IMMS MATHEMATICS TUTORIALS

  Perimeters

Factoring in Algebra Factors Numbers have factors: And expressions (like  x 2 +4x+3 ) also have factors: Factoring Factoring is the process of  finding the factors : Factoring: Finding what to multiply together to get an expression. It is like "splitting" an expression into a multiplication of simpler expressions. Example: factor  2y+6 Both 2y and 6 have a common factor of 2: 2y is 2×y 6 is 2×3 So we can factor the whole expression into: 2y+6 = 2(y+3) So  2y+6  has been "factored into"  2  and  y+3 Factoring is also the opposite of expanding: Common Factor In the previous example we saw that 2y and 6 had a common factor of  2 But to do the job properly we need the  highest common factor , including any variables Example: factor  3y 2 +12y Firstly, 3 and 12 have a common factor of  3 . So we could have: 3y 2 +12y = 3(y 2 +4y) But we can do better! 3y 2  and  12y  also share the variable  y . Together that mak...

  Sequences A sequence is a list of numbers in a specified order. The different numbers occurring in a sequence are called the terms of the sequence. Let the terms of a sequence be a 1 , a 2 , a 3 , …, a n , …, etc. The subscripts indicate the position of the term. That means, First term = a 1 Second term = a 2 Third term = a 3 …. The nth term is the number at the nth position of the sequence and is denoted by a n . This term is also called the general term of the sequence. Sequences Rules All the terms of a sequence possess a certain rule upon which the successive terms are defined. In the above examples, the sequence 3, 9, 27, 81…,59049 indicates the rule as the powers of 3. Here, each succeeding number will be obtained by multiplying the previous term with 3. Similarly, the set of even numbers also contains a rule that each successive term is obtained by adding 2 to the previous term. Based on these rules, we can define the general term of the sequences. This can be done as give...

  Functions A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. Let A & B be any two non-empty sets; mapping from A to B will be a function only when every element in set A has one end and only one image in set B. Example: Another definition of functions is that it is a relation “f” in which each element of set “A” is mapped with only one element belonging to set “B”. Also, in a function, there can’t be two pairs with the same first element. Condition for a Function Set  A  and Set  B  should be non-empty. In a function, a particular input is given to get a particular output. So, a function  f: A->B  denotes that f is a function from  A  to  B , where  A  is a domain, and  B  is a co-domain. For an element,  a , which belongs to  A , a  ∈ A,  a unique element  b , b  ∈ B  is there...

  Probability Formula for Probability The probability formula is defined as the possibility of an event to happen is equal to the ratio of the number of favorable outcomes and the total number of outcomes. Probability of event to happen P(E) = Number of favorable outcomes/Total Number of outcomes NB:  Sometimes Learners get mistaken for “favorable outcome” with “desirable outcome”. This is the basic formula. But there are some more formulas for different situations or events. Solved Examples 1) There are 6 pillows in a bed, 3 are red, 2 are yellow and 1 is blue. What is the probability of picking a yellow pillow? Answer: The probability is equal to the number of yellow pillows in the bed divided by the total number of pillows, i.e. 2/6 = 1/3. 2) There is a container full of colored bottles, red, blue, green and orange. Some of the bottles are picked out and displaced. Sumit did this 1000 times and got the following results: No. of blue bottles picked out: 300 No. of ...

  Logarithms logarithms  are the other way of writing the exponents. A logarithm of a number with a base is equal to another number. A logarithm is just the opposite function of exponentiation. For example, if 10 2  = 100 then log 10  100 = 2. Hence, we can conclude that,  Log b  x = n or   b n  = x Where b is the base of the logarithmic function. This can be read as “Logarithm of x to the base b is equal to n”. we are going to learn the definition of logarithms, two types of logarithms such as common logarithm and natural logarithm, and different properties of logarithms with many solved examples. “The logarithm of a positive real number a with respect to base b, a positive real number not equal to 1, is the exponent by which b must be raised to yield a”. e.g. b y = a  ⇔ log b a =y “a” and “b” are two positive real numbers y is a real number “a” is called argument, which is inside the log “b” is called the base, which is at the bottom o...