FORMULAE. 8. a 2 + b 2 + c 2 ab bc ca = 1 2 [(a b)2 + (b c) 2 + (c a) 2 ] 10. (a b) 3 = a 3 b 3 3ab (a b) = a 3 3a 2 b + 3ab 2 b 3

Transcription

1 FORMULAE Algeba 1. (a + b) = a + b + ab = (a b) + 4ab. (a b) = a + b ab = (a + b) 4ab 3. a b = (a b) (a + b) 4. a + b = (a + b) ab = (a b) + ab 5. (a + b) + (a b) = (a + b ) 6. (a + b) (a b) = 4ab 7. (a + b + c) = a + b + c + ab + bc + ca 8. a + b + c ab bc ca = 1 [(a b) + (b c) + (c a) ] 9. (a + b) 3 = a 3 + b 3 + 3ab (a + b) = a 3 + 3a b + 3ab + b (a b) 3 = a 3 b 3 3ab (a b) = a 3 3a b + 3ab b a 3 + b 3 = (a + b) 3 3ab (a + b) = (a + b) (a ab + b ) 1. a 3 b 3 = (a b) 3 + 3ab (a b) = (a b) (a + ab + b ) 13. (a + b + c) 3 = a 3 + b 3 + c 3 + 3(a + b) (b + c) (c + a) 14. a 3 + b 3 + c 3 3abc = (a + b + c) (a + b + c ab bc ca) If a + b + c = 0 ten a 3 + b 3 + c 3 = 3abc

2 Linea Equations a1 b1 15. Unique solution a b a b c 16. No solution = a b c a1 b1 c1 17. Infinite many solution = = a b c If unit digit of a two digit numbe is x and ten s digit is y ten te numbe is 10y + x. Quadatic Equations 19. Disciminant D = b 4ac 0. Fo D > 0, oots ae eal & unequal 1. Fo D = 0, oots ae eal & equal. Fo D < 0, no eal oots. 3. Fo linea facto D > 0 4. Factoisation of Q.E. = a (x α ) (x β ) 5. Sum of oots (α + β) = b/a 6. Poduct of oots (α β) = c/a 7. b + b 4ac Root α = a b b 4ac 8. Root β = a

3 9. Foming of quadatic equation x (sum of te oots) x + (poduct of te oot) 30. Fo two consecutive natual numbes we take x and x Fo two consecutive even numbes we take x and x Fo two consecutive odd numbes we take x + 1 and x x + ( x ) ( x ) x = + x = x ( x + ) ( x ) 4 x = x ( x ) = ( x+ ) 4 x x Aitmetic popessions 36. Sequence of seies denoted by T n o a n. Fist tem t 1 o a 1 second tem t o a, tid tems t 3 o a 3... last tem T n o a n. 37. Last tem of an A.P. is l o T n o A n = a + (n 1) d, wee n is te numbe of tems and n N. 38. Tee consecutive tems of A.P. take as (a d), a, (a + d) 39. Fou consecutive tems of A.P. take as (a 3d), (a d), (a + d), (a + 3d) 40. d = T T 1 = T 3 T = T 4 T 3 = T n T n 1

4 41. Sum of n tems of an A.P. = T 1 + T + T T n n = Sn = ( a+(n 1)d) Instalments 4. Amount = pincipal t n = ( a+l) 1+,A = p t Income Tax A p= Admissible Deduction: Deductions wic ae fee fom Income tax ae called admissible deduction. (i) Standad Deduction [unde section 16(i)]. Geneally, it is one tid of te Goss Income subject to maximum of Rs. 30,000. It vaies fom yea to yea. (ii) House Rent Allowance (HRA) [unde section 10 (13A)]. House Rent Allowance is geneally fee fom Income tax. t

5 (iii) Donations [unde section 80G]: Donations given to appoved caitable tusts ae fee fom Income tax unde section Rs. 80G, up to cetain pecentage. (iv) Inteests on Deposit [unde section 80L]: Inteests on te deposit in cetain govenment secuity ae fee fom Income tax subject to cetain limit. Tax Rebates [unde section 88]: 0% of te savings unde cetain scemes ae deducted fom te Income tax payable. Income tax Rates : Rates fo te pesent financial yea is given below : (i) up to Rs. 50,000 No tax (ii) Fom Rs to Rs. 60,000 10% of te amount exceeding Rs. 50,000 (iii) Fom Rs. 60,000 to Rs. 1,50,000 Rs % of te amount exceeding Rs. 60,000 (iv) Rs. 1,50,000 and above Rs. 19, % of te amount exceeding Rs. 1,50,000. Mensuation 44. Aea of tiangle = b = absin C = bc sin A = casinb 45. Aea of isosceles igt tiangle = 1 a 46. Altitude and base of isosceles igt tiangle = a

6 47. Hypotenuse of isosceles igt tiangle = a 48. Aea of equilateal tiangle = 3 a Heo s Fomula fo aea of scalence tiangle. a a a Aea of tiangle = s(s a)(s b)(s c) (a +b+c) wee s = 50. Aea of ectangle = lengt beadt 51. Peimete of ectangle = (l + b) 5. Aea of a squae = (side) 53. Peimete of squae = 4 side c a b 54. Hypotenuse of a squae = side 55. Side of squae = aea 56. Aea of ombus = 1 d 1 d = side coesponding altitude. 57. Peimete of ombus = 4 side d Side of ombus = 1 d d + d

7 59. Aea of paallelogam = base coesponding altitude. 60. Aea of tapezium = 1 (Sum of paallel sides) eigt. 61. Aea of quadilateal = 1 bigge ypotenuse Sum of two altitudes dawn on tis ypotenuse fom opposite vetex. 6. Aea of scalene quadilateal = (s a)(s b)(s c)(s d) a+b+c+d s= 63. Eac angle of a egula polygon n = 180 n 64. Aea of egula polygon = 65. Peimete of egula polygon = Na. 1 n π N a= a cot 4 n 66. Aea of cicle = π = > = aea π 67. Cicumfance of cicle = π = > = c π

8 68. Radius of cicle = diamete/ θ 69. Ac lengt (l) = π Aea of secto (S.A) = θ π l S. A = 7. Aea of segment = θ 1 π sin θ Peimete of secto = + l 74. Aea of ing = π(r ) 75. Aea of semi cicle = 1 π 76. Peimete of semi cicle = π Volume of cuboid = l b 78. Suface aea of cuboid = (lb + l + b) 79. Aea of fou walls = (l + b) 80. Diagonal of cuboid = l +b +

9 81. Volume of cube = s 3 8. Suface aea of cube = 6s 83. Side of cube = 3 Volume 84. Diagonal of cube = 3a Tigonomety 85. Tignometical atio : In a igt angled tiangled te atio of any two sides is known as Tignometical atio. A p 86. p sinθ = = 1/cosec θ B b C 87. b cosθ = =1/secθ 88. p tanθ = =1/cotθ b 89. b cot θ = =1/tanθ p 90. sec θ = =1/cosθ b

10 91. cosec θ = = 1/sinθ p 9. tan θ = sin θ /cos θ 93. cot θ = cos θ / sin θ 94. sin θ + cos θ = 1 sin θ = 1 cos θ cos θ = 1 sin θ 95. sec θ tan θ = 1 sec θ 1 = tan θ 1 + tan θ = sec θ 96. cosec θ cot θ = cot θ = cosec θ cosec θ 1 = cot θ tanθ 97. tan θ = 1 tan θ 98. cos θ = cos θ sin θ = cos θ 1 = 1 sin θ 99. sin (90 θ) = cos θ 100. cos (90 θ) = sin θ 101. tan (90 θ) = cot θ 10. cot (90 θ) = tan θ

11 103. sec (90 θ) = cosec θ sin θ = (1 + sin θ) (1 sin θ) cos θ = (1 + cos θ) (1 cos θ) 106. AB = sin θ/ (AB = Cod) 107. Cuved suface aea of cylinde = π 108. Total suface aea of cylinde = π ( + ) 109. Volume of cylinde = π volumeof cylinde 110. Heigt of cylinde () = π Volume of cone = 3 π l 11. Cuved suface aea of cone = π l, l= Total suface aea of cone = π (l + ) 114. Lateal eigt of cone ( l ) = Volume of spee = π 116. Total suface aea of spee = 4 p 117. Volume of te emi spee = 3 3 π

12 118. Total suface aea of emi spee = 3π 119. Cuved suface aea of emispee = π 10. Volume of speical cell = Fustum of a cone 11. Volume of fustum = 4 (R 3 3 ) 3 π 1 (R + +R) 3 π Lateal suface aea = π l (R + ) Total suface aea = π l (R + ) + πr + π 1. Aitmetic mean wen fequency is not given Σx x=, Σx = sumof items,n = No.of items) n 13. Aitmetic mean wen fequency is given Σfx x= Σ f 14. Aitmetic mean fom assumd mean: Σ fd x=a+ i Σ f (A = assumed mean, d = deviation, Σ f = sum of all fequency, i = class inteval) nx +n x 15. Combined mean x= n +n 1 1 1

13 Pobability contol angle of an item = 16. Pobability of event P (A) = P(A) = 1 P(A) = P(A) + P(A) = 1 Co-odinate geomety Volume of item 360 sumof tevaluesof items fomuable outcomes Totalnumbe of outcomes 17. Distance fomula = (x x 1) + (y y 1) 18. Section Fomulae (Intenally) lx + mx ly + my x=,y= l+m l+m 1 1 x y 1 1 x y 19. Mid point fomulae = x + x y + y x=,y= Section Fomulae (extenally) lx 1 mx ly 1 my x=,y= l m l m 131. Co-odinate of centoid x + x + x y + y + y =, ax + bx + cx ay + by + cy 13. Incente of a tiangle =, a+b+c a+b+c