Page 1 I COMPLEX NUMBERS AND DEMOIVRES THEOREM 1. General form of Complex numbers x + iy where x is Real part and y is Imaginary part. 2. Sum of n th root of unity is zero 3. Product of n th root of unity (–1) n–1 4. Cube roots of unity are 1, ?, ? 2 5. 1 + ? + ? 2 = 0, ? 3 = 1, 6. Arg principle value of ? is –p??=p 7. Arg of x + iy is for every x > 0, y > 0 8. Arg of x – iy is for every x > 0 , y > 0 9. Arg of –x + iy is for every x > 0, y > 0 10. Arg of –x – iy is for every x > 0, y > 0 11. 12. 13. 14. where 15. 16. 17. If three complex numbers Z 1 , Z 2 , Z 3 are collinear then 18. Area of triangle formed by Z, IZ, Z + Zi is 19. Area of triangle formed by Z, ?Z, Z + ?Z is 20. If then origin, Z 1 , Z 2 forms an equilateral triangle 21. If Z 1 , Z 2 , Z 3 forms an equilateral triangle and Z 0 is circum center then 22. If Z 1 , Z 2 , Z 3 forms an equilateral triangle and Z 0 is circum center then 23. Distance between two vertices Z 1 , Z 2 is 24. = is a circle with radius p and center z 0 25. Represents circle With radius where a is nonreal complex and ß is const ant 26.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(z 1 ),B(z 2 ),C(z 3 ), and ? is angle between AB, AC then 29. e i? = Cos? + iSin? = Cos?, e ip = –1, 30. (Cos? + iSin?) n = Cosn? + iSinn? 31. Cos?+iSin?=CiS?, Cisa. Cisß=Cis (a+ ß), 32. If x=Cos?+iSin? then =Cos?–iSin? 33. If SCosa = SSina = 0 SCos2a = SSin2a = 0 SCos2 n a = SSin2 n a = 0, SCos 2 a = SSin 2 a = 3/2 SCos3a = 3Cos(a + ß + ?), SSin3a = 3Sin(a + ß + ?) SCos(2a – ß – ?) = 3, SSin(2a – ß – ?) = 0, 34. a 3 + b 3 + c 3 – 3abc = (a + b + c) (a + b? + c? 2 ) (a + b? 2 + c?) Quadratic Expressions 1. Standard form of Quadratic equation is ax 2 + bx +c = 0 Sum of roots = product of roots discriminate = b 2 – 4ac If a, ß are roots then Quadratic equation is x 2 –x(a + ß) + aß = 0 2. If the roots of ax 2 + bx + c = 0 are 1, then a + b + c = 0 3. If the roots of ax 2 + bx + c = 0 are in ratio m : n then mnb 2 = (m + n) 2 ac 4. If one root of ax 2 + bx + c = 0 is square of the other then ac 2 + a 2 c + b 3 = 3abc 5. If x > 0 then the least value of is 2 6. If a 1 , a 2 ,....., a n are positive then the least value of 1 x x + c a c , a b , a - n n 1 x2Sinn x ?- = a n n 1 x2Cosn x ?+ = a 11 x2Cos x 2Sin xx ?+ = a?- = a 1 x Cis Cis( ) Cis ß =a+ß ß i 2 ei,logi i 2 p p == i 12 13 zz AB e zz AC ? - = - 12 kz z <- 12 12 zz zz k,k z z -+ - = > - 21 kz z k1 ± ± 1 2 zz k zz - = - 2 a-ß zz z z 0 +a+ a+ß= 0 zz - 12 .z z - 22 2 12 3 12 23 31 ZZ Z ZZZZ ZZ ++ = + + 22 2 2 12 3 0 3, ++ = ZZ Z Z 22 112 2 ZZZ Z 0 -+ = 2 3 Z 4 2 1 Z 2 11 22 33 0 zz 1 zz 1 zz 1 ?? ?? = ?? ?? ?? 12 1 2 zz z z -= - 12 1 2 zz z z ; += - 12 1 2 zz z z ; += + n 1 nn 2 n (1 i) (1 i) 2 Cos 4 + p ++ - = nnn1 n (1 3i) (1 3i) 2 Cos 3 + p ++- = 22 xa b =+ xa x a i 22 +- =- xa x a aib i , a ib 22 +- += + - 22 i,(1 i) 2i,(1 i) 2i =- + = - =- 1i 1 i i1, i, 1i 1 i +- =- = -+ zArgz Arg =- 1 21 2 z zzArgz Arg Arg - = 12 1 2 zz z Argz Arg Arg + = 1 y tan x - ?= -p+ 1 y tan x - ?= p- 1 y tan x - ?= - 1 y tan x - ?= 1 b ztan a - = 2 13i 1 3i , 22 -+ -- ?= ? = Page 2 I COMPLEX NUMBERS AND DEMOIVRES THEOREM 1. General form of Complex numbers x + iy where x is Real part and y is Imaginary part. 2. Sum of n th root of unity is zero 3. Product of n th root of unity (–1) n–1 4. Cube roots of unity are 1, ?, ? 2 5. 1 + ? + ? 2 = 0, ? 3 = 1, 6. Arg principle value of ? is –p??=p 7. Arg of x + iy is for every x > 0, y > 0 8. Arg of x – iy is for every x > 0 , y > 0 9. Arg of –x + iy is for every x > 0, y > 0 10. Arg of –x – iy is for every x > 0, y > 0 11. 12. 13. 14. where 15. 16. 17. If three complex numbers Z 1 , Z 2 , Z 3 are collinear then 18. Area of triangle formed by Z, IZ, Z + Zi is 19. Area of triangle formed by Z, ?Z, Z + ?Z is 20. If then origin, Z 1 , Z 2 forms an equilateral triangle 21. If Z 1 , Z 2 , Z 3 forms an equilateral triangle and Z 0 is circum center then 22. If Z 1 , Z 2 , Z 3 forms an equilateral triangle and Z 0 is circum center then 23. Distance between two vertices Z 1 , Z 2 is 24. = is a circle with radius p and center z 0 25. Represents circle With radius where a is nonreal complex and ß is const ant 26.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(z 1 ),B(z 2 ),C(z 3 ), and ? is angle between AB, AC then 29. e i? = Cos? + iSin? = Cos?, e ip = –1, 30. (Cos? + iSin?) n = Cosn? + iSinn? 31. Cos?+iSin?=CiS?, Cisa. Cisß=Cis (a+ ß), 32. If x=Cos?+iSin? then =Cos?–iSin? 33. If SCosa = SSina = 0 SCos2a = SSin2a = 0 SCos2 n a = SSin2 n a = 0, SCos 2 a = SSin 2 a = 3/2 SCos3a = 3Cos(a + ß + ?), SSin3a = 3Sin(a + ß + ?) SCos(2a – ß – ?) = 3, SSin(2a – ß – ?) = 0, 34. a 3 + b 3 + c 3 – 3abc = (a + b + c) (a + b? + c? 2 ) (a + b? 2 + c?) Quadratic Expressions 1. Standard form of Quadratic equation is ax 2 + bx +c = 0 Sum of roots = product of roots discriminate = b 2 – 4ac If a, ß are roots then Quadratic equation is x 2 –x(a + ß) + aß = 0 2. If the roots of ax 2 + bx + c = 0 are 1, then a + b + c = 0 3. If the roots of ax 2 + bx + c = 0 are in ratio m : n then mnb 2 = (m + n) 2 ac 4. If one root of ax 2 + bx + c = 0 is square of the other then ac 2 + a 2 c + b 3 = 3abc 5. If x > 0 then the least value of is 2 6. If a 1 , a 2 ,....., a n are positive then the least value of 1 x x + c a c , a b , a - n n 1 x2Sinn x ?- = a n n 1 x2Cosn x ?+ = a 11 x2Cos x 2Sin xx ?+ = a?- = a 1 x Cis Cis( ) Cis ß =a+ß ß i 2 ei,logi i 2 p p == i 12 13 zz AB e zz AC ? - = - 12 kz z <- 12 12 zz zz k,k z z -+ - = > - 21 kz z k1 ± ± 1 2 zz k zz - = - 2 a-ß zz z z 0 +a+ a+ß= 0 zz - 12 .z z - 22 2 12 3 12 23 31 ZZ Z ZZZZ ZZ ++ = + + 22 2 2 12 3 0 3, ++ = ZZ Z Z 22 112 2 ZZZ Z 0 -+ = 2 3 Z 4 2 1 Z 2 11 22 33 0 zz 1 zz 1 zz 1 ?? ?? = ?? ?? ?? 12 1 2 zz z z -= - 12 1 2 zz z z ; += - 12 1 2 zz z z ; += + n 1 nn 2 n (1 i) (1 i) 2 Cos 4 + p ++ - = nnn1 n (1 3i) (1 3i) 2 Cos 3 + p ++- = 22 xa b =+ xa x a i 22 +- =- xa x a aib i , a ib 22 +- += + - 22 i,(1 i) 2i,(1 i) 2i =- + = - =- 1i 1 i i1, i, 1i 1 i +- =- = -+ zArgz Arg =- 1 21 2 z zzArgz Arg Arg - = 12 1 2 zz z Argz Arg Arg + = 1 y tan x - ?= -p+ 1 y tan x - ?= p- 1 y tan x - ?= - 1 y tan x - ?= 1 b ztan a - = 2 13i 1 3i , 22 -+ -- ?= ? = SAKSHI is n 2 7. If a 2 + b 2 + c 2 = K then range of ab + bc + ca is 8. If the two roots are negative, then a, b, c will have same sign 9. 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 is increased by 'K' is f(x – K), multiplied by K is f(x/K) 11. For a, b, h ? R the roots of (a – x) (b – x) = h 2 are real and unequal 12. For a, b, c ? R the roots of (x – a) (x – b) + (x – b) (x – c) + (x – c) (x – a) = 0 are real and unequal 13. Three roots of a cubical equation are A.P, they are taken as a – d, a, a + d 14. Four roots in A.P, a–3d, a–d, a+d, a+3d 15. If three roots are in G.P are taken as roots 16. If four roots are in G.P are taken as roots 17. For ax 3 + bx 2 + cx + d = 0 (i) Sa 2 ß = (aß + ß? + ?a) (a + ß + ?) –3aß ? = s 1 s 2 – 3s 3 (ii) (iii) (iv) (v) In to eliminate second term roots are diminished by Binomial Theorem And Partial Fractions 1. Number of terms in the expansion (x + a) n is n + 1 2. Number of terms in the expansion is 3. In 4. For independent term is 5. In above, the term containing x s is 6. (1 + x) n – 1 is divisible by x and (1 + x) n – nx –1 is divisible by x 2 . 7. Coefficient of x n in (x+1) (x+2)...(x+n)=n 8. Coefficient of x n–1 in (x+1) (x+2)....(x+n) is 9. Coefficient of x n–2 in above is 10. If f(x) = (x + y) n then sum of coefficients is equal to f(1) 11. Sum of coefficients of even terms is equal to 12. Sum of coefficients of odd terms is equal to 13. If are in A.P (n–2r) 2 =n + 2 14. For (x+y) n , if n is even then only one middle term that is term. 15. For (x + y) n , if n is odd there are two mid- dle terms that is term and term. 16. In the expansion (x + y) n if n is even greatest coefficient is 17. In the expansion (x + y) n if n is odd great- est coefficients are if n is odd 18.For expansion of (1+ x) n General notation 19. Sum of binomial coefficients 20. Sum of even binomial coefficients 21. Sum of odd binomial coefficients MATRICES 1. A square matrix in which every element is equal to '0', except those of principal diagonal of matrix is called as diagonal matrix 2. A square matrix is said to be a scalar matrix if all the elements in the principal diagonal are equal and Other elements are zero's 3. A diagonal matrix A in which all the elements in the principal diag- onal are 1 and the rest '0' is called unit matrix 4. A square matrix A is said to be Idem-potent matrix if A 2 = A, 5. A square matrix A is said to be Involu-ntary matrix if A 2 = I 6. A square matrix A is said to be Symm-etric matrix if A = A T A square matrix A is said to be Skew symmetric matrix if A=-A T 7. A square matrix A is said to be Nilpotent matrix If their exists a positive integer n such that A n = 0 'n' is the index of Nilpotent matrix 8. If 'A' is a given matrix, every square mat-rix can be expressed as a sum of symme-tric and skew symmetric matrix where Symmetric part unsymmetric part 9. A square matrix 'A' is called an ortho-gonal matrix if AA T = I or A T = A -1 10. A square matrix 'A' is said to be a singular matrix if det A = 0 11. A square matrix 'A' is said to be non singular matrix if det A ? 0 12. If 'A' is a square matrix then det A=det A T 13. If AB = I = BA then A and B are called inverses of each other 14. (A -1 ) -1 = A, (AB) -1 = B -1 A -1 15. If A and A T are invertible then (A T ) -1 = (A -1 ) T 16. If A is non singular of order 3, A is invertible, then 17. If if ad-bc ? 0 18. (A -1 ) -1 =A, (AB) -1 =B -1 A -1 , (A T ) -1 =(A -1 ) T (ABC) -1 = C -1 B -1 1 ab d b 1 AA cd c a ad bc - - ?? ? ? =? = ?? ? ? - - ?? ? ? 1 AdjA A det A - = T AA 2 + = T AA 2 + = n1 13 5 C C C .... 2 - +++ = n1 o2 4 C C C .... 2 - +++ = n o1 2 n C C C ........ C 2 ++ + + = nn n 0o 1 1 r r CC,C C,C C == = nn n1 n1 22 C,C -+ n n 2 C th n3 2 + th n1 2 + th n 1 2 ?? + ?? ?? nn n r1 r r1 CC C -+ () ( ) f1 f 1 2 +- () ( ) f1 f 1 2 -- ()( )( ) nn 1 n 1 3n 2 24 +- + () nn 1 2 + np s 1 pq - + + np 1 pq + + n p q b ax x ?? + ?? ?? () n r1 r Tnr1 xa , Tr + -+ += nr 1 r1 C +- - () n 12 r x x ... x ++ + b na - nn1 n2 ax bx cx ............ 0 -- ++ = 33 3 3 112 3 s3ss 3s a+ß +? = - + 42 2 112 13 2 s4ss 4ss 2s =- + + 44 4 a+ß +? 22 2 2 12 s2s a+ß +? = - 3 3 aa ,,ar,ar rr a ,a,ar r 1 f0 x ?? = ?? ?? K ,K 2 -?? ?? ?? () 12 n 12 n 11 1 a a .... a .... aa a ?? ++ + + + + ?? ?? Page 3 I COMPLEX NUMBERS AND DEMOIVRES THEOREM 1. General form of Complex numbers x + iy where x is Real part and y is Imaginary part. 2. Sum of n th root of unity is zero 3. Product of n th root of unity (–1) n–1 4. Cube roots of unity are 1, ?, ? 2 5. 1 + ? + ? 2 = 0, ? 3 = 1, 6. Arg principle value of ? is –p??=p 7. Arg of x + iy is for every x > 0, y > 0 8. Arg of x – iy is for every x > 0 , y > 0 9. Arg of –x + iy is for every x > 0, y > 0 10. Arg of –x – iy is for every x > 0, y > 0 11. 12. 13. 14. where 15. 16. 17. If three complex numbers Z 1 , Z 2 , Z 3 are collinear then 18. Area of triangle formed by Z, IZ, Z + Zi is 19. Area of triangle formed by Z, ?Z, Z + ?Z is 20. If then origin, Z 1 , Z 2 forms an equilateral triangle 21. If Z 1 , Z 2 , Z 3 forms an equilateral triangle and Z 0 is circum center then 22. If Z 1 , Z 2 , Z 3 forms an equilateral triangle and Z 0 is circum center then 23. Distance between two vertices Z 1 , Z 2 is 24. = is a circle with radius p and center z 0 25. Represents circle With radius where a is nonreal complex and ß is const ant 26.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(z 1 ),B(z 2 ),C(z 3 ), and ? is angle between AB, AC then 29. e i? = Cos? + iSin? = Cos?, e ip = –1, 30. (Cos? + iSin?) n = Cosn? + iSinn? 31. Cos?+iSin?=CiS?, Cisa. Cisß=Cis (a+ ß), 32. If x=Cos?+iSin? then =Cos?–iSin? 33. If SCosa = SSina = 0 SCos2a = SSin2a = 0 SCos2 n a = SSin2 n a = 0, SCos 2 a = SSin 2 a = 3/2 SCos3a = 3Cos(a + ß + ?), SSin3a = 3Sin(a + ß + ?) SCos(2a – ß – ?) = 3, SSin(2a – ß – ?) = 0, 34. a 3 + b 3 + c 3 – 3abc = (a + b + c) (a + b? + c? 2 ) (a + b? 2 + c?) Quadratic Expressions 1. Standard form of Quadratic equation is ax 2 + bx +c = 0 Sum of roots = product of roots discriminate = b 2 – 4ac If a, ß are roots then Quadratic equation is x 2 –x(a + ß) + aß = 0 2. If the roots of ax 2 + bx + c = 0 are 1, then a + b + c = 0 3. If the roots of ax 2 + bx + c = 0 are in ratio m : n then mnb 2 = (m + n) 2 ac 4. If one root of ax 2 + bx + c = 0 is square of the other then ac 2 + a 2 c + b 3 = 3abc 5. If x > 0 then the least value of is 2 6. If a 1 , a 2 ,....., a n are positive then the least value of 1 x x + c a c , a b , a - n n 1 x2Sinn x ?- = a n n 1 x2Cosn x ?+ = a 11 x2Cos x 2Sin xx ?+ = a?- = a 1 x Cis Cis( ) Cis ß =a+ß ß i 2 ei,logi i 2 p p == i 12 13 zz AB e zz AC ? - = - 12 kz z <- 12 12 zz zz k,k z z -+ - = > - 21 kz z k1 ± ± 1 2 zz k zz - = - 2 a-ß zz z z 0 +a+ a+ß= 0 zz - 12 .z z - 22 2 12 3 12 23 31 ZZ Z ZZZZ ZZ ++ = + + 22 2 2 12 3 0 3, ++ = ZZ Z Z 22 112 2 ZZZ Z 0 -+ = 2 3 Z 4 2 1 Z 2 11 22 33 0 zz 1 zz 1 zz 1 ?? ?? = ?? ?? ?? 12 1 2 zz z z -= - 12 1 2 zz z z ; += - 12 1 2 zz z z ; += + n 1 nn 2 n (1 i) (1 i) 2 Cos 4 + p ++ - = nnn1 n (1 3i) (1 3i) 2 Cos 3 + p ++- = 22 xa b =+ xa x a i 22 +- =- xa x a aib i , a ib 22 +- += + - 22 i,(1 i) 2i,(1 i) 2i =- + = - =- 1i 1 i i1, i, 1i 1 i +- =- = -+ zArgz Arg =- 1 21 2 z zzArgz Arg Arg - = 12 1 2 zz z Argz Arg Arg + = 1 y tan x - ?= -p+ 1 y tan x - ?= p- 1 y tan x - ?= - 1 y tan x - ?= 1 b ztan a - = 2 13i 1 3i , 22 -+ -- ?= ? = SAKSHI is n 2 7. If a 2 + b 2 + c 2 = K then range of ab + bc + ca is 8. If the two roots are negative, then a, b, c will have same sign 9. 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 is increased by 'K' is f(x – K), multiplied by K is f(x/K) 11. For a, b, h ? R the roots of (a – x) (b – x) = h 2 are real and unequal 12. For a, b, c ? R the roots of (x – a) (x – b) + (x – b) (x – c) + (x – c) (x – a) = 0 are real and unequal 13. Three roots of a cubical equation are A.P, they are taken as a – d, a, a + d 14. Four roots in A.P, a–3d, a–d, a+d, a+3d 15. If three roots are in G.P are taken as roots 16. If four roots are in G.P are taken as roots 17. For ax 3 + bx 2 + cx + d = 0 (i) Sa 2 ß = (aß + ß? + ?a) (a + ß + ?) –3aß ? = s 1 s 2 – 3s 3 (ii) (iii) (iv) (v) In to eliminate second term roots are diminished by Binomial Theorem And Partial Fractions 1. Number of terms in the expansion (x + a) n is n + 1 2. Number of terms in the expansion is 3. In 4. For independent term is 5. In above, the term containing x s is 6. (1 + x) n – 1 is divisible by x and (1 + x) n – nx –1 is divisible by x 2 . 7. Coefficient of x n in (x+1) (x+2)...(x+n)=n 8. Coefficient of x n–1 in (x+1) (x+2)....(x+n) is 9. Coefficient of x n–2 in above is 10. If f(x) = (x + y) n then sum of coefficients is equal to f(1) 11. Sum of coefficients of even terms is equal to 12. Sum of coefficients of odd terms is equal to 13. If are in A.P (n–2r) 2 =n + 2 14. For (x+y) n , if n is even then only one middle term that is term. 15. For (x + y) n , if n is odd there are two mid- dle terms that is term and term. 16. In the expansion (x + y) n if n is even greatest coefficient is 17. In the expansion (x + y) n if n is odd great- est coefficients are if n is odd 18.For expansion of (1+ x) n General notation 19. Sum of binomial coefficients 20. Sum of even binomial coefficients 21. Sum of odd binomial coefficients MATRICES 1. A square matrix in which every element is equal to '0', except those of principal diagonal of matrix is called as diagonal matrix 2. A square matrix is said to be a scalar matrix if all the elements in the principal diagonal are equal and Other elements are zero's 3. A diagonal matrix A in which all the elements in the principal diag- onal are 1 and the rest '0' is called unit matrix 4. A square matrix A is said to be Idem-potent matrix if A 2 = A, 5. A square matrix A is said to be Involu-ntary matrix if A 2 = I 6. A square matrix A is said to be Symm-etric matrix if A = A T A square matrix A is said to be Skew symmetric matrix if A=-A T 7. A square matrix A is said to be Nilpotent matrix If their exists a positive integer n such that A n = 0 'n' is the index of Nilpotent matrix 8. If 'A' is a given matrix, every square mat-rix can be expressed as a sum of symme-tric and skew symmetric matrix where Symmetric part unsymmetric part 9. A square matrix 'A' is called an ortho-gonal matrix if AA T = I or A T = A -1 10. A square matrix 'A' is said to be a singular matrix if det A = 0 11. A square matrix 'A' is said to be non singular matrix if det A ? 0 12. If 'A' is a square matrix then det A=det A T 13. If AB = I = BA then A and B are called inverses of each other 14. (A -1 ) -1 = A, (AB) -1 = B -1 A -1 15. If A and A T are invertible then (A T ) -1 = (A -1 ) T 16. If A is non singular of order 3, A is invertible, then 17. If if ad-bc ? 0 18. (A -1 ) -1 =A, (AB) -1 =B -1 A -1 , (A T ) -1 =(A -1 ) T (ABC) -1 = C -1 B -1 1 ab d b 1 AA cd c a ad bc - - ?? ? ? =? = ?? ? ? - - ?? ? ? 1 AdjA A det A - = T AA 2 + = T AA 2 + = n1 13 5 C C C .... 2 - +++ = n1 o2 4 C C C .... 2 - +++ = n o1 2 n C C C ........ C 2 ++ + + = nn n 0o 1 1 r r CC,C C,C C == = nn n1 n1 22 C,C -+ n n 2 C th n3 2 + th n1 2 + th n 1 2 ?? + ?? ?? nn n r1 r r1 CC C -+ () ( ) f1 f 1 2 +- () ( ) f1 f 1 2 -- ()( )( ) nn 1 n 1 3n 2 24 +- + () nn 1 2 + np s 1 pq - + + np 1 pq + + n p q b ax x ?? + ?? ?? () n r1 r Tnr1 xa , Tr + -+ += nr 1 r1 C +- - () n 12 r x x ... x ++ + b na - nn1 n2 ax bx cx ............ 0 -- ++ = 33 3 3 112 3 s3ss 3s a+ß +? = - + 42 2 112 13 2 s4ss 4ss 2s =- + + 44 4 a+ß +? 22 2 2 12 s2s a+ß +? = - 3 3 aa ,,ar,ar rr a ,a,ar r 1 f0 x ?? = ?? ?? K ,K 2 -?? ?? ?? () 12 n 12 n 11 1 a a .... a .... aa a ?? ++ + + + + ?? ?? SAKSHI A -1 . If A is a n x n non- singular matrix, then a) A(AdjA)=|A|I b) Adj A = |A| A -1 c) (Adj A) -1 = Adj (A -1 ) d) Adj A T = (Adj A) T e) Det (A -1 ) = ( Det A) -1 f) |Adj A| = |A| n -1 g) lAdj (Adj A ) l= |A| (n - 1)2 h) For any scalar 'k' Adj (kA) = k n -1 Adj A 19. If A and 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 Echolon form i.e. Echolon form of No of non zero rows=n= Rank of a matrix If the system of equations AX=B is consistent if the coeff matrix A and augmented matrix K are of same rank Let AX = B be a system of equations of 'n' unknowns and ranks of coeff matrix = r 1 and rank of augmented matrix = r 2 If r 1 ? r 2 , then AX = B is inconsistant, i.e. it has no solution If r 1 = r 2 = n then AX=B is consistant, it has unique solution If r 1 = r 2 < n then AX=B is consistant and it has infinitely many number of solutions Random Variables- Distributions & Statistics 1. For probability distribution if x=x i with range (x 1 , x 2 , x 3 ----) and P(x=x i ) are their probabilities then mean µ= Sx i P(x-x i ) Variance =s 2 =Sx i 2 p(x=x i ) -µ 2 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) n i) probability of occurrence = p ii) probability of non occurrence = q iii) p + q = 1 iv) probability of 'x' successes v) Mean = µ = np vi) Variance = npq vii) Standard deviation = 3. If number of trials are large and probab-ility of success is very small then poisson distribution is used and given as 4. i) If x 1 ,x 2 ,x 3 ,.....x n are n values of variant x , then its Arithmetic Mean ii) For individual series If A is assumed average then A.M iii) For discrete frequency distribution: iv) Median = where l = Lower limit of Median class f = frequency N = Sf i C = Width of Median class F = Cumulative frequency of class just preceding to median class v) First or lower Quartile deviation where f = frequency of first quarfile class F = cumulative frequency of the class just preceding to first quar- tile class vi) upperQuartiledeviation vii) Mode where l = lower limit of modal class with maximum frequency f 1 = frequency preceding modal class f 2 = frequency successive modal class f 3 = frequency of modal class viii) Mode = 3Median - 2Mean ix) Quartile deviation = x) coefficient of quartile deviation = xi) coefficient of Range = VECTORS 1. A system of vectors are said to be linearly independent if are exists scalars Such that 2. Any three non coplanar vectors are linea-rly independent A system of vectors are said to be linearly dependent if there atleast one of x i ?0, i=1, 2, 3….n And determinant = 0 3. Any two collinear vectors, any three coplanar vectors are linearly dependent. Any set of vectors containing null vectors is linearly independent 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 a is or 6. are ends of diameter then equation of sphere 7. If are unit vectors then unit vector along bisector of ? AOB is or 8. Vector along internal angular bisector is 9. If 'I' is in centre of ABC then, le ? ab b a ? ?? ?? ±+ ?? ?? () ab ab + ±+ ab ab + + , ab ()() .0 ra r b --= , ab 22 2 2. rrcc a -+ = () 2 2 rc a -= c 11 2 2 ... 0 nn xa x a x a +++ = 12 , ,..... n aa a 12 3 ........ 0 n xx x x ?= = = = 11 2 2 ... 0 nn xa xa xa +++ = 12 , .... . n xx x 12 , ,..... n aa a Range Maximum Minimum + 31 31 QQ QQ - + 31 2 QQ - 1 12 . 2 m m ff Zl C ff f ?? - =+ ?? -- ?? 3 3 4 . N F Ql C f ?? - ?? =+?? ?? ?? ?? 1 4 . N F Ql C f ?? - ?? =+?? ?? ?? ?? 2 N F lC f ?? - ?? ?? +× () ii where d x A =- ii i fd xA f =+ ? ? () i xA xA n - =+ ? i x x n = ? () k e Px k k ? ? - == npq () nx x ix Px x nC q p - == variance 12 3 4 A0 x y z 00 k l ?? ?? = ?? ?? ?? 1 234 A2 3 1 2 32 1 0 ?? ?? = ?? ?? ?? Page 4 I COMPLEX NUMBERS AND DEMOIVRES THEOREM 1. General form of Complex numbers x + iy where x is Real part and y is Imaginary part. 2. Sum of n th root of unity is zero 3. Product of n th root of unity (–1) n–1 4. Cube roots of unity are 1, ?, ? 2 5. 1 + ? + ? 2 = 0, ? 3 = 1, 6. Arg principle value of ? is –p??=p 7. Arg of x + iy is for every x > 0, y > 0 8. Arg of x – iy is for every x > 0 , y > 0 9. Arg of –x + iy is for every x > 0, y > 0 10. Arg of –x – iy is for every x > 0, y > 0 11. 12. 13. 14. where 15. 16. 17. If three complex numbers Z 1 , Z 2 , Z 3 are collinear then 18. Area of triangle formed by Z, IZ, Z + Zi is 19. Area of triangle formed by Z, ?Z, Z + ?Z is 20. If then origin, Z 1 , Z 2 forms an equilateral triangle 21. If Z 1 , Z 2 , Z 3 forms an equilateral triangle and Z 0 is circum center then 22. If Z 1 , Z 2 , Z 3 forms an equilateral triangle and Z 0 is circum center then 23. Distance between two vertices Z 1 , Z 2 is 24. = is a circle with radius p and center z 0 25. Represents circle With radius where a is nonreal complex and ß is const ant 26.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(z 1 ),B(z 2 ),C(z 3 ), and ? is angle between AB, AC then 29. e i? = Cos? + iSin? = Cos?, e ip = –1, 30. (Cos? + iSin?) n = Cosn? + iSinn? 31. Cos?+iSin?=CiS?, Cisa. Cisß=Cis (a+ ß), 32. If x=Cos?+iSin? then =Cos?–iSin? 33. If SCosa = SSina = 0 SCos2a = SSin2a = 0 SCos2 n a = SSin2 n a = 0, SCos 2 a = SSin 2 a = 3/2 SCos3a = 3Cos(a + ß + ?), SSin3a = 3Sin(a + ß + ?) SCos(2a – ß – ?) = 3, SSin(2a – ß – ?) = 0, 34. a 3 + b 3 + c 3 – 3abc = (a + b + c) (a + b? + c? 2 ) (a + b? 2 + c?) Quadratic Expressions 1. Standard form of Quadratic equation is ax 2 + bx +c = 0 Sum of roots = product of roots discriminate = b 2 – 4ac If a, ß are roots then Quadratic equation is x 2 –x(a + ß) + aß = 0 2. If the roots of ax 2 + bx + c = 0 are 1, then a + b + c = 0 3. If the roots of ax 2 + bx + c = 0 are in ratio m : n then mnb 2 = (m + n) 2 ac 4. If one root of ax 2 + bx + c = 0 is square of the other then ac 2 + a 2 c + b 3 = 3abc 5. If x > 0 then the least value of is 2 6. If a 1 , a 2 ,....., a n are positive then the least value of 1 x x + c a c , a b , a - n n 1 x2Sinn x ?- = a n n 1 x2Cosn x ?+ = a 11 x2Cos x 2Sin xx ?+ = a?- = a 1 x Cis Cis( ) Cis ß =a+ß ß i 2 ei,logi i 2 p p == i 12 13 zz AB e zz AC ? - = - 12 kz z <- 12 12 zz zz k,k z z -+ - = > - 21 kz z k1 ± ± 1 2 zz k zz - = - 2 a-ß zz z z 0 +a+ a+ß= 0 zz - 12 .z z - 22 2 12 3 12 23 31 ZZ Z ZZZZ ZZ ++ = + + 22 2 2 12 3 0 3, ++ = ZZ Z Z 22 112 2 ZZZ Z 0 -+ = 2 3 Z 4 2 1 Z 2 11 22 33 0 zz 1 zz 1 zz 1 ?? ?? = ?? ?? ?? 12 1 2 zz z z -= - 12 1 2 zz z z ; += - 12 1 2 zz z z ; += + n 1 nn 2 n (1 i) (1 i) 2 Cos 4 + p ++ - = nnn1 n (1 3i) (1 3i) 2 Cos 3 + p ++- = 22 xa b =+ xa x a i 22 +- =- xa x a aib i , a ib 22 +- += + - 22 i,(1 i) 2i,(1 i) 2i =- + = - =- 1i 1 i i1, i, 1i 1 i +- =- = -+ zArgz Arg =- 1 21 2 z zzArgz Arg Arg - = 12 1 2 zz z Argz Arg Arg + = 1 y tan x - ?= -p+ 1 y tan x - ?= p- 1 y tan x - ?= - 1 y tan x - ?= 1 b ztan a - = 2 13i 1 3i , 22 -+ -- ?= ? = SAKSHI is n 2 7. If a 2 + b 2 + c 2 = K then range of ab + bc + ca is 8. If the two roots are negative, then a, b, c will have same sign 9. 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 is increased by 'K' is f(x – K), multiplied by K is f(x/K) 11. For a, b, h ? R the roots of (a – x) (b – x) = h 2 are real and unequal 12. For a, b, c ? R the roots of (x – a) (x – b) + (x – b) (x – c) + (x – c) (x – a) = 0 are real and unequal 13. Three roots of a cubical equation are A.P, they are taken as a – d, a, a + d 14. Four roots in A.P, a–3d, a–d, a+d, a+3d 15. If three roots are in G.P are taken as roots 16. If four roots are in G.P are taken as roots 17. For ax 3 + bx 2 + cx + d = 0 (i) Sa 2 ß = (aß + ß? + ?a) (a + ß + ?) –3aß ? = s 1 s 2 – 3s 3 (ii) (iii) (iv) (v) In to eliminate second term roots are diminished by Binomial Theorem And Partial Fractions 1. Number of terms in the expansion (x + a) n is n + 1 2. Number of terms in the expansion is 3. In 4. For independent term is 5. In above, the term containing x s is 6. (1 + x) n – 1 is divisible by x and (1 + x) n – nx –1 is divisible by x 2 . 7. Coefficient of x n in (x+1) (x+2)...(x+n)=n 8. Coefficient of x n–1 in (x+1) (x+2)....(x+n) is 9. Coefficient of x n–2 in above is 10. If f(x) = (x + y) n then sum of coefficients is equal to f(1) 11. Sum of coefficients of even terms is equal to 12. Sum of coefficients of odd terms is equal to 13. If are in A.P (n–2r) 2 =n + 2 14. For (x+y) n , if n is even then only one middle term that is term. 15. For (x + y) n , if n is odd there are two mid- dle terms that is term and term. 16. In the expansion (x + y) n if n is even greatest coefficient is 17. In the expansion (x + y) n if n is odd great- est coefficients are if n is odd 18.For expansion of (1+ x) n General notation 19. Sum of binomial coefficients 20. Sum of even binomial coefficients 21. Sum of odd binomial coefficients MATRICES 1. A square matrix in which every element is equal to '0', except those of principal diagonal of matrix is called as diagonal matrix 2. A square matrix is said to be a scalar matrix if all the elements in the principal diagonal are equal and Other elements are zero's 3. A diagonal matrix A in which all the elements in the principal diag- onal are 1 and the rest '0' is called unit matrix 4. A square matrix A is said to be Idem-potent matrix if A 2 = A, 5. A square matrix A is said to be Involu-ntary matrix if A 2 = I 6. A square matrix A is said to be Symm-etric matrix if A = A T A square matrix A is said to be Skew symmetric matrix if A=-A T 7. A square matrix A is said to be Nilpotent matrix If their exists a positive integer n such that A n = 0 'n' is the index of Nilpotent matrix 8. If 'A' is a given matrix, every square mat-rix can be expressed as a sum of symme-tric and skew symmetric matrix where Symmetric part unsymmetric part 9. A square matrix 'A' is called an ortho-gonal matrix if AA T = I or A T = A -1 10. A square matrix 'A' is said to be a singular matrix if det A = 0 11. A square matrix 'A' is said to be non singular matrix if det A ? 0 12. If 'A' is a square matrix then det A=det A T 13. If AB = I = BA then A and B are called inverses of each other 14. (A -1 ) -1 = A, (AB) -1 = B -1 A -1 15. If A and A T are invertible then (A T ) -1 = (A -1 ) T 16. If A is non singular of order 3, A is invertible, then 17. If if ad-bc ? 0 18. (A -1 ) -1 =A, (AB) -1 =B -1 A -1 , (A T ) -1 =(A -1 ) T (ABC) -1 = C -1 B -1 1 ab d b 1 AA cd c a ad bc - - ?? ? ? =? = ?? ? ? - - ?? ? ? 1 AdjA A det A - = T AA 2 + = T AA 2 + = n1 13 5 C C C .... 2 - +++ = n1 o2 4 C C C .... 2 - +++ = n o1 2 n C C C ........ C 2 ++ + + = nn n 0o 1 1 r r CC,C C,C C == = nn n1 n1 22 C,C -+ n n 2 C th n3 2 + th n1 2 + th n 1 2 ?? + ?? ?? nn n r1 r r1 CC C -+ () ( ) f1 f 1 2 +- () ( ) f1 f 1 2 -- ()( )( ) nn 1 n 1 3n 2 24 +- + () nn 1 2 + np s 1 pq - + + np 1 pq + + n p q b ax x ?? + ?? ?? () n r1 r Tnr1 xa , Tr + -+ += nr 1 r1 C +- - () n 12 r x x ... x ++ + b na - nn1 n2 ax bx cx ............ 0 -- ++ = 33 3 3 112 3 s3ss 3s a+ß +? = - + 42 2 112 13 2 s4ss 4ss 2s =- + + 44 4 a+ß +? 22 2 2 12 s2s a+ß +? = - 3 3 aa ,,ar,ar rr a ,a,ar r 1 f0 x ?? = ?? ?? K ,K 2 -?? ?? ?? () 12 n 12 n 11 1 a a .... a .... aa a ?? ++ + + + + ?? ?? SAKSHI A -1 . If A is a n x n non- singular matrix, then a) A(AdjA)=|A|I b) Adj A = |A| A -1 c) (Adj A) -1 = Adj (A -1 ) d) Adj A T = (Adj A) T e) Det (A -1 ) = ( Det A) -1 f) |Adj A| = |A| n -1 g) lAdj (Adj A ) l= |A| (n - 1)2 h) For any scalar 'k' Adj (kA) = k n -1 Adj A 19. If A and 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 Echolon form i.e. Echolon form of No of non zero rows=n= Rank of a matrix If the system of equations AX=B is consistent if the coeff matrix A and augmented matrix K are of same rank Let AX = B be a system of equations of 'n' unknowns and ranks of coeff matrix = r 1 and rank of augmented matrix = r 2 If r 1 ? r 2 , then AX = B is inconsistant, i.e. it has no solution If r 1 = r 2 = n then AX=B is consistant, it has unique solution If r 1 = r 2 < n then AX=B is consistant and it has infinitely many number of solutions Random Variables- Distributions & Statistics 1. For probability distribution if x=x i with range (x 1 , x 2 , x 3 ----) and P(x=x i ) are their probabilities then mean µ= Sx i P(x-x i ) Variance =s 2 =Sx i 2 p(x=x i ) -µ 2 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) n i) probability of occurrence = p ii) probability of non occurrence = q iii) p + q = 1 iv) probability of 'x' successes v) Mean = µ = np vi) Variance = npq vii) Standard deviation = 3. If number of trials are large and probab-ility of success is very small then poisson distribution is used and given as 4. i) If x 1 ,x 2 ,x 3 ,.....x n are n values of variant x , then its Arithmetic Mean ii) For individual series If A is assumed average then A.M iii) For discrete frequency distribution: iv) Median = where l = Lower limit of Median class f = frequency N = Sf i C = Width of Median class F = Cumulative frequency of class just preceding to median class v) First or lower Quartile deviation where f = frequency of first quarfile class F = cumulative frequency of the class just preceding to first quar- tile class vi) upperQuartiledeviation vii) Mode where l = lower limit of modal class with maximum frequency f 1 = frequency preceding modal class f 2 = frequency successive modal class f 3 = frequency of modal class viii) Mode = 3Median - 2Mean ix) Quartile deviation = x) coefficient of quartile deviation = xi) coefficient of Range = VECTORS 1. A system of vectors are said to be linearly independent if are exists scalars Such that 2. Any three non coplanar vectors are linea-rly independent A system of vectors are said to be linearly dependent if there atleast one of x i ?0, i=1, 2, 3….n And determinant = 0 3. Any two collinear vectors, any three coplanar vectors are linearly dependent. Any set of vectors containing null vectors is linearly independent 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 a is or 6. are ends of diameter then equation of sphere 7. If are unit vectors then unit vector along bisector of ? AOB is or 8. Vector along internal angular bisector is 9. If 'I' is in centre of ABC then, le ? ab b a ? ?? ?? ±+ ?? ?? () ab ab + ±+ ab ab + + , ab ()() .0 ra r b --= , ab 22 2 2. rrcc a -+ = () 2 2 rc a -= c 11 2 2 ... 0 nn xa x a x a +++ = 12 , ,..... n aa a 12 3 ........ 0 n xx x x ?= = = = 11 2 2 ... 0 nn xa xa xa +++ = 12 , .... . n xx x 12 , ,..... n aa a Range Maximum Minimum + 31 31 QQ QQ - + 31 2 QQ - 1 12 . 2 m m ff Zl C ff f ?? - =+ ?? -- ?? 3 3 4 . N F Ql C f ?? - ?? =+?? ?? ?? ?? 1 4 . N F Ql C f ?? - ?? =+?? ?? ?? ?? 2 N F lC f ?? - ?? ?? +× () ii where d x A =- ii i fd xA f =+ ? ? () i xA xA n - =+ ? i x x n = ? () k e Px k k ? ? - == npq () nx x ix Px x nC q p - == variance 12 3 4 A0 x y z 00 k l ?? ?? = ?? ?? ?? 1 234 A2 3 1 2 32 1 0 ?? ?? = ?? ?? ?? SAKSHI 10. If 'S' is circum centre of ABC then, 11.If 'S' is circum centre, 'O' is orthocenter of ABC then, 12. If & if axes are rotated through an i) x - axis ii) y - axis iii) z - axis If 'O' is circumcentre of ABC then (Consider equilateral ) 13. where i) is acute ii) is obtuse iii) two vectors are to each other. 14. In a right angled ABC, if AB is the hypotenuse and AB = P then 15. is equilateral triangle of side 'a' then 16. 17. Vector equation. of a line passing through the point A with P.V . and parallel to 'b' is 18. Vector equation of a line passing through is r =(1-t)a +tb 19. Vector equation. of line passing through & to 20. Vector equation. of plane passing through a pt and- parallel to non-collinear vectors is . s,t?R and also given as 21. Vector equation. of a plane passing through three non-collinear Points. is i.e = = 22. Vector equation. of a plane passing through pts and parallel to is 23. Vector equation of plane, at distance p (p >0) from origin and to is 24. Perpendicular distance from origin to plane passing through a,b,c 25. Plane passing through a and parallel to b,c is [r - a, b - c] = and [r b 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] = abc 27. 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 is ii) the projection of b on a is iii) the projection of b on a vector perpe-ndicular to' a' in the plane generated by a,b is 29. If a,b are two nonzero vectors then 30. 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 then and hence 33. If a is any vector then a×a = 0 34. 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 is 38. If a,b,c are the position vectors of the vertices of a triangle, then the vector area of the triangle 39. If ABCD is a parallelogram and then the vector area of ABCD is la×bl 40. The length of the projection of b on a vector perpendicular to a in the plane generated by a,b is 41. The perpendicular distance from a point P to the line joining the points A,B is 42. Torque: The torque or vector moment or moment vector M of a force F about a point P is defined as M = r×F where r is the vector from the point P to any point A on the line of action L of F. 43. a,b,c are coplanar then [abc]=0 44. 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 is 49. If i,j,k are unit vectors then [i j k] = 1 50. If a,b,c are vectors then [a+b, b+c, c+a] = 2[abc] [] , acbd bd -- × 0 AB AC AD ?? = ?? [] 1 6 ab c ± 1 6 AB AC AD ?? ± ?? AP AB AB × ab a × , AB a BC b == () 1 2 ab bc c a =×+×+× [] 1 2 ab × () 1 2 ab × ABC ? , AB a AC b == () sin , ab ab ab × = () sin . ab ab ab ×= () . cos , ab ab ab = () 2 . ba a b a - () 2 . ba a a . ba a () . ba a . ba 0 a ? abc bc ca a b ?? ?? ?? ×+ × + × ?? . rn p = n r ? 0 AP ABC ?? = ?? () Cc () Bb () Aa ,, rab ac a ?? =- - - ?? () 1 sta sb sc -- + + () () ra sb a tc a =+ - + - 0 AB AC AP ?? = ?? () () () ,, Aa B b C c r a bc rbc abc ????? ? -= = ????? ? ra sb tc =+ + & bc () Aa () ra tb c =+ × , bc r ? a () () , Aa B b ra tb =+ a ()()() 22 2 2 2 ai a j a k a ×+ × + × = () () () 22 2 2 .. . ; ai a j ak a ++ = 2 3 2 a - .. . AB BC BC CA CA AB ++= . AB BC ABC ? 2 .. . AB BC BC CA CA AB P ++= le ? r ? .0 90 ab ? =? = °? . 0 90 180 ab ?? <? °< < °? .0 0 90 ab ?? >? < < °? 0 180 ? °= = ° .cos ab a b ? = le ?() 3 sin 2 2 OA A OA OB OC S= ++ le ? () ( ) ()) 12 3 , cos 90 sin 90 , aa a aa ++ + (12 cos sin , aa aa + )) 23 1 , ,( cos sin aa a aa + () ( (31 cos 90 ) sin 90 , aa a aa ++ + 12 3 2 1 ( , cos sin , cos sin(90 ) aa a a a aa a a ++ - () 12 3 ,, aaaa = 2 OA OB OC OS ++ = le ? SA SB SC SO ++ = le ? 0 BC IA CA IB AB IC ++ = Page 5 I COMPLEX NUMBERS AND DEMOIVRES THEOREM 1. General form of Complex numbers x + iy where x is Real part and y is Imaginary part. 2. Sum of n th root of unity is zero 3. Product of n th root of unity (–1) n–1 4. Cube roots of unity are 1, ?, ? 2 5. 1 + ? + ? 2 = 0, ? 3 = 1, 6. Arg principle value of ? is –p??=p 7. Arg of x + iy is for every x > 0, y > 0 8. Arg of x – iy is for every x > 0 , y > 0 9. Arg of –x + iy is for every x > 0, y > 0 10. Arg of –x – iy is for every x > 0, y > 0 11. 12. 13. 14. where 15. 16. 17. If three complex numbers Z 1 , Z 2 , Z 3 are collinear then 18. Area of triangle formed by Z, IZ, Z + Zi is 19. Area of triangle formed by Z, ?Z, Z + ?Z is 20. If then origin, Z 1 , Z 2 forms an equilateral triangle 21. If Z 1 , Z 2 , Z 3 forms an equilateral triangle and Z 0 is circum center then 22. If Z 1 , Z 2 , Z 3 forms an equilateral triangle and Z 0 is circum center then 23. Distance between two vertices Z 1 , Z 2 is 24. = is a circle with radius p and center z 0 25. Represents circle With radius where a is nonreal complex and ß is const ant 26.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(z 1 ),B(z 2 ),C(z 3 ), and ? is angle between AB, AC then 29. e i? = Cos? + iSin? = Cos?, e ip = –1, 30. (Cos? + iSin?) n = Cosn? + iSinn? 31. Cos?+iSin?=CiS?, Cisa. Cisß=Cis (a+ ß), 32. If x=Cos?+iSin? then =Cos?–iSin? 33. If SCosa = SSina = 0 SCos2a = SSin2a = 0 SCos2 n a = SSin2 n a = 0, SCos 2 a = SSin 2 a = 3/2 SCos3a = 3Cos(a + ß + ?), SSin3a = 3Sin(a + ß + ?) SCos(2a – ß – ?) = 3, SSin(2a – ß – ?) = 0, 34. a 3 + b 3 + c 3 – 3abc = (a + b + c) (a + b? + c? 2 ) (a + b? 2 + c?) Quadratic Expressions 1. Standard form of Quadratic equation is ax 2 + bx +c = 0 Sum of roots = product of roots discriminate = b 2 – 4ac If a, ß are roots then Quadratic equation is x 2 –x(a + ß) + aß = 0 2. If the roots of ax 2 + bx + c = 0 are 1, then a + b + c = 0 3. If the roots of ax 2 + bx + c = 0 are in ratio m : n then mnb 2 = (m + n) 2 ac 4. If one root of ax 2 + bx + c = 0 is square of the other then ac 2 + a 2 c + b 3 = 3abc 5. If x > 0 then the least value of is 2 6. If a 1 , a 2 ,....., a n are positive then the least value of 1 x x + c a c , a b , a - n n 1 x2Sinn x ?- = a n n 1 x2Cosn x ?+ = a 11 x2Cos x 2Sin xx ?+ = a?- = a 1 x Cis Cis( ) Cis ß =a+ß ß i 2 ei,logi i 2 p p == i 12 13 zz AB e zz AC ? - = - 12 kz z <- 12 12 zz zz k,k z z -+ - = > - 21 kz z k1 ± ± 1 2 zz k zz - = - 2 a-ß zz z z 0 +a+ a+ß= 0 zz - 12 .z z - 22 2 12 3 12 23 31 ZZ Z ZZZZ ZZ ++ = + + 22 2 2 12 3 0 3, ++ = ZZ Z Z 22 112 2 ZZZ Z 0 -+ = 2 3 Z 4 2 1 Z 2 11 22 33 0 zz 1 zz 1 zz 1 ?? ?? = ?? ?? ?? 12 1 2 zz z z -= - 12 1 2 zz z z ; += - 12 1 2 zz z z ; += + n 1 nn 2 n (1 i) (1 i) 2 Cos 4 + p ++ - = nnn1 n (1 3i) (1 3i) 2 Cos 3 + p ++- = 22 xa b =+ xa x a i 22 +- =- xa x a aib i , a ib 22 +- += + - 22 i,(1 i) 2i,(1 i) 2i =- + = - =- 1i 1 i i1, i, 1i 1 i +- =- = -+ zArgz Arg =- 1 21 2 z zzArgz Arg Arg - = 12 1 2 zz z Argz Arg Arg + = 1 y tan x - ?= -p+ 1 y tan x - ?= p- 1 y tan x - ?= - 1 y tan x - ?= 1 b ztan a - = 2 13i 1 3i , 22 -+ -- ?= ? = SAKSHI is n 2 7. If a 2 + b 2 + c 2 = K then range of ab + bc + ca is 8. If the two roots are negative, then a, b, c will have same sign 9. 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 is increased by 'K' is f(x – K), multiplied by K is f(x/K) 11. For a, b, h ? R the roots of (a – x) (b – x) = h 2 are real and unequal 12. For a, b, c ? R the roots of (x – a) (x – b) + (x – b) (x – c) + (x – c) (x – a) = 0 are real and unequal 13. Three roots of a cubical equation are A.P, they are taken as a – d, a, a + d 14. Four roots in A.P, a–3d, a–d, a+d, a+3d 15. If three roots are in G.P are taken as roots 16. If four roots are in G.P are taken as roots 17. For ax 3 + bx 2 + cx + d = 0 (i) Sa 2 ß = (aß + ß? + ?a) (a + ß + ?) –3aß ? = s 1 s 2 – 3s 3 (ii) (iii) (iv) (v) In to eliminate second term roots are diminished by Binomial Theorem And Partial Fractions 1. Number of terms in the expansion (x + a) n is n + 1 2. Number of terms in the expansion is 3. In 4. For independent term is 5. In above, the term containing x s is 6. (1 + x) n – 1 is divisible by x and (1 + x) n – nx –1 is divisible by x 2 . 7. Coefficient of x n in (x+1) (x+2)...(x+n)=n 8. Coefficient of x n–1 in (x+1) (x+2)....(x+n) is 9. Coefficient of x n–2 in above is 10. If f(x) = (x + y) n then sum of coefficients is equal to f(1) 11. Sum of coefficients of even terms is equal to 12. Sum of coefficients of odd terms is equal to 13. If are in A.P (n–2r) 2 =n + 2 14. For (x+y) n , if n is even then only one middle term that is term. 15. For (x + y) n , if n is odd there are two mid- dle terms that is term and term. 16. In the expansion (x + y) n if n is even greatest coefficient is 17. In the expansion (x + y) n if n is odd great- est coefficients are if n is odd 18.For expansion of (1+ x) n General notation 19. Sum of binomial coefficients 20. Sum of even binomial coefficients 21. Sum of odd binomial coefficients MATRICES 1. A square matrix in which every element is equal to '0', except those of principal diagonal of matrix is called as diagonal matrix 2. A square matrix is said to be a scalar matrix if all the elements in the principal diagonal are equal and Other elements are zero's 3. A diagonal matrix A in which all the elements in the principal diag- onal are 1 and the rest '0' is called unit matrix 4. A square matrix A is said to be Idem-potent matrix if A 2 = A, 5. A square matrix A is said to be Involu-ntary matrix if A 2 = I 6. A square matrix A is said to be Symm-etric matrix if A = A T A square matrix A is said to be Skew symmetric matrix if A=-A T 7. A square matrix A is said to be Nilpotent matrix If their exists a positive integer n such that A n = 0 'n' is the index of Nilpotent matrix 8. If 'A' is a given matrix, every square mat-rix can be expressed as a sum of symme-tric and skew symmetric matrix where Symmetric part unsymmetric part 9. A square matrix 'A' is called an ortho-gonal matrix if AA T = I or A T = A -1 10. A square matrix 'A' is said to be a singular matrix if det A = 0 11. A square matrix 'A' is said to be non singular matrix if det A ? 0 12. If 'A' is a square matrix then det A=det A T 13. If AB = I = BA then A and B are called inverses of each other 14. (A -1 ) -1 = A, (AB) -1 = B -1 A -1 15. If A and A T are invertible then (A T ) -1 = (A -1 ) T 16. If A is non singular of order 3, A is invertible, then 17. If if ad-bc ? 0 18. (A -1 ) -1 =A, (AB) -1 =B -1 A -1 , (A T ) -1 =(A -1 ) T (ABC) -1 = C -1 B -1 1 ab d b 1 AA cd c a ad bc - - ?? ? ? =? = ?? ? ? - - ?? ? ? 1 AdjA A det A - = T AA 2 + = T AA 2 + = n1 13 5 C C C .... 2 - +++ = n1 o2 4 C C C .... 2 - +++ = n o1 2 n C C C ........ C 2 ++ + + = nn n 0o 1 1 r r CC,C C,C C == = nn n1 n1 22 C,C -+ n n 2 C th n3 2 + th n1 2 + th n 1 2 ?? + ?? ?? nn n r1 r r1 CC C -+ () ( ) f1 f 1 2 +- () ( ) f1 f 1 2 -- ()( )( ) nn 1 n 1 3n 2 24 +- + () nn 1 2 + np s 1 pq - + + np 1 pq + + n p q b ax x ?? + ?? ?? () n r1 r Tnr1 xa , Tr + -+ += nr 1 r1 C +- - () n 12 r x x ... x ++ + b na - nn1 n2 ax bx cx ............ 0 -- ++ = 33 3 3 112 3 s3ss 3s a+ß +? = - + 42 2 112 13 2 s4ss 4ss 2s =- + + 44 4 a+ß +? 22 2 2 12 s2s a+ß +? = - 3 3 aa ,,ar,ar rr a ,a,ar r 1 f0 x ?? = ?? ?? K ,K 2 -?? ?? ?? () 12 n 12 n 11 1 a a .... a .... aa a ?? ++ + + + + ?? ?? SAKSHI A -1 . If A is a n x n non- singular matrix, then a) A(AdjA)=|A|I b) Adj A = |A| A -1 c) (Adj A) -1 = Adj (A -1 ) d) Adj A T = (Adj A) T e) Det (A -1 ) = ( Det A) -1 f) |Adj A| = |A| n -1 g) lAdj (Adj A ) l= |A| (n - 1)2 h) For any scalar 'k' Adj (kA) = k n -1 Adj A 19. If A and 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 Echolon form i.e. Echolon form of No of non zero rows=n= Rank of a matrix If the system of equations AX=B is consistent if the coeff matrix A and augmented matrix K are of same rank Let AX = B be a system of equations of 'n' unknowns and ranks of coeff matrix = r 1 and rank of augmented matrix = r 2 If r 1 ? r 2 , then AX = B is inconsistant, i.e. it has no solution If r 1 = r 2 = n then AX=B is consistant, it has unique solution If r 1 = r 2 < n then AX=B is consistant and it has infinitely many number of solutions Random Variables- Distributions & Statistics 1. For probability distribution if x=x i with range (x 1 , x 2 , x 3 ----) and P(x=x i ) are their probabilities then mean µ= Sx i P(x-x i ) Variance =s 2 =Sx i 2 p(x=x i ) -µ 2 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) n i) probability of occurrence = p ii) probability of non occurrence = q iii) p + q = 1 iv) probability of 'x' successes v) Mean = µ = np vi) Variance = npq vii) Standard deviation = 3. If number of trials are large and probab-ility of success is very small then poisson distribution is used and given as 4. i) If x 1 ,x 2 ,x 3 ,.....x n are n values of variant x , then its Arithmetic Mean ii) For individual series If A is assumed average then A.M iii) For discrete frequency distribution: iv) Median = where l = Lower limit of Median class f = frequency N = Sf i C = Width of Median class F = Cumulative frequency of class just preceding to median class v) First or lower Quartile deviation where f = frequency of first quarfile class F = cumulative frequency of the class just preceding to first quar- tile class vi) upperQuartiledeviation vii) Mode where l = lower limit of modal class with maximum frequency f 1 = frequency preceding modal class f 2 = frequency successive modal class f 3 = frequency of modal class viii) Mode = 3Median - 2Mean ix) Quartile deviation = x) coefficient of quartile deviation = xi) coefficient of Range = VECTORS 1. A system of vectors are said to be linearly independent if are exists scalars Such that 2. Any three non coplanar vectors are linea-rly independent A system of vectors are said to be linearly dependent if there atleast one of x i ?0, i=1, 2, 3….n And determinant = 0 3. Any two collinear vectors, any three coplanar vectors are linearly dependent. Any set of vectors containing null vectors is linearly independent 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 a is or 6. are ends of diameter then equation of sphere 7. If are unit vectors then unit vector along bisector of ? AOB is or 8. Vector along internal angular bisector is 9. If 'I' is in centre of ABC then, le ? ab b a ? ?? ?? ±+ ?? ?? () ab ab + ±+ ab ab + + , ab ()() .0 ra r b --= , ab 22 2 2. rrcc a -+ = () 2 2 rc a -= c 11 2 2 ... 0 nn xa x a x a +++ = 12 , ,..... n aa a 12 3 ........ 0 n xx x x ?= = = = 11 2 2 ... 0 nn xa xa xa +++ = 12 , .... . n xx x 12 , ,..... n aa a Range Maximum Minimum + 31 31 QQ QQ - + 31 2 QQ - 1 12 . 2 m m ff Zl C ff f ?? - =+ ?? -- ?? 3 3 4 . N F Ql C f ?? - ?? =+?? ?? ?? ?? 1 4 . N F Ql C f ?? - ?? =+?? ?? ?? ?? 2 N F lC f ?? - ?? ?? +× () ii where d x A =- ii i fd xA f =+ ? ? () i xA xA n - =+ ? i x x n = ? () k e Px k k ? ? - == npq () nx x ix Px x nC q p - == variance 12 3 4 A0 x y z 00 k l ?? ?? = ?? ?? ?? 1 234 A2 3 1 2 32 1 0 ?? ?? = ?? ?? ?? SAKSHI 10. If 'S' is circum centre of ABC then, 11.If 'S' is circum centre, 'O' is orthocenter of ABC then, 12. If & if axes are rotated through an i) x - axis ii) y - axis iii) z - axis If 'O' is circumcentre of ABC then (Consider equilateral ) 13. where i) is acute ii) is obtuse iii) two vectors are to each other. 14. In a right angled ABC, if AB is the hypotenuse and AB = P then 15. is equilateral triangle of side 'a' then 16. 17. Vector equation. of a line passing through the point A with P.V . and parallel to 'b' is 18. Vector equation of a line passing through is r =(1-t)a +tb 19. Vector equation. of line passing through & to 20. Vector equation. of plane passing through a pt and- parallel to non-collinear vectors is . s,t?R and also given as 21. Vector equation. of a plane passing through three non-collinear Points. is i.e = = 22. Vector equation. of a plane passing through pts and parallel to is 23. Vector equation of plane, at distance p (p >0) from origin and to is 24. Perpendicular distance from origin to plane passing through a,b,c 25. Plane passing through a and parallel to b,c is [r - a, b - c] = and [r b 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] = abc 27. 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 is ii) the projection of b on a is iii) the projection of b on a vector perpe-ndicular to' a' in the plane generated by a,b is 29. If a,b are two nonzero vectors then 30. 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 then and hence 33. If a is any vector then a×a = 0 34. 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 is 38. If a,b,c are the position vectors of the vertices of a triangle, then the vector area of the triangle 39. If ABCD is a parallelogram and then the vector area of ABCD is la×bl 40. The length of the projection of b on a vector perpendicular to a in the plane generated by a,b is 41. The perpendicular distance from a point P to the line joining the points A,B is 42. Torque: The torque or vector moment or moment vector M of a force F about a point P is defined as M = r×F where r is the vector from the point P to any point A on the line of action L of F. 43. a,b,c are coplanar then [abc]=0 44. 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 is 49. If i,j,k are unit vectors then [i j k] = 1 50. If a,b,c are vectors then [a+b, b+c, c+a] = 2[abc] [] , acbd bd -- × 0 AB AC AD ?? = ?? [] 1 6 ab c ± 1 6 AB AC AD ?? ± ?? AP AB AB × ab a × , AB a BC b == () 1 2 ab bc c a =×+×+× [] 1 2 ab × () 1 2 ab × ABC ? , AB a AC b == () sin , ab ab ab × = () sin . ab ab ab ×= () . cos , ab ab ab = () 2 . ba a b a - () 2 . ba a a . ba a () . ba a . ba 0 a ? abc bc ca a b ?? ?? ?? ×+ × + × ?? . rn p = n r ? 0 AP ABC ?? = ?? () Cc () Bb () Aa ,, rab ac a ?? =- - - ?? () 1 sta sb sc -- + + () () ra sb a tc a =+ - + - 0 AB AC AP ?? = ?? () () () ,, Aa B b C c r a bc rbc abc ????? ? -= = ????? ? ra sb tc =+ + & bc () Aa () ra tb c =+ × , bc r ? a () () , Aa B b ra tb =+ a ()()() 22 2 2 2 ai a j a k a ×+ × + × = () () () 22 2 2 .. . ; ai a j ak a ++ = 2 3 2 a - .. . AB BC BC CA CA AB ++= . AB BC ABC ? 2 .. . AB BC BC CA CA AB P ++= le ? r ? .0 90 ab ? =? = °? . 0 90 180 ab ?? <? °< < °? .0 0 90 ab ?? >? < < °? 0 180 ? °= = ° .cos ab a b ? = le ?() 3 sin 2 2 OA A OA OB OC S= ++ le ? () ( ) ()) 12 3 , cos 90 sin 90 , aa a aa ++ + (12 cos sin , aa aa + )) 23 1 , ,( cos sin aa a aa + () ( (31 cos 90 ) sin 90 , aa a aa ++ + 12 3 2 1 ( , cos sin , cos sin(90 ) aa a a a aa a a ++ - () 12 3 ,, aaaa = 2 OA OB OC OS ++ = le ? SA SB SC SO ++ = le ? 0 BC IA CA IB AB IC ++ = SAKSHI 51. [a×b, b×c, c×a] = (abc) 2 52. 53. . 54. 55. If A,B,C,D are four points, and 56. are called reciprocal system of vectors 57. If a,b,c are three vectors then [a b c] = [b c a]= [c a b] = -[b a c] = -[c b a] = -[a c b] 58.Three vectors are coplanar if det = 0 If ai + j + k, i + bj + k, i + j + ck where are coplanar then i) ii) Preparation Tips - Mathematics Memorizing land mark problems (rememb-ering standard formulae, concepts so that you can apply them directly) and being strong in mental calculations are essential (Never use the calculator during your entire AIEEE preparation. Try to do first and sec-ond level of calculations mentally You are going to appear for AIEEE this year, you must be very con- fident, don't pa-nic,it is not difficult and tough. You need to learn some special tips and tricks to solve the AIEEE questions to get the top rank. Don't try to take up new topics as they con-sume time, you will also lose your confide-nce on the topics that you have already pre-pared. Don't try to attempt 100% of the paper unl-ess you are 100% confi- dent: It is not nece-ssary to attempt the entire question paper, Don't try if you are not sure and confident as there is negative marking. If you are confident about 60% of the questions, that will be enough to get a good rank. Never answer questions blindly. Be wise, preplanning is very impor- tant. There are mainly three difficulty levels, si-mple, tough and average. First try to finish all the simple questions to boost your Conf-idence. Don't forget to solve question papers of previous years AIEEE before the examinat-ion. As you prepare for the board examinat-ion, you should also prepare and solve the last year question papers for AIEEE. You also need to set the 3 hours time for each and every pre- vious year paper, it will help you to judge yourself, and this will let you know your weak and strong areas. You will gradually become confident. You need to cover your entire syllabus but don't try to touch any new topic if the exa-mination is close by. Most of the questions in AIEEE are not dif-ficult. They are just dif- ferent & they requi-re a different approach and a different min-dset. Each question has an element of sur-prise in it & a student who is adept in tack-ling 'surprise questions' is most likely to sail through successfully. It is very important to understand what you have to attempt and what you have to omit. There is a limit to which you can improve your speed and strike rate beyond which what becomes very important is your selec-tion of question. So success depends upon how judi- ciously one is able to select the questions. To optimize your per- formance you should quickly scan for easy questions and come back to the difficult ones later. Remember that cut-off in most of the exa-ms moves between 60 to 70%. So if you fo-cus on easy and average question i.e. 85% of the questions, you can easily score 70% marks without even attempting difficult qu-estions. Try to ensure that in the initial 2 hours of the paper the focus should be clea-rly on easy and average questions, After 2 hours you can decide whether you want to move to difficult questions or revise the ones attempted to ensure a high strike rate. Topic-wise tips Trigonometry: In trigonometry, students usually find it diffi-cult to memorize the vast number of formul-ae. Understand how to derive formulae and then apply them to solving problems.The mo-re you practice, the more ingrained in your br-ain these formulae will be, enabling you to re-call them in any situation. Direct questions from trigonometry are usually less in number, but the use of trigonometric concepts in Coor-dinate Geometry & Calculus is very profuse. Coordinate Geometry: This section is usually considered easier than trigonometry. There are many common conc-epts and formulae (such as equations of tang-ent and normal to a curve) in conic sections (circle, parabola, ellipse, hyperbola). Pay att-ention to Locus and related topics, as the und- erstanding of these makes coordinate Geome-try easy. Calculus: Calculus includes concept-based problems which require analytical skills. Functions are the backbone of this section. Be thorough with properties of all types of functions, such as trigonometric, algebraic, inverse trigonom-etric, logarithmic, exponential, and signum. Approximating sketches and graphical interp-retations will help you solve problems faster. Practical application of derivatives is a very vast area, but if you understand the basic concepts involved, it is very easy to score. Algebra: Don't use formulae to solve problems in topi-cs which are logic-ori- ented, such as permuta-tions and combinations, probability, location of roots of a quadratic, geometrical applicati-ons of complex numbers, vectors, and 3D-geometry. AIEEE 2009 Mathematics Section Analysis of CBSE syllabus Of all the three sections in the AIEEE 2009 paper, the Mathematics section was the toughest. Questions were equally divided between the syllabi of Class XI and XII. Many candidates struggled with the Calcu- lus and Coordinate Geometry portions. Class XI Syllabus Topic No. of Questions Trigonometry 1 Algebra (XI) 6 Coordinate Geometry 5 Statistics 3 3-D (XI) 1 Class XII Syllabus Topic No. of Questions Calculus 8 Algebra (XII) 2 Probability 2 3-D (XII) 1 Vectors 1 11 1 2 ab bc ca ++ = 11 1 1 11 1 abc ++ = -- - 1 abc ?? ? [] [] [] 11 1 ,, bc c a ab abc abc abc abc - ×× × == = () 4 AB CD BC AD CA BD ABC ×+ × + × = ? ()() .. . .. ac ad ab c d bc bd ×× = 22 22 . ab ab a b ×+ = () 2 ix a i a S×=Read More

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