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Jee Mains Formula of Maths
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    S  A   K  S   H   I COMPLEX NUMBERS AND DEMOIVRES THEOREM 1.General form of Complex numbers x + iy where x is Real part andy is Imaginary part.2.Sum of nthroot of unity is zero3.Product of nthroot of unity (–1)n–14.Cube roots of unity are 1, ω  , ω  25.1 + ω  + ω  2= 0, ω  3= 1,6.Arg principle value of θ is – π∠θ≤π 7.Arg of x + iy isfor every x > 0, y > 08.Arg of x – iy is for everyx > 0 , y > 09.Arg of –x + iy is for every x > 0, y > 010.Arg of –x – iy is for every x > 0, y > 011.12.13.14.where15.16.17.If three complex numbersZ1, Z2, Z3are collinear then18.Area of triangle formed by Z, IZ, Z + Ziis 19.Area of triangle formed by Z,  ω  Z, Z + ω  Z is20.Ifthensrcin, Z1, Z2forms an equilateral triangle21.If Z1, Z2, Z3forms an equilateral triangle and Z0is circum centerthen22.If Z1, Z2, Z3forms an equilateral triangle and Z0is circum centerthen23.Distance between two verticesZ1, Z2is24.= is a circle with radius p andcenter z025.Represents circleWith radius where α is nonreal complex and β is constant26.If (k  ≠ 1) represents circle with ends of diameter If k = 1 the locus of z represents a line or perpendicular bisector.27.then locus of z represents Ellipse and if it is less, then it represents hyperbola 28.A(z1),B(z2),C(z3), and θ is angle between AB, AC then29.ei θ = Cos θ + iSin θ = Cos θ , ei π = –1,30.(Cos θ + iSin θ )n= Cosn θ + iSinn θ 31.Cos θ +iSin θ =CiS θ ,Cis α . Cis β =Cis ( α + β ),32.If x=Cos θ +iSin θ then=Cos θ –iSin θ 33.If Σ Cos α = Σ Sin α = 0 Σ Cos2 α = Σ Sin2 α = 0 Σ Cos2n α = Σ Sin2n α = 0, Σ Cos2 α = Σ Sin2 α = 3/2 Σ Cos3 α = 3Cos( α + β + γ  ), Σ Sin3 α = 3Sin( α + β + γ  ) Σ Cos(2 α –  β – γ  ) = 3, Σ Sin(2 α – β – γ  ) = 0,34.a3+ b3+ c3– 3abc = (a + b + c)(a + b ω  + c ω  2) (a + b ω  2+ c ω  ) Quadratic Expressions 1.Standard form of Quadratic equation is ax2+ bx +c = 0Sum of roots =product of rootsdiscriminate = b2– 4acIf α , β are roots then Quadratic equation is x2–x( α + β ) + αβ = 02.If the roots of ax2+ bx + c = 0 are1,then a + b + c = 03.If the roots of ax2+ bx + c = 0 are in ratio m : n then mnb2= (m+ n)2ac4.If one root of ax2+ bx + c = 0 is square of the other then ac2+ a2c+ b3= 3abc5.If x > 0 then the least value ofis 26.If a1, a2,....., anare positive then the least value of  1xx + ca c,ab,a − nn 1x 2Sinnx ⇒ − = α nn 1x 2Cosnx ⇒ + = α 1 1x 2Cos x 2Sinx x ⇒ + = α ⇒ − = α 1x CisCis()Cis β= α +ββ   i2 e i,logi i2 π π= = i1 21 3 z z ABez z AC θ −=− 1 2 k z z < − 1 2 1 2 z z z z k,k z z − + − = > − 2 1 kz zk 1 ±± 12 zzk zz −=− 2 α −β zzzz0 + α+ α+β= 0 z z − 1 2 .zz − 2 2 21 2 3 1 2 2 3 3 1 ZZZZZZZZZ + + = + + 2 2 2 21 2 3 0 3 , + + =  Z Z Z Z  2 21 1 2 2 Z Z Z Z 0 − + = 2 3Z4 2 1Z2 1 12 23 3 0 z z 1z z 1z z 1     =        1 2 1 2 z z z z − ≥ − 1 2 1 2 z z z z ; + ≥ − 1 2 1 2 z z z z ; + ≤ + n1n n2 n(1i)(1i)2Cos4 +  π+ + − = n n n 1  n(1 3i) (1 3i) 2 Cos3 +  π+ + − = 2 2 x a b = + xaxai22 + −= − x a x aa ib i , a ib2 2 + −+ = + − 2 2 i,(1 i) 2i,(1 i) 2i = − + = − = − 1 i 1 ii 1, i,1 i 1 i + −= − =− + z Argz Arg  = − 1212 zz z Argz Arg Arg  − = 1212 z z z Argz Arg Arg  + = 1  ytanx − θ= −π+ 1  ytanx − θ= π− 1 ytanx − θ = − 1  ytanx − θ = 1 bztana − = 2 13i13i,22 − + − −ω= ω = www.sakshieducation.com AIEEE Mathematics Quick Review    S  A   K  S   H   I is n27.If a2+ b2+ c2= K then range of ab + bc + ca is8.If the two roots are negative, then a, b, c will have same sign9.If the two roots are positive, then the sign of a, c will have differ-ent sign of 'b'10.f(x) = 0 is a polynomial then the equation whose roots are recipro-cal of the roots of f(x) = 0 isincreased by 'K' isf(x – K), multiplied by K is f(x/K)11.For a, b, h ∈ R the roots of (a – x) (b – x) = h2 are real and unequal12.For a, b, c ∈ R the roots of (x –a) (x – b)+ (x – b) (x –c) + (x – c)(x – a) = 0 are real andunequal13.Three roots of a cubical equation are A.P, they are taken as a –d, a,a + d14.Four roots in A.P, a–3d, a–d, a+d, a+3d15.If three roots are in G.Pare taken as roots16.If four roots are in G.Pare taken as roots17.For ax3+ bx2+ cx + d = 0(i) Σα 2 β = ( αβ + βγ  + γα )( α + β + γ  ) –3 αβ γ  = s1s2– 3s3(ii)(iii)(iv)(v) Into eliminate second term roots arediminished by Binomial Theorem And Partial Fractions 1.Number of terms in the expansion(x + a)nis n + 12.Number of terms in the expansionis 3.In 4.Forindependent term is5.In above, the term containing xs is6.(1 + x)n– 1 is divisible by x and(1 + x)n– nx –1 is divisible by x2.7.Coefficient of xnin (x+1) (x+2)...(x+n)=n8.Coefficient of xn–1in (x+1) (x+2)....(x+n) is9.Coefficient of xn–2in above is10.If f(x) = (x + y)nthen sum of coefficients is equal to f(1) 11.Sum of coefficients of even terms is equal to12.Sum of coefficients of odd terms is equal to13.Ifare in A.P(n–2r)2=n + 214.For (x+y)n, if n is even then only onemiddle term that is term.15.For (x + y)n, if n is odd there are two mid-dle terms that isterm andterm.16.In the expansion (x + y)nif n is evengreatest coefficient is17.In the expansion (x + y)nif n is odd great-est coefficients areif n is odd 18.For expansion of (1+ x)n General notation19.Sum of binomial coefficients20.Sum of even binomial coefficients21.Sum of odd binomial coefficients MATRICES 1.Asquare matrix in which every element is equal to '0', except thoseof principal diagonal of matrix is called as diagonal matrix2.Asquare matrix is said to be a scalar matrix if all the elements inthe principal diagonal are equal and Other elements are zero's3.Adiagonal matrix Ain which all the elements in the principal diag-onal are 1 and the rest '0' is called unit matrix4.Asquare matrix Ais said to be Idem-potent matrix if A2= A,5.Asquare matrix Ais said to be Involu-ntary matrix if A2= I6.Asquare matrix Ais said to be Symm-etric matrix if A= ATAsquare matrix Ais said to be Skew symmetric matrix if A=-AT7.Asquare matrix Ais said to be Nilpotent matrix If their exists apositive integer n such that An= 0 'n' is the index of Nilpotentmatrix8.If 'A' is a given matrix, every square mat-rix can be expressed as asum of symme-tric and skew symmetric matrix whereSymmetric partunsymmetric part9.Asquare matrix 'A' is called an ortho-gonal matrix if AAT= I or AT= A-110.Asquare matrix 'A' is said to be a singular matrix if det A= 011.Asquare matrix 'A' is said to be non singular matrix if det A  ≠ 012.If 'A' is a square matrix then det A=det AT13.If AB = I = BAthen Aand B are called inverses of each other14.(A-1)-1= A, (AB)-1= B-1A-115.If Aand ATare invertible then (AT)-1= (A-1)T16.If Ais non singular of order 3, Ais invertible, then17.If if ad-bc ≠ 018.(A-1)-1=A, (AB)-1=B-1A-1, (AT)-1=(A-1)T(ABC)-1= C-1B-1 1 a b d b1A Ac d c aad bc −  −   = ⇒ =   −−    1  AdjAAdetA − = T A A2 += T A A2 += n 11 3 5 C C C .... 2  − + + + = n 1o 2 4 CCC....2  − + + + = no 1 2 n CCC........C2 + + + + = n n n0 o 1 1 r r CC,CC,CC = = = n nn 1 n 12 2 C , C − + nn2 C th n32 + th n12 + th n12   +   n n nr 1 r r 1 CCC − + ( ) ( ) f 1 f 12 + − ( ) ( ) f 1 f 12 − − ( )( )( ) nn1n13n224 + − + ( ) nn12 + nps1pq −++ np1pq ++ npq baxx   +     ( ) nr1r Tnr1xa,Tr +  − ++ = n r 1r 1 C + −− ( ) n1 2 r x x ... x + + + bna − n n 1 n 2 ax bx cx ............ 0 − − + + = 3 3 3 31 1 2 3 s3ss3s α +β + γ = − + 4 2 21 1 2 1 3 2 s 4s s 4s s 2s = − + + 4 4 4 α +β + γ  2 2 2 21 2 s2s α +β + γ = − 33 a a, ,ar,arr r a,a,arr1f0x   =     K,K2 −    ( ) 12n12n 1 1 1a a .... a ....a a a   + + + + + +     www.sakshieducation.comwww.sakshieducation.com    S  A   K  S   H   I A-1. If Ais a n  x n non- singular matrix, thena) A(AdjA)=|A|Ib) Adj A= |A| A-1c) (Adj A)-1= Adj (A-1)d) Adj AT= (Adj A)Te) Det (A-1) = ( Det A)-1f) |Adj A| = |A|n -1g) lAdj (Adj A) l= |A|(n - 1)2h) For any scalar 'k'Adj (kA) = k n -1Adj A19.If Aand B are two non-singular matrices of the same type then (i) Adj (AB) = (Adj B) (Adj A)(ii) |Adj (AB) | = |Adj A| |Adj B | = |Adj B| |Adj A|20.To determine rank and solution first con-vert matrix into Echolonformi.e.Echolon form of No of non zero rows=n= Rank of a matrixIf the system of equations AX=B is consistent if the coeff matrix Aand augmented matrix K are of same rank Let AX = B be a system of equations of 'n' unknowns and ranks of coeff matrix = r1and rank of augmented matrix = r2If r1 ≠ r2, then AX = B is inconsistant,i.e. it has no solutionIf r1 = r2= n then AX=B is consistant, it has unique solutionIf r1 = r2< n then AX=B is consistant and it has infinitely manynumber of solutions Random Variables-Distributions & Statistics 1.For probability distribution if  x =  xi with range (  x 1,  x 2,  x 3----) and P (  x=xi ) are their probabilities thenmean µ = Σ  xi P (  x -  xi )Variance = σ 2 =Σ  xi 2  p (  x =  xi ) - µ2 Standard deviation = 2.If n be positive integer p be a real number such that 0 ≤ P  ≤ 1 a ran-dom variable X with range (0,1,2,----n) is said to follows binomi-al distribution.For a Binomial distribution of (q+p)ni) probability of occurrence = pii) probability of non occurrence = qiii) p + q = 1iv) probability of 'x' successesv) Mean = µ = npvi) Variance = npqvii) Standard deviation = 3.If number of trials are large and probab-ility of success is verysmall then poisson distribution is used and given as4.i) If  x 1,  x 2,  x 3,.....xnare n values of variant x , then its Arithmetic Meanii) For individual series If Ais assumed average then A.Miii) For discrete frequency distribution:iv) Median =where l = Lower limit of Median classf = frequencyN = Σ f iC = Width of Median classF = Cumulative frequency of class just preceding to median class v) First or lower Quartile deviation where f = frequency of first quarfile classF = cumulative frequency of the class just preceding to first quar-tile classvi) upperQuartiledeviationvii) Modewherel = lower limit of modal class with maximum frequencyf 1= frequency preceding modal classf 2= frequency successive modal classf 3= frequency of modal classviii) Mode = 3Median - 2Meanix) Quartile deviation =x) coefficient of quartile deviation =xi) coefficient of Range= VECTORS 1.Asystem of vectors are said to be linearly independentif are exists scalars Such that 2.Any three non coplanar vectors are linea-rly independentAsystem of vectors are said to be linearly dependentif thereatleast one of  x i ≠0, i=1, 2, 3….nAnd determinant = 03.Any two collinear vectors, any three coplanar vectors are linearlydependent. Any set of vectors containing null vectors is linearlyindependent 4.If ABCDEF is regular hexagon with center 'G' then AB + AC + AD+ AE + AF = 3AD = 6AG.5.Vector equation of sphere with center at and radius aisor 6.are ends of diameter then equation of sphere7.If are unit vectors then unit vector along bisector of ∠  AOB is or8.Vector along internal angular bisector is9.If 'I' is in centre of ABC then, le ∆ a bba λ      ± +    ( )  a ba b +± +  a ba b ++ , a b ( )   ( ) . 0 r a r b − − = , a b 2 22 2. r r c c a − + = 22 r c a − = c 1 1 2 2 ...0 n n  x a x a x a + + + = 1 2 , ,..... n a a a 1 2 3 ........ 0 n  x x x x ⇒ = = = = 1 1 2 2 ...0 n n  x a x a x a + + + = 12 , .... . n  x x x 1 2 , ,..... n a a a  Range Maximum Minimum + 3 13 1 Q QQ Q −+ 31 2 Q Q − 11 2 .2 mm  f f  Z l C  f f f  −= +  − −   3 34.  N F Q l C  f    −  = +         1 4.  N F Q l C  f    −  = +       2  N F l C  f    −       + × ( ) i i where d x A = − i ii  f d  x A f  = + ∑∑ () i  x A x An −= + i  x xn = ∑ ( )  k  eP x k k  λ  λ  − = = npq ( )  n x xi x P x x nC q p − = = var iance 1 2 3 4A 0 x y z0 0 k l   =    1 2 3 4A 2 3 1 23 2 1 0   =    www.sakshieducation.comwww.sakshieducation.com    S  A   K  S   H   I 10.If 'S' is circum centre of ABC then, 11.If 'S' is circum centre, 'O' is orthocenter of ABCthen,12.If & if axes are rotated through an i) x - axis ii) y - axisiii) z - axisIf 'O' is circumcentre of ABC then(Consider equilateral)13.where i) is acuteii) is obtuseiii) two vectors are to each other.14.In a right angled ABC, if AB is the hypotenuse and AB = Pthen15.is equilateral triangle of side 'a' then 16.17.Vector equation. of a line passing through the point Awith P.V. and parallel to 'b' is18.Vector equation of a line passing throughis r =(1-t)a+tb19.Vector equation. of line passing through &to 20.Vector equation. of plane passing through a pt and- parallelto non-collinear vectors is .s,t ∈ R and also given as21.Vector equation. of a plane passing through three non-collinearPoints.isi.e == 22.Vector equation. of a plane passing through ptsandparallel to is23.Vector equation of plane, at distance p (p >0) from srcin and to is 24.Perpendicular distance from srcin to plane passing through a,b,c 25.Plane passing through a and parallel to b,c is [r - a, b - c] = and [rb c] = [abc]26.Vector equation of plane passing through A,B,C with position vec-tors a,b,c is [ r - a, b-a, c-a] =0 and r.[b×c + c×a+a×b] = abc27.Let, b be two vectors. Then i) The component of b on a is ii) The projection of b on a is 28.i) The component of b on a isii) the projection of b on a isiii) the projection of b on a vector perpe-ndicular to' a' in the planegenerated bya,b is29.If a,b are two nonzero vectors then30.If a,b are not parallel then a×b is perpendicular to both of the vec-tors a,b.31.If a,b are not parallel then a.b, a×b form a right handed system.32.If a,b are not parallel thenand hence 33.If a is any vector then a×a = 034.If a,b are two vectors then a×b = - b×a.35.a×b = -b×a is called anticommutative law. 36.If a,b are two nonzero vectors, then 37.If ABC is a triangle such that then the vector area of is and scalar area is38.If a,b,c are the position vectors of the vertices of a triangle, then thevector area of the triangle 39.If ABCD is a parallelogram and then the vector areaof ABCD is la×bl40.The length of the projection of b on a vector perpendicular to a in the plane generated by a,b is41.The perpendicular distance from a point Pto the line joining the points A,B is42.Torque: The torque or vector moment or moment vector M of aforce F about a point Pis defined as M = r×F where r is the vectorfrom the point Pto any point Aon the line of action Lof F.43.a,b,c are coplanar then [abc]=044.Volume of parallelopiped = [abc] with a, b, c as coterminus edges.45.The volume of the tetrahedron ABCD is 46.If a,b,c are three conterminous edges of a tetrahedron then the vol-ume of the tetrahedron =47.The four points A,B,C,D are coplanar if 48.The shortest distance between the skew lines r = a +s b and r = c+ td is49.If i,j,k are unit vectors then [i j k] = 150.If a,b,c are vectors then [a+b, b+c, c+a] = 2[abc] [ ] , a c b d b d  − −× 0  AB AC AD   =     [ ] 16 ab c ± 16  AB AC AD  ±      AP AB AB ×   a ba × ,  AB a BC b = =   ( ) 12 a b b c c a = × + × + × [ ] 12 a b × ( ) 12 a b ×  ABC  ∆ ,  AB a AC b = =    ( ) sin , a ba b   a b ×= ( ) sin . a b a b ab × = ( )  .cos ,  a ba ba b = ( ) 2 . b a aba − ) 2 . b a aa . b aa   . ba a  . ba 0 a  ≠ abcb c c a a b    × + × + ×   . r n p =  n r  ⊥ 0  AP ABC    =  ( C c (  B b ( )  A a , , r a b a c a  = − − −  ( ) 1  s t a sb sc − − + + ( ) ( ) r a s b a t c a = + − + − 0  AB AC AP   =  ( ) ( ) () ,,  A a B b C cr a bc rbc abc      − = =      r a sb tc = + + & b c  A a r a t b c = + × , b c r  ⊥ a ( ) ( ) ,  A a B b r a tb = + a ( ) ( ) ( ) 2 2 22 2 a i a j a k a × + × + × = ( ) ( ) ( ) 2 2 2 2 . . . ; ai a j ak a + + = 2 32 a − . . .  AB BC BC CA CA AB + + = .  AB BC  ABC  ∆ 2 . . .  AB BC BC CA CA AB P + + = le ∆ r  ⊥ . 0 90 ab  θ  = ⇒ = ° ⇒ . 0 90 180 ab  θ θ  < ⇒ ° < < ° ⇒ . 0 0 90 ab  θ θ  > ⇒ < < ° ⇒ 0 180 θ  ° ≤ ≤ ° .cos ab a b  θ  = le ∆ ( 3sin22 OA A OA OB OC  Σ = + + le ∆ ( ) ( ) ( ) ) 1 2 3 cos 90 sin 90 , a a a α α  + + + (  1 2 cos sin , a a α α  + ) ) 2 3 1 ,( cos sin a a a α α  + ( ) ( (  3 1 cos 90 ) sin 90 , a a α α  + + + 1 2 3 2 1 ( , cos sin , cos sin(90 ) a a a a a α α α α  + + − ( ) 1 2 3 , , a a a a = 2 OA OB OC OS  + + = le ∆ SA SB SC SO + + = le ∆ 0  BC IA CA IB AB IC  + + = www.sakshieducation.comwww.sakshieducation.com
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