Mathematics

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Mathematics

  1. 1. MATHEMATICS FUNDAE
  2. 2. <ul><li>A game of numbers. </li></ul><ul><li>It is also a tool used in day to day life. </li></ul><ul><li>Needed in Physics and Chemistry also. </li></ul><ul><li>Helps in getting a good rank. </li></ul>What is Maths?
  3. 3. HOW TO ENJOY MATHEMATICS <ul><li>Focus On Basics . </li></ul><ul><li>Try To Understand The Formulae. </li></ul><ul><li>Predict Its Application. </li></ul><ul><li>Match With The Applications As In Books. </li></ul><ul><li>Think WHY? </li></ul><ul><li>Write The Concepts Which You Are Going To Use On The Paper Before Starting The Solution Of Problem. </li></ul>
  4. 4. Important Chapters??
  5. 6. TRIGONOMETRY
  6. 7. Trigonometric Functions and Equations <ul><li>Widely used in other chapters of mathematics. </li></ul><ul><li>Requires lot of practice. </li></ul><ul><li>It has only formulae. </li></ul><ul><li>Try to see some pattern in the formulae and then learned. </li></ul>
  7. 8. Inverse Trigonometric Functions <ul><li>Very conceptual </li></ul><ul><li>Requires the use of graphs </li></ul><ul><li>Should be studied after studying functions. </li></ul>
  8. 9. <ul><li>Topics are interlinked </li></ul><ul><li>Visualization of problems </li></ul><ul><li>Follow ‘read & draw strategy’ </li></ul><ul><li>Some formulae should be learned </li></ul>CO – ORDINATE GEOMETRY
  9. 10. STRAIGHT LINE Normal Form : x cos α + y sin α = p Parametric Form : x – x 1 x – x 2 r cos θ sin θ
  10. 11. Equation of angle bisectors a 1 x + b 1 y +c 1 a 2 x + b 2 y + c 2 √ a 1 2 + b 1 2 √a 2 2 + b 2 2 If c 1 , c 2 > 0, then bisector containing origin is given by +ve sign ACUTE & OBTUSE ANGLE BISECTOR
  11. 12. Use of Reflection
  12. 14. ax 2 + 2hxy + by 2 + 2gx + 2fy + c = 0 Angle between the two lines : θ = tan -1 2 √(h 2 – ab) │ a + b│ Point of intersection of 2 lines : a h g h b f g f c ax + hy + g = 0 hx + by + f = 0 PAIR OF STRAIGHT LINES
  13. 15. <ul><li>CIRCLE </li></ul><ul><li>(x – a) 2 + (y – b) 2 = r 2 </li></ul><ul><li>Equation of the tangent : </li></ul><ul><li>Point Form </li></ul><ul><li>Parametric Form </li></ul><ul><li>Slope Form </li></ul>Replace x 2 xx 1 y 2 yy 1 2x (x + x 1 ) 2y (y + y 1 ) 2xy (xy 1 + x 1 y) <ul><li>Only valid for a </li></ul><ul><li>two degree </li></ul><ul><li>equation </li></ul>
  14. 16. Normal To A Circle Equation for a 2 nd order conic : ax 2 + 2hxy + by 2 + 2gx + 2fy + c = 0 <ul><li>Normal always passes through the centre of the circle. </li></ul>Then normal is x – x 1 y – y 1 ax 1 + hy 1 + g hx 1 + by 1 + f a h g h b f g f c ax 1 + hy 1 + g , hx 1 + by 1 + f
  15. 17. Radical Axis <ul><li>S – S’ = 0 </li></ul><ul><li>Perpendicular to the line joining the centres. </li></ul><ul><li>Bisects the direct common tangents. </li></ul><ul><li>For 3 circles, taking 2 at a time, they are concurrent. </li></ul>
  16. 18. The curve y = x 3 – 3x + 2 and x + 3y = 2 intersect in points (x 1 ,y 1 ), (x 2 ,y 2 ) and (x 3 ,y 3 ). Then the point P(A,B) where A = Σ x i and B = Σ y i lies on the line (A) x – 3y = 5 (B) x + y = 1 (C) 3x – 7 = y (D) 2x + y = 2
  17. 19. Conic Section ax 2 + 2hxy + by 2 + 2gx + 2fy + c = 0
  18. 20. How a conic section is formed
  19. 21. PARABOLA Focal Chord t 1 t 2 = -1 Length (PQ) = a*(t 2 – t 1 ) 2 Tangents at P & Q will be perpendicular to each other Length of Latus Rectum : 4 * PS * QS PS + QS
  20. 22. Tangents Slope Form : Point of intersection of tangents: ( at 1 t 2 , a(t 1 + t 2 ) ) y = mx + a m Remembering Method : G O A (GOA rule) GM of at 1 2 & at 2 2 , AM of 2at 1 & 2at 2 i.e. at 1 t 2 i.e. a(t 1 + t 2 )
  21. 23. Equation of tangent to parabolas of different form :
  22. 24. <ul><li>Co – Normal Points </li></ul><ul><li>Properties : </li></ul><ul><li>m 1 + m 2 + m 3 = 0 </li></ul><ul><li>y 1 + y 2 + y 3 = 0 </li></ul><ul><li>For the normals to be real h > 2a </li></ul><ul><li>For the normals to be real and distinct 27ak 2 < 4(h-2a) 3 </li></ul>
  23. 25. ELLIPSE Tangent in slope form : y = mx + √ (a 2 m 2 + b 2 ) - Normal in slope form : y = mx - m (a 2 – b 2 ) + √ (a 2 + b 2 m 2 )
  24. 26. Co – Normal Points <ul><li>4 Normals can be drawn from any point to an ellipse. </li></ul><ul><li>α + β + λ + δ = (2n + 1) π </li></ul>Properties:
  25. 27. <ul><li>Sin ( α + β ) + Sin ( β + λ ) + Sin ( λ + α ) = 0 </li></ul><ul><li>Co – Normal Points lie on a fixed curve called Apollonian Rectangular Hyperbola </li></ul><ul><li>A.R.H (a2 – b2)xy + b2kx – a2hy = 0 </li></ul>
  26. 28. Director Circle : Locus of the points from which perpendicular tangents can be drawn x 2 + y 2 = a 2 + b 2
  27. 29. Reflection Property : Ray passing through a focus, passes through the other focus after reflection.
  28. 30. HYPERBOLA Asymptotes Tangent to the hyperbola at infinity <ul><li>Properties : </li></ul><ul><li>Difference between hyperbola and pair of asymptotes is constant. </li></ul><ul><li>Hyperbola and its conjugate hyperbola have the same asymptotes. </li></ul>x 2 - y 2 = 0 a 2 b 2
  29. 31. <ul><li>Angle between two asymptotes is </li></ul><ul><li>2tan -1 (b/a) </li></ul><ul><li>Asymptotes pass through the centre of </li></ul><ul><li>the hyperbola. </li></ul><ul><li>Co – Ordinate axes are angular bisector </li></ul><ul><li>of the two asymptotes. </li></ul><ul><li>Hyperbola, Asymptotes and Conjugate </li></ul><ul><li>Hyperbola are in A.P </li></ul><ul><li>i.e. C + H = 2A </li></ul>
  30. 32. If normal at P (2, 1.5√3) meets the major axis of the ellipse x 2 + y 2 = 1 at Q. S and S’ are 16 9 foci of given ellipse, then SQ:S’Q is (A) 8 – √7 8 + √7 (B) 4 + √7 4 – √7 (C) 8 + √7 8 – √7 <ul><li>4 – √7 </li></ul><ul><li>4 + √7 </li></ul>
  31. 33. DIFFERENTIAL CALCULUS
  32. 34. FUNCTIONS <ul><li>Most Important Is Concept Of DOMAIN And RANGE . </li></ul><ul><li>Knowledge About Some Important Functions Like LOGARATHMIC, TRIGONOMETRIC, GREATEST INTEGER etc. </li></ul><ul><li>Focus On Domain And Range Of These Functions. </li></ul>
  33. 35. <ul><li>Give Attention To COMPOSITE FUNCTIONS . </li></ul><ul><li>For Finding PERIOD, Be Careful About CONSTANT FUNCTION. </li></ul><ul><li>HOW TO SOLVE PROBLEMS IN EXAMS ? </li></ul><ul><li>Identify The Nature Of Function . </li></ul><ul><li>Coordinate With Its Domain And Range. </li></ul>
  34. 36. <ul><li>Think Practically. </li></ul><ul><li>Go What The Question Says……. </li></ul><ul><li>Get The Knowledge Of Symmetry About Any Point Or Line. </li></ul>
  35. 37. Find the period of y = log Cos(x) Sin(x) Answer : 2 π
  36. 38. LIMIT <ul><li>Condition For EXISTANCE OF LIMIT </li></ul><ul><li>Remember Some Important Expansions. </li></ul><ul><li>L’HOSPITAL RULE </li></ul><ul><li>Try To Simplify The Question If You Get A Hard And Tough Looking Problem. 95% Of Such Problems Becomes Easy After Simplification </li></ul>
  37. 39. <ul><li>In Case When x Tends To Infinity In Algebraic /Algebraic Function Be Careful About Constants If Given In Question. </li></ul>
  38. 40. CONTINUITY AND DIFFERENTIABILITY <ul><li>Concept Of LIMIT Should Be Clear Before Attempting The Questions Of Continuity. </li></ul><ul><li>Remember The Approach Of Continuity At End Points . </li></ul><ul><li>For Differentiability Of Function, Careful Where To Use Basic Funda And Where Direct Differntiation . </li></ul>
  39. 41. <ul><li>Differentiability At End Points In A Closed Or Open Interval. </li></ul><ul><li>Differentiability Implies Contuinity . </li></ul>
  40. 42. APPLICATIONS OF dy/dx <ul><li>ROLLE’S THEOREM . </li></ul><ul><li>LAGRANGE’S MEAN VALUE THEOREM. </li></ul><ul><li>Use Of dy/dx In Deciding The Nature Of Curve. </li></ul><ul><li>Try To Co-Relate Max. And Min. With The Help Of Graph. </li></ul>
  41. 43. <ul><li>Use nth Derivative Test. </li></ul><ul><li>Concentrate On Maxima And Minima Of Discontinuous Function. </li></ul>
  42. 44. INTEGRAL CALULUS
  43. 45. INDEFINITE INTEGRAL <ul><li>Get Some Important Results. </li></ul><ul><li>Use By Parts Method Whenever There Is Any Scope. </li></ul><ul><li>Rearrangement Is Your Motto . </li></ul><ul><li>Get The Approach Of Some Important Forms. </li></ul>
  44. 46. DEFINITE INTEGRALS <ul><li>An Easy Version Of Indefinite Integrals. </li></ul><ul><li>Use Of Properties Are Very Useful. </li></ul><ul><li>No Need Of Gamma Function , Walli’s Function etc. </li></ul><ul><li>Maximum And Minimum Value of Integral Are Very Helpful During Exams. </li></ul>
  45. 47. <ul><li>Newton Method Of Differentiation Of Integrals. </li></ul><ul><li>For Area , First Draw The Curve. </li></ul><ul><li>Find Symmetrical Parts. </li></ul><ul><li>Be Careful About Sign Of Integration. </li></ul>
  46. 48. (A) f(2012) + f(-2012) (B) f(2012) – f(-2012) (C) 0 (D) 2012
  47. 49. DIFFERENTIAL EQUATION <ul><li>Order And Degree. </li></ul><ul><li>Degree Is Defined For Polynomial Equation Only. </li></ul><ul><li>First Step Is To Check If Equation Can Be Solved By Rearranging. </li></ul><ul><li>If Equation Is Of Form f(ax+by+c) , Then Solve It By Taking ax+by+c=t. </li></ul>
  48. 50. <ul><li>For Homogeneous Equation , Use y=vx. </li></ul><ul><li>Concept Of Exact Equations. </li></ul><ul><li>Linear Equation And Conversion Into Linear Equations. </li></ul>
  49. 51. ALGEBRA
  50. 52. COMPLEX NUMBER <ul><li>De Moivre’s Theorem </li></ul><ul><li>(Cos θ + i Sin θ ) n = Cos n θ + i Sin n θ </li></ul><ul><li>If z = (Cos θ 1 + i Sin θ 1 ) (Cos θ 2 + i Sin θ 2 ) (Cos θ 3 + i Sin θ 3 )……………..(Cos θ n + I Sin θ n ) </li></ul><ul><li>then z = Cos ( θ 1 + θ 2 + θ 3 +………+ θ n) + </li></ul><ul><li>i Sin ( θ 1 + θ 2 + θ 3 +………+ θ n ) </li></ul><ul><li>If z = r(Cos θ + i Sin θ ) and n is a positive integer, then </li></ul><ul><li>then z 1/n = r 1/n Cos (2k π + θ ) + i Sin (2k π + θ ) </li></ul><ul><li>n n </li></ul><ul><li>where k = 0,1,2,3………,(n-1) </li></ul>
  51. 53. Coni Method (Rotation Theorem) z 3 – z 1 OQ ( Cos α + i Sin α ) z 2 – z 1 OP CA . e i α BA │ z 3 – z 1 │ . e i α │ z 2 – z 1 │ or arg z 3 – z 1 α z 2 – z 1
  52. 54. Co – Ordinate in terms of Complex Equation of Straight Line : z – z 1 = z – z 1 z 2 – z 1 z 2 – z 1 Circle : zz + az + az + b = 0 , Centre is ‘-a’ radius = √aa - b
  53. 55. SEQUENCE & SERIES Identifying whether the sequence is A.P, G.P, H.P If, a – b a A.P b – c a a – b a G.P b – c b a – b a H.P b – c c Arithmetic Mean A = a + b 2 G 2 = AH Geometric Mean G = √ab A > G > H Harmonic Mean H = 2ab a + b
  54. 56. Some tips : <ul><li>If first common difference is in A.P take the General Term as ‘ax 2 + bx +c’ and determine a, b, c by solving for known values. </li></ul><ul><li>S n = 1 + 2 + 4 + 7 + 11 + ……. </li></ul><ul><li>T n = an 2 + bn + c , </li></ul><ul><li>a = 0.5 , b = -0.5 , c = 1 </li></ul><ul><li>S n = Σ T n </li></ul>
  55. 57. If second common difference is in A.P then take the cubic expression as the General Term and solve for constants.
  56. 58. Solution : S n = cn 2 T n = S n – S n-1 = cn 2 – c(n-1) 2 = c(2n – 1) T n 2 = c 2 (2n – 1) 2 S n = Σ T n * Shortcut Method : Put n = 1 in the question If the sum of first n terms of an A.P is cn 2 , then the sum of squares of these n terms is (A) n (4n 2 – 1)c 2 6 (B) n(4n 2 + 1)c 2 3 <ul><li>n(4n 2 – 1)c 2 </li></ul><ul><li>3 </li></ul><ul><li>n(4n 2 + 1)c 2 </li></ul><ul><li>6 </li></ul>
  57. 59. QUADRATIC EQUATION ax 2 + bx +c = 0 Conditions For A Common Root : ax 2 + bx + c = 0 , a’x 2 + b’x +c’ = 0 a = b = c a’ b’ c’ Note : To find the common root between the two equations, make the same coefficient of x 2 in both the equations and then subtract the 2 equations.
  58. 60. Graph of Quadratic Expression f(x) = ax 2 + bx + c
  59. 61. Location Of Roots 1. If both the roots are less than k (i) D >= 0 , (ii) a*f(k) > 0 , (iii) k > -b 2a 2 . If both the roots are greater than k (i) D >= 0 , (ii) a*f(k) > 0 , (iii) k < -b 2a
  60. 62. 3. If k lies between the roots (i) D > 0 , (ii) a*f(k) < 0 4 . If one of the roots lie in the interval (k 1 , k 2 ) (i) D > 0 ,(ii) f(k 1 )*f(k 2 ) < 0
  61. 63. 5 . If both the roots lie in the interval (k 1 ,k 2 ) (i) D >= 0 6. If k1,k2 lie between the roots (i) D > 0 (ii) a*f(k 1 ) > 0 (iii) a*f(k 2 ) > 0 (iv) k 1 < -b < k 2 2a (iii) a*f(k 2 ) > 0 (ii) a*f(k1) > 0
  62. 64. PERMUTATION & COMBINATION 1. Permutation of n different things taking r at a time = n P r 2. Permutation of n things taken all at a time, p are alike of one kind, q are alike of 2 nd kind, r are alike of 3 rd kind, rest are different n! p! q! r! 3. Number of permutations of n different things taken r at a time, when each thing may be repeated any no. of times n r Circular Permutation <ul><li>When anticlockwise and clockwise are treated as different : </li></ul><ul><li>(n – 1)! </li></ul><ul><li>When anticlockwise and clockwise are treated as same : </li></ul><ul><li>(n – 1)! </li></ul><ul><li>2 </li></ul>
  63. 65. COMBINATION 1. Combination of n different things taking r at a time : n C r 2 . Combination of n different things taking r at a time, when k particular objects occur is: n-k C r-k When k particular objects never occur : n-k C r 3. Combination of n different things selecting at least one of them : n C 1 + n C 2 + n C 3 + …………. + n C n = 2 n – 1 4. If out of (p+q+r+t) things, p are alike of one kind , q are alike of 2 nd kind, r are alike of 3 rd kind, and t are different, then the total number of combinations is : (p+1)(q+1)(r+1)*2 t – 1 5. Number of ways in which n different things can be arranged into r different groups is : n+r-1 P n
  64. 66. PROBABILITY <ul><li>Different types of events like Mixed Event, Independent Events, Complimentary Events. </li></ul><ul><li>Learn some formulas like </li></ul><ul><li>P(A U B) = P(A) + P(B) – P( A ∩ B) </li></ul><ul><li>Conditional Probability. </li></ul><ul><li>Baye’s Theorem or Inverse Probability. </li></ul><ul><li>Binomial Theorem. </li></ul><ul><li>Multinomial Theorem. </li></ul>
  65. 67. VECTORS AND 3D GEOMETRY
  66. 68. VECTORS <ul><li>You Must Give More Concentration On Vectors As It Is Also Required In Physics. </li></ul><ul><li>You All Are Aware Of Simple Applications. </li></ul><ul><li>Linearly Dependent And Linearly Independent Vectors. </li></ul>
  67. 69. <ul><li>Combination Of DOT And CROSS In Problems. </li></ul><ul><li>For Solving Vector Equation, Just Try To Simplify It And Use The Conditions Given. </li></ul><ul><li>If Sol. Requires r In Form Of Two Vectors a & b, Take </li></ul><ul><li>r = λ a + µ b + σ (a×b) </li></ul>
  68. 70. <ul><li>In Case Of Three Non-Coplanar Vectors a, b And c, </li></ul><ul><li>r = λ a + µb + σ c </li></ul><ul><li>And Then Use/Apply The Conditions Given. </li></ul>
  69. 71. 3D-GEOMETRY <ul><li>Projection Of Segment Joining Two Points On Line. </li></ul><ul><li>Angle Between Two Lines. </li></ul><ul><li>Different Forms Of Straight Line Including Vector Form. </li></ul><ul><li>Perpendicular Distance Of A Point From Line. </li></ul><ul><li>Be Careful From Skew Lines . </li></ul>
  70. 72. <ul><li>Shortest Distance Between Two Skew Lines(Its Better To Use Detailed Approach). </li></ul><ul><li>Equation Of Plane In Different Forms. </li></ul><ul><li>Angle Between Line And Plane.Angle Between Two Planes. </li></ul><ul><li>Just Try To Visualize Problem And Use Examination Hall To Put The Conditions In That Frame. </li></ul>
  71. 73. <ul><li>Derive The Following Yourself: </li></ul><ul><li>Equation Of Plane Containing The Given Lines. </li></ul><ul><li>Shortest Distance Between Two Lines In All Three Forms. </li></ul><ul><li>Condition For Lines To Intersect. </li></ul>
  72. 74. TIPS SECTION
  73. 75. TIPS FOR EXAMINATION <ul><li>First Of All , Keep Faith In Yourself. </li></ul><ul><li>On Entering Examination Hall, Your Confidence Level Is Like That You Are JEE-2012 Topper But Don’t Let This Confidence To Become Over-Confidence. </li></ul><ul><li>During Examination, First Make A Quick View On Q. Paper. </li></ul>
  74. 76. <ul><li>Then Select The Question Which You Think ,You Will Solve Easily. </li></ul><ul><li>Don’t Lose Your Confidence When You Are Not Able To Solve Any Problem. </li></ul><ul><li>Keep In Mind, Some Questions Of JEE Are Not Given For Solving But Are Given To Leave . You Must Develop A Sense About Selection Of Question. </li></ul>
  75. 77. CONTACTS PHONE NO : 07501541135 EMAIL – ID : [email_address] Facebook.com/bilalshakir1 THANK YOU PHONE NO : 08927482599 EMAIL – ID : [email_address] Facebook.com : Ashnil Kumar

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