Basic Mathematics

Quadratic Equation

Roots of ax2+bx+c=0 are x=b±b24ac2a
Sum of roots x1+x2=ba;
Product of roots x1x2=ca
For real roots, b24ac0
For imaginary roots, b24ac<0

Logarithm

log10N=x10x=NlogbN=logbalogaNlogb1=0,logaa=1logmn=logm+lognlogmn=logmlognlogmn=nlogmloge m=2.303log10 mlog2=0.3010log3=0.4771

Logarithmic Expansion

ln(1+x)=x12x2+13x3(|x|<1)

Arithmetic progression-AP

a,a+d,a+2d,a+3d,a+(n1)d, here d= common difference

Sum of n terms Sn=n2[2a+(n1)d], nth  term: an=a+(n1)d

(i) 1+2+3+4+5+n=n(n+1)2
(ii) 12+22+32++n2=n(n+1)(2n+1)6
(iii) 13+22+32++n3=[n(n+1)2]2

Binomial Theorem

(1+x)n=1+nx+n(n1)2x2+n(n1)(n2)6x3+
(1x)n=1nx+n(n1)2x2n(n1)(n2)6x3+(x2<1)

(1x)n=1nx1!+n(n+1)x22!+..(x2<1)
 If x<<1 then (1+x)n1+nx&(1x)n1nx

Exponential Expansion

ex=1+x+x22!+x33!+

Trigonometric Expansion (θ in radians)

sinθ=θθ33!+θ55!.cosθ=1θ22!+θ44!.tanθ=θ+θ33+2θ515.

Componendo and dividendo theorem

 If pq=ab then p+qpq=a+bab

Geometrical progression-GP

a,ar,ar2,ar3, here, r=common ratio, nth  term, an=arn1
Sum of n terms Sn=a(1rn)1r
Sum of terms S=a1r [where |r|<1 ]

Trigonometry

Pythagorean Theorem: In this right triangle, a2+b2=c2

2π radian =3601rad=57.3sinθ= perpendicular  hypotenuse cosθ= base  hypotenuse tanθ= perpendicular  base cotθ= base  perpendicular secθ= hypotenuse  base cosecθ= hypotenuse  perpendicular sinθ=aa2+b2cosθ=ba2+b2tanθ=abcosecθ=1sin2secθ=1cos2θcotθ=1tan2θsin2θ+cos2θ=11+tan2θ=sec2θ1+cot2θ=cosec2θ

 

sin(A±B)=sinAcosB±cosAsinBcos(A±B)=cosAcosBsinAsinBtan(A±B)=tanA±tanB1tanAtanBsin2A=2sinAcosAcos2A=cos2Asin2A=12sin2A=2cos2A1tan2A=2tanA1tan2Asin3α=3sinα4sin3αcos3α=4cos3α3cosα2sinAsinB=cos(AB)cos(A+B)2cosAcosB=cos(AB)+cos(A+B)2sinAcosB=sin(A+B)+sin(AB)

 

θ0(0)30(π/6)45(π/4)60(π/3)90(π/2)120(2π/3)135(3π/4)150(5π/6)180(π)270(3π/2)360(2π)sinθ01212321321212010cosθ13212120121232101tanθ01313311300

 

sin(90+θ)=cosθsin(180θ)=sinθsin(θ)=sinθsin(90θ)=cosθcos(90+θ)=sinθcos(180θ)=cosθcos(θ)=cosθcos(90θ)=sinθtan(90+θ)=cotθtan(180θ)=tanθtan(θ)=tanθtan(90θ)=cotθsin(180+θ)=sinθsin(270θ)=cosθsin(270+θ)=cosθsin(360θ)=sinθcos(180+θ)=cosθcos(270θ)=sinθcos(270+θ)=sinθcos(360θ)=cosθtan(180+θ)=tanθtan(270θ)=cotθtan(270+θ)=cotθtan(360θ)=tanθ

sine law

Trlangles
Angles are A,B,C
Opposite sides are a,b,c
Angles A+B+C=180
sinAa=sinBb=sinCc
c2=a2+b22abcosC
Exterior angle D=A+C

cosine law

cosA=b2+c2a22bc,cosB=c2+a2b22ca,cosC=a2+b2c22ab

For small θ

sinθθcosθ1tanθθsinθtanθ

Differentiation

y=xndydx=nxn1y=nxdydx=1x
y=sinxdydx=cosxy=cosxdydx=sinx
y=eαx+βdydx=αeαx+βy=uvdydx=udvdx+vdudx
y=f(g(x))dydx=df(g(x))dg(x)×d(g(x))dx
y=k( constant )dydx=0
y=uvdydx=vdudxudvdxv2

Integration

C= Arbitrary constant, k= constant
f(x)dx=g(x)+C
ddx(g(x))=f(x)
kf(x)dx=kf(x)dx
(u+v+w)dx=udx+vdx+wdx
exdx=ex+C
xndx=xn+1n+1+C,n1
1xdx=lnx+C
sinxdx=cosx+C
cosxdx=sinx+C
eαx+βdx=1αeαx+β+C
(αx+β)ndx=(αx+β)n+1α(n+1)+C

Definite Integration

baf(x)dx=|g(x)|ba=g(b)g(a)
Area under the curve y=f(x) from x=a to a=b is 
A=baf(x)dx

Maxima & Minima of a function y=f(x)

For maximum value dydx=0&d2ydx2=ve
For minimum value dydx=0&d2ydx2=+ve

Average of a varying quantity

 If y=f(x) then y=ˉy=x2x1ydxx2x1dx=x2x1ydxx2x1

Formulae For Determination Of Area

Area of a square =( side )2
Area of rectangle = length × breadth

Area of a triangle =12× base × height

Area of a trapezoid =12×( distance between parallel sides )×( sum of parallel sides )
Area enclosed by a circle =πr2(r= radius )
Surface area of a sphere =4πr2 ( r= radius )
Area of a parallelogram = base × height

Area of curved surface of cylinder =2πr, where r= radius and = length

Area of whole surface of cylinder =2πr(r+) where = length

Area of ellipse =πab, (a & b are semi-major and semi-minor axis respectively)

Surface area of a cube =6( side )2
Total surface area of a cone =πr2+πr
where πr=πrr2+h2= lateral area

Arc length s=r.θ
Area of sector =r2θ2
Plane angle, θ=sr radian

Solid angle, Ω=Ar2 steradian

Formulae for Determination of Volume

Volume of a rectangular slab= length × breadth × height
=lwh
Volume of a cube =( side )3
Volume of a sphere =43πr3 (r= radius)

Volume of a cylinder =πr2
( r= radius and = length)

Volume of a cone =13πr2h
( r= radius and h= height )

Key Points to Remember

To convert an angle from degree to radian, we have to multiply it by π180 and to convert an angle from radian to degree, we have to multiply it by 180π.

By help of differentiation, if y is given, we can find dydx and by help of integration, if dydx is given, we can find y.

The maximum and minimum values of the function
Acosθ+Bsinθ are A2+B2 and A2+B2 respectively. (a+b)2=a2+b2+2ab(ab)2=a2+b22ab(a+b)(ab)=a2b2(a+b)3=a3+b3+3ab(a+b)(ab)3=a3b33ab(ab)

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