Area : Triangles

1.Sum of the angles of a triangle is 180 degrees.
2.The sum of any two sides of a triangle is greater
than third side.
3.PYTHAGORAS Theorem:
In a right angled triangle (Hypotenuse)2 = (Base)2 +(Height)2
4.The line joining the mid point of a side of a triangle
to the opposite vertex is called the MEDIAN.
5.The point where the three medians of a triangle meet,
is called CENTROID. The centroid divides each of the
medians in the ratio 2:1
6.In an isosceles triangle, the altitude from the
vertex bisects the base
7.The median of a triangle divides it into two triangles
of the same area.
8.The area of the triangle formed by joining the mid points
of the sides of a given triangle is one-fourth of the area
of the given triangle.

ALLIGATION OR MIXTURES

Important Facts and Formula:
1.Allegation:It is the rule that enables us to find the
ratio in which two of more
ingredients at the given price must be
mixed to produce a mixture of a desired price.

2.Mean Price:The cost price of a unit quantity of the mixture
is called the mean price.
3.Rule of Allegation:If two ingredients are mixed then
Quantity of Cheaper / Quantity of Dearer =
(C.P of Dearer - Mean Price) /(Mean Price - C.P of Cheaper).

C.P of a unit quantity of cheaper(c)    C.P of unit quantity of dearer(d)


                              Mean Price(m)

            (d-m)                                   (m-c) 

Cheaper quantity:Dearer quantity = (d-m):(m-c)

4.Suppose a container contains x units of liquid from which y units
are taken out and replaced by water. After n operations the
quantity of pure liquid = x (1 - y/x)^n units. To make it little easy to learn :
Quantity of Pure Liquid after n operation = L (1 – W/L) ^ n

Cubes upto 30

Following are the cubes of first 30 numbers :

1	1
2	8
3	27
4	64
5	125
6	216
7	343
8	512
9	729
10	1000
11	1331
12	1728
13	2197
14	2744
15	3375
16	4096
17	4913
18	5832
19	6859
20	8000
21	9261
22
23
24
25
26
27
28
29
30	27000

Theory of Equation notes for CAT

Some Rules for finding property of roots
(1) If an equation (i:e f(x)=0 ) contains all positive co-efficients of any powers of x , it has no positive roots then.
eg: x^4+3x^2+2x+6=0 has no positive roots .

(2) For an equation , if all the even powers of x have some sign coefficients and all the odd powers of x have the opposite sign coefficients , then it has no negative roots .

(3)Summarising DESCARTES RULE OF SIGNS:
For an equation f(x)=0 , the maximum number of positive roots it can have is the number of sign changes in f(x) ; and the maximum number of negative roots it can have is the number of sign changes in f(-x) .
Hence the remaining are the minimum number of imaginary roots of the equation(Since we also know that the index of the maximum power of x is the number of roots of an equation.)

(4) Complex roots occur in pairs, hence if one of the roots of an equation is 2+3i , another has to be 2-3i and if there are three possible roots of the equation , we can conclude that the last root is real . This real roots could be found out by finding the sum of the roots of the equation and subtracting (2+3i)+(2-3i)=4 from that sum. (More about finding sum and products of roots next time )

Sum and product of roots
(1) For a cubic equation ax^3+bx^2+cx+d=o
sum of the roots = – b/a
sum of the product of the roots taken two at a time = c/a
product of the roots = -d/a

(2) For a biquadratic equation ax^4+bx^3+cx^2+dx+e = 0
sum of the roots = – b/a
sum of the product of the roots taken three at a time = c/a
sum of the product of the roots taken two at a time = -d/a
product of the roots = e/a

(3) If an equation f(x)= 0 has only odd powers of x and all these have the same sign coefficients or if f(x) = 0 has only odd powers of x and all these have the same sign
coefficients then the equation has no real roots in each case(except for x=0 in the second case.

(4) Besides Complex roots , even irrational roots occur in pairs. Hence if 2+root(3) is a root , then even 2-root(3) is a root .
(All these are very useful in finding number of positive , negative , real ,complex etc roots of an equation )

Vedic mathematics : Easy way of finding square of a number ending with 5

quick way to square numbers that end in 5 using the formula BY ONE MORE THAN THE ONE BEFORE.

• 75² = 5625 75² means 75 x 75.
  The answer is in two parts: 56 and 25.
  The last part is always 25.
  The first part is the first number, 7, multiplied by the number "one more", which is 8:
  so 7 x 8 = 56
• Similarly 85² = (8 * 9) 25 = 7225
  

Square of 2 digit number having same digit, AA

N     N^2
11   121
22   484
33   1089
44   1936
55   2916
66   4356
77   5776
88   7744
99   9801


Now suppose that number is AA
than AA = 10A+A
We know,
(a+b)^2 = a^2 + 2ab + b^2
AA ^ 2 = (10A +A) ^2 = 100*(A^2 ) + (A^2 ) + 2*10A*A= 121*(A^2 )
So all these numbers are divided by 121 :P

So if you want to find 99^2, you can do 121 * (9^2), though it may look tough this way.
=121 * 81
But for 22 ^2 = 121 * 4 = 484 , i.e. little easier

Squares upto 100

Number	Square
1	1
2	4
3	9
4	16
5	25
6	36
7	49
8	64
9	81
10	100
11	121
12	144
13	169
14	196
15	225
16	256
17	289
18	324
19	361
20	400
21	441
22	484
23	529
24	576
25	625
26	676
27	729
28	784
29	841
30	900
31	961
32	1024
33	1089
34	1156
35	1225
36	1296
37	1369
38	1444
39	1521
40	1600
41	1681
42	1764
43	1849
44	1936
45	2025
46	2116
47	2209
48	2304
49	2401
50	2500
51	2601
52	2704
53	2809
54	2916
55	3025
56	3136
57	3249
58	3364
59	3481
60	3600
61	3721
62	3844
63	3969
64	4096
65	4225
66	4356
67	4489
68	4624
69	4761
70	4900
71	5041
72	5184
73	5329
74	5476
75	5625
76	5776
77	5929
78	6084
79	6241
80	6400
81	6561
82	6724
83	6889
84	7056
85	7225
86	7396
87	7569
88	7744
89	7921
90	8100
91	8281
92	8464
93	8649
94	8836
95	9025
96	9216
97	9409
98	9604
99	9801
100	10000