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Paper 11 Quadratics • carry out the process of completing the square for a quadratic
polynomial ax2 + bx + c, and use this form, e.g. to locate the vertex ofthe graph of y = ax2 + bx + c or to sketch the graph;• find the discriminant of a quadratic polynomial ax2 + bx + c and usethe discriminant, e.g. to determine the number of real roots of theequation ax2 + bx + c = 0;• solve quadratic equations, and linear and quadratic inequalities, in oneunknown;• solve by substitution a pair of simultaneous equations of which one islinear and one is quadratic;• recognise and solve equations in x which are quadratic in somefunction of x, e.g. x4 – 5x2 + 4 = 0.
2 Functions • understand the terms function, domain, range, one-one function,inverse function and composition of functions;• identify the range of a given function in simple cases, and find thecomposition of two given functions;• determine whether or not a given function is one-one, and find theinverse of a one-one function in simple cases;• illustrate in graphical terms the relation between a one-one functionand its inverse.
3 Coordinate Geometry
• find the length, gradient and mid-point of a line segment, given thecoordinates of the end-points;• find the equation of a straight line given sufficient information (e.g. thecoordinates of two points on it, or one point on it and its gradient);• understand and use the relationships between the gradients of paralleland perpendicular lines;• interpret and use linear equations, particularly the forms y = mx + cand y – y1 = m(x – x1);• understand the relationship between a graph and its associatedalgebraic equation, and use the relationship between points ofintersection of graphs and solutions of equations (including, in simplecases, the correspondence between a line being tangent to a curveand a repeated root of an equation).
4 Circular Measure • understand the definition of a radian, and use the relationshipbetween radians and degrees;• use the formulae s = r θ and A =21 r2θ in solving problems concerningthe arc length and sector area of a circle.
5 Trigonometry • sketch and use graphs of the sine, cosine and tangent functions (for
angles of any size, and using either degrees or radians);• use the exact values of the sine, cosine and tangent of 30°, 45°, 60°,and related angles, e.g. cos 150° = –21 3 ;• use the notations sin−1x, cos−1x, tan−1x to denote the principal values ofthe inverse trigonometric relations;• use the identitiescossin ii≡ tan θ and sin2θ + cos2θ ≡ 1;• find all the solutions of simple trigonometrical equations lying in aspecified interval (general forms of solution are not included).
6 Vectors • use standard notations for vectors, i.e. yx, xi + yj, zyx, xi + yj + zk,AB , a;• carry out addition and subtraction of vectors and multiplication of avector by a scalar, and interpret these operations in geometrical terms;• use unit vectors, displacement vectors and position vectors;• calculate the magnitude of a vector and the scalar product of twovectors;• use the scalar product to determine the angle between two directionsand to solve problems concerning perpendicularity of vectors.
7 Series • use the expansion of (a + b)n , where n is a positive integer (knowledgeof the greatest term and properties of the coefficients are notrequired, but the notations
rn and n! should be known);• recognise arithmetic and geometric progressions;• use the formulae for the nth term and for the sum of the first n termsto solve problems involving arithmetic or geometric progressions;• use the condition for the convergence of a geometric progression,and the formula for the sum to infinity of a convergent geometricprogression.
8 Differentiation • understand the idea of the gradient of a curve, and use the notationsf’(x), f’’(x),xyddand 22ddxy (the technique of differentiation from firstprinciples is not required);• use the derivative of xn (for any rational n), together with constantmultiples, sums, differences of functions, and of composite functionsusing the chain rule;• apply differentiation to gradients, tangents and normals, increasingand decreasing functions and rates of change (including connectedrates of change);• locate stationary points, and use information about stationary pointsin sketching graphs (the ability to distinguish between maximumpoints and minimum points is required, but identification of points ofinflexion is not included).
9 Integration • understand integration as the reverse process of differentiation,and integrate (ax + b)n (for any rational n except –1), together withconstant multiples, sums and differences;• solve problems involving the evaluation of a constant of integration,e.g. to find the equation of the curve through (1, –2) for whichxydd= 2x + 1;• evaluate definite integrals (including simple cases of ‘improper’integrals, such as x dx and x dx012121 3
y - y - );• use definite integration to findthe area of a region bounded by a curve and lines parallel to theaxes, or between two curves,a volume of revolution about one of the axes.
Topic 1 Quadratics a x2+bx+c
The general form of a quadratic function is f ( x )=a x2+bx+c , where a≠0.
Completing the square form is a (x+h)2+k
As we know, the expansion of ( x+a )2
x2+2ax+a2=( x+a )2
Minus both sides of the equation with a2
x2+2ax= (x+a )2−a2
Example 1
x2+4 x= (x+2 )2−22
Example 2
x2−5 x=( x−52 )
2
−(−52 )
2
=( x−52 )
2
− 254
Example 3
x2+4 x+5= ( x+2 )2−22+5
Example 4
If a≠1, we must factorise first
2 x2+4 x+5=2(x¿¿2+2 x)+5¿
Apply completing the square for x2+2 x
2 x2+4 x+5=2 [ ( x+1 )2−12 ]+5
x2+4 x+5= ( x+2 )2−22+5
Locate the vertex and Sketch the graph
The discriminant and the nature of the roots
b2−4 ac>0; 2 distinct real roots (2 different real roots)
b2−4 ac=0; 2 equal real roots
b2−4 ac<0; no real roots
Solve quadratic equation, a x2+bx+c=0
1. Factorization 2. Completing the square 3. Formulae x=−b±√b2−4ac2a
Quadratic inequalities
a x2+bx+c<0
a x2+bx+c≤0
a x2+bx+c>0
a x2+bx+c≥0
Simultaneous equations (one linear and one curve)
Recognize and solve equations which can be reduced to quadratic in function of x.
Topic 2 Functions
Define function
Domain and range
Composite function
Inverse function one-one function
Sketch graph of a function and its inverse (reflection on the line y=x)
Topic 3 Coordinate geometry
a (x+h)2+k
The length of two points
Gradient of a line segment
Mid-point of a line segment
Equation of a straight line given sufficient information (two points or one point and its gradient)
y=mx+c and y− y1=m(x−x1)
Parallel lines m1=m2 and Perpendicular lines m1×m2=−1
Use the relationship between points of intersection of graphs and solutions of equations
4. Circular measure
Understand the definition of a radian and use the relationship between radians and degrees
The arc length, s=rθ, where θ in radian.
The sector area, A=12r2θ, where θ in radian.
The area of triangle, A=12ab sinC
The sine rule a
sin A= b
sinB= c
sinC
The cosine rule a2=b2+c2−2bc cosA
5. Trigonometry
Graphs of the sine, cosine and tangent functions (and its transformation)
Exact values of the sine, cosine and tangent
θ 0 ° 30 ° 45 ° 60 ° 90 °sin θ 0 1
21√2
=√22
√32
1
cosθ 1 √32
1√2
=√22
12
0
tanθ 0 1√3
=√33
1 √3 Undefined
Simple trigonometric equation
sin x=k
Basic identities
sin2θ+cos2θ≡1 tanθ ≡ sinθcosθ
6. Vectors
Use standard notations for vectors ( xy )=x i+ y j, ( xyz )=x i+ y j+z k , A⃗B, a
Carry out addition and subtraction of vectors and multiplication of a vector by a scalar, and interpret these operations in geometrical terms
Use unit vectors, displacement vectors and position vectors
Calculate the magnitude of a vector and the scalar product of two vectors
Use the scalar product to determine the angle between two directions and to solve problems concerning perpendicularity of vectors.
7. Series
Use the expansion of (a+b)n, where n is a positive integer
The notation nCr=(nr ) and n !
Recognise arithmetic and geometric progressions
Use the formulae for the nth term and for the sum of the first n terms
Use the condition for the convergence of a geometric progression, and the formula for the sum to infinity of a convergent geometric progression
8. Differentiation
Understand the idea of the gradient of a curve
Use the notations f ' ( x ) , f(x ) , {dy } over {dx } and {{d} ^ { 2} y } over {d { x} ^ { 2}(the technique of differentiation from first principles is not required)
Use the derivative of xn(for any rational n), together with constant multiples, sums, differences of functions and of composite functions using chain rule
Apply differentiation to gradients, tangents and normals, increasing and decreasing functions and rates of change (including connected rates of change)
Locate stationary points, and use information about stationary points in sketching graphs (the ability to distinguish between maximum and minimum points is required, but identification of points of inflexion is not included).
9. Integration
Understand integration as the reverse process of differentiation, and integrate (ax+b)n(for any rational n except -1), together with constant multiples, sums and differences
Solve problems involving the evaluation of a constant of integration
e.g. to find the equation of the curve through (1, -2) for which
dydx
=2 x+1
evaluate definite integrals (including simple cases of ‘improper’ integrals, such as ∫0
1
x−1
2 dx and ∫1
∞
x−2dx
use definite integration to find the area of a region bounded by a curve and lines parallel to the axes, or between two curves
find a volume of revolution about one of the axes
Paper 6 Statistics 11 Representation of
data• select a suitable way of presenting raw statistical data, and discussadvantages and/or disadvantages that particular representations mayhave;• construct and interpret stem-and-leaf diagrams, box-and-whisker plots,histograms and cumulative frequency graphs;• understand and use different measures of central tendency (mean,
median, mode) and variation (range, interquartile range, standarddeviation), e.g. in comparing and contrasting sets of data;• use a cumulative frequency graph to estimate the median value, thequartiles and the interquartile range of a set of data;• calculate the mean and standard deviation of a set of data (includinggrouped data) either from the data itself or from given totals such asΣx and Σ x 2, or Σ(x – a) and Σ(x – a)2.
2 Permutation and combination
• understand the terms permutation and combination, and solve simpleproblems involving selections;• solve problems about arrangements of objects in a line, includingthose involving:repetition (e.g. the number of ways of arranging the letters of theword ‘NEEDLESS’),restriction (e.g. the number of ways several people can stand ina line if 2 particular people must — or must not — stand next toeach other).
3 Probability • evaluate probabilities in simple cases by means of enumerationof equiprobable elementary events (e.g. for the total score whentwo fair dice are thrown), or by calculation using permutations orcombinations;• use addition and multiplication of probabilities, as appropriate, insimple cases;• understand the meaning of exclusive and independent events, andcalculate and use conditional probabilities in simple cases, e.g.situations that can be represented by means of a tree diagram.
4 Discrete Random variable
• construct a probability distribution table relating to a given situationinvolving a discrete random variable variable X, and calculate E(X) andVar(X);• use formulae for probabilities for the binomial distribution, andrecognise practical situations where the binomial distribution is asuitable model (the notation B(n, p) is included);• use formulae for the expectation and variance of the binomialdistribution.
5 Normal Distribution
• understand the use of a normal distribution to model a continuousrandom variable, and use normal distribution tables;• solve problems concerning a variable X, where X ~ N( μ, σ2), includingfinding the value of P(X > x1), or a related probability, given thevalues of x1, μ, σ ,
finding a relationship between x1, μ and σ given the value ofP(X > x1) or a related probability;• recall conditions under which the normal distribution can be usedas an approximation to the binomial distribution (n large enough toensure that np > 5 and nq > 5), and use this approximation, with acontinuity correction, in solving problems.
Permutation and Combination
n factorial
n !=n× (n−1 )× (n−2 )×…×3×2×1
0 !=1
Arrangements in a line
The number of different arrangements in a line of n distinct objects is _____________
The number of different arrangements in a line of n items of which p are alike is ______________
The number of different arrangements in a line of n items of which p are alike, q of another type are alike, r of another type are alike, and so on, is ___________________
Arrangements when there are restrictions
The word ARGENTINA includes the four consonants R, G, N, T and the three vowels A, E, I.
a) Find the number of different arrangements using all nine letters.
b) How many of these arrangements have a consonant at the beginning, then a vowel, then another consonant, and so on alternately?
Arrangements when repetitions are allowed
Permutations =………………….. Combinations = ………………………The number of permutations of r items taken from n distinct items is
In permutations, order matters.
The number of combinations of r items taken from n distinct items is
In combinations, order does not matter.Three committee members are to be chosen from 6 students for the position of president,
Three committee members are to be chosen from 6 students. Find the number of ways
vice president and secretary. Find the number of ways committee can be chosen.
committee can be chosen.
With condition: With condition:Find the number of different ways the letters YOUTUBE can be arranged if
a) there are no restrictions.
b) it must be begin with a vowel.
c) all vowels must be next to each other.
d) no two vowels are next to each other.
Find the number of ways of choosing a team of 5 students from 6 boys and 4 girls if
a) there are more boys than girls.
b) there are more girls than boys.
c) 2 of the boys are cousins and are either all in the team or all not in the team.
Find the number of different ways the letters INTRAGRAM can be arranged if
a) there are no restriction
b) it must be begin with a vowel.
c) all vowels must be next to each other.
d) no two vowels are next to each other.
Find the total number of selections
a) if four letters are selected at random from the letters of the word BRAZIL. (all distinct items)
b) if four letters are selected at random from the letters of the word FACEBOOK. (it consists of identical items)
6 boys and 4 girls are standing in a line.Find the number of ways that all these ten people can be arranged if
a) 6 boys must be next to each other.
b) no two girls stand next to each other.
Probability
A sample space is the set of all possible outcomes of an experiment. It is denoted by the letter S and the number of outcomes is n(S).
The probability of an event is a measure of the likelihood that it will happen.
Probability of an event A= number of outcomes∈event Atotal number of possible outcomes
Notation:
P(A)= n(A)n(S)
, where 0≤P(A)≤1
A probability of 0 indicates that the event is impossible (which does not occur).
A probability of 1 (or 100%) indicates that the event is certain (sure) to happen.
All other events have a probability between 0 and 1.
For example:
If a fair coin is tossed, the probability of getting a head is 0.5, 50% or 12 .
If a die is thrown, the probability that it lands on 5 is 16 .
If you select a counter from a box of green counters, the probability that you will select a green counter is 1 (certain).
The probability that you will select a yellow counter is 0 (impossible).
Complement
The complement of A is the event that A does not occur and it is denoted by A '. An alternative notation is A.
The probability that event A does not occur is denoted by P (A ' ) .
P (A ' )=1−P(A)
Combined events
For events A and B,
The probability of both events A and B (intersection) occurring is represented by:
P (A∧B )=P ( A∩B )=n(A∩B)n(S )
The probability of both events A or B (union) occurring is represented by:
P (A∨B )=P ( A∪B )=n(A∪B)n(S)
Addition rule of probability
P (A∪B )=P ( A )+P (B )−P (A ∩B)
For example:
A fair dice is rolled once. Find the probability of getting a prime or even number.
Prime number = {2, 3, 5}
Even number = {2, 4, 6}
Even and Prime number = {2}
∴P ( prime∨even)=P ( prime )+P ( even )−P ( prime∧even )
∴P ( prime∨even)=36+ 3
6−1
6=5
6
Alternative method: List down all the elements of even or prime numbers.
Even or Prime number = {2, 3, 4, 5, 6}
∴P ( prime∨even )=56
Mutually exclusive events (Addition rule)
Events are mutually exclusive if they cannot occur at the same time.
P (A ∩B )=0
∴P ( A∪B )=P (A )+P (B )=n ( A )+n(B)
n(S)
For example:
A bag contains 5 blue balls, 4 red balls and 1 black ball. A ball is drawn randomly from the bag. What is the probability that the ball is either red or blue?
P (red )= 410
,P (blue )= 510
∴P (red∨blue )= 410
+ 510
= 910
Independent events (Multiplication rule)
Two events are independent if the outcome of one does not affect the outcome of the other.
For independent events A and B
P (A∧B )=P ( A )×P(B)
In set notation: P (A∩B )=P (A )× P(B)
For example:
A fair die is rolled twice. Find the probability of obtaining the number 2 on the first roll and an odd number on the second roll.
Let A = event of obtaining the number 2 on the first roll.
Let B = event of obtaining an odd number on the second roll.
Both A and B are independent events.
∴P ( A∧B )=16× 3
6= 1
12
Conditional probability
The probability that event A occurs, given that event B has already occurred is denoted by P (A|B ).
P (A givenB )=P (A|B )=P ( A∧B )P (B )
Or alternatively,
P (A|B )=n(A∧B)n(B)
For example:
A fair die is thrown. Find the probability of getting a prime number given that the score is an even number.
Prime number = {2, 3, 5}, Even number = {2, 4, 6}
Prime and even = {2}
∴P (Prime|Even )=P(Prime∧even)P(even)
=
1636
=13
Tree diagrams
A diagram that shows all the possible outcomes of an event is called tree diagram.
For two events A and B, each with two outcomes:
For example:
A) Selection with replacement (independent events)
A box contains 4 red marbles and 6 blue marbles. A marble is selected at random. It is then replaced back into the box and a second marble is selected randomly from the box. Find the probability that the first marble is red and the second marble is blue.
∴P (First is red and second is blue )= 410
× 610
= 625
B) Selection without replacement (dependent events)
A box contains 4 red marbles and 6 blue marbles. Two marbles are selected at random. Find the probability that there are one red and one blue marbles.
∴P (one red and one blue )=P (RB )+P (BR )= 410
× 69+ 6
10× 4
9=24
45
Or alternatively,
We can use nCr to solve this question.
∴P (one red and one blue )=4C1×6C110C2
=2445
Discrete random variables
Random variables
A random variable usually written X, is a variable whose possible values are numerical outcomes which could occur for some random experiment. There are two types of random variables, discrete and continuous.
A discrete random variable is a variable which can take individual values each with a given probability. The values of the variable are the outcome of an experiment.
For example:
Experiment Possible outcomesThe score when you throw a fair die 1, 2, 3, 4, 5, 6
The number of winning badminton matches by Charles in 3 matches 0, 1, 2, 3
The number of heads when you toss a coin 5 times 0, 1, 2, 3, 4, 5
The number of children in a family in your housing area 0, 1, 2, 3, 4, 5, 6, 7, 8
A continuous random variable is a variable which has all possible values in some interval. Continuous random variables are usually measurements. For examples, the amount of sugar in an apple (10g < X < 20g), the time required to run a 100 meter (10s < X < 25s), the heights of students in a class (140cm < X < 200cm).
Probability distributions
Probability distribution is a list of all possible values of the discrete random variable X together with their associated probabilities.
The sum of the probabilities of all possible values of a discrete random variable X is 1.
∑ P (X=x )=1 or∑ p=1
The expectation, or expected value, of a random variable X is written as E(X). It is also called the expected mean or the mean μ.
E (X )=μ=∑ xp
The variance of a discrete random variable X is written as Var (X) and is denoted by σ 2. It is a measure of the spread of X about the expected mean μ.
Var (X )=∑ x2 p−[E (X )]2=∑ x2 p−μ2
Example: 9709_s02_6 Q3
A fair cubical die with faces numbered 1 ,1 ,1 ,2 ,3 ,4 is thrown and the scored noted. The area A of a square of side equal to the score is calculated, so, for example, when the score on the die is 3, the value of A is 9.
i) Draw up a table to show the probability distribution of A.ii) Find E ( A ) and Var (A).
Solution:
i) Die:1 ,1 ,1 ,2 ,3 ,4∴ A :1,1 ,1 ,4 ,9 ,16
The possible outcomes of A are 1, 4, 9, 16.
P (A=1 )=36, P ( A=4 )=1
6,P ( A=9 )=1
6∧P ( A=16 )=1
6
a 1 4 9 16P(A=a) 3
616
16
16
ii) Formulae: E (X )=μ=∑ xp and Var (X )=∑ x2 p−[E (X )]2=∑ x2 p−μ2
E ( A )=1× 36+4× 1
6+9× 1
6+16× 1
6
∴E (A )=163
Var (A )=12× 36+42× 1
6+92× 1
6+162× 1
6−( 16
3 )2
∴Var ( A )=30.9(3 s . f .)
Example: 9709_s03_6 Q2
A box contains 10 pens of which 3 are new. A random sample of two pens is taken.
i) Show that the probability of getting exactly one new pen in the sample is 715 .
Possible outcomes: NN’ or N’N
P(exactly one new pen)∴ = 310
× 79+ 7
10× 3
9= 7
15
ii) Construct a probability distribution table for the number of new pens in the sample.
Let X be the discrete random variable representing the number of new pens in the random sample of two pens.
The possible outcomes of X are 0, 1 and 2.
∴P (X=0 )= 710
× 69= 7
15,P (X=1 )= 3
10× 7
9+ 7
10× 3
9= 7
15, P (X=2 )= 3
10× 2
9= 1
15
x 0 1 2P(X=x) 7
15715
115
iii) Calculate the expected number of new pens in the sample.
∴E (X )=0× 715
+1× 715
+2× 115
=35
Binomial Distribution
A discrete random variable X follows a binomial distribution when all the following conditions are satisfied.
The distribution of X is written as
X B(n , p)
where n is number of trials and p is the probability of a successful outcome in each trial.
Conditions:
1) There are a fixed number of trials, n.2) The trials are independent.3) Each trial results in two possible outcomes. (success and failure)4) The probability of success, p, is constant for each trial. The probability of failure, q is also
constant. (where q=1−p)
The probability of r successes in n trials is
P (X=r )=(nr ) prqn−r
The expectation of the binomial is
E (X )=μ=np
The variance of the binomial is
Var (X )=σ2=npq
Example:
A biased coin is tossed 3 times and the probability of obtaining head is 0.4.
i) Construct the probability distribution table of number of heads.
Let X be the discrete random variable of the number of heads obtained in 3 tosses.
The number of trials, n =3 (fixed) and the probability of getting head, p is 0.4 (constant).
∴ X B(3,0.4)
HHH⏟3heads⏟
(33)=1
, HHT , HTH ,THH⏟2heads⏟
(32)=3
, HTT ,THT ,TTH⏟1head⏟
(31)=3
, TTT⏟0heads⏟
(30)=1
∴P (X=0 )=(30) (0.4 )0 (0.6 )3=0.216
∴P (X=1 )=(31) (0.4 )1 (0.6 )2=0.432
∴P (X=2 )=(32)(0.4 )2 (0.6 )1=0.288
∴P (X=3 )=(33)(0.4 )3 (0.6 )0=0.064
x 0 1 2 3
P(X=x) 0.216 0.432 0.288 0.064
ii) Find the expected number and the variance of heads thrown.
∴E (X )=np=3×0.4=1.2
∴Var (X )=npq=3×0.4×0.6=0.72
Normal Distributions
The normal distribution is the most important distribution for a continuous random variable. Many naturally occurring phenomena have a distribution that is normal, or approximately normal.
Some examples are the heights of adults, the lengths of leaves, the weights of apples, the IQ scores.
A function is used to specify the probability distribution for a continuous random variable. The function is called the probability density function.
If X is normally distributed then its probability density function is given by
f ( x )= 1σ √2π
e−12 ( x−μ
σ )2
for−∞<x<∞.
The normal distribution has two parameters which are the mean, μ and the variance, σ 2. It can be written as X N (μ ,σ2).
Characteristics of the normal probability density function
1. The curve is symmetrical about the vertical line x=μ.
2. The area under the curve is 1 unit2. Hence, ∫−∞
∞
f (x)dx=1.
3. More scores are distributed closer to the mean than further away. It is a bell shaped curve.
From the diagram above, we can see that
1. Approximately 2×34.13 %≈68 % of the values lie within one standard deviation of the mean.2. Approximately 2× (34.13 %+13.59% )≈95% of the values lie within two standard deviations
of the mean.
The standard normal distribution
A normal distribution X N (μ ,σ2) can be converted to the standard normal distribution or Z-distribution Z N (0,1) with a mean of 0 and a standard deviation of 1. The continuous random variable
X of a normal distribution is converted to Z, the standardized variable of a standard normal distribution. The values of Z are called z-scores. The z-scores can be found using a normal distribution table. A normal distribution table giving the lower probabilities of Z N (0,1) lists all the values of P(Z≤a).
If Z has standard normal distribution, find using tables and a sketch
(i) P(Z≤0.533)
∴P (Z ≤0.533 )=0.7029
(ii) P(Z≥1.327)
∴P (Z ≥1.327 )=1−P (Z<1.327)∴P (Z ≥1.327 )=1−0.9077∴P (Z ≥1.327 )=0.0923
(iii) P(Z←0.533)
∴P (Z←0.533 )=P (Z>0.533 ) symmetrical∴P (Z←0.533 )=1−P(Z<0.533)
∴P (Z←0.533 )=1−0.7029∴P (Z←0.533 )=0.2971
(iv) P(Z>−1.327)
∴P (Z>−1.327 )=P (Z<1.327 ) symmetrical∴P (Z ≥1.327 )=0.9077
(v) P(0.533≤Z<1.327)
∴P (0.533≤Z<1.327 )=P (Z<1.327 )−P(Z<0.533)∴P (0.533≤Z<1.327 )=0.9077−0.7029
∴P (0.533≤Z<1.327 )=0.2048(v) P(−0.533≤Z<1.327)
∴P (−0.533≤Z<1.327 )=P (Z<1.327 )−P(Z←0.533)
∴P (−0.533≤Z<1.327 )=0.9077−(1−0.7029)∴P (−0.533≤Z<1.327 )=0.6106
Find the value of a if Z has standard normal distribution and
(i) P (Z ≤a )=0.6595
∴P (Z ≤0.411)=0.6595∴a=0.411
(ii) P (Z ≥a )=0.1112
∴P (Z<1.22 )=0.8888∴a=1.22
(iii) P (Z<a )=0.3764
1−0.3764=0.6236∴P (Z<0.315 )=0.6236
∴P (Z←0.315 )=1−P (Z<0.315 )=0.3764∴a=−0.315
(iv) P (Z>a )=0.918
∴P (Z<1.392 )=0.918∴P (Z>−1.392 )=P (Z<1.392 ) symmetrical
∴a=−1.392
Standardising any normal distribution
To find probabilities for normally distributed random variable X we can follow these steps:
Step 1: Convert X values to Z using Z= X−μσ .
Step 2: Sketch a standard normal distribution curve and shade the required region.
Step 3: Use the standard normal table to find the probability.
Example:
The masses of workers of a factory follow a normally distribution with mean 62 kg and standard deviation 5 kg.
i) Find the probability that a worker chosen at random from this factory has a mass of more than 72 kg.
X N (62 ,52)
Step 1: Convert X values to Z using Z= X−μσ
P (X>72 )=P(Z> 72−625 )
Step 2: Sketch a standard normal distribution curve and shade the required region
P (X>72 )=P (Z>2 )
Step 3: Use the standard normal table to find the probability
∴P (Z>2 )=1−P(Z<2)
∴P (Z>2 )=1−0.9772=0.0228
∴P (X>72 )=0.0228
ii) A random sample of 500 workers is chosen. Find the number of workers from this sample has a mass of less than 72 kg.
P (X>72 )=0.0228
∴P (X<72 )=1−0.0228=0.9772
500×0.9772=488.6
Number of workers has a mass of less than 72 kg∴ =489(3 s . f .)
iii) Given 258 out of 500 workers have a mass of more than m kg, find the value of m.
Step 1: Convert X values to Z using Z= X−μσ
P (X>m )=258500
=0.516 P(Z>m−625 )=0.516
Step 2: Sketch a standard normal distribution curve and shade the required region
P(Z>m−625 )=0.516
Step 3: Use the standard normal table to find the z-score
From normal distribution table,
P (Z<0.04 )=0.516
∴P (Z>−0.04 )=0.516 symmetrical
∴m−625
=−0.04 →∴m=61.8
Example: 9709_w09_62 Q7
The weights, X grams, of bars of soap are normally distributed with mean 125 grams and standard deviation 4.2 grams.
i) Find the probability that a randomly chosen bar of soap weighs more than 128 grams.
Step 1: Convert X values to Z using Z= X−μσ
P (X>128 )=P (Z> 128−1254.2 )
Step 2: Sketch a standard normal distribution curve and shade the required region
P (X>128 )=P (Z>0.714 )
Step 3: Use the standard normal table to find the probability
P (Z>0.714 )=1−P (Z<0.714)
P (Z>0.714 )=1−0.7623=0.2377
∴P (X>128 )=0.2377
ii) Find the value of k such that P (k<X<128 )=0.7465.
Step 1: Convert X values to Z using Z= X−μσ
P (k<X<128 )=0.7465
P( k−1254.2
<Z<128−1254.2 )=0.7465
P( k−1254.2
<Z<0.714)=0.7465
∴P (Z<0.714 )−P(Z< k−1254.2 )=0.7465
∴0.7623−P(Z< k−1254.2 )=0.7465
∴P(Z< k−1254.2 )=0.7623−0.7465=0.0158
Step 3: Use the standard normal table to find the z-score
1−0.0158=0.9842
P (Z<2.15 )=0.9842
P (Z←2.15 )=1−0.9842=0.0158
∴ k−1254.2
=−2.15→∴ k=115.97 ≈116(3 s . f .)
iii) Five bars of soap are chosen at random. Find the probability that more than two of the bars each weigh more than 128 grams.
From part i), P (X>128 )=0.2377
Let Y be discrete random variable of the number of bars of soap weighs more than 128 grams in random sample of five bars of soap. It is a binomial distribution with n=5 and p=0.2377.
Y B (5 ,0.2377)
P (Y >2 )=P (Y=3 )+P (Y=4 )+P(Y=5)
P (Y >2 )=(53)(0.2377 )3 (0.7623 )2+(54) (0.2377 )4 (0.7623 )1+(55) (0.2377 )5 (0.7623 )0
∴P (Y >2 )=0.0910(3 s . f .)
The normal approximation to the binomial distribution
If the binomial distribution X B(n , p) has np>5 and nq>5, then its distribution is symmetrical and bell shaped. It can be approximated by normal distribution with mean μ=np and variance σ 2=npq where q=1−p.
X B (n , p )→X N (np ,npq ) , if both np>5 and nq>5
Example 1: X B(20 ,0.5) (vertical line diagram is symmetrical)
Since both np>5 and nq>5, it can be approximated as normal distribution with mean μ=np=10 and variance
σ 2=npq=5.
Example 2: X B(20 ,0.2) (Vertical line diagram is not symmetrical)
Since only nq>5 but np<5, it cannot be approximated by normal distribution.
To calculate the probabilities using normal approximation to binomial distribution, we need to apply continuity correction.
Example:
If X B(20 ,0.5), then it can be approximated by X N (10 ,5).
Find the following probabilities using normal approximation:
Case 1: P(X>14)
P (X>14 )→P ( X>14.5 ) continuity correction ,doesnot include X=14
P (X>14 )→P(Z> 14.5−10√5 )
P (X>14 )→P (Z>2.012 )
P (X>14 )→1−P (Z<2.012 )
P (X>14 )→1−0.9779=0.0221
Case 2: P(X≥14)
P (X ≥14 )→P (X>13.5 ) continuity correction, include X=14
P (X>14 )→P(Z> 13.5−10√5 )
P (X>14 )→P (Z>1.565 )
P (X>14 )→1−P (Z<1.565 )
P (X>14 )→1−0.9412=0.0588
Case 3: P(X<14)
P (X<14 )→P ( X<13.5 ) continuity correction ,doesnot include X=14
P (X<14 )→P(Z< 13.5−10√5 )
P (X<14 )→P (Z<1.565 )=0.9412
Case 4: P(X≤14)
P (X ≤14 )→P (X<14.5 ) continuity correction , include X=14
P (X ≤14 )→P(Z< 14.5−10√5 )
P (X ≤14 )→P (Z<2.012 )=0.9779
P (X<8 )
Case 5: P(8<X<14)
Case 6: P(8<X ≤14)
Case 7: P(8≤ X<14)
Case 8: P(8≤ X ≤14 )
Paper 3
Paper 4 Mechanics 1Topic 1: Forces and equilibrium
• identify the forces acting in a given situation;
• understand the vector nature of force, and find and use componentsand resultants;
• use the principle that, when a particle is in equilibrium, the vectorsum of the forces acting is zero, or equivalently, that the sum of thecomponents in any direction is zero;
• understand that a contact force between two surfaces can berepresented by two components, the normal component and thefrictional component;
• use the model of a ‘smooth’ contact, and understand the limitations ofthis model;
• understand the concepts of limiting friction and limiting equilibrium;recall the definition of coefficient of friction, and use the relationshipF = μR or F ≤ μR, as appropriate;
• use Newton’s third law.
Topic 2: Kinematics of motion along a straight line
• understand the concepts of distance and speed as scalar quantities,and of displacement, velocity and acceleration as vector quantities (inone dimension only);• sketch and interpret displacement-time graphs and velocity-timegraphs, and in particular appreciate thatthe area under a velocity-time graph represents displacement,the gradient of a displacement-time graph represents velocity,the gradient of a velocity-time graph represents acceleration;• use differentiation and integration with respect to time to solvesimple problems concerning displacement, velocity and acceleration(restricted to calculus within the scope of unit P1);• use appropriate formulae for motion with constant acceleration in astraight line.
Topic 3: Newton’s laws of motion
• apply Newton’s laws of motion to the linear motion of a particle ofconstant mass moving under the action of constant forces, which mayinclude friction;• use the relationship between mass and weight;• solve simple problems which may be modelled as the motion ofa particle moving vertically or on an inclined plane with constantacceleration;• solve simple problems which may be modelled as the motion of twoparticles, connected by a light inextensible string which may pass overa fixed smooth peg or light pulley.
Topic 4: Energy, work and power
• understand the concept of the work done by a force, and calculate thework done by a constant force when its point of application undergoesa displacement not necessarily parallel to the force (use of the scalarproduct is not required);• understand the concepts of gravitational potential energy and kineticenergy, and use appropriate formulae;• understand and use the relationship between the change in energyof a system and the work done by the external forces, and use inappropriate cases the principle of conservation of energy;• use the definition of power as the rate at which a force does work,and use the relationship between power, force and velocity for a forceacting in the direction of motion;• solve problems involving, for example, the instantaneous accelerationof a car moving on a hill with resistance.
1. calculate probabilities for the distribution Po( μ) ;
In general, if X is a Poisson distribution, then
and this is denoted by X Po(λ).
2. use the fact that if X Po(λ) then the mean and variance of X are each equal to μ ;
3. understand the relevance of the Poisson distribution to the distribution of random events, and use the Poisson distribution as a model;
4. use the Poisson distribution as an approximation to the binomial distribution where appropriate ( n > 50 and np < 5 , approximately) ;
5. use the normal distribution , with continuity correction, as an approximation to the poisson distribution where appropriate (λ > 15 , approximately) .
E(X)=λ and Var (X)= λ
E(X)=np, X Po(np)
X N (λ , λ)
Conditions for a Poisson modela. events occur independently and at random in a given interval of time or space,b. λ, the mean number of occurrences in the given interval is known and is finite.
Topic 2: Linear Combinations of Random Variables
1. For independent X and Y,
The sum of the random variables
The different of the random variables
Multiples of the random variables
2. If X has a normal distribution, then so does aX+b.
3. If X and Y have independent normal distributions, then aX+bY has a normal distribution.
4. If X and Y have independent Poisson distributions, then X+Y has a Poisson distribution.
E(aX±b)=a E(X )±b
Var (aX±b)=a2Var (X)
E(aX±bY )=aE (X )±b E (Y )
Var (aX±bY )=a2Var (X)+b2Var (Y )
E(X+Y )=E(X)+E (Y )
Var (X+Y )=Var (X )+Var (Y )
E (X−Y )=E (X )−E(Y )
Var (X−Y )=Var (X)+Var (Y )
E(X1+X2+...+Xn)=nE (X )
Var (X1+X2+ ...+Xn)=nVar (X)
Topic 3: Continuous Random Variables
1. understand the concept of a continuous random variable, and recall and use properties of a probability density function, (p.d.f.), (restricted to functions defined over a single interval);
A continuous random variable takes on an uncountably infinite number of possible values. For a discrete random variable X that takes on a finite or countably infinite number of possible values, we determined P(X = x) for all of the possible values of X, and called it the probability mass function ("p.m.f."). For continuous random variables, as we shall soon see, the probability that X takes on any particular value x is 0. That is, finding P(X = x) for a continuous random variable X is not going to work. Instead, we'll need to find the probability that X falls in some interval (a, b), that is, we'll need to find P(a < X < b). We'll do that using a probability density function ("p.d.f.").
2. use a probability density function to solve problems involving probabilities, and to calculate the mean and variance of a distribution (explicit knowledge of the cumulative distribution function is not included, but location of the median, for example, in simple cases by direct consideration of an area may be required).
The mean of a distribution
The variance of a distribution
Location of the median
E (X )=∫−∞
∞
xf (x)dx
Var (X )=E ( X2 )−μ2
∫−∞
m
f (x)dx=∫m
∞
f (x )dx=0.5
Topic 4: Sampling and estimation
• understand the distinction between a sample and a population, and appreciate the necessity for randomness in choosing samples;
• explain in simple terms why a given sampling method may be unsatisfactory (knowledge of particular sampling methods, such as quota or stratified sampling, is not required, but candidates should have an elementary understanding of the use of random numbers in producing random samples);
• recognise that a sample mean can be regarded as a random variable, and use the facts that E(X) = μ and that Var(X)=σ2/n ;
• use the fact that X has a normal distribution if X has a normal distribution;
• use the Central Limit theorem where appropriate;
• calculate unbiased estimates of the population mean and variance from a sample, using either raw or summarised data (only a simple understanding of the term ‘unbiased’ is required);
• determine a confidence interval for a population mean in cases where the population is normally distributed with known variance or where a large sample is used;
• determine, from a large sample, an approximate confidence interval for a population proportion.
Topic 5: Hypothesis tests
• understand the nature of a hypothesis test, the difference between one-tail and two-tail tests, and the terms null hypothesis, alternative hypothesis, significance level, rejection region (or critical region), acceptance region and test statistic;
• formulate hypotheses and carry out a hypothesis test in the context of a single observation from a population which has a binomial or Poisson distribution, using either direct evaluation of probabilities or a normal approximation, as appropriate;
• formulate hypotheses and carry out a hypothesis test concerning the population mean in cases where the population is normally distributed with known variance or where a large sample is used;
• understand the terms Type I error and Type II error in relation to hypothesis tests;
• calculate the probabilities of making Type I and Type II errors in specific situations involving tests based on a normal distribution or direct evaluation of binomial or Poisson probabilities.
