How to find first root of cubic equation or first zero of cubic polynomial?

Please find attached the proof and method with example.
Your views about usefulness or preference to other methods is most welcome.

How to write prime numbers up to a certain number?

(1) Suppose you want to write prime numbers  upto a= 100.
Take square root of a=100. It is 10. Take integer part of it if a is not a perfect square.
In other words find the largest perfect square b less than or equal to this number a.  Find square root of this perfect square, say c.
(2) Select the greatest prime number e less than or equal to this square root i.e. c here 10, so e is 7 since 10, 9 & 8 are composite. 
(3) Now start writing. In first step write 2 and 3.
(4) This method is based on the fact that every prime number which is greater than 3 can be expressed as 6K + 1.
(5) Divide a= 100 by 6.
(6) Take integer part of quotient,  i.e. answer of division say d.
(7) Write those many (d) multiples of 6 living some space on the left side and right of it one  below the other.
(8) Write natural numbers before and after the multiples of 6.
(9) According to me, multiples of 2 and 3 are the most occurring numbers so 2x3=6.
(10) Cancel all multiples of prime number to next 3 i.e. multiples of 5.
(11) After this step you are able to see some prime numbers.
(12) Cancel all the multiples of prime number next to 5 i.e. 7. Here you can use trick. Find answers of multiplication (products) of 7 with prime numbers next to 7. i.e. 7 x 11=77, 7 x 13 = 91. For this step you can use prime numbers obtained so far in previous step.
(13) Continue the process of cancellation till you reach e. In other words if c is a composite number then continue previous step till you get greatest prime number less than c. If c is a prime then continue till c.
(14) You are ready with the required prime numbers. 

10 CBSE Co-ordinate Geometry Ex. 7.1 Q. 3

10 CBSE Co-ordinate Geometry Ex. 7.1 sum ( question) no. 3. After getting the distances using distance formula. The sum is about how to decide whether the points are collinear. Finding square roots of 5, 212 & 265 by long division will take lot of time, so the alternative method which is the best short cut. 

You may even find by geometrical construction 9 CBSE by taking suitable scale (by dividing by a suitable perfect square say 25). However this method also is time-consuming and needs lot of practice to master it.

How to know if a given number is a perfect square or not?

For a perfect square following numbers should be last 2 right-most digits:

0	1	4	9	16	25
	21	24	29	36
	41	44	49	56
	61	64	69	76
	81	84	89	96

write squares of numbers

you can keep on writing squares of natural numbers from 1. Write 1 and to its right sq. of 1. Go to next line write next number i.e. 2 below first '1' written. Now add to 1 written in square column number written in natural number column. Add the latest number written in natural number column to the previous sum. Continue like this till you want.
algebra behind:
( a + 1) sq = a sq + 2a +1 = a sq + a + ( a + 1 )

Trigonometry

Trigonometry:
Old Hindustani Singer Adores His Collection Of Albums, Tunes.
The first letter of every word can be used to remember the 3 basic trigonometric ratios of sine, cosine & tangent.
Opposite side / Hypotenuse = Sin
Adjacent side / Hypotenuse = Cos
Opposite side / Adjacent side = Tan

Cotangent is complementary tangent. So reciprocal of Cot is Tan.
Reciprocal of Secant is Cosine.
Reciprocal of Cosec is Sine.
C & C do not go together.
so also S & S.

In right angled triangle,
(1) Side opposite to 90 deg. is longest and it is called Hypotenuse. In short hyp.
In triangle ABC, if /_ B = 90 deg, then
AB sq + BC sq = AC sq
(2) Note letter B gets repeated. angle B is right angle.
(3) On R.H.S. B disappears and remaining letters get connected.