User:Rehan nasir

Rana Rehan Nasir Ali Math Formulas Algebra (a+b)^2=a^2+ 2ab+b^2 (a-b)^2=a^2- 2ab+b^2 a^2- b^2= (a+b) (a-b) (a+b)^2=(a-b)^2+ 4ab (a-b)^2= (a+b)^2-4ab a^3+b^3= (a+b)(a^2-ab+b^2 ) a^3-b^3= (a-b)(a^2+ab+b^2 ) (x+a )(x+b)= x^2+(a+b)  x+ab (a+b+c )^2= a^2+ b^2+ c^2+2ab+2bc+2ca (a+b)^3= a^3+ 〖3a〗^2 b+3ab^2+b^3=a^3+3ab(a+b)+b^3 (a-b)^3= a^3- 〖3a〗^2 b+3ab^2-b^3=a^3-3ab(a-b)-b^3 (a+b+c)(a^2+b^2+c^2-ab-bc-ca)=a^3+b^3+c^3-3abc α+β= - b/a 		;	αβ=   c/a α^2+β^2=(α+β)^2-2αβ α^2-β^2=(α+β)(α-β)=[(α+β)((α+β)^2-4αβ) ] (α-β)^2=(α+β)^2-4αβ α^3+β^3=(α+β)^3-3αβ(α+β) √n/n=1/√n	(Note: Here n = n and is a real number.) Quadratic Fromula;  x= (-b±√(b^2-4ac))/2a ω=(-1+√3i)/2               ;  ω^2=(-1-√3i)/2 Laws of Exponents (Law of sum of powers) x^m.x^n=x^(m+n) (Law of Power of Product) (x∙y)^n=x^n∙y^n (Law of Power of Power) (X^m )^n=X^mn (Law of Quotient of Powers with Common Base) X^m/X^n =X^(m-n),m>n X^m/X^n =1/X^(n-m)     ,m<n X^m/X^n =X^(m-m)=X^0=1 ,m= n	Laws of Power of fraction ∀ x,y ∈R,y≠0,n∈N,   (x/y)^n= x^n/y^n Rational Exponent a^□(m/n)=(a^(1⁄n) )^m=(√(n&a))^m   or  a^□(m/n)=(a^m )^(1⁄n)=√(n&a^m ) a^□(-m/n)= √(n&a^(-m) )  ,     1/a^□(m/n) =1/√(n&a^m ) If x, y ∈ R and m, n, p, q ∈ N then x^□(m/n).x^□(p/q)=x^(□(m/n)+p/q) (x^□(m/n) )^□(p/q)=x^□(mp/nq) (x.y)^□(m/n)=x^□(m/n).y^□(m/n) (x/y)^□(m/n)=x^□(m/n)/y^□(m/n), y≠0 (x^□(m/n)/x^□(p/q) )=x^□(m/n-□(p/q)) , x≠0 x^□(-m/n)=1/x^□(m/n), x≠0 Radical √a=b ⇒a=b^2 ,  a, b ≥0 (√a)^2=√a .√a=a  ,  a ≥0 √ab=√a .√b √a/√a =1  , a >0 a/√a=√a ,    a >0 √a/a=1/√a, a > 0 √(a/b)=√a/√b   , a ≥ 0 ,  b >0 a√b+c√b=(a+c) √b, b ≥ 0 Nature of the roots of a quadratic equation If b^2-4ac=0 then the roots will be - b/2aand - b/2a So, the roots are real and repeated. If b^2-4ac<0 then√(b^2-4ac) will be imaginary. So, the roots are complex/imaginary and distinct/unequal. If b^2-4ac>0 then√(b^2-4ac) will be real. So, the roots are real and distinct / unequal. However; If b^2-4ac is a perfect square then√(b^2-4ac) will be rational, and so the roots are rational, else irrational. Result for Real numbers x, y and Natural number q	√(q&x)=x^(1⁄q) √(q&x) = y ⇒  x = y^q (√(q&x))^q=x √(q&xy)=√(q&x) .√(q&y) √(q&x)/√(q&x) =1,x≠0 √(q&x/y)=√(q&x)/√(q&y) y≠0 √(q&x^p )=x^(p⁄q) Percentage Percentage:             [ O/T  ×100]   Example     [945/1050  ×100=90%] Discount:                         [(T ×P)/100]   Example     [(1050 ×80)/100  =840] If we know that 80% of a certain value x is 840. To know 100% value of 840 following formula will be used: [T= 100/GP ×GVP]   Example     [100/80  ×840=1050] Where: O= Obtained Marks	T = Total Marks	P = Percentage GP = Given Value	GVP = Given Value of Percentage Matrices Inverse of a Matrix	A^(-1)=1/|A| adj A	Cofactor of an Element:          A_ij=(-1)^(i+j)  × M_ij Minor of a_ij = a_ij.A_ij = a_ij  (-1)^(i+J)  M_ij Geometry & Trigonometry: Circle: Area of circle:		A=πr^2 Diameter: Radios: Triangle: Pythagorean Thyreom =a^2+b^2=c^2

Formulas of Trigonometric Cos(α+β) =cos α cos β- sin α sin β cos(α+β) =cos α cos β- sin α sin β sin(α-β) =sin α cos β- cos α sin β sin(α+β) =sin α cos β+ cos α sin β tan(α+β) =tan α+ tan β/1-tanαtanβ tan(α-β) =tnaα-tanβ/1+tanαtanβ sin 2α =2sinα cos α cos 2α =cos²α-sin²α cos 2α =2cos²α-1 cos 2α =1-2sin²α tan2α =2tanα/1-tan²α cosα/2 =±√1+cosα/2 sinα/2 =±√1-cosα/2 2sinαcosβ =sin(α+β) +sin(α-β) 2cosαsibβ =sin(α+β) –sin(α-β) 2cosαcosβ =sin(α+β) –sin(α-β) -2sinαsinβ =cos(α+β) –cos(α-β) 2sinαsinβ =cos(α-β) –cos(α+β) sin P -sin Q =2cos P+Q/2 sin P-Q/2 cos P +cos Q=2cosP+Q/ sinP-Q/2 cos P-cos Q =2sin P+Q/2 cosP-Q/2 cos P –cos Q =2sin P+Q/2 sinP-Q/2 Sin (π/2-θ) =Cos θ,  cos (π/2-θ) =Sin θ, Tan (π/2-θ) =cotθ Sin (π/2+θ) =cos θ,  cos (π/2+θ) =-sin θ, Tan (π/2+θ) =-cotθ Sin (π-θ) =sinθ,       cos (π-θ) =-cos θ, Tan (π-θ) =-tanθ Sin (π+θ) =-sinθ,     cos (π+θ) =-cos θ, Tan (π+θ) =tanθ Sin (3π\2-θ) =-cos θ, cos (3π\2-θ) =-sin θ, Tan (3π\2-θ) =cot θ Sin (3π\2+θ) =-cos θ, cos (3π\2+θ) =sin θ, Tan (3π\2+θ) =-cotθ Sin (2π-θ) =-sin θ,     cos (2π-θ) =cos θ, Tan (2π-θ) =-tanθ Sin (2π+θ) =sin θ,     cos (2π+θ) =cos θ, Tan (2π+θ) =tan θ