a constant force is acting on an object what can you say about its momentum?
4. Calculate the gravitational force due to a person sitting a meter away from you. If they exert a force on you, why don’t you move towards them?
5. Look up the masses and distances of various planets in our solar system. Calculate the gravitational force due to them on a person on Earth. What is the order of magnitude of this force? Calculate the force of an average 10 floor building 100 meters away from you. What do you notice?
6. Based on your answer to question four, do you think that these planets can have any significant effect on you?