CO-ORDINATE GEOMETRY

Transcription

1 CO-ORDINATE GEOMETRY 1 To change from Cartesian coordinates to polar coordinates, for X write r cos θ and for y write r sin θ. 2 To change from polar coordinates to cartesian coordinates, for r 2 write X 2 + y 2 ; for r cos θ write X, for r sin θ. Write y and for tan θ write. 3 Distance between two points (X 1, Y 1 ) and (X 2, Y 2 ) is x 2 x 1 y 2 y 1 4 Distance of ( x 1, y 1 ) from the origin is x 2 1 y Distance between (r 1, θ 1 ) and (r 2, θ 2 ) is r 2 1 r r 1 r 2 cos θ 2 θ 1 6 Coordinates of the point which divides the line joining (X 1, Y 1 ) and (X 2, Y 2 ) internally in the ratio m 1 : m 2 are :-, ( m 1 + m 2 0 ) 7. Coordinates of the point which divides the line joining (X 1, Y 1 ) and (X 2, Y 2 ) externally in the ratio m 1 : m 2 are :-, (m 1 m 2 0) 8. Coordinates of the mid-point (point which bisects) of the seg. Joining (X 1, y 1 ) and (X 2 y 2 ) are :

2 , 9. (a) Centriod is the point of intersection of the medians of triangle. (b) In-centre is the point of intersection of the bisectors of the angles of the triangle. (c) Circumcentre is the point of intersection of the right (perpendicular) bisectors of the sides of a triangle. (d) Orthocentre is the point of intersection of the altitudes (perpendicular drawn from the vertex on the opposite sides) of a triangle. 10. Coordinates of the centriod of the triangle whose vertices are (x 1, y 1 ) ; (x 2, y 2 ) ; ( x 3, y 3 ) are 11. Coordinates of the in-centre of the triangle whose vertices are A (x 1,y 1 ) ; B (x 2,y 2, ) ; C (x 3,y 3 ) and 1 (BC ) a, 1 (CA) b, 1 (AB) c. are. 12 Slope of line joining two points (x 1,y 1 ) and (x 2,y 2 )is m 13. Slope of a line is the tangent ratio of the angle which the line makes with the positive direction of the x-axis. i.e. m tan θ 14. Slope of the perpendicular to x-axis (parallel to y axis) does not exist, and the slope of line parallel to x-axis is zero.

3 15. Intercepts: If a line cuts the x-axis at A and y-axis at B then OA is Called intercept on x-axis and denoted by a and OB is called intercept on y-axis and denoted by b. 16. X a is equation of line parallel to y-axis and passing through (a, b) and y b is the equation of the line parallel to x-axis and passing through (a, b). 17. X 0 is the equation of y-axis and y 0 is the equation of x-axis. 18. Y mx is the equation of the line through the origin and whose slope is m. 19. Y mx +c is the equation of line in slope intercept form is the equation of line in the Double intercepts form, where a is x-intercept and b is y-intercept. 21. X cos a + y sin a p is the equation of line in normal form, where p is the length of perpendicular from the origin on the line and α is the angle which the perpendicular (normal) makes with the positive direction of x-axis. 22. Y Y 1 m (x x 1 ) is the slope point form of line which passes through (x 1, y 1 )and whose slope is m. 23. Two points form: - y-y 1 (x x 1 ) is the equation of line which Passes through the points (x 1, y 1 ) and (x 2, y 2 ). 24. Parametric form :- r is the equation of line which

4 passes through the point (x 1 y 1 )makes an angle θ with the axis and r is the distance of any point (x, y) from ( x 1, y 1 ). 25. Every first degree equation in x and y always represents a straight line ax + by + c 0 is the general equation of line whose. (a) Slope - - (b) X - intercept - (c) Y- intercept Length of the perpendicular from (x 1, y 1 ) on the line ax + by + c 0 is 27. To find the coordinates of point of intersection of two curves or two lines, solve their equation simultaneously. 28. The equation of any line through the point of intersection of two given lines is (L.H.S. of one line) +K (L.H.S. of 2nd line) 0 (Right Hand Side of both lines being zero)

5 TRIGONOMETRY 29. SIN 2 + Cos 2 1; Sin Cos 2, Cos 2 1 Sin tan θ ; cot ; sec ; Cosec ; cot tan 2 sec 2 ; tan 2 sec 2-1 ; Sec 2 - tan cot 2 cosec 2 ; cot 2 cosec 2-1; Cosec 2 - cot Y Only sine and cosec are positives all trigonometric ratios are positives O X X 1 III IV Only tan and cot are positives only cos and sec are positives Y 1

6 34. angle ratio 0 0 O Sin Cos Tan Sin (- ) = - Sin ; cos (-) = cos ; tan (- ) = - tan. 36. sin (90 ) cos cos (90 ) sin tan (90 ) cot cot (90 ) tan sec (90 ) cosec cosec (90 ) sec sin (90 + ) cos cos (90 + sin tan (90 + ) cot cot (90+ ) tan sec (90 + ) cosec cosec (90 + ) = sec sin (180 ) sin cos (180 cos tan ( 180 ) tan cot (180 ) cot sec (180 ) sec cosec (180 ) cosec

7 37. Sin (A + B) = SinA CosB + CosA SinB Sin (A- B) = CosA SinB - SinA CosB Cos (A + B) = CosA CosB - SinA CosB Cos (A B) = CosA CosB + SinA SinB tan (A + B) = tan (A - B) = 38. tan tan A A 39. SinC + SinD = 2 sin SinC - SinD = 2 cos CosC + CosD = 2 cos CosC - CosD = 2 sin cos sin cos sin sin A cos B = sin (A + B) + sin (A-B) 2 cos A sin B = sin (A + B) - sin (A-B) 2 cos A COS B cos ( A +B) + cos (A-B) 2 sin A sin B cos (A-B)- cos (A + B) 41. Cos (A +B). cos ( A - B ) = cos 2 A - sin 2 B Sin (A +B). sin (A B) = sin 2 A - sin 2 B

8 42. Sin 2θ = 2 sinθ cosθ = 43. Cos2 θ =cos 2 θ - sin 2- θ = 2cos 2 θ -1 = 1 2 sin 2 θ = ; cos 2θ = 2 cos 2 θ; 1 cos 2 θ = 2 sin 2 θ 45. tan 2 θ = ; 46. sin 3 = 3 sin - 4 sin 3 ; cos 3 = 4 cos 3-3 cos ; tan 3 = 47. = = 48. Cos A = ; Cos B ; Cos C ; 49. a = b cos C + c cos B; b = c cos A + a cos C ; c = a cos B + b cos A 50. Area of triangle =

9 bc sin A = ca sin B = ab sin c sin A = (cos A/ 2 sin A/ 2 ) sec A tan A = tan /2 53. Cosec A - cot A = tan A/ Cosec A + cot A = cot A/ 2 P A I R O F L I N E S 1. A homogeneous equation is that equation in which sum of the powers of x and y is the same in each term. 2. If m 1 and m 2 be the slopes of the lines represented by ax 2 + 2hxy + by 2 = 0, then m 1 + m = - and m 1 +m 2 = = 3. If be the acute angle between the lines represented by ax 2 + 2hxy + by 2 = 0, then tan = These lines will be co incident (parallel) if h 2 = ab and perpendicular if a +b = The condition that the general equation of the second degree viz ax 2 + 2hxy + by 2 +2gx +2fy + c = 0 may represent a pair of straight line is abc + 2fgh af 2 bg 2 - ch 2 = 0

10 i.e. = Ax 2 + 2hxy + by 2 = 0 and ax 2 + 2hxy + by 2 +2gx +2fy + c = 0 are pairs of parallel lines. 6. The point of intersection of lines ax 2 + 2hxy + by 2 + 2gx + 2fy + c = 0 is obtained by solving the equation ax + hy + g = 0 and hx + by + f = Joint equation of two lines can be obtained by multiplying the two equations of lines and equating to zero. (UV =0, where u = 0, v = 0). 8. If the origin is changed to (h,k) and the axis remain parallel to the original axis then for x and y put x + h and y + k respectively. C I R C L E 1. X 2 + y 2 = a 2 is the equation of circle whose centre is (0, 0) and radius is a. 2. (x h) 2 + (y - k) 2 = a 2 is the equation of a circle whose centre is (h, k) and radius is a.

11 3. X 2 + y 2 + 2gx + 2fy + c = 0 is a general equation of circle, its centre is (-g,-f) and radius is g f c. 4. Diameter form: - (x x 1) (x x 2 ) + (y y 1 ) (y- y 2 ) = 0 is the equation of a circle whose (x 1, y 1 ) and (x 2, y 2 ) are ends of a diameter. 5. Condition for an equation to represent a circle are : (a) Equation of the circle is of the second degree in x and y. (b) The coefficient of x 2 and y 2 must be equal. (c) There is no xy term in the equation (coefficient of xy must be zero). 1. To find the equation of the tangent at (x 1, y 1 ) on any curve rule is: In the given equation of the curve for x 2 put xx 1 ; for y 2 put yy 1 ; for 2x put x+ x 1 and for 2y put y +y 1 2. For the equation of tangent from a point outside the circle or given slope or parallel to a given line or perpendicular to a given line use y = mx + c or y y 1 = m (x x 1 ). 3. For the circle x 2 + y 2 = a 2 (a) Equation of tangent at (x 1, y 1 ) is xx 1 + yy 1 = a 2 (b) Equation of tangent at (a cos, a sin ) is x cos + y sin = a. (C) Tangent in terms of slope m is

12 Y = mx a 1 4. For the circle x 2 + y 2 + 2gx + 2fy + c = 0 (a) Equation of tangent at (x 1, y 1 ) is Xx 1 + yy 1 + g (x + x 1) + f ( y + y 1 ) + c = 0 (b) Length of tangent from (x 1, y 1 ) is For the point P (x, y), x is abscissa of P and y is ordinate of P. P A R A B O L A 1. Distance of any point P on the parabola from the focus S is always equal to perpendicular distance of P from the directrix i.e. SP = PM. 2. Parametric equation of parabola y 2 = 4ax is x = at 2, y = 2at. Coordinates of any point (t) is (at 2, 2at) 3. Different types of standard parabola Parabola Focus Directrix Latus rectum Axis of Parabola (axis of symmetry)

13 Y 2 = 4ax (a, 0) X = - a 4a Y = 0 Y 2 = - 4ax (-a, 0) X = a 4a Y = 0 X 2 = 4by (0, b) Y = - b 4b X = 0 X 2 = - 4by (0, -b) Y = b 4b X = 0 4. For the parabola y 2 = 4ax (a) Equation of tangent at (x 1, y 1 ) is Yy 1 = 2a (x + x1 ). (b) Parametric equation of tangent at (at 2 1, 2at 1) is yt 1 = x + at 2 1 (c) Tangent in term of slope m is y = mx + and its point of contact is (a/m 2, 2a/m) (d) If P (t 1 ) and Q (t 2 ) are the ends of a focal chord then t 2 t 1 = -1 (e) Focal distance of a point P (x 1, y 1 ) is x 1 + a. E L L I P S E Ellipse Foci Directrices Latus Rectum Equation of axis Ends of L.R

14 + =1 (a b) + =1 (a b ) ( ae, 0) (0, be) X = 1. Distance of any point on an ellipse from the focus = e (Perpendi cular distance of the point from the correspon ding Directrix) i.e. SP = e PM. 2a b major axis Y = 0 minor axis x = 0 major axis x = 0 minor axis y = 0 (ae, ) (ae, ) (, be ) ( ),be 2. Different types of ellipse Y = 3 Parametric equation of ellipse + = 1 (a b) is x = a cos θ

15 and y = b sin θ. 4. For the ellipse + = 1, a b, b2 =a 2 (1 =e 2 ) And + = 1, a b, a2 = b 2 (1 e) 5. For the ellipse + =1 (a b ) (a) Equation of tangent at x 1, y 1 ) is + = 1. (b ) Equation of tangent in terms of its slope m is y = mx a m b (c) Tangent at (a cos, b sin θ) is + = 1 6. Focal distance of a point P (x 1, y 1 ) is SP = a ex 1 and SP = ex 1 a H Y P E R B O L A 1. Distance of a point on the hyperbola from the focus = e (Perpendicular distance of the point from the corresponding directrix) i.e. SP =epm 2. Different types of Hyperbola

16 Hyperbola Foci Directrices L.R End of L.R Eqn of axis - = 1 ( ae, 0) X= 2b a (ae, ) (ae, - ) Transverse axis y= 0 conjugate axis x = o =1 (0, be) Y = (,be) (-,be) Transverse axis x=0 conjugate axis y =0 3. For the hyperbola - = 1, b2 = a 2 (e 2-1) and for = 1, a2 = b2 (e 2 1). 4. Parametric equations of hyperbola X = a sec, y = b tan - 5. For the hyperbola - = 1 (a) Equation of tangent at (x 1, y 1 ) are - = 1 = 1 are Equation of tangent in terms of its slope m is Y = mx

17 (c) Equation of tangent at (a sec, b tan ) is - = 1 (d) Focal distance of P (x 1, y 1 ) is S P = ex 1 a and S P = ex 1 + a S O L I D G E O M E T R Y 1. Distance between ( x 1, y 1, z 1 ) and ( x 2, y 2, z 2 ) is Distance of (x 1, y 1, z 1 ) from origin Coordinates of point which divides the line joining (x 1, y 1, z 1 ) and ( x 2, y 2, z 2 ) internally in the ratio m:n are,, m + n O (x 1,y 1, z 1 ) m n (x 2, y 2, z 2 ) 4. Coordinates of point which divides the joint of (x 1, y 1, z 1 ) and (x 2,y 2, z 2 ) externally in the ratio m:n are,, m - n O

18 5. Coordinates of mid point of join of ( x 1, y 1, z 1 ) and ( x 2, y 2, z 2 ) are,,. 6. Coordinates of centriod of triangle whose vertices are (x 1, y 1, z 1 ), (x 2, y 2, z 2 ) and (x 3, y 3, z 3 ) are,, 7. Direction cosines of x axis are 1, 0, 0 8. Direction cosines of y axis are 0, 1, 0 9. Direction cosines of z axis are 0, 0, If OP = r, and direction cosines of OP are l, m, n, then the coordinates of P are ( l r, mr, nr) 11. If 1, m, n are direction cosines of a line then l 2 + m 2 + n 2 = If l, m, n, are direction cosines and a,b, c, are direction ratios of a line then l = n =,, m =, 13. If l, m, n, are direction cosines of a line then a unit vector along the line is l ı + m + n k 14. If a, b, c are direction ratio of a line, then a vector along the line is a ı+ b + c k

19 V E C T O R S 1. a b = ab cos θ = a 1 a 2 + b 1 b 2 + c 1 c projection of a on b = and projection of b on a = 3. a b = ab sin θ ^ n a b = - ( b a ) ı k a 1 b 1 c 1 a 2 b 2 c 2 a 1 b 1 c 1 4. a b c = a b c = a 2 b 2 c 2 a 3 b 3 c 3 5. Vector area of ABC is (AB AC ) = ( a b + b c + c a ) And area of ABC = AB AC

20 6. Volume of parallelepiped : a b c = AB AC AD Volume of Tetrahedram ABCD is = AB AC AD 8. Work done by a force F in moving a particle from A to B = AB F 9. Moment of force F acting at A about a point B is M = BA F P R O B A B I L T Y 1. Probability of an event A is P (A) = 0 p () 1 2. p ( AUB ) = P (A) + P (B) - P (AB). IF A and B are mutually exclusive then P (AB) = 0 and P (AB) = P(A) + P(B) 3 P (A) = 1 P (A) = 1 - P (A) 4. P(AB) = P(A) P(B/A) = P(B) P(A/B). IF A and B are independent events P(A B) = P(A) P(B) 5. P(A) = P(AB) + P(AB)

21 6. P(B) = P(AB) + P(AB) 7. lim θ 0 = 1 ; lim x 0 = 1 lim θ 0 = lim θ 0 lim θ 0 cos. = 1; lim x a m = m = na n 8. lim x 0 (1 + x) = e ; lim x 0 (1 + kx) = lim x 0 1 kx = e K. D I F F E R E N T I A L C A L C U L A S lim 1. F(x) = ; where f (x) is derivative of h 0 function f (x) with respect to x. F (a) = lim h 0 2. (a) = 0, where a is constant ; (x) = 1, (ax) = a, = ; =

22 =. x = ; u =.. Where u = f(x) 3. x = n x n-1 ; u = nu n-1 ; = nyn-1 4. logx = ; (logu) = log a x = ; log a u = 5. a = a x log a ; a =a u log a 6. e = e x ; e = e u 7. sin x =cos x ; sin u =cos u, e. g. sin (4x) = cos 4x 4x = cos 4x 4 = 4 cos 4x 8. cos x = - sin x ; cos u = - sin u

23 9. tan x = sec2 x ; tan u = sec2 u 10. cot x = - cosec2 x ; cot u = - cosec2 u 11. sec x = sec x tan x ; sec u = sec u tan u cosec x = - cosec x cot x ; cosec u = - cosec u cot u sin2 x = 2 sin x (sin x) = 2 sinx cos x = sin 2x sinn x = n sin n-1 sin x = n sinn-1 x cos x 14. sin-1 x = ; (sin-1 u) = 15. cos-1 x = ; (cos-1 u) = 16. tan-1 x = ; (tan -1 u) =

24 17. cot-1 x = ; cot-1 u = 18. sec-1 x = ; sec-1 u = cosec-1 x = ; cosec-1 u = (uv) = u + v (uvw) = vw + uw + uv 21. =, v = 23. F ( x + h ) = f (x) + h f (x) 24. Error in y is δy = δ x, Relative error in Y is = and percentage error in y = Velocity =, acceleration a = v

25 I T N T E G R A L C A L C U L U S 1. u v w... ) dx = u dx + vdx + wdx + 2. afx = a fx dx, where a is a constant. 3. x dx = +c, ( n -1 ) ; ax b = + c 4. fx n f (x) dx = + c, (n -1) 5. dx = log x + c ; dx = log ax b + c ; dx = log f (x) + c ; the integral of a function in which the numerator is the differential coefficient of the denominator is log (Denominator). 6. x dx = x + c ;

26 ax b dx = (ax + b)3/2 + c 7. a dx = + c ; a +c dx = + c 8. e dx = e x + c ; e +b dx = eax+b + c. 9. sinax b dx = sin x dx = - cos x + c cos (ax + b) +c ; 10. cosax b dx = sin (ax +b) + c ; cos x dx = sin x + c 11. tanax b dx = log sec (ax+b) + c ; tan x dx = log sec x + c 12. cotax b dx = log sin (ax+b) +c ; cot x dx = log sin x + c 13. secax b dx

27 = log sec (ax+ b ) + tan (ax + b) + c = log tan + c sec x dx = log sec x tan x + c = log tan + c 14. cosec ax bdx = = log cosec ax b cotax b + c log tan + c cosec x dx log cosec x cot x + c = log tan ( ) + c 15. sec x dx = tan x + c ; sec ax b dx = tan (ax + b) + c 16. cosec (ax +b) dx = cot (ax +b) + c ; cosec x dx = - cot 17. secax b tan (ax +b) dx = sec x tan x dx = sec x + c 18. cosec (ax +b) cot (ax +b) dx = sec (ax +b) + c; cosec (ax +b) +c ; cosec x cot x dx = - cosec x + c 19. To integrate sin 2 x, tan 2 x, cot 2 x change to (1 cos2x);

28 (1 cos2x); (1 + cos2x); sec2 x - 1 and cosec 2 x 1 Respectively 20. = sin-1 x + c = - cos -1 x + c 21 = tan-1 x + c = - cot -1 x + c 22 = sec-1 + c ; = sec -1 x + c = -cosec -1 x N I N E I M P O R T A N T R E S U L T S 1. = sin-1 + c = - cos-1 + c 2. = log x x a + c 3. = log x x a + c 4. a x dx = a x + sin -1 + c

29 5. x a dx = x a + log x x a + c 6. x a dx = x a log x x a + c 7., = log + c 8. = tan-1 + c 9. = log + c I N T E G R A T I O N B Y S U B S T I T U T I O N If the integrand contain Proper substitution to be used

30 a x x a x a e f(x) Any odd power of sin x Any odd power of cos x Odd powers of both sin x and cos x Any inverse function Any even power of sec x Any even power of cosec x Function of e x X = a sin θ X = a tan θ X= a sec θ F(x) = t Cos x = t Sin x = t Put that function = t which is of the higher power. Inverse function = t Tan x = t Cot x = t e x = t 12,, tan = t then dx = 1 a b cos x c sin x sin x = cosx = 13, tan x = t then dx =

31 sin 2t = cos 2x = 14 1 a sin x b cos x divide numerator and denominator by cos 2 x and put tan x = t 15 x m = t 16 Expression containing fractional power of x or (ax +b) x or ax +b = t k where k is the L.C.M of the denominators of the fractional indices. I N T E G R A T I O N B Y P A R T S 1. Integral of the product of two function = First function Integral of 2 nd - differential coeficient of 1st integral of 2nd dx i.e. I II dx I II dx I IIdx dx

32 Note : 1. The choice of first and second function should be according to the order of the letters of the word LIATE. Where L = Logarithmic; I = Inverse; A = Algebric; T =Trignometric ; E = Exponential 2. If the integrand is product of same type of function take that function as second which is orally integrable. 3. If there is only one function whose integral is not known multiply it by one and take one as the 2 nd function. D E F I N I T E I N T E G R A L S 1. f (x) dx = gxb = g(b) g(a), where fx a dx = g(x) 2 b a f(x)dx = b a f(t) dt = b a f(m) dm 3 a b f(x) dx = - a b f (x) dx 4 b f(x) = a c f(x) dx + a b f(x) dx, a < c < b. c

33 5 a 0 f(x) dx = a 0 f (a - x) dx ; b a f(x) dx = b a f ( a+ b - x ) dx 6 a a f(x) dx = 2 a 0 a a f(x) dx if f is even f(x) dx = 0 if f is odd 7 2a 0 f(x) dx = If f (2a - x ) = f (x) then e. g. π 0 a a f(x) dx + f (2a x) dx 0 0 2a 0 π sin n 2 x dx = 2 0 sin n x = sin n (π - x ) f(x) dx = 2 sin n x dx as a f (x) dx 0 N U M E R I C A L M E T H O D S 1. Simpson s Rule : According to Simpson s rule the value y dx is approximately given by y dx

34 = y 3 y 5 y n 1 2y 2 y 4 y 6 y n 2 y n Where h =, and y 0, y 1, y 2, y 3, y n are the values of y when x = a, a + h, a + 2h, , b In words : y dx = X 2. Trapezoidal rule : According to Trapezoidal rule the value of y dx is approximately given by y dx = In words : X y dx = 3. Finite Differences : 2 f (a) = f (a + h) f (a) = f (a +h ) - f(a)

35 n f (a) = 1 + = E = E - 1 E f (a) = f ( a +h ) E 2 f (a) = f ( a + 2h ) E n f(a) = f ( a + nh ) n-1 f (a + h ) - n-1 f(a) In words : To obtain of any function, for a write a + h In the function and subtract the function. If interval of differencing is 1, than f(a) = f( a + 1 ) -f(a) 2 f(a) = f(a + 1 ) - f(a) 4. Interpolation : Newton s Forward formula of interpolation. t = f (x 0 + th) = f (x 0 ) +t f (x 0 ) +! f (x 0 ) +! f(x 0 ) + Y =y 0 + t y 0 +! 2 y 0

36 +! y 0 + Newton s Backward formula of Interpolation. t = F(x n + th) = f (x n ) + t f ( x n ) +! f( x n ) +! f(x n ) + or y = y n + t y n +! y n +! y n + Bisection Method : If y = f(x) is an algebraic function and any a and b such that f (a) > 0 and f (b) < 0, then one root of the function f(x) = 0 lies between a and b, we take c 1 = and check f ( c 1 ) If f (c 1 ) = 0, c 1 is the exact root if not and if f ( c 1 ) > 0, f (c 1 ). f (b) < 0 a root c 2 lies between c 1 and b. If not and if (c 1 ) < 0, f (c 1 ). f (a) < 0, a root c 2 lies between c 1 and a. Keep on repeating till the desired accuracy of the root is reached.

37 False Position Method: If y = f(x) is an algebraic function and for any x 0 and x 1 such that f(x 0 ) > 0 and f(x 1 ) < 0 have opposite signs, then a root of f(x) = 0 lies between x 0 and x 1 Let it be x 2 x 2 = x 1 - f (x 1 ). Check f(x 2 ) if (fx 2 ) = 0 then x 2 is exact root, if not and if f(x 2 ) < 0, f(x 0 ). f(x 2 ) < 0, then a root x 3 lies between x 0 and x 2, then X 3 = x 2 f(x 2 ). Keep on repeating till the desired accuracy of the root is reached. Newton Raphson Method: The interactive formula in Newton - Raphson method is X i + 1 = x i -, i 1 Keep on repeating till the desired accuracy of the root is reached. F O R C O M M E R C E

38 Lagrange s Interpolation formula : This is used when interval of differencing is not same. If f(a), f(b), f(c), f(d), bethe corresponding value of f(x) when x = a, b, c, d then F(x) = f(a) + f(b) + f(c) + f(d) + 6 Difference Equations Let the equation be (E) y n = f(n) The complete solution = complimentary function (C.F.) +Particular Integral (P.I.)

39 When R.H.S. is zero, then only C.F. is required Method to find C.F. (1) Write the given equation in E. (2) Form the auxiliary equation. This is obtained by equating to zero the coefficient of y n. (3) Solve the auxiliary equation. Following are the different cases Case (1) If all the roots of the auxiliary equation are real and different. Let the roots be m 1, m 2, m 3, then C.F. is (solution is ) Y n = C 1 (m 1 ) x + C 2 (m 2 ) x + C 3 (m 3 ) x Case (ii) (1) Let two roots be real and equal, suppose the roots are m 1 and m 1 then general solution is Y n = (C 1 + C 2 x ) (m 1 ) x (2) If three roots be equal and real suppose the roots are m 1, m 1, m 1, Then the general solution is Y n = (C 1 + C 2 x + C 3 x 2 ) (m 1 ) x

40 Case (iii) One pair of complex roots. Let the roots be α β i where I = 1 general solution is then the Y n = r n (C 1 cos nθ + C 2 sin nθ) where r = a β, θ = tan -1 ( β x) Statistics : (I) Arithmeic mean or simply mean is denoted by I.e. x is the mean of the x s (II) Methods for finding the arithmetic mean for individual items. (a) x = (b) x = a + Where a is assumed mean and D i = x i - a (c) x = a + I Where D i = I is the length of class interval.

41 (2) Methods for finding the arithmetic Mean for frequency distribution. (a) Direct Method x = (B) Method of assumed mean x = a + Where D i = x i - a (C) Step deviation method, shift of origin method. x = a + h Where D i =, and h is length of class interval. (II) Median - If the variates are arranged in accending or

42 descending order of magnitude, the middle value is called the median. If there are two middle values then the mean of the variate is median. Method of finding Median for a Group data Find the cumulative frequencies. Find the median group. Median group is the group corresponding to (n + 1)th frequency. The formula for the median is Median = l +. I where l is the lower limit of median group.. i is the length of class interval f is the frequency of median group Cf is the cumulative frequency

43 preceeding the median class. (iii) Standard deviation (σ) (a) S.D. = σ = = Where di = xi - x (b) Assumed mean method S.D. = σ = Where Di = xi a, and a is assumed mean. (c) S.D. = σ = When the variates are small numbers.

44 For Grouped Data : (a) Directed method σ = S.D. = = Where i = N (b) Method of assumed mean S.D. = σ = Where D 1 = x 1 = a, a is assumed mean. (c) Step deviation or shift of origin method σ = S.D. = i

45 Where D i =, i is length of class interval. Correlation and Regression. (1) Coefficient of Correlation or Karl Pearson s coefficient of correlation. r = = where d 1 = x - x and d2 = y - y this is used when x and y are integers (2) Correlation coefficientis independent of the origin of reference and unit of measurement if U = & V = Than r xy = r uv xy - r = x y

46 For bi variate frequency table r =. = Karl person coefficient of correlation can also be expressed as r = If the correlation is perfect then r = 1, if the correlation is negative perfect, then r = - 1, if there is no correlation, then r = 0-1 r 1, r lies between -1& 1

47 Regression lines (1) The equation of the line of regression of y on x is Y - y = r (x x i.e. y - y = b yx x x where byx = (2) The equation of line of regression of x and y is x - x = r ( y - y ) i.e. x - x = b xy (y - y ) bxy = (3) b yx = r is called regression coefficient of y and x (4) b xy = r is called regression coefficient of x and y (5) r = byx bxy (6) In the case of line of regression of y on x, its slope and regression cofficient are equal

48 (7) The regression line of y on x is used to find the value of y when the value of x is given (8) In case of line of regression of x on y, its regression cofficient is reciprocal of its slope (9) The regression line of x on y is used to find the value of x when the value of y is given (10) (x, y ) is the point of intersection of two regression lines (11) If the line is written in the form y = a + bx, then this is the line of regression of y on x If the line is written in the form x = a + by, then this is the line of regression of x on y If both the lines are written in the form ax + by + c = 0, and nothing is mentioned, then take first equation as the equation of line of regression of y on x and second as the equation of line of regression of x on y Error of prediction (a) y on x δ yx = σ y 1 r (b) x on y δ xy = σ x 1 r

49 C H E M I S T R Y C H E M I C A L T H E R M O D Y N A M I C S A N D E N E R G E T I C S (1) q = E + W (2) W = P (V 2 - V 1 ) joule (3) N =.. (4) q = W max = n RT x log joule. = n RT log joule (5) H = H P - H R (6) H = E + nrt (7) H 2 = H 1 + C p ( T 2 - T 1 )

50 (1) K = α 2. C I O N I C E Q U I L I B R I A (2) α = (3) H = a. C = K a. C mole / dm 3 (4) OH = a. C = K b. C mole / dm 3 (5) P H = - log 10 H, P OH = - log 10 OH (6) P H + P OH = 14 (7) K h = h 2. C = (8) K h = = h 2 =.

51 (9) Molarity =.. (10) K sp = S 2 E L E C T R O C H E M I S T R Y (1) W = Z. Q = Z. I.t (2) = (3) W = = (4) C. E. = E. C. E. x (5) E = E + E = E - E (6) Equivalent weight =..

52 (7) One Faraday = coulombs. N U C L E A R A N D R A D I O C H E M I S T R Y (1) Mass defect = Z mh A Z mn - M a.m.u. (2) Mass defect = mass of reactants mass of products. (3) Binding energy = Mass defect 931 Me V (4) Binding energy per nucleon = Me V (5) λ =. log per unit time (6) T =. P H Y S I C S

53 C I R C U L A R M O T I O N ω = ; v = r ; v = r ω ; ω = 2πn ; T = ; n = ; a = r α ; a = = rω C.P. force = = m r ω ; v = µ r g ; tanθ = G R A V I T A T I O N V = ; V c = = g h R h T = 2π = 2π ; T 2 r 3 V e = = 2gR ; B.E. = ; For orbiting satellite; B.E. = R O T A T I O N A L M O T I O N I = m r = r d m ; I = M K 2 ; τ = I α

54 KE = I ω2 ; For rolling body, K.E. = MV2 1 Conservation of angular momentum I 1 ω 1 = I 2 ω 2 M.I.of ( i ) ring = Mr 2, ( ii ) disc =, (iii) hollow sphere = Mr2 (iv) solid sphere = Mr2, (v) thin rod =, (vi) rect.bar = M Equation of motion, ( i ) ω = ω 0 + αt ; (ii ) θ = ω 0 t + α t2 ; (iii) ω = ω α θ O S C I L L A T I O N S Differential Equation, ( i ) of Lin. S.H.M. + x = 0 or + ω2 x = 0 ( ii ) of Ang. S.H.M. :- + θ = 0, = - ω2 x ; = ω a x ; x = a sin ( ω t + α ) T = = 2π =, =2π

55 K.E. = m ω (a 2 - x 2 ); P.E. = M ω2 x 2 ; Total Energy = m a2 ω = 2π m a 2 n 2 For simple pendulum, T = 2π For oscillating magnet, T = 2π ; R = a 2 1 a2 2 2a 1 a 2 cosα 1 α 2 ; = E L A S T I C I T Y A N D P R O P E R T I E S O F F L U I D S Tensile Strain = ; Tensile stress = ; Y = Volume Strain = ; Volume stress = = dp ; K = - V Shearing strain = = θ ; Shearing stress = ;

56 n = ; σ = = Work done in stretching a wire = x load x extension. Work done per unit volume = x stress x strain Cos θ h = W A V E M O T I O N Equation of progressive wave :- In + ve x - direction, y = a sin 2 π In - ve x - direction, y = a sin 2 π Phase difference between two points x apart = Number of beats per sec. = n 1 n 2 Doppler effect : n = n when both are approaching each other. n = n When both are receeding away from each other. n = n when source is approaching towards stationary listner

57 n = n when source is receeding from stationary listner n = n when listner is approaching stationary source n = n when listner is receeding from stationary source S T A T I O N A R Y W A V E S Transverse Waves along a string, V =, n = Melde s Experiment : Parallel position, N = 2n =. Perpendicular position, N = n =

58 For both positions, Tp 2 = a constant Air columns : closed at one end, n = and odd harmonics. Open at both ends, n = and integer multiples of n. Resonance tube : V = 4n I 0.3 d R A D I A T I O N a + r + t + 1 ; Stefan s law, = σ T 4 Newton s law, = k θ θ 0 Radiation correction θ θ θ KINETIC THEORY Regnault s method: m o c p θ = m w (θ 1 - θ 2 )

59 C p - C v =, c p - c v =, = = γ L = Li + Le, Le = c =, c =, R.M.S. vel, C = c = P = ρ C = C 2 = K.E. per unit vol. = p ; K.E. per mole = RT C = ; K.E. PER MOLECULE = = Kt T H E R M O D Y N A M I C S Van der Waals equation, P (V - b) = RT covolume, b = 4 actual volume occupied by molecules. W A V E T H E O R Y A N D

60 n = I N T E R F E R E N C E O F L I G H T = ; n = Bright Point :- Path Difference = n λ ; x n = n λ Dark Point :- Path Difference = (2n 1), x n = (2n - 1 ) X = λ ; λ = X ; d = d 1 d 2 E L E C T R O S T A T I C S T.N.E.I. = q ; E due to (i) charged sphere = (ii) charged cylinder = = (iii) any charged conductor at the point near it = Mech. Force per unit area of charged conductor =

61 Energy per unit volume = k ε 0 E2 C = ; For parallel plate condenser, C = Energy of a charged condenser = QV = CV2 = In series,. In parallel, C = C 1 + C 2 + C3 +.+ C n C U R R E N T E L E C T R I C I T Y Wheatstone s Net Work, = Meter Bridge, Potentiometer, While assistin & opposing, Internal resistance of a cell, r = R

62 M A G N E T I C E F F E C T O F C U R R E N T Moving coil Galvanometer : I = θ AMMETER, s = ; voltmeter, R = Tangent Galvanometer, I = µ tan θ = k tan θ M A G N E T I S M M= 2ml; B axil = ; B eqa = For any point, B = 3 cos 1 ; = tan -1 tan OR tan = tan V axial =, V eqn = 0, Any point, V = E L E C T R O M A G N E T I C I N D U C T I O N

63 e = - ; charge induced = Straight conductor, e = B l V Earth Coil B H = α 1, B v = α 2 tan θ = e = e 0 sinωt = 2 π fnab sin2πnt I = = I 0 sinωt; e rms =, I rms = X L = ω L = 2 π f L X c = Z = = R ω L A T O M S, M O L E C U L E S A N D N U C L E I r n =, E n =, v = = µ χ η = P

64 = - λ N = N 0e - λ t T = =. ; λ =. ; λ = E L E C T R O N S A N D P H O T O N S A photon = hv = ; w = hv 0 = h m V2 max = h (v - v 0 ) = hc