Maths 2A

Q.No:11 SAQ complex numbers

Complex Numbers

complex number  is a number of the form a+ib , where  a  and  b  are real numbers and i is the Imaginary number, the square root of i is −1 .

In a complex number  z=a+biz=a+bi , aa is called the “real part” of  zz  and  bb  is called the “imaginary part.” If  b=0b=0 , the complex number is a real number; if  a=0a=0 , then the complex number is “purely imaginary.”

We can graph a complex number on the Cartetian plane using the horizontal axis as the real axis and the vertical axis as the imaginary axis. When we use the Cartesian plane this way, we call it the complex plane .

The complex number  a+bia+bi can be plotted as the ordered pair (a,b)(a,b) on the complex plane.

The absolute value or modulus of a complex number z=a+biz=a+bi can be interpreted as the distance of the point (a,b)(a,b) from the origin on a complex plane.

Using the Distance Formula,

|z|=|a+bi|=(a0)2+(b0)2−−−−−−−−−−−−−−−√=a2+b2−−−−−−√|z|=|a+bi|     =(a−0)2+(b−0)2     =a2+b2

 

 

 

Q.No:12 SAQ

 

 

Q 12 mathematics 2A

 

Q.No:13 & 14 SAQF

 

Important Concepts and Formulas – Permutations and Combination .Multiplication Theorem (Fundamental Principles of Counting)If an operation can be performed in mm different ways and following which a second operation can be performed in nn different ways, then the two operations in succession can be performed in m×nm×n different ways.

2. Addition Theorem (Fundamental Principles of Counting)

If an operation can be performed in mm different ways and a second independent operation can be performed in nn different ways, either of the two operations can be performed in (m+n)(m+n) ways.

3. Factorial

Let nn be a positive integer. Then nn factorial can be defined as
n!=n(n1)(n2)1n!=n(n−1)(n−2)⋯1

Examples

5!=5×4×3×2×1=120 3!=3×2×1=65!=5×4×3×2×1=120 3!=3×2×1=6

Special Cases

0!=1 1!=10!=1 1!=1

4. PermutationsPermutations are the different arrangements of a given number of things by taking some or all at a time.

Examples

All permutations (or arrangements) that can be formed with the letters a, b, c by taking three at a time are (abc, acb, bac, bca, cab, cba)

All permutations (or arrangements) that can be formed with the letters a, b, c by taking two at a time are (ab, ac, ba, bc, ca, cb)

5. CombinationsEach of the different groups or selections formed by taking some or all of a number of objects is called a combination.

Examples

Suppose we want to select two out of three girls P, Q, R. Then, possible combinations are PQ, QR and RP. (Note that PQ and QP represent the same selection.)

Suppose we want to select three out of three girls P, Q, R. Then, only possible combination is PQR

6. Difference between Permutations and Combinations and How to identify them

Sometimes, it will be clearly stated in the problem itself whether permutation or combination is to be used. However if it is not mentioned in the problem, we have to find out whether the question is related to permutation or combination.

Consider a situation where we need to find out the total number of possible samples of two objects which can be taken from three objects P, Q, R. To understand if the question is related to permutation or combination, we need to find out if the order is important or not.

If order is important, PQ will be different from QP, PR will be different from RP and QR will be different from RQ

If order is not important, PQ will be same as QP, PR will be same as RP and QR will be same as RQ

Hence,
If the order is important, problem will be related to permutations.
If the order is not important, problem will be related to combinations.

For permutations, the problems can be like “What is the number of permutations the can be made”, “What is the number of arrangements that can be made”, “What are the different number of ways in which something can be arranged”, etc.

For combinations, the problems can be like “What is the number of combinations the can be made”, “What is the number of selections the can be made”, “What are the different number of ways in which something can be selected”, etc.

pq and qp are two different permutations, but they represent the same combination.

Mostly problems related to word formation, number formation etc will be related to permutations. Similarly most problems related to selection of persons, formation of geometrical figures, distribution of items (there are exceptions for this) etc will be related to combinations.

7. Repetition

The term repetition is very important in permutations and combinations. Consider the same situation described above where we need to find out the total number of possible samples of two objects which can be taken from three objects P, Q, R.

If repetition is allowed, the same object can be taken more than once to make a sample. i.e., PP, QQ, RR can also be considered as possible samples.

If repetition is not allowed, then PP, QQ, RR cannot be considered as possible samples.

Normally repetition is not allowed unless mentioned specifically.

8. Number of permutations of n distinct things taking r at a time

Number of permutations of n distinct things taking r at a time can be given by

nPr = n!(nr)!n!(n−r)! =n(n1)(n2)...(nr+1)=n(n−1)(n−2)…(n−r+1) where 0rn0≤r≤n

Special Cases
nP0 = 1
nPr = 0 for r>nr>n

nPr is also denoted by P(n,r). nPr has importance outside combinatorics as well where it is known as the falling factorial and denoted by (n)r or nr

Examples

8P2 = 8 × 7 = 56
5P4= 5 × 4 × 3 × 2 = 120

9. Number of permutations of n distinct things taking all at a time

Number of permutations of n distinct things taking them all at a time
= nPn = n!

10. Number of Combinations of n distinct things taking r at a time

Number of combinations of n distinct things taking r at a time ( nCr) can be given by
nCr = n!(r!)(nr)!n!(r!)(n−r)! =n(n1)(n2)(nr+1)r!=n(n−1)(n−2)⋯(n−r+1)r! where 0rn0≤r≤n

Special Cases
nC0 = 1
nCr = 0 for r>nr>n

nCr is also denoted by C(n,r). nCr occurs in many other mathematical contexts as well where it is known as binomial coefficient and denoted by (nr)(nr)

Examples

8C2 = 8×72×18×72×1 = 28

5C4= 5×4×3×24×3×2×15×4×3×24×3×2×1 = 5

Factorial Notation:

  1. Let n be a positive integer. Then, factorial n, denoted n! is defined as:

    n! = n(n – 1)(n – 2) … 3.2.1.

    Examples:

    1. We define 0! = 1.
    2. 4! = (4 x 3 x 2 x 1) = 24.
    3. 5! = (5 x 4 x 3 x 2 x 1) = 120.
  2. Permutations:The different arrangements of a given number of things by taking some or all at a time, are called permutations.Examples:
    1. All permutations (or arrangements) made with the letters a, b, c by taking two at a time are (ab, ba, ac, ca, bc, cb).
    2. All permutations made with the letters a, b, c taking all at a time are:
      ( abc, acb, bac, bca, cab, cba)
  3. Number of Permutations:Number of all permutations of n things, taken rat a time, is given by:
    nPr = n(n – 1)(n – 2) … (nr + 1) = n!
    (nr)!

    Examples:

    1. 6P2 = (6 x 5) = 30.
    2. 7P3 = (7 x 6 x 5) = 210.
    3. Cor. number of all permutations of nthings, taken all at a time = n!.
  4. An Important Result:If there are n subjects of which p1 are alike of one kind; p2 are alike of another kind; p3 are alike of third kind and so on and pr are alike of rth kind,
    such that (p1 + p2 + … pr) = n.

    Then, number of permutations of these n objects is = n!
    (p1!).(p2)!…..(pr!)
  5. Combinations:Each of the different groups or selections which can be formed by taking some or all of a number of objects is called a combination.Examples:
    1. Suppose we want to select two out of three boys A, B, C. Then, possible selections are AB, BC and CA.Note: AB and BA represent the same selection.
    2. All the combinations formed by a, b, ctaking ab, bc, ca.
    3. The only combination that can be formed of three letters a, b, c taken all at a time is abc.
    4. Various groups of 2 out of four persons A, B, C, D are:

      AB, AC, AD, BC, BD, CD.

    5. Note that ab ba are two different permutations but they represent the same combination.
  6. Number of Combinations:The number of all combinations of n things, taken r at a time is:
    nCr = n! = n(n – 1)(n – 2) … to r factors .
    (r!)(nr)! r!

    Note:

    1. nCn = 1 and nC0 = 1.
    2. nCr = nC(n – r)

    Examples:

    i.   11C4 = (11 x 10 x 9 x 8) = 330.
    (4 x 3 x 2 x 1)
    ii.   16C13 = 16C(16 – 13) = 16C3 = 16 x 15 x 14 = 16 x 15 x 14 = 560

 

mathematics 2A Q 13 & 14

 

Q.No:15 SAQ

 

mathematic Q 15 2A

 

Q.No:16 & 17 SAQ

 

mathematics Q 16 & 17 2A

 

Q.No:18 LAQ

 

mathematics Q 18 2A

 

Q.No:19  LAQ

 

mathematics Q 19 2A

 

Q.No:20 &  21  LAQ

 

mathematics Q 20 & 21 2A

 

Q.No:22  LAQ

 

mathematics 2A Q.No 22

 

Q.No:23  LAQ

mathematics Q 23 2A

 

Q.No:24  LAQ

 

mathematics Q 24 2A

 

Probability

How likely something is to happen.

Many events can’t be predicted with total certainty. The best we can say is how likely they are to happen, using the idea of probability.

head tails coin

Tossing a Coin

When a coin is tossed, there are two possible outcomes:

  • heads (H) or
  • tails (T)

We say that the probability of the coin landing H is ½.

And the probability of the coin landing T is ½.

pair of dice

Throwing Dice

When a single die is thrown, there are six possible outcomes: 1, 2, 3, 4, 5, 6.

The probability of any one of them is 1/6.

Probability

In general:

Probability of an event happening

= Number of ways it can happen/Total number of outcomes

 

Example: the chances of rolling a “4” with a die

Number of ways it can happen: 1 (there is only 1 face with a “4” on it)

Total number of outcomes: 6 (there are 6 faces altogether)

So the probability = 1/6

Example: there are 5 marbles in a bag: 4 are blue, and 1 is red. What is the probability that a blue marble gets picked?

Number of ways it can happen: 4 (there are 4 blues)

Total number of outcomes: 5 (there are 5 marbles in total)

So the probability = 4/5 = 0.8

Probability Line

We can show probability on a  :Probability line

probability line

Probability is always between 0 and 1

Probability is Just a Guide

Probability does not tell us exactly what will happen, it is just a guide

Example: toss a coin 100 times, how many Heads will come up?

Probability says that heads have a ½ chance, so we can expect 50 Heads.

But when we actually try it we might get 48 heads, or 55 heads … or anything really, but in most cases it will be a number near 50.

Words

Some words have special meaning in Probability:

Experiment or Trial: an action where the result is uncertain.

Tossing a coin, throwing dice, seeing what pizza people choose are all examples of experiments.

Sample Space: all the possible outcomes of an experiment

Example: choosing a card from a deck

There are 52 cards in a deck (not including Jokers)

So the Sample Space is all 52 possible cards: {Ace of Hearts, 2 of Hearts, etc… }

The Sample Space is made up of Sample Points:

Sample Point: just one of the possible outcomes

Example: Deck of Cards

  • the 5 of Clubs is a sample point
  • the King of Hearts is a sample point

“King” is not a sample point. As there are 4 Kings that is 4 different sample points.

 

Event: a single result of an experiment

Example Events:

  • Getting a Tail when tossing a coin is an event
  • Rolling a “5” is an event.

An event can include one or more possible outcomes:

  • Choosing a “King” from a deck of cards (any of the 4 Kings) is an event
  • Rolling an “even number” (2, 4 or 6) is also an event

 

probability sample space The Sample Space is all possible outcomes.

A Sample Point is just one possible outcome.

And an Event can be one or more of the possible outcomes.

 

Hey, let’s use those words, so you get used to them:

pair of dice

Example: ramu wants to see how many times a “double” comes up when throwing 2 dice.

Each time ramu throws the 2 dice is an Experiment.

It is an Experiment because the result is uncertain.

 

The Event ramu is looking for is a “double”, where both dice have the same number. It is made up of these 6 Sample Points:

{1,1} {2,2} {3,3} {4,4} {5,5} and {6,6}

 

The Sample Space is all possible outcomes (36 Sample Points):

{1,1} {1,2} {1,3} {1,4} … {6,3} {6,4} {6,5} {6,6}

 

These are ramu, Results:

Experiment Is it a Double?
{3,4} No
{5,1} No
{2,2} Yes
{6,3} No

 

After 100 Experiments, ramu has 19 “double” Events … is that close to what you would expect?

Activity: An Experiment with a Die

You will need:

  • A single Die
Diesingle die

Interesting point

Many people think that one of these cubes is called “a dice”. But no!

The plural is dice, but the singular is die. (i.e. 1 die, 2 dice.)

The common die has six faces:

dice faces 1 to 6

We usually call the faces 1, 2, 3, 4, 5 and 6.

High, Low, and Most Likely

Before we start, let’s think about what might happen.

Question: If you roll a die:

  • 1. What is the least possible score?
  • 2. What is the greatest possible score?
  • 3. What do you think is the most likely score?

The first two questions are quite easy to answer:

  • 1. The least possible score must be 1
  • 2. The greatest possible score must be 6
  • 3. The most likely score is … ???

Are they all just as likely? Or will some happen more often?

Let us see which is most likely …

An experiment gives results.

When done again it may give different results!

So it is important to know when results are good quality, or just random.

Probability

On the page Probability you will find a formula:

Probability of an event happening

= Number of ways it can happen/Total number of outcomes

Example: Probability of a 2

We know there are 6 possible outcomes.

And there is only 1 way to get a 2.

So the probability of getting 2 is:

Probability of a 2 = 16

Doing that for each score gets us:

Score Probability
1 1/6
2 1/6
3 1/6
4 1/6
5 1/6
6 1/6
Total = 1

The sum of all the probabilities is 1

For any experiment:

The sum of the probabilities of all possible outcomes is always equal to 1