Most of the students are don’t realize what is Universal Set U
is all about ,Let’s see an example problem which explains us more ….

Question:-

Out of forty students, 14 are taking English Composition and 29 are taking Chemistry. If five students are in both classes, how many students are in neither class? How many are in either class? What is the probability that a randomly-chosen student from this group is taking only the Chemistry class?

Answer:-

This online help with math gives the solution of the above problem.
Here the total number of students in class is what is Universal Set U ,which is given as 40 students

Here is the venn diagram template for the complete question

Two students are taking neither class.
There are 38 students in at least one of the classes.
There is a 24/40 = 0.6 = 60%  probability that a randomly-chosen student in this
group is taking Chemistry but not English.

Topic:-Trigonometry

Trigonometry is a branch of mathematics that deals with triangles, particularly those plane triangles in which one angle has 90 degrees (right triangles). Trigonometry deals with relationships between the sides and the angles of triangles and with the trigonometric functions, which describe those relationships.

This trigonometry help show us how to solve a problem using Tan formulas.

Question:-

 Solve 

  Cos2x=cosx

Answer:-

Lets write 

cos2x as 2 cos2x-1

So we have 

   2 cos2x-1=cosx

  Then subtract cosx on both sides

2cos2x-1-cosx=cosx-cosx

2cos2x-1-cosx= 0

Lets take cosx as y

So the equaton become

  2y2-y-1= 0

  (y-1)(2y+1)=0

  y-1=0 or 2y+1=0
                 -1
  So y = 1 or y= ------
                  2

Since y=cosx
                    -1
 Cosx=1   or  cosx= ----
                     2
We know that cos0=0 

So cosx=cos0

 x=0 ,

Similarly for the second equation 

 x=120 or 240

For more help on this ,Please reply me.

Topic : Solve for x

Problem : Solve (x+2)/(x-3) = 42/(x+7)

Solution :

Cross multiply,

(x + 2) (x + 7) = 42 (x – 3)

x + 2x + 7x + 14 = 42x -126

x + 9x + 14 = 42x – 126

   – 9x  – 14 = -9x – 14(adding -9x and -14 on both sides)

x = 33x – 140

x – 33x + 140 = 0

x – 28x - 5x + 140 = 0

x(x – 28) - 5(x + 28) = 0

(x-28)(x-5) = 0

x-28 = 0 or x-5=0

x = 28 or x = 5

Topic : Simultaneous Equation

Problem : Solve 3x -2y = 3 and x = y + 4

Solution :

3x -2y = 3 ——- (1)

x = y + 4 ——- (2)

Use equation (2) in (1) for substitution method

So, 3x -2y = 3

3(y+4) – 2y = 3

3y + 12 – 2y = 3

y + 12 = 3

Subtract 12 from both sides

y = – 9

Substitute y = -9 in equation (2)

x = y + 4

x = -9 + 4 

x = - 5