Additional Mathematics

These revision cards are to help with the revision of OCR FSMQ Additional Mathematics. They cover the four topics of Algebra, Coordinate Geometry, Trigonometry and Calculus.

HideShow resource information
  • Created by: Gianamar
  • Created on: 10-07-12 17:10

Algebra Review

Quadratic Formula:

x=\frac{-b \pm \sqrt {b^2-4ac}}{2a}, (http://upload.wikimedia.org/wikipedia/en/math/8/e/4/8e4fef5352eb498b3534af481c8c4fd4.png)  where ax^2 + bx + c = 0

Completing the Square:

(http://t0.gstatic.com/images?q=tbn:ANd9GcTsE8U0711KHBn5m_JnSUfYO3CC3bLyDLGqQcGPfVwgJWUNhFh23A)

1 of 8

Binomial Theorem

Combinations - Number of possible outcomes of an event where the ORDER in which the events occur DOES NOT MATTER!

Permutations - Number of possible outcomes of an event where the ORDER in which the events occur DOES MATTER!

Binomial Expansion - Describes the algebraic expansion of exponents of a binomial. It expands the power (x + y)^n into a sum involving terms in the form of ax^b.y^c.  The Binomial expansion is performed as following:

(x+y)^n = {n \choose 0}x^n y^0 + {n \choose 1}x^{n-1}y^1 + {n \choose 2}x^{n-2}y^2 + \cdots + {n \choose n-1}x^1 y^{n-1} + {n \choose n}x^0 y^n,  (http://upload.wikimedia.org/wikipedia/en/math/d/4/5/d454f8aff0f034629550981e2646d2c8.png)

where \tbinom nk  (http://upload.wikimedia.org/wikipedia/en/math/3/8/2/382c5908d125a08662b2fedc22f4d02c.png) is a binomial and can also be written as nCk. It is equal to the number of combinations in which n events result in k outcomes (where the ORDER DOES'NT MATTER!!!)

2 of 8

Binomial Expansion

(http://hyperphysics.phy-astr.gsu.edu/hbase/imgmth/algf.gif)

3 of 8

Binomial Probability

The probability of event X occurring k times out of n trials where p is theprobability of the event X occurring is represented by the following mathematical formula:

(http://www.allianthawk.org/images/stats%20binomial%20probability%20formula.gif)

Furthermore, the probability of an event X occurring k times is equal to p^k. Note that \tbinom nk  (http://upload.wikimedia.org/wikipedia/en/math/3/8/2/382c5908d125a08662b2fedc22f4d02c.png) can also be written as nCk.

4 of 8

Example Questions and Answers

Q1The probability that a pen drawn at random from a large box of pens is defective is 0.1. A sample of eight of these pens are taken. Find the probability (to 2 d.p.) that it contains (i) no defective pens (ii) one defective pen (iii) at least two defective pens A1:n=8, p=0.1, in (i) k=0, (ii) k=1, (iii) k≥2,  
(i) P(X=0) = 8C0*(0.1^0)(0.9^8) = 0.43 
(ii) P(X=1) = 8C1
(0.1^1)(0.9^7) = 0.38 
(iii) P(X≥2) = P(X=2) + P(X=3) +...+P(X+8) = 1 - [P(X=0)+P(X=1)] = 1 - [(8C0(0.1^0)
(0.9^8))+(8C1(0.1^1)*(0.9^7))] = 1 - 0.43 - 0.38 = 0.18.

5 of 8

Calculus

Differentiation - used to find the rate of change of a function (the gradient). For a function f(x) = axⁿ, f'(x) = naxⁿ⁻¹. For example, the function 3x²+ 24+ 10 differentiates into dy/dx = 6x + 24. The rule is: Multiply the coefficient by the exponent, then decrease the exponent by 1.

Integration - used to find the area under a curve. By integrating a differentiated function, you return to the original function. Therefore, the method of integration is quite logically following. Increase the exponent by 1, then divide by the new exponent.

f'(x) dx = f(x) + C

 where C is the constant of integration. It cannot be determined unless we know one coordinate through which f(x) passes.

This is called the fundamental theorem of calculus

6 of 8

Application of Calculus to Kinematics

Calculus can be applied to almost anything in the entire universe which contains rates of change. One example in the FSMQ Add Maths syllabus is Kinematics. Kinematics is a branch of mathematical physics concerned with the equations of motion. These are used to find:

u - Initial velocity

v - Final velocity

a - Acceleration

s - Displacement

t - Time

Please note that all of these parameters except for time are in bold because they are vectors. They have both magnitude (length etc.), and direction (e.g. 24° above of horizontal).

7 of 8

The Constant Acceleration Equations

Below are the equations of motion:

\begin{align} v & = at+u \quad [1]\\ s & = ut + \frac{at^2}{2} \quad [2]\\ s & = \left( \frac{v+u}{2} \right )t \quad [3]\\ v^2 & = u^2 + 2as \quad [4]\\ s & = vt - \frac{at^2}{2} \quad [5]\\ \end{align} (http://upload.wikimedia.org/wikipedia/en/math/5/1/c/51c322ae72e8e2ccb6844178ae4f6426.png)

8 of 8

Comments

No comments have yet been made

Similar Mathematics resources:

See all Mathematics resources »See all Algebra resources »