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fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM/IISc, JEST, TIFR and GRE in PHYSICAL SCIENCES Basic Mathematics Formula Sheet for Physical Sciences Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 1 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES Basic Mathematics Formula Sheet for Physical Sciences 1. Trigonometry (3-9) 1.1 Trigonometrical Ratios and Identities ... (3-7) 1.2 Inverse Circular Functions ... . (8-9) 2. Differential and integral Calculus . (10-20) 2.1 Differentiation ... . ... (10-12) 2.2 Limits .. (13-14) 2.3 Tangents and Normal .. .(15-16) 2.4 Maxima and Minima ..(16) 2.5 Integration ....(17-19) 2.5.1 Gamma integral ..(19) 3. Differential Equations . ....(20-22) 4. Vectors . .................................................................................(23-25) 5. Algebra . .................................................................................(26-32) 5.1 Theory of Quadratic equations ...(26) 5.2 Logarithms ..(27) 5.3 Permutations and Combinations ......(28-29) 5.4 Binomial Theorem ......................(30) 5.5 Determinants ....................................(31-32) 6. Conic Section . ..........................................................................(33) 7. Probability . ...........................................................................(34-35) Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 2 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 1. Trigonometry 1.1 Trigonometrical Ratios and Identities 1. sin 2 cos 2 1 2. sec 2 1 tan 2 3. cos ec 2 1 cot 2 4. tan sin cos 5. cot cos sin 6. sin 1 cos ec 7. cos 1 sec 8. tan 1 cot Addition and Subtraction Formulae For any two angles A and B 1. Sin A B sin A cos B cos A sin B 2. Sin A B sin A cos B cos A sin B 3. cos A B cos A cos B sin A sin B 4. cos A B cos A cos B sin A sin B 5. tan A B tan A tan B 1 tan A. tan B 6. tan A B tan A tan B 1 tan A. tan B Double Angle Formulae 2. cos 2 cos 2 sin 2 1 2 sin 2 2 cos 2 1 1. sin 2 2 sin cos , 3. tan 2 2 tan 1 tan 2 Triple angle Formulae 1. sin 3 3 sin 4 sin 3 3. tan 3 2. cos 3 4 cos3 3 cos 3 tan tan 3 1 3 tan 2 Trigonometric Ratios of /2 1. sin 2 sin cos , 2 2 2. cos cos 2 sin 2 2 cos 2 1 2 sin 2 2 2 2 2 2 3. tan 2 1 tan 2 2 tan Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 3 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES Formulae for sin2 & cos2 in terms of tan 1. sin 2 2 tan 1 tan 2 2. cos 2 1 tan 2 1 tan 2 Formulae for sin & cos in terms of tan /2 2 1. sin 1 tan 2 2 2 2. cos 1 tan 2 2 1 tan 2 2 tan Transformation of sum/differences into Products C D C D 1. sin C sin D 2 sin cos 2 2 C D C D 2. sin C sin D 2 cos sin 2 2 C D C D 3. cos C cos D 2 cos cos 2 2 C D C D C D D C 4. cos C cos D 2 sin sin 2 sin sin 2 2 2 2 Transformations of Products into sum/difference 1. 2 SinA cos B Sin( A B) Sin( A B) 2. 2 cos A sin B Sin( A B) Sin( A B) 3. 2 cos A cos B cos( A B) cos( A B) 4. 2 sin A sin B cos( A B) cos( A B) Trigonometric Ratios of (- ) 1. sin sin 2. cos cos 3. tan tan 4. cot 5. sec cot 6. cos ec cos ec Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 4 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES Trigonometric Ratio of : (All Positive) 2 1. cos sin 2 2. sin cos 2 3. tan cot 2 4. cot tan 2 5. cos ec sec 2 6. sec cos ec 2 Trigonometric Ratio of :( Only sin and cos ec is Positive) 2 1. cos sin 2 2. sin cos 2 3. tan cot 2 4. cot tan 2 5. cos ec sec 2 6. sec cos ec 2 Trigonometric Ratios of :( Only sin and cos ec is Positive) 1. cos cos 2. sin sin 3. tan tan 4. cot cot 5. cos ec cos ec 6. sec sec Trigonometric Ratios of :( Only tan and cot is Positive) 1. cos cos 2. sin sin 3. tan tan 4. cot cot 5. cos ec cos ec 6. sec sec Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 5 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 3 Trigonometric Ratio of :( Only tan and cot is Positive) 2 3 sin 1. cos 2 3 cos 2. sin 2 3 3. tan cot 2 3 4. cot tan 2 3 sec 5. cos ec 2 3 cos ec 6. sec 2 3 Trigonometric Ratio of :( Only cos and sec is Positive) 2 3 sin 1. cos 2 3 cos 2. sin 2 3 3. tan cot 2 3 4. cot tan 2 3 sec 5. cos ec 2 3 cos ec 6. sec 2 Trigonometric Ratios of 2 :( Only cos and sec is Positive) 1. cos 2 cos 2. sin 2 sin 3. tan 2 tan 4. cot 2 cot 5. cos ec 2 cos ec 6. sec 2 sec Trigonometric Ratios of 2 : (All Positive) 1. cos 2 cos 2. sin 2 sin 3. tan 2 tan 4. cot 2 cot 5. cos ec 2 cos ec 6. sec 2 sec Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 6 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES Short-cut method to remember the Trigonometric ratios n 1. sin sin 2 n cos 2. cos 2 when n is an even integer n tan 3. tan 2 n cos 4. sin 2 n 5. cos sin 2 when n is an odd integer n cot 6. tan 2 0o 30o sin 0 cos tan 45 o 60o 1 3 2 1 2 1 2 1 3 2 0 1 2 1 0 1 2 1 90o 120o 3 3 2 135o 2 1 1 2 3 -1 2 3 cot 3 1 1 0 sec 1 2 2 -1 0 3 2 -1 0 1 1 0 0 -1 3 0 -2 2 -1 1 -1 3 cosec 2 3 3 2 0 2 1 3 180o 270o 360 o 1 2 1 150o 3 2 2 3 1 2 2 2 3 Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 7 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 1.2 Inverse Circular Functions 1. sin 1 sin x x 2. cos 1 cos x x 3. tan 1 tan x x 4. cot 1 cot x x 5. sec 1 sec x x 6. cos ec 1 cos ec x 9. sec sec x x 8. cos cos 1 x x 10. cos ec cos ec 1 x x 1 11. sin 1 cos ec 1 x x 1 12. cos 1 sec 1 x x 1 13. tan 1 cot 1 x x 1 14. cot 1 tan 1 x x 1 15. sec 1 cos 1 x x 1 16. cos ec 1 sin 1 x x 17. sin 1 x sin 1 x 18. cos 1 x cos 1 x 7. sin sin 1 x x 1 19. tan 1 x tan 1 x 20. tan 1 x cot 1 x 19. sin 1 x cos 1 x 2 2 21. sec 1 x cos ec 1 x 2 22. sin 1 1 x 2 cos 1 x 23. cos 1 1 x 2 sin 1 x 24. tan 1 x 2 1 sec 1 x 25. cot 1 x 2 1 cos ec 1 x 26. sec 1 1 x 2 tan 1 x 27. cos ec 1 1 x 2 cot 1 x 28. sin 1 2 x 1 x 2 2 sin 1 x 29. sin 1 3 x 4 x 3 3 sin 1 x 30. cos 1 4 x 3 3x 3 cos 1 x 2x 31. tan 1 2 tan 1 x 2 1 x 3x x 3 1 32. tan 1 1 3x 2 3 tan x x y 1 1 33. tan 1 1 xy tan x tan y x y 1 1 34. tan 1 1 xy tan x tan y Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 8 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES Some Important Expansions: 1. sin x x x3 x5 ......... 3! 5! 2. sinh x x x 3 x5 ......... 3! 5! 3. cos x 1 4. cosh x 1 x2 x4 ......... 2! 4! 1 2 5. tan x x x 3 x 5 ......... 3 15 x 2 x3 6. e 1 x ......... 2! 3! x 7. e x x2 x4 ......... 2! 4! x 2 x3 1 x ......... 2! 3! Some useful substitutions:Expressions Substitution Formula Result 3x 4 x 3 x sin 3 sin 4 sin 3 Sin3 4 x 3 3x x cos 4 cos 3 3 cos cos3 3x x 3 1 3x 2 x = tan 3 tan tan 3 1 3 tan 2 tan3 2x 1 x2 x = tan 1 x2 1 x2 x = tan 1 tan 2 1 tan 2 cos2 2x 1 x2 x = tan 2 tan 1 tan 2 tan2 1 2 x2 x = sin 1 2 sin 2 cos2 2x2 1 x = cos 2 cos 2 1 cos2 1 x2 x = sin 1 sin 2 cos2 1 x2 x = cos 1 cos 2 sin2 x2 1 x = sec sec 2 1 tan2 x2 1 x = cosec cos ec 2 1 cot2 1 x2 x = tan 1 tan 2 sec2 1 x2 x = cot 1 cot 2 cosec2 2 tan 1 tan 2 Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com sin2 Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 9 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 2. Differential and integral Calculus 2.1 Differentiation f x h f x h 1. f x lim h 0 2. f ' a lim h 0 f a h f a h 3. dy y lim dx x 0 x 4. d k 0 ; k is constant function dx 5. d x 1 dx 6. d dx 7. d 1 dx x n 9. d 1 1 dx x x 2 n n 1 x x 2 1x 8. d n x nx n 1 ; n N dx 10. d sin x cos x dx 11. d cos x sin x dx 12. d tan x sec 2 x dx 13. d sec x sec x. tan x dx 14. d cos ecx cos ecx. cot x dx 15. d x a a x log a; a 0, a 1 dx 16. d x e ex dx 17. d log x 1 dx x 18. d 1 sin 1 x ; 1 x 1 dy 1 x2 19. d 1 cos 1 x ; 1 x 1 dx 1 x2 20. d 1 tan 1 x ; x R dx 1 x2 21. d 1 cot 1 x ; x R dx 1 x2 22. d 1 sec 1 : dx x x2 1 23. d 1 cos ec 1 x ; dx x x2 1 x 1 x 1 Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 10 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES Rules of Differentiations 1. Addition Rule: If y = (u + v) then dy du dv dx dx dx 2. Substations Rule: If y = (u - v) then 3. Product Rule: If y = uv then dy dv du u v dx dx dx u dy 4. Quotient Rule: If y then v dx 5. If y = f(u) is u = g(x) then 6. If u = f(y) , then dy du dv dx dx dx v du dv u dx dx 2 v dy dy du . dx du dx du du dy dy . f y dx dy dx dx dy dx dy 1 dx . 1 or where 0 dx dx dy dx dy dy 7. Derivatives of composite functions 1. d f x n n f x n 1 . d f x dx dx 2. d dx 3. d 1 1 d . f x 2 dx f x f x dx 4. d sin f x cos f x . d f x dx dx 5. d cos f x sin f x . d f x dx dx 6. d tan f x sec 2 f x . d f x dx dx 7. d cot f x cos ec 2 f x . d f x dx dx 8. d sec f x sec f x tan f x . d f x dx dx 9. d cos ecf x cos ecf x cot f x . d f x dx dx 10. d log f x 1 . d f x dx f x dx 11. f x 1 d f x 2 f x dx d f x d a a f x log a. f x dx dx Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 11 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 12. d f x d e e f x f x dx dx 13. d f g x n n f g x n 1 f g x d g x dx dx Derivatives of composite functions 1. d 1 d sin 1 f x . f x 2 dx 1 f x dx 2. d 1 d cos 1 f x . f x 2 dx 1 f x dx 3. d 1 d tan 1 f x . f x 2 dx 1 f x dx 4. d 1 d cot 1 f x . f x 2 dx 1 f x dx 5. d sec 1 f x dx f x 6. d 1 d cos ec 1 f x . f x 2 dx f x f x 1 dx 1 f x 2 d f x dx 1 . Implicit functions:Take the derivatives of these functions directly and find dy/dx Parametric functions:- dy dy / dx dx dx / dt If x f t & y g t then where dx 0 dt g x Logarithemic Differentiation:- If the function is in the form of f x Then taking Logarithm on both sides 15 & then find dy/dx Higher order Derivatives of composite functions:- y2 d2y f " x dx 2 In General; y n IInd order, dny f dx n n x y3 d3y f " ' x dx 3 IIIrd order nth order Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 12 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 2.2 Limits Limits of function If for every 0 there exist > 0 such that if f x l whenever 0 x a then we say lim of f(x) as x a is l i.e. lim f x l x a Theorem of limits If f(x) and g(x) are two functions then 1. lim f x g x lim f x lim g x x a x a x a 2. lim f x g x lim f x lim g x x a x a x a 3. lim f x .g x lim f x . lim g x x a 4. lim x a x a x a f x f x lim x a g x lim g x x a 5. lim kf x k lim f x where k is constant x a 6. lim x a x a f x lim f x x a 7. lim f x p/q x a lim f x x a p/q : where p & q are integers Some Important standard limits 1. lim x a 2. lim c c : where c is constant c R x a 3. lim x n a n ; x a x a 4. lim n R x a sin 1 0 n N,a 0 6. lim tan 1 0 xn an na n 1 ; x a 1 0 sin 8. lim 5. lim 1 0 tan 7. lim 9. lim 0 sin k k 10. lim 0 tan k k Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 13 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES x2 2 x 0 1 cos x 1 cos x 1 x 0 2 x 11. lim 12. lim sin x 0 sin mx 0 m ; lim x 0 x 18 x 0 sin nx 0 n 1 cos kx k 2 x 0 2 x2 13. lim 15. lim cos x 1; x 0 14. lim a x 1 log a where a > 0 x a x ex 1 1 x a x 20. lim 1 x 1/ x x 0 18. lim log a 1 x log a e x a x log 1 x 1; x a x a 0 lim lim 1 kx 1/ x e; x / 2 x 0 17. lim 19. lim lim sin / 2 1 16. lim sin x 0; lim cos x 0 x / 2 x 0 ek log 1 kx 21. lim k x 0 x cos ax cos bx a 2 b 2 22. lim 2 x 0 cos cx cos dx c d2 2 2 cos ax cos bx b a 23. lim x 0 2 x 1 ax 24. lim x 0 1 bx a bx 25. lim x 0 a cx 1/ x e b c a 26. lim x 1 1 0; lim 2 0 2 x x x x 28. lim x 30. lim sin x sin a 31. lim cos x cos a x a x 1 1 1 0; lim 0 x x x lim k k where k is constant x 32. lim e a b 1 0 where k >0 x x k 27. lim 29. lim k k ; 1/ x x a x x 2 x 3 ...x n n n n 1 x 1 2 ax bx a log ; 33. lim x 0 x b a, b 0 sec x 1 1 cos ecx 1 ; lim 1 2 x 0 2 x 0 x x2 34. lim x n 1 1 35. lim 1 2 e x ; lim 1 e x n x n h a 36. lim 1 e a h h Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 14 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 2.3 Tangents and Normal Tangent at x, y to y f x Let y f x be a given curve and P x, y and x x, y y Q x x, y y be two neighbouring points on it. Q Equation of the line PQ is Y y y y y X x or Y y y X x x x x x P x, y This line will be tangent to the given curve at P if Q P which in tern means that x 0 and we know that lim x 0 y dy x dx Therefore the equation of the tangent is Y y dy X x dx Normal at x, y The normal at x, y being perpendicular to tangent will have its slope as 1 and dy dx hence its equation is Y y 1 X x dy dx Geometrical meaning of dy dx dy dx represents the slope of the tangent to the given curve y f x at any point x, y dy tan dx where is the angle which the tangent to the curve makes with +ve direction of x-axis. dy dy In case we are to find the tangent at any point x1 , y1 then i.e. the value of dx dx x1 , y1 at x1 , y1 will represent the slope of the tangent and hence its equation in this case will be Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 15 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES dy y y1 x x1 dx x1 , y1 y y1 Normal 1 x x1 dy dx x1 , y1 Condition for tangent to be parallel or perpendicular to x-axis If tangent is parallel to x-axis or normal is perpendicular to x-axis then dy 0 dx If tangent is perpendicular to x-axis or normal is parallel to x-axis then dx dy 0. or dx dy 2.4 Maxima and Minima For the function y f x at the maximum as well as minimum point the tangent is parallel to x-axis so that its slope is zero. Calculate dy dy 0 and solve for x. Suppose one root of 0 is at x=a. dx dx If d2y ve for x=a, then maximum at x=a. d 2x If d2y ve for x=a, then minimum at x=a. d 2x If d2y d3y 0 at x=a, then find . d 2x d 3x If d3y 0 at x=a, neither maximum nor minimum at x=a. d 3x d3y d4y If 3 0 at x=a, then find 4 . d x d x If d4y d4y 0 i.e +ve at x=a, then y is minimum at x=a and if 0 i.e -ve at x=a, then y d 4x d 4x is maximum at x=a and so on. Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 16 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 2.5 Integration Indefinite Integration If d F x c f x , then we say that F x c is an indefinite integral or dx antiderivative of f x and we write f x dx F x c Some standard Integrals x n 1 1. x dx c n 1 n 3. 1 x n 1 1 1 c n 1 n 1 x n 1 x 4. dx 2 x c 2. x dx log x c n dx 1 ax c log a 5. e x dx e x c 6. a x dx 7. cos x dx sin x c 8. sin x dx cos x c 9. sec 2 xdx tan x c 10. cos ec 2 x dx cot x c 11. sec x tan x dx sec x c 12. cos ecx cot x dx cos ecx c dx dx tan 1 x c sin 1 x c 14. sec 1 x c 16. cosh x dx sinh x c 17. sinh x dx cosh x c 18. sec h 2 xdx tanh x c 13. 15. 1 x 2 dx 2 x x 1 19. cos ech 2 x dx coth x c 1 x 2 20. sec hx tanh x dx sec hx c 21. cos echx coth x dx cos echx c 22. tan xdx log sec x c 23. cot x dx log sin x c 24. sec x dx log sec x tan x c x 25. cos ecx dx log tan c 2 26. a 2 dx 1 x tan 1 c 2 a a x Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 17 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 27. x 28. 2 dx 1 x a log c; 2 2a a x a if x a dx 1 a x log c ; if x a 2 2a a x a x 2 x log x x 2 a 2 c or sinh 1 c a x2 a2 dx 30. 31. dx 29. 2 x a 2 dx 2 a x 2 x or cosh 1 c a log x x 2 a 2 c sin 1 x c a x 2 a 2 dx x 2 a2 x a2 log x x 2 a 2 2 2 x 2 a 2 dx x 2 a2 x a2 log x x 2 a 2 2 2 34. a 2 x 2 dx x a2 x a2 x2 sin 1 c 2 2 a 35. x 32. 33. dx x2 a2 a 36. sin 0 1 x 1 x sec 1 c cos ec 1 c a a a a m x n x a sin dx mn L L 2 0, m n a , 2 m n Rules of Integration 1. f1 x f 2 x dx f1 x dx f 2 x dx 2. k f x dx k f x dx , where k is constant 3. k f x k f x dx k f x dx k f x dx , where k1 and k2 are constants 1 1 2 2 1 1 2 2 Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 18 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES Rule of integration by substitution 1. If x t , dx f x dx f x dt dt f t ' t dt g ax b c a 2. f ax b dx 3. f x f x f ' x dx 4. f x dx log f x c n 1 n c : n 1 n 1 f ' x Rules of integration by partial fraction This method can be used to evaluate an integral of the type P x Q x dx where (i) P(x) & Q(x) are Polynomials in x (ii) Degree of P(x) < degree of Q(x) (iii) Q(x) contains two/more distinct linear/quadratic factors i.e. P x A B C Q x a1 x b1 a 2 x b2 a3 x b3 du dx uvdx u vdx vdx dx 2.5.1 Gamma integral (i) Gamma integral is given by (n) x n 1e x dx = (n 1) . 0 1 2 (ii) n Bx 2 x e 0 dx 1 2B n 1 2 n 1 2 Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 19 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 3. Differential Equations Order and degree of a differential equation The order of a differential equation is the order of the highest differential co-efficient present in the equation. d2y dy Example: 2 2 3 y 0 is a second order differential equation. dx dx The degree of a differential equation is the degree of the highest derivative after removing the radical sign and fraction. 2 3 d2y dy Example: dx 2 2 dx 3 y 0 has degree of 3. D.E. of the first order and first degree 1. Separation of the variables: f y dy x dx 2. Homogeneous Equation dy f x, y x, y is of the same degree. dx x, y if each term of f(x,y) and 3. Equations reducible to homogeneous form x X h dy ax by c dy dY aX bY ah bk , let y Y k dx Ax By C dx dX AX BY Ah Bk dy aX bY ah bk c 0 Choose h, k so that Ah Bk C 0 dx AX BY a b 1 Case of failure: A B m 4. dy ax by c dx m ax by C Linear Differential Equations dy Py Q where P and Q are function of x (but not y) or constant. dx I .F . e Pdx y I .F . Q I .F . dx c Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 20 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 5. Equations reducible to the linear form dy Py Qy n dx 1 divide by yn and put y n 1 z 1 dy 1 n 1 P Q n y dx y 1 n dy dz yn dx dx 1 dz Pz Q 1 n dx 6. Exact differential Equation M N Mdx + Ndy = 0 if y x Mdx terms of N not containing x dy C y constant 7. Equations reducible to the exact form M N x is a function of x alone, say f(x) then .F. e f x dx multiply with a) If y N different equation. M N x is a function of y alone, say f(y) then .F. e f y dy . b) If y N 1 c) If M = yf1 (xy) and N = xf2 (xy), then .F . Mx Ny Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 21 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES Linear D.E. of second order with constant coefficients d2y dy P Qy R where P, Q and R are function of x or constant. 2 dx dx y = C.F. + P.I. C.F. a) roots, real and different y C1 e m1x C 2 e m2 x b) roots, real and equal y C1 C 2 x e m2 x c) roots imaginary y C1e x C 2 e x e x A cos x B sin x P.I. 1 1 ax ax a) f D e f a e if f a 0 then 1 1 e ax x e ax f D f a 1 1 n n b) f D x f D x 1 1 1 1 c) f D 2 sin ax f a 2 sin ax and f D 2 cos ax f a 2 cos ax 1 1 If f(-a2) = 0 then f D 2 sin ax f a 2 sin ax 1 1 ax ax d) f D 2 e x e f D a x e) 1 x e ax e ax x dx D a 1 1 n ax n f) f D x sin ax Im e f D a x Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 22 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 4. Vectors Cartesian coordinate system dy y dz z Infinitesimal displacement dl dxx Volume element d dxdydz f f f Gradient: f x y z x y z Divergence: A A A . A x y z x y z Curl: A A A A x z A z y x z x z y Laplacian: 2 f A A y x z y y x 2 f 2 f 2 f x 2 y 2 z 2 Spherical Polar Coordinate System( r , , ) x r sin cos , y r sin sin , z r cos z y r x 2 y 2 z 2 , cos 1 , tan 1 r x r sin d rd Infinitesimal displacement dl drr 2 Volume element d r sin drd d r range from 0 to , from 0 to , and from 0 to 2 . Gradient: f 1 f 1 f f r r r r sin Divergence: 1 1 1 A . A 2 r 2 Ar sin A r r r sin r sin Curl: r 1 A 2 r sin r A r r rA r sin r sin A Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 23 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES Laplacian: 2 f 1 2 f 1 f 1 r 2 sin 2 2 2 r r r r sin r sin 2 f 2 Cylindrical Coordinate System ( r , , z ) r x2 y 2 , x r cos , y r sin , z z and tan 1 y x dz z rd Infinitesimal displacement d l drr Volume element d rdrd dz r range from 0 to , from 0 to 2 , and z from to . Gradient: f 1 f f f r z r r z Divergence: 1 1 A A . A rAr z r r r z Curl: r 1 A r r A r Laplacian: 2 f r rA z z Az 1 f 1 2 f 2 f r r r r r 2 2 z 2 Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 24 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES VECTOR IDENTITIES Triple Product (1) A.( B C ) B.(C A) C.( A B) (2) A ( B C ) B( A.C ) C ( A.B) Product Rules (3) ( fg ) f ( g ) g ( f ) (4) A.B A ( B) B ( A) A. B ( B. ) A (5) .( f A) f ( . A) A.( f ) (6) .( A B) B.( A) A.( B) (7) ( f A) f ( A) A ( f ) (8) A B ( B. ) A ( A. ) B A( .B) B( . A) Second Derivative (9) .( A) 0 i.e. divergence of a curl is always zero. (10) ( f ) 0 i.e. curl of a gradient is always zero. (11) ( A) . A 2 A FUNDAMENTAL THEOREMS b Gradient Theorem: f .dl f b f a a Divergence Theorem: Curl Theorem: . A d A.d a A .d a A.dl Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 25 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 5. Algebra 5.1 Theory of Quadratic equations 1. Roots of the equation b b 2 4ac 2a ax 2 bx c 0 are x Sum and Product of the roots If and be the roots, then b c and . a a 2. To find the equation whose roots are and . The required equation will be x x 0 or x 2 x . 0 or x 2 Sx P 0 where S is the sum and P is the product of the root. 3. Nature of the roots. Roots of the equation ax 2 bx c 0 are x b b 2 4ac . 2a The expression b 2 4 ac is called discriminant. (a) If b 2 4 ac 0 , roots are real. (i) If b 2 4 ac 0 , then roots are real and unequal. b (ii) If b 2 4 ac 0 , then roots are real and equal . 2a (b) If b 2 4 ac 0 , then b 2 4ac is imaginary. Therefore roots are imaginary and unequal. Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 26 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 5.2 Logarithms Properties of Logarithms ( a 0, a 1, m 0, n 0 ) 1. a x y then x log a y 2. log a a 1 3. log a 1 0 4. log b a 5. Base changing formula log b a log c a. log b c 6. log a mn log a m log a n , 1 or log b a. log a b 1 log a b log c a log c b n log log a m a m log a n 7. log a m n n log a m Or in particular log a a n n p p q 8. log a q n q q log a n Or in particular log n q n q 9. a log a n n Rules of indices 1. a m a n a m n 3. a m n 2. am a m n n a m 4. a b a m b m a mn m am a 5. m b b 6. a m 1 am 7. a 0 1 Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 27 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 5.3 Permutations and Combinations Permutation Each of the different arrangements which can be made by taking some or all of a number of things is called permutation. Combination Each of the different groups or selections which can be made by taking some or all of a number of things (irrespective of the order) is called combination. Fundamental Theorem If there are m ways of doing a thing and for each of the m ways there are associated n ways of doing a second thing then the total number of ways of doing the two things will be mn. Important Results (a) Number of permutations of n dissimilar things taken r at a time. n Pr n! n n 1 n 2 ....... n r 1 n r ! where n! 1.2.3...............n . Note that n! n. n 1 ! n. n 1 . n 2 ! (b) Number of permutations of n dissimilar things taken all at a time. n Pn n! n n 1 n 2 ....... n n 1 n r ! n n 1 n 2 ..........3.2.1 n! (c) Number of combinations of n dissimilar things taken r at a time. n Cr n P n! r n r !r! r! (d) Number of combinations of n dissimilar things taken all at a time. n Cn n! 1 1 n n !n! 0! 0! 1 Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 28 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES (e) If out of n things p are exactly alike of one kind, q exactly alike of second kind and r exactly alike of third kind and the rest all different, then the number of permutations of n things taken all at a time n! p!.q!.r! (f) If some or all of n things be taken at a time then the number of combinations will be 2n 1 n C1 n C 2 .......... n C n 2 n 1 (g) n C r n C n r (h) n C r1 n C r2 r1 r2 or r1 r2 n . (i) n C r n C r 1 n 1C r Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 29 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 5.4 Binomial Theorem (a) Statement of binomial theorem for positive and negative integral index x a n x n n C1 x n 1 a 1 n C 2 x n 2 a 2 ........ n C r x n r a r ...... n C n 1 xa n 1 n C n a n x a n x n n C1 x n 1 a 1 n C 2 x n 2 a 2 ........ n C r x n r a r ................. (b) Number of terms and middle term n The number of terms in the expansion of x a is n 1 . n If n is even there will be only one middle term i.e. 1 th . 2 n 1 n 3 If n is odd there will be two middle terms i.e. th and th . 2 2 Expansion 2 2. a b a 2 2ab b 2 2 3 4. a b a 3 3ab a b b 2 1. a b a 2 2ab b 2 3 3. a b a 3 3ab a b b 3 Factorization 1. a 2 b 2 a b a b a b a a b a 3. a 3 b 3 a b a 2 ab b 2 5. a n b n 6. a n b n n 1 n 1 2. a 3 b 3 a b a ab b 2 4. a 4 b 4 a b a b a b 2 .... a n 2 b a n 3 b 2 .... a n 2 b a n 3 b 2 Sterling s formula Using summation notation, binomial expansion can be written as x y Sterling s n n n x k y n k k 0 k approximation (or Sterling s formula) is an approximation for large factorials. ln n n ln n n where n is very large Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 30 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 5.5 Determinants In linear algebra the determinant is a value associated with a square matrix. The determinant of a matrix A is denoted by det( A) , or A . For instance, the determinant of the matrix a b a b If A then ad bc c d c d a b If A d e g h c a b f then det( A) = d e i g h c e f a h i f i b d f g i c d e g h Properties (a) The values of determinant is not altered by changing rows into columns and columns into rows. 1 y 1 1 z =1 x y x2 z x2 e.g. 1 x y2 z2 z z2 1 (b) If any two adjacent rows or two adjacent columns of a determinant are interchanged the determinant retains its absolute value but changes its sign. 1 y 1 x z = 1 y 1 z 1 x2 e.g. 1 x y2 z2 y2 z2 x2 (c) If any two rows or two columns of determinant are identical then the determinant vanishes. Thus a1 c1 c1 a2 c2 c2 a3 c3 c3 Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 31 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES (d) If each constituent in any row or in any column be multiplied by the same factor then the determinant is multiplied by that factor pa1 b1 c1 a1 b1 c1 pa 2 b2 c 2 p a2 b2 c2 and qa 2 pa3 b3 c3 b3 c3 a3 a1 ra 3 b1 c1 a1 b1 c1 qb2 q c2 qr a 2 b2 c2 rb3 rc3 b3 c3 a3 (e) If each constituent in any row or in any column consists of r terms then the determinant can be expressed as the sum of r determinants. a1 1 b1 c1 a1 1 c2 2 c3 3 b1 Thus a 2 2 b2 c2 a2 b2 a 3 3 b3 c3 c1 b3 a3 b1 c1 b2 c2 b3 c3 (f) If one row or column is k times the other row or columns respectively then determinant of matrix will be 0 . k .a k .d c a b c f 0 and k .a k.b k.c 0 g e.g. a d k .g i g h i Some basic properties of determinants are: 1. det( I n ) I where I n is the n n identity matrix. 2. det( AT ) det( A) where AT is transpose of A . 3. det( A 1 ) 1 where A 1 is inverse of A . det( A) 4. For square matrices A and B of equal size, det( AB) det( A) det( B) 5. det(cA) c n det( A) for an n n matrix Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 32 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 6. Conic Section In the Cartesian coordinate system the graph of a quadratic equation of two variables represent a conic section which is given by Ax 2 Bxy Cy 2 Dx Ey F 0 . The conic sections described by this equation can be classified with the discriminant D B 2 4 AC if D 0 , the equation represents an ellipse if D 0 , A C and B 0 , the equation represents a circle which is a special case of an ellipse; if D 0 , the equation represents a parabola if D 0 the equation represents a hyperbola if we also have D 0 , A C 0 , the equation represents a rectangular hyperbola Note that A and B are polynomial coefficients, not the lengths of semi-major/minor axis as defined in some sources. Conic Equation Eccentricity section x2 y 2 a2 Ellipse x2 y 2 1 a 2 b2 Hyperbola Polar equation Parametric form rectum Circle Parabola Semi-lactus a 0 e 1 b2 a2 y 2 4ax e 1 x2 y 2 1 a2 b2 b2 e 1 2 a l b2 a 2a l b2 a Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com r a x a cos , y a sin l 1 e cos r 0 e 1 x a cos , y b sin l 1 cos r x at 2 , y 2at l 1 e cos r e 1 x a tan , y b sec Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 33 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 7. Probability Probability The probability pr of occurrence of an event r in a system is defined with respect to statistical ensemble of N such a systems. If N r systems in the ensemble exhibit the event r then pr Nr N Probability density The probability density (u ) is defined by the property that (u )du yields the probability of finding the continuous variable u in the range between u and u du . Mean value The mean value of u is denoted by u as defined as u pr ur where the sum is over r all possible value values u r of the variable u and pr is denotes the probability of occurrence of the particular value u r .Above definition is for discrete variable . For continuous variable u , u = u (u )du Dispersions or variance The dispersion of u is defined as 2 ( u ) 2 pr (ur u )2 which is equivalent r to 2 ( u ) 2 ( u 2 u 2 ) r Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 34 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES Joint probability If both events A and B occur on a single performance of an experiment, this is called the intersection or joint probability of A and B, denoted as p ( A B) . Independent probability If two events, A and B are independent then the joint probability is p ( A B) p ( A).P( B) Mutually exclusive If either event A or event B or both events occur on a single performance of an experiment this is called the union of the events A and B denoted as p ( A B ) . If two events are mutually exclusive then the probability of either occurring is p ( A B) p ( A) P ( B) Not mutually exclusive If the events are not mutually exclusive then p ( A B ) p ( A) P ( B ) p ( A B ) Conditional probability Conditional probability is the probability of some event A, given the occurrence of some other event B. Conditional probability is written p ( A / B ) , and is read "the probability of A, given B". It is defined by p ( A / B) p ( A B) p ( B) Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 35

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