Maths and Physics Tuition/Tests/Notes

Proof That Tangents Drawn From a Point to a Circle are Equal in Length

The theorem is illustrated below. Proof: Construct the triangles and by drawing radii as below. since both are radii of the circle and is common to both. Further, angle since these are between a tangent and a radius. From Pythagoras theorem, so

Proof That the Angle Subtended by a Chord or Arc at the Center of a Circle is Twice the Angle Subtended at the Circumference

Triangle is isosceles since is isosceles similarly. We can labels the angles as below. Then and as below. Then the remaining angles as below (since is a straight line, so and similarly for BOC) Then

The Copenhagen Interpretation

at a particular point and a particular time is given by then the probability of finding the particle in a small volume centred at that point is given by In general the particle will exist in many possible states simultaneously – the wavefunction...

Using Gershgorin's Theorem to Estimate Eigenvalues

for the entry on the diagonal - entry \[a_{ii}\] and let \[D_i = \{ z: \| z -a+{ii} \| \le r_i \}\] be the closed disk centred at \[a_{ii}\] with radius \[r_i\] . Gershgorin's Theorem says that the eigenvalues are to be found in the union of the \[D_i\]...

Parametric Equations

along the curve from a given point, and this is useful for some purposes. Example: We can define a circle radius \[5\] centred at the point \[(2,3)\] in terns of the parameter \[\theta\] , the angle between the horizontal and a radius from the centre of...

Volume of a Torus

a torus by considering it as a volume of revolution of a circle rotated about the origin. Let a circle of radius \[a\] be centred at \[(0,b)\] with \[b \gt a\] . The circle is rotated about the \[x\] axis. The upper half of the circle has equation...

Planes of Symmetry

the cylinder, we can cut the cylinder lengthwise in may different ways, as long as the plane of symmetry passes though the centre of the circle at either end. Cubes have many planes of symmetry. Two are shown below to the right.

Proof That a Triangle Drawn in a Semicircle is Right Angled

the diameter (angle ) is a right angle. The proof follows straight from the fact that the angle subtended by an arc at the centre of a circle is twice the angle subtended at the circumference, together with the fact that angle so

Proof That Opposite Angles in a Cyclic Quadrilateral Add to 180 Degrees

The diagram below illustrates the theorem. and From C and B draw lines to the centre of the circle. If then as below and if then so

A Level Maths Notes - S3

Multiple Images

and image distances from the mirror are made equal. Geometrically the object and the images all lie on a circle whose centre is at the intersection of the two mirrors. More images may be formed by decreasing the angle between the mirrors. The they are...

The Inverse Square Law for Electromagnetic Waves

an area of which forms part of the surface of a sphere of radius receives wave energy from a point source placed at the centre. If is the energy passing through this area of in Watts then we say that is the wave intensity in in the diagram below since...

The Projector

a carbon electric arc or quartz iodine lamp which both give a small but high intensity light. The source is situated at the centre of curvature of a small concave mirror which reflects light that would otherwise be wasted on the projector housing. In...

Tuning a Keyboard Instrument

such as a piano or organ there is a difficulty. The diagram below shows a middle C, given by a white key near the centre of the keyboard. The rest of the scale (to the right of middle C) is given by the next seven white keys, which have relative...

Essential Formulae

Geometry If at a point them the equation of the tangent at is and the equation of the normal is A circle with equation has centre and radius

Loci

The set of all points distance r from O is the circle, centre O, radius r. The set of points the same distance from B as from C is the perpendicular bisector of the line BA.

Venn Diagrams

Dirty Digger and Daily Trash and 2 read all three. We have to fill out the diagram above. It is best to work out from the centre. 2 people read all three newspapers, so the entry in the central region is 2. Complete the regions in the order shown:1) 2)...

Star Birth

it heats up. The collapse speeds up as it proceeds, and the gas heats up more and more... Eventually the gas at the centre because hot enough so that the nuclei – mostly protons/hydrogen nuclei – have enough energy to overcome their mutual repulsion,...

The Structure of the Earth - The Four Spheres

As we move out from the centre of the Earth we pass through several sharply defined regions: The core, consisting of liquid iron and nickel surrounding a solid inner core. The core is the origin of the Earth's magnetic field. The mantle, consisting of...

Constructing an Angle of 45 Degrees

edge of which is a diameter, is always a right angled triangle. Therefore, start by drawing a circle with a compass. The centre ofthe circle will be at the position of the compass needle. Draw a diameter of the circle. Put the compass needle on one...