1. Algebra
Common Formulas:
1. (a+b)2=a2+2ab+b2
2. (a+b)3=a3+b3+3a2b+3ab2
3. (a2−b2)=(a+b)(a−b)
4. (a+b+c)2=a2+b2+c2+2ab+2bc+2ca
5. (a+b)2+(a−b)2=2(a2+b2)
6. (a+b)2−(a−b)2=4ab
7. (a−b)3=a3−b3−3a2b+3ab2
Solving Quadratic Equation: Let any quadratic equation be ax2+bx+c=0. Roots of the equation are given by, x=−b±√b2−4ac2a
If b2−4ac=0 then roots are real and equal.
If b2−4ac>0 then roots are distinct and real.
If b2−4ac<0 then roots are imaginary.
If α and β are two roots of the equations, then
Sum of roots: α+β=−ba
Product of roots: α×β=ca
Difference of roots: α−β=√b2−4aca
For example, Let x2+x+1=0 is a quadratic equation and we need to find the roots of the equation.
For the given equation, b2−4ac=1−(4×1×1)=1−4=−3<0
This means that the roots of the equation are imaginary.
Roots will be, x=−1±√−32=−1−√−32,−1+√−32
For
√−1=i, then roots will be,
x=−1±√−32=−1−√3i2,−1+√3i2
Binomial Expansion:
If we need to expend (1+x)n in powers of x where n is positive integer, we expand it binomially.
Expansion will be
(1+x)n=1+n1x+n2x2+.......+nixi+.....+xn
It can be written as
(1+x)n=Σnj=0(njC)(xj)
where
njC=n!(n−j)!j!,n!=n(n−1)(n−2).......3.2.1
and
0!=1.
The number of terms in the expansion of (1+x)n are (n+1)
Binomial expansion for any index, i.e. if n is not a positive integer.
(1+x)n=1+n1!x+n(n−1)2!x2+n(n−1)(n−2)3!x3+..........∞terms
If |x| << 1 then (1+x)n=1+nx i.e. we can ignore highest power of the expansion.
For example: Expand (1+x)−2.
(1+x)−2=1+−21!x+−2(−2−1)2!x2+−2(−2−1)(−2−2)3!x3+..........∞=1−2x−3x2−4x3+.......
Try Yourself:
Q1. Find the roots of the equations:
(a)
x2+2x+3=0(b) x2−2x−3=0
(c) x2+30x+1=0
(d) 2x2+x+1=0
Q2. Expand following:
(a) (1+x)7
(b) (1+x)−7
(c) (1+y)−1
(d) (1+z)−10
***Solutions of the above problems will be uploaded soon.....
2. Trigonometry
Relation between arc length, l, radius of the circle, r, and angle
θ subtended by the arc at the center,
l=rθ
Usefull Trigonometric Formulas for Right Angle Triangle
Let any right angle triangle with Right angle at B, as shown in the figure
click here to see large image.
1. sinA=ab
2. cosA=cb
3. tanA=sinAcosA=ac
4. cosecA=1sinA=ba
5. secA=1cosA=bc
6. cotA=1tanA=cosAsinA=ca
7. sin2A+cos2A=1
8. 1+tan2A=sec2A
9. 1+cot2A=cosec2A
Above formulas are valid only for Right angle triangle.
In general, for any triangle ABC where A, B and C are the angles, and a, b, and c are the sides opposite to angle A, B and C respectively.
1. sinAa=sinBb=sinCc
2. cosA=b2+c2−a22bc
Value of some trigonometric functions:
|
0
|
30
|
45
|
60
|
90
|
sin
|
0
|
12
|
1√2
|
√32
|
1
|
cos
|
1
|
√32
|
1√2
|
12
|
0
|
tan
|
0
|
1√3
|
1
|
√3
|
∞
|
Compound Formula:
1. sin(A±B)=sinAcosB±sinBcosA
2. cos(A±B)=cosAcosB∓sinAsinB
3. tan(A±B)=tanA±tanB1∓tanAtanB
4. sin2A=2sinAcosA
5. cos2A=cos2A−sin2A=1−2sin2A=2cos2A−1
6. tan2A=2tanA1−tan2A
In the first quadrant, all trigonometric functions have positive values. In second quadrant, sine and cosec are positive and all others are negative. In third quadrant, tan and cot are positive and all others are negative. In fourth quadrant, cos and sec are positive and all other are negative.
1. sin(−θ)=−sinθ
2. cos(−θ)=cosθ
3. tan(−θ)=−tanθ