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 r2 write X2 + y2 ; for r cos θ write X, for r sin θ
. Write y and for tan θ write
.
3 Distance between two points (X1, Y1 ) and (X2 , Y2 ) is
x 2 x 1 y2 y1
4 Distance of ( x1 , y1 ) from the origin is x 2
1 y 2 1 5 Distance between (r1 , θ 1 ) and (r2 , θ2 ) is
r 21 r 22 2 r1 r2 cos θ2 θ1
6 Coordinates of the point which divides the line joining (X1 , Y1 ) and
(X2, Y2 ) internally in the ratio m1 : m2 are :-
, ( m1 + m 2 0 )
7. Coordinates of the point which divides the line joining (X1 , Y1 ) and
(X2 ,Y2 ) externally in the ratio m1 : m2 are :-
, (m1 – m2 0)
8. Coordinates of the mid-point (point which bisects) of the seg. Joining (X1, y1) and (X2 y2 ) are :
,
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 (x1 , y1 ) ; (x2 , y2 ) ; ( x3 , y3 ) are
! "
11. Coordinates of the in-centre of the triangle whose vertices are A
(x1 ,y1) ; B (x2 ,y2 , ) ; C (x3 ,y3 ) and 1 (BC ) # a, 1 (CA) # b, 1 (AB)
# c.
are$% & '
%&' !
% & ' %&' (.
12 Slope of line joining two points (x1 ,y1) and (x2 ,y2 )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.
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. 20.
% + )
& # 1 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 – Y1 # m (x –x1 ) is the slope point form of line which passes through (x1 , y1)and whose slope is m.
23. Two points form: - y-y1 # )
(x –x1) is the equation of line which
Passes through the points (x1, y1) and (x2, y2).
24. Parametric form :-
'+, - #
passes through the point (x1 y1 )makes an angle θ with the axis and r is
the distance of any point (x, y) from ( x1, y1 ).
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 # - 0 1 # - '+234.'.2/5 +3 '+34.'.2/5 +3 (b) X - intercept # - 6 0 (c) Y- intercept # - 6 1
26. Length of the perpendicular from (x1, y1 ) on the line ax + by + c # 0 is !%&'
√%8 &8 !
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)
TRIGONOMETRY
29. SIN29 + Cos29 # 1; Sin29 # 1 - Cos29 ,
Cos29 # 1 – Sin29 30. tan θ # =>? @ 6A= @ ; cot 9 # 6A= @ =>? @ ; sec 9 # 6A= @ ; Cosec 9 # =>? @ ; cot 9 # B0? @
31. 1 + tan29 # sec2 9 ; tan29 # sec29 - 1 ; Sec2 9 - tan2 9 # 1
32. 1 + cot29 # cosec2 ; cot29 # cosec2 9-1; Cosec29 - cot2 9 # 1
33. Y
Only sine and cosec all trigonometric are positives ratios are positives
O X X1 III IV
Only tan and cot only cos and sec
are positives are positives
Y1
34. angle ratio 00 O 300 C 6 450 E F 600 C 3 900 H 1200 2C 3 1350 3C 4 1500 JH K 1800 C Sin 0 1 2 √ √ 1 √ √ 0 Cos 1 √ 12 0 - √ - √ -1 Tan 0 √ 1 √3 ∞ -√3 -1 1 √3 0
35. Sin (- 9 ) = - Sin 9; cos (-9) = cos 9 ; tan (- 9) = - tan 9 . 36. sin (90 – 9 ) # cos 9 cos (90 – 9) # sin 9 tan (90 – 9) # cot 9 cot (90 – 9) # tan 9 sec (90 – 9) # cosec 9 cosec (90 – 9) #sec 9 sin (90 + 9 ) # cos 9 cos (90 +9 N # sin 9 tan (90 +9 ) # cot 9 cot (90+ 9 ) # tan 9 sec (90 +9 ) # cosec9 cosec (90 +9 ) = sec 9 sin (180 – 9 ) # sin 9 cos (180 – 9N # cos 9 tan ( 180 – 9) # tan 9 cot (180 – 9 ) # cot 9 sec (180 – 9 ) # sec9 cosec (180 – 9) # cosec9
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) = 5%/ O5%/ P 5%/ O 5%/ P tan (A - B) = 5%/ O 5%/ P 5%/ O 5%/ P 38. tan QE F AS # 5%/ O 5%/ O tanQE F AS # 5%/ O 5%/ O
39. SinC + SinD = 2 sin TU V
W cos T U V
W
SinC - SinD = 2 cos TU V
W sin T U V
W
CosC + CosD = 2 cos TU V
W cos T U V
W
CosC - CosD = 2 sin TU V
W sin T V U
W
40. 2 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 ) = cos2A - sin2B Sin (A +B). sin (A – B) = sin2A - sin2B
42. Sin 2θ = 2 sinθ cosθ = 5%/
-5%/8
-43. Cos2 θ =cos2θ - sin2-θ = 2cos2 θ -1 = 1 – 2 sin2θ
= 5%/8
- 5%/8 - ;
44. 1 + cos 2θ = 2 cos2 θ; 1 – cos 2 θ = 2 sin2 θ
45. tan 2 θ = 5%/
-5%/8 - ;
46. sin 3 9 = 3 sin 9 - 4 sin 39; cos 3 9 = 4 cos 39 - 3 cos 9; tan 3 9 = 5%/ @5%/X @ 5%/8 @ 47. 0 ,./ Y = 1 ,./ Z = 6 ,./ U 48. Cos A = &8'8%8 &' ; Cos B# 680818 60 ; Cos C# %8&8'8 %& ;
49. a = b cos C + c cos B; b = c cos A + a cos C ; c = a cos B + b cos A
bc sin A = ca sin B = ab sin c
51. 1 [ sin A = (cos A/2 [ sin A/2)2 52. sec A [ tan A = tan TH
F [ \/2W
53. Cosec A - cot A = tan A/2
54. 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 m1 and m2 be the slopes of the lines represented by ax2 + 2hxy
+ by2 = 0, then
m1 + m2 + - ^& = - T'+234.'.2/5 +3 '+234.'.2/5 +3 8W
and m1 +m2 = %& = '+234.'.2/5 +3 8
'+234.'2/5 +3 8
3. If 9 be the acute angle between the lines represented by ax2 +
2hxy + by2 = 0, then tan 9 = _ √^8%&
%& `
These lines will be co –incident (parallel) if h2 = ab and
perpendicular if a +b = 0.
4. The condition that the general equation of the second degree viz
ax2 + 2hxy + by2 +2gx +2fy + c = 0 may represent a pair of straight
line is
abc + 2fgh – af2 –bg2 - ch2 = 0
i.e. a b c d c e f d f ga = 0.
5. Ax2 + 2hxy + by2 = 0 and ax2 + 2hxy + by2 +2gx +2fy + c = 0 are pairs of parallel lines.
6. The point of intersection of lines ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 is obtained by solving the equation ax + hy + g = 0 and hx + by + f = 0.
7. 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. X2 + y2 = a2 is the equation of circle whose centre is (0, 0) and radius is a.
2. (x – h) 2 + (y - k) 2 = a2 is the equation of a circle whose centre is (h, k) and radius is a.
3. X2 + y2 + 2gx + 2fy + c = 0 is a general equation of circle, its centre is (-g ,-f) and radius is hg f c.
4. Diameter form: - (x – x1) (x – x2) + (y – y1) (y- y2) = 0 is the
equation of a circle whose (x1, y1) and (x2 , y2) 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 x2 and y2 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 (x1 , y1 ) on any curve rule
is:
In the given equation of the curve for x2 put xx1 ; for y2put yy1 ;
for 2x put x+ x1 and for 2y put y +y1
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 – y1 = m (x –x1).
3. For the circle x2 + y2 = a2
(a) Equation of tangent at
(x1, y1) is xx1 + yy1 = a2
(b) Equation of tangent at (a cos 9, a sin 9 ) is x cos 9 + y sin
9 = a.
Y = mx [ a √l 1
4. For the circle x2 + y2 + 2gx + 2fy + c = 0 (a) Equation of tangent at (x1, y1 ) is
Xx1 + yy1 + g (x + x1) + f ( y + y1 ) + c = 0
(b) Length of tangent from (x1, y1) is
m21 n2
1 2dm1 2fn1 g
10. 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 y2 = 4ax is x = at2, y = 2at.
Coordinates of any point (t) is (at2 , 2at)
3. Different types of standard parabola
Parabola Focus Directrix Latus
rectum
Axis of Parabola (axis of symmetry)
Y2 = 4ax Y2 = - 4ax X2 = 4by X2 = - 4by (a, 0) (-a, 0) (0, b) (0, -b) X = - a X = a Y = - b Y = b 4a 4a 4b 4b Y = 0 Y = 0 X = 0 X = 0
4. For the parabola y2 = 4ax
(a) Equation of tangent at (x1, y1) is
Yy1 = 2a (x + x1).
(b) Parametric equation of tangent at (at2
1, 2at1) is
yt1 = x + at21
(c) Tangent in term of slope m is y = mx + %
and its point of
contact is (a/m2, 2a/m)
(d) If P (t1) and Q (t2) are the ends of a focal chord then t2 t1 = -1
(e) Focal distance of a point P (x1, y1) is x1 + a.
E L L I P S E
Ellipse Foci Directrices Latus
Rectum
Equation of axis
Ends of L.R
8 %8 + 8 &8 =1 (a o b) 8 %8 + 8 &8 =1 (ap b ) ([ ae, 0) (0, [ be) X = [ % 2 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. 2.Different types of ellipse Y = [ & 2 &8 % 2a b major axis Y = 0 minor axis x = 0 major axis x = 0 minor axis y = 0 (ae, &8 % ) (ae, &8 % ) (%8 & , be ) (%8 & ,be )
3 Parametric equation of ellipse 8
%8 + 8
and y = b sin θ . 4. For the ellipse 8
8 + 8 &8 = 1, ao b, b2 =a2 (1 =e2) And 8 %8 + 8 &8 = 1, ap b, a2 = b2 (1 – e)
5. For the ellipse 8
%8 + 8
&8 =1 (a o b )
(a) Equation of tangent at x1, y1) is
%8 + &8 = 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 (x1 , y1) is SP = sa ex1s
and SP = sex1 as
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
Hyperbola Foci Directrices L.R End of L.R Eqn of axis 8 %8 - 8 &8 = 1 8 &8 – 8 %8 =1 ([ ae, 0) (0, [ be) X= [ % 2 Y = [ & 2 2b a %8 & (ae, &8 % ) (ae, - &8 % ) (%8 & ,be) (-%8 & ,be) Transverse axis y= 0 conjugate axis x = o Transverse axis x=0 conjugate axis y =0
3. For the hyperbola u8
08 - v 8 18 = 1, b 2 = a2 (e2 -1) and for v8 18 – u 8 08 = 1, a2 = b2 (e2 – 1).
4. Parametric equations of hyperbola u8
08 - v 8
18 = 1 are
X = a sec 9 , y = b tan 9
5. For the hyperbola u8
08 - v 8
18 = 1
(a) Equation of tangent at (x1 , y1 ) are
uu
08 - vv18 = 1
weN Equation of tangent in terms of its slope m is Y = mx [ √b l e
(c) Equation of tangent at (a sec, b tan 9 ) is u ,2' @
0 -
v 5%/ @
1 = 1
(d) Focal distance of P (x1, y1) is S P = | ex1 – a | and
S P = |ex1 + a |
S O L I D G E O M E T R Y
1. Distance between ( x1 , y1 , z1 ) and ( x2 , y2, z2 ) is
m2 m1 n2 n1 x2 x1
2. Distance of (x1 , y1, z1 ) from origin hm 1 n 1 x 1
3. Coordinates of point which divides the line joining (x1, y1, z1)
and ( x2, y2, z2) internally in the ratio m:n are
/ / , / / , y /y / m + n O (x1 ,y1 , z1 ) m n (x2 , y2 , z2)
4. Coordinates of point which divides the joint of (x1, y1, z1) and
(x2 ,y2, z2) externally in the ratio m:n are
Q / / , / / , y /y / S m - n O
5. Coordinates of mid point of join of ( x1 , y1 , z1 ) and ( x2 , y2 , z2 ) are , , y y .
6. Coordinates of centriod of triangle whose vertices are (x1, y1, z1 ) ,
(x2 , y2 , z2 ) and (x3, y3, z3 ) are
, ,y y y
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, 1
10. 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 l2 + m2 + n2 = 1 12. If l, m, n, are direction cosines and a ,b, c, are direction ratios of a line then l = %
[ √%8&8'8, m = [ √%8&& 8'8 ,
n = '
[ √%8&8'8 ,
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~
V E C T O R S
1. a~ · b~ = ab cos θ = a1 a2 + b1 b2 + c1 c2.2. projection of a~ on b~ = %~ · &
|&| and projection of b on a = %~ · & | % | 3. a~ b~ = ab sin θ ^n a ı{ |{ k~ a1 b1 c1 a2 b2 c2 a a~ b~ = - ( b~ a~ ) 4. a~ · b~ c~ = a~ b~ c~ = a1 b1 c1 a2 b2 c2 a3 b3 c3
5. Vector area of ∆ ABC is
(AB~~~~ AC~~~~ ) = ( a b~ + b~ c~ + c a~ ) And area of ∆ ABC = | AB~~~~ AC~~~~ |
6. Volume of parallelepiped : | a~ b~ c~ | b1 e1 g1 b2 e2 g2 b3 e3 g3 = |AB~~~~ AC~~~~ AD~~~~ |
7. Volume of Tetrahedram ABCD is =
K |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) = / wON/wN 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)
6. P(B) = P(AB) + P(AB) 7. lim θ 0 ,./ -- = 1 ; limx 0 +w N = 1 lim θ 0 ,./ -- = limθ 0 ,./ - - m = m lim θ 0 cos . = 1; limx a –% % = na n 8 . lim x 0 (1 + x) = e ; lim x 0 (1 + kx) = limx 0 w1 kxN = eK.
D I F F E R E N T I A L C A L C U L A S
1. F(x) = lim h 0 3 w ^ N 3 wN^ ; where f ‘ (x) is derivative offunction f (x) with respect to x. F (a) = lim
h 0
3 w% ^ N 3w%N ^
2.
(a) = 0, where a is constant ; (x) = 1, (ax) = a, T W = 8 ; T W = 8
T W = / . √x = √ ; √u = .√ . Where u = f(x) 3. ¢x/£ = n ¢x£ n-1 ; ¢u/£ = nu n-1 ; = ny n-1 4. logx = ; (logu) = loga x = + % ; loga 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
9. tan x = sec 2 x ; tan u = sec 2 u 10. cot x = - cosec 2 x ; cot u = - cosec 2 u 11.
sec x = sec x tan x ;
sec u = sec u tan u
12.
cosec x = - cosec x cot x ; cosec u = - cosec u cot u 13. sin 2 x = 2 sin x
(sin x) = 2 sinx cos x = sin 2x
sin n x = n sin n-1 sin x = n sin n-1 x cos x 14. sin -1 x = √8 ; (sin -1 u) = √ 8 15. cos -1 x = √8 ; (cos -1 u) = √ 8 16. tan -1 x = 8 ; (tan-1 u) = 8
17. cot -1 x = 8 ; cot-1 u = 8 18. sec -1 x = √8 ; sec -1 u = √+¤/ 8 19. cosec -1 x = √8 ; cosec -1 u = √ 8 20. (uv) = u ¥ + v (uvw) = vw + uw ¥ + uv ¤ 21. T ¥W = ¥¦§¦¨ ¦©¦¨ ¥8 , v 0. 22. = 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 =
« 100 25. Velocity = , 5 , acceleration a = ¥ 5 # v ¥ , # 8, 58
I T N T E G R A L C A L C U L U S
1. wu v w . . . ) dx = u dx + vdx + wdx + …
2. afwxN = a fwxN dx, where ‘a’ is a constant.
3. x/ dx = / +c, ( n -1 ) ; wax bN/ = % w%& N° / + c 4. ± fwxN²n f (x) dx = ±3wN ² / + c, (n -1) 5. dx = log x + c ; %& dx = % log ¢ax b£ + c ; 3³wN 3wN dx = log | f (x) | + c ;
the integral of a function in which the numerator is the differential coefficient of the denominator is log
(Denominator).
√ax b dx = % (ax + b) 3/2 + c 7. a dx = %¨ + % + c ; a&+c dx = & %µ¨°¶ + % + c 8. e dx = ex + c ; e%+b dx = % e ax+b + c. 9. sinwax bN dx = % cos (ax + b) +c ; sin x dx = - cos x + c
10. coswax bN dx = % sin (ax +b) + c ; cos x dx = sin x + c
11. tanwax bN dx =
% log sec (ax+b) + c ;
tan x dx = log sec x + c 12. cotwax bN dx =
% log sin (ax+b) +c ;
cot x dx = log sin x + c 13. secwax bN dx
=
% log | sec (ax+ b ) + tan (ax + b) | + c
= % log tan ! %& E F! + c
sec x dx = log |sec x tan x| + c = log tan T E
FW + c
14. cosec wax bNdx =
% log |cosec wax bN cotwax bN| + c
=
% log tan ! %&
! + c
cosec x dx # log |cosec x cot x| + c
= log tan ( ) + c 15. sec x dx = tan x + c ; sec wax bN dx = % tan (ax + b) + c 16. cosec (ax +b) dx = % cot (ax +b) + c ; cosec x dx = - cot
17. secwax bN tan (ax +b) dx =
% sec (ax +b) + c;
sec x tan x dx = sec x + c 18. cosec (ax +b) cot (ax +b) dx =
% cosec (ax +b) +c ;
cosec x cot x dx = - cosec x + c
19. To integrate sin2 x, tan2x, cot2 x change to
(1 – cos2x); (1 + cos2x); sec 2 x - 1 and cosec2x – 1 Respectively 20. √8 = sin -1 x + c = - cos-1 x + c 21 8 = tan-1 x + c = - cot -1 x + c 22 √8%8 = % sec -1 T %W + c ; √8 = 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. √%88 = sin -1 % + c = - cos -1 T%W + c 2. √8%8 = log x √x a + c 3. 8 %8 = log x √x a + c 4. √a x dx = √a x + %8 sin -1 T %W + c5. √x a dx = √x a + %8 log sx √x a s + c 6. √x a dx = √x a – %8 log ·x √x a ¸ + c 7. %8,8 = % log !%%! + c 8. 8%8 = % tan-1T%W + c 9. 8%8 = % 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
1 2 3 4 5 6 7 8 9 10 11 12 13 √a x √x a √x a ef(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 ex %& ,./ , %& '+, , 1 a b cos x c sin 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 ex = t tan = t then dx = 5 58 sin x = 5 58 cosx = 5 8 58 tan x = t then dx = 5 58
14 15 16 1 a sin x b cos x w¹º »N Expression containing fractional power of x or (ax +b) sin 2t = 5 58 cos 2x = 5 8 58
divide numerator and denominator by cos2 x and put tan x = t
xm = t
x or ax +b = tk 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 2nd -
¢differential coef4icient of 1st integral of 2nd£ dx i.e. ¢I II £ dx # I II dx
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 2nd
function.
D E F I N I T E I N T E G R A L S
1. f%& (x) dx = ¢ gwxN£ba = g(b) –g(a), where fwxN dx = g(x) 2 ba f(x)dx = b a f(t) dt = ba f(m) dm 3 a b f(x) dx = - a b f (x) dx 4 b af(x) = c f a (x) dx + b c f(x) dx , a < c < b.
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 af(x) dx = 2 a 0 f(x) dx if f is even a a f(x) dx = 0 if f is odd 7 2a 0 f(x) dx = a 0f(x) dx + a 0f (2a – x) dx If f (2a - x ) = f (x) then 2a 0 f(x) dx = 2 a 0f (x) dx e. g. π 0 sin n x dx = 2 π 2 ´ 0 sin n x dx as sinnx = sinn (π - x )
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
= ¾ n0 4 n 1 y3 y5 … yn 1 2y2 y4
y6 Á yn 2 yn
Where h = 10
? , and y0, y1, y2, y3, --- yn are the
values of y when x = a, a + h, a + 2h, ---, b
In words : 01 y dx =ÂÃ?B¾ AÄ B¾Ã =Å1 >?BÃÆÇ0Â
X ¢ wÈÉl Êf ËcÌ ÍÈË bÎÏ ÐbÈË ÊÑÏÒÎbËÌN
fÊÉÑ wËcÌ ÈÉl Êf ËcÌ ÑÌlbÒÎÒÎd ÊÏÏ ÊÑÏÒÎbËÌÈN ËÓÒgÌ wËcÌ ÈÉl Êf bÐÐ ÌÔÌÎ ÊÑÏÒÎbËÌÈ N £
2. Trapezoidal rule : According to Trapezoidal rule the
value of 01 y dx is approximately given by 01 y dx = ¾ n0 nÎ 2 n1 n2 n3 Á nÎ 1 In words : 01 y dx = ÂÃ?B¾ AÄ =Å1 >?BÃÆÇ0Â X ¢ ÈÉl Êf ËcÌ fÒÑÈË bÎÏ ÐbÈË ÊÑÏÒÎbËÌÈ ËÓÊ ËÒlÌÈ ÑÌlbÒÎÒÎd bÐÐ ÊÑÏÒÎbËÑÈ £ 3. Finite Differences : f (a) = f (a + h) 2 f (a) = ∆ f (a +h ) - f(a)
n f (a) = n-1 f (a + h ) - n-1 f(a) 1 + = E = E - 1 E f (a) = f ( a +h ) E2 f (a) = f ( a + 2h ) En f(a) = f ( a + nh )
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 = Õ ^ f (x0 + th) = f (x0 ) +t ∆ f (x0) + 5w5N ! ∆ f (x0) + 5w5 Nw5 N ! ∆ f(x0) + _____ Y =y0 + t y0 + 5w5N ! 2 y0
+ 5 w5N w5 N
! ∆ y0 + _____
Newton’s Backward formula of Interpolation.
t = / ^ F(xn + th) = f (xn) + t f ( xn ) + 5 w5N ! × f( xn ) + 5 w5 Nw5 N ! × f(xn) + _____ or y = yn + t yn + 5w5N ! yn + 5 w5Nw5 N ! yn +
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 c1 = 0 1 and check f ( c1)
If f (c1) = 0, c1 is the exact root if not and if f ( c1 ) > 0,
f (c1) . f (b) < 0 a root c2 lies between c1 and b. If
not and if (c1) < 0, f (c1 ). f (a) < 0, a root c2 lies
between c1 and a.
Keep on repeating till the desired accuracy of the root is reached.
False Position Method: If y = f(x) is an algebraic function and for any x0 and x1 such that f(x0) > 0 and
f(x1) < 0 have opposite signs, then a root of f(x) = 0 lies
between x0 and x1
Let it be x2
x2 = x1 - f (x1) . Ø 3Õ
3Õ Ù
Check f(x2) if (fx2) = 0 then x2 is exact root, if not and if
f(x2) < 0, f(x0) . f(x2) < 0, then a root x3 lies between x0
and x2, then
X3 = x2 – f(x2) . Ø 3 –Õ
3Õ Ù
Keep on repeating till the desired accuracy of the root is reached.
Newton – Raphson Method: The interactive formula in Newton - Raphson method is
Xi + 1 = xi - 3w3w.N
.N , i 1
Keep on repeating till the desired accuracy of the root is reached.
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) = w& N w'N wN __________ w%&N w%'N w%N_________ f(a) + w%Nw'NwN_____________ w&%Nw&'Nw&N_____________ f(b) + w%Nw&NwN__________ w'%Nw'&Nw'N__________ f(c) + w%Nw&Nw'N____________ w%Nw&Nw'N____________ f(d) + _____________ 6 Difference Equations
Let the equation be (E) yn = f(n)
The complete solution = complimentary function (C.F.) +Particular Integral (P.I.)
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 yn.
(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 m1, m2, m3, then
C.F. is (solution is )
Yn = C1 (m1)x + C2 (m2) x + C3 (m3)x
Case (ii) (1) Let two roots be real and equal,
suppose the roots are m1 and m1 then
general solution is
Yn = (C1 + C2 x ) (m1) x
(2) If three roots be equal and real suppose the roots are m1, m1, m1,
Then the general solution is
Case (iii) One pair of complex roots.
Let the roots be α [ β i where I = √ 1 then the
general solution is
Yn = rn (C1 cos nθ + C2 sin nθ)
where r = ha β , θ = tan-1 (β´ ) x
Statistics :
(I) Arithmeic mean or simply mean is denoted by Ü~ 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 + ∑ Þ.
/
Where a is assumed mean and Di = xi - a
(c) x~ = a + T∑ Þ.
/ W I
Where Di = .%ß
(2) Methods for finding the arithmetic Mean for frequency distribution.
(a) Direct Method
x~ = ∑ 3. .
∑ 3.
(B) Method of assumed mean
x~ = a + ∑ 3. Þ.
∑ 3.
Where Di = xi - a
(C) Step deviation method, shift of origin method.
x~ = a + T∑ 3. Þ.
∑ 3. W h
Where Di = . %^ , and h is length of class interval.
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 + à/´ '3
3 á. I where l is the
lower limit of median group.. i is the length of class interval f is the frequency of median
preceeding the median class.
(iii) Standard deviation (σ)
(a) S.D. = σ = ∑ w.~N8
/ =
∑ .8
/
Where di = xi - x~
(b) Assumed mean method
S.D. = σ = ∑ Þ.8 / T ∑ Þ. / W Where Di = xi – a , and a is assumed mean.
(c) S.D. = σ = ∑ .8 / T ∑ . / W
When the variates are small numbers.
For Grouped Data :
(a) Directed method σ = S.D. = ∑ 4. .8
∑ 4. T ∑ 4. . ∑ 4. W = ∑ 38 ã T ∑ 3 ã W Where ∑ 4i = N
(b) Method of assumed mean
S.D. = σ = ∑ 4..8 ã T ∑ 4.. ã W Where D1 = x1 = a, a is assumed mean.
(c) Step deviation or shift of origin method
σ = S.D. = i ∑ 3Þ. 8 ã T ∑ 3Þ. ã W
Where Di = .% . , i is length of class interval.
Correlation and Regression .
(1) Coefficient of Correlation or Karl Pearson’s coefficient of correlation. r = ∑wN~~~ w N~~~ h ∑w~N8 ∑ wN8 = ∑ ∑ ∑ where d1 = 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 rxy = ruv ∑ xy - ∑ ∑ ã r = ∑ x ∑wN8 ã Ø∑ y ∑wN ã Ù
For bi variate frequency table r = ∑ w∑ ä¨N . w∑ äåN æ ç∑ è8 w∑ ä¨N8 æ ∑ 38∑ äå8 æ = ∑ éê ∑ é ∑ ê ë çì∑ éí w∑ éN ë í î ì∑ êí T∑ ê ëW í î
Karl person coefficient of correlation can also be expressed as
r = ∑ / ~
∑ 8 /~~~ ∑8 8 /~~~~8
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
Regression lines
(1) The equation of the line of regression of y on x is
Y - y~ = r ð
ð (x –xN
i.e. y - y~ = byx wx x~N where byx = ð
ð
(2) The equation of line of regression of x and y is
x - x~ = r ð ð ( y - y~ ) i.e. x - x~ = bxy (y - y~ ) bxy = ðð (3) byx = r ð
ð is called regression coefficient of y and x
(4) bxy = r ð
ð is called regression coefficient of x and y
(5) r = hbyx bxy
(6) In the case of line of regression of y on x , its slope and
(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
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 (V2 - V1) joule (3) N = ñ2.^5 ./ ò.ñ../ (4) q = Wmax = 2.303 n RT x log óó joule. = 2.303 n RT log ô ô joule (5) H = ∑ HP - ∑ H R (6) ∆ H = E + nRT (7) H2 = H1 + Cp ( T2 - T1)I O N I C E Q U I L I B R I A
(1) K = α2 . C (2) α = ô2ø'2/5%2 +3 .+/.,%5.+/ ÕÕ (3) ¢H£ = a . C = Ka . C mole / dm3 (4) ¢OH£ = a . C = Kb . C mole / dm3(5) PH = - log 10 ¢H£ , POH = - log10 ¢OH£
(6) PH + POH = 14 (7) Kh = h2 . C = ûû¤ % # û¤ û& (8) Kh = ^ 8 w ^N = h 2 = û¤ û% . û&
(9) Molarity = ¹2ø X ò.ñ. ./ (10) Ksp = S2
E L E C T R O C H E M I S T R Y
(1) W = Z. Q = Z. I .t (2) ñ ñ = ü ü (3) W = ý ü è = ß 5 ü è (4) C. E. = E. C. E. x 96500 (5) E'2Õ = Ew+.NÕ + E wø2NÕ = Ew+.N Õ - E w+.NÕ (6) Equivalent weight = O5.ñ5.(7) One Faraday = 96500 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 wA ZN 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 = ò%,, 232'5
ò%,, / &2ø Me V
(5) λ = . Õ
5 log
ãÕ
ã per unit time
(6) T = Õ.K
C I R C U L A R M O T I O N
ω = -5 ; v = r -5 ; v = r ω ; ω = 2πn ; T = E ; n = # E ; a = r α ; a = ¥8 ø = rω C.P. force = ¥8 ø = m r ω ; v = hµ r g ; tanθ = ¥8 øG R A V I T A T I O N
V = ò ø ; V c = ò ^ = gh wR hN T = 2π w ^NX ò = 2π w ^N ^ ; T 2 r3 Ve = ò = 2gR ; B.E. = ò ;For orbiting satellite; B.E. = ò
w ^N
R O T A T I O N A L M O T I O N
KE =
I ω
2
; For rolling body, K.E. =
MV
2
T1 û8
ø8W
Conservation of angular momentum I1 ω1 = I2 ω2
M.I.of ( i ) ring = Mr2 , ( ii ) disc = òø8
,
(iii) hollow sphere = Mr2 (iv) solid sphere =
J Mr 2 , (v) thin rod = òß8 , (vi) rect.bar = M T ß8 &8 W
Equation of motion, ( i ) ω = ω0 + αt ; (ii ) θ = ω0 t + α t2 ;
(iii) ω = ω02 + 2 α θ
O S C I L L A T I O N S
Differential Equation, ( i ) of Lin. S.H.M. 8 58 + x = 0 or 8 58 + ω 2 x = 0 ( ii ) of Ang. S.H.M. :- 8 - 58 + ûß θ = 0 , 8 58 = - ω2 x ; 5 = ω √a x ; x = a sin ( ω t + α ) T = E = 2π = E h%'',¹2ø /.5 .,¹%'22/5 =2π %,, 3+ø'2 ¹2ø /.5 .,¹%'22/5
K .E. = m ω (a 2 - x2) ; P.E. = M ω 2 x2 ; Total Energy = m a 2 ω = 2π m a2 n2
For simple pendulum, T = 2π
;
For oscillating magnet, T = 2π
òP R = a21 a2 2 2a1 a2 cosα1 – α2 ; ËbÎ = 00 =>? 0 =>? 6A= 0 6A=
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 = ß ; Tensile stress = è O ; Y = ò E ø8ß Volume Strain = ó ó ; Volume stress = è O = dP ; K = - V ô ó Shearing strain = ∆ ß = ∆ θ ; Shearing stress = è O ;n = è O ∆- ; σ = ø ´ ´ =
Work done in stretching a wire =
x load x extension.
Work done per unit volume =
x stress x strain Cos θ # – h = '+, -ø
W A V E M O T I O N
Equation of progressive wave :- In + ve x - direction, y = a sin 2 π T5 W In - ve x - direction , y = a sin 2 π T5 W
Phase difference between two points x apart = E
Number of beats per sec. = n1 n2
Doppler effect : n = n àó +
ó ,á when both are approaching each
other.
n = n àó +
ó ,á When both are receeding away from each other.
n = n à ó
ó ,á when source is approaching towards stationary
n = n à ó
ó ,á when source is receeding from stationary listner
n = n Tó +
ó W when listner is approaching stationary source
n = n Tó – +
ó W 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 = ô ß Melde’s Experiment : Parallel position, N = 2n = ô ß . Perpendicular position , N = n = ô ß
For both positions , Tp2 = a constant
Air columns : closed at one end, n = ó
F ß and odd harmonics.
Open at both ends , n = ó
ß and integer multiples of n.
Resonance tube : V = 4n wI 0.3 dN
R A D I A T I O N
a + r + t + 1 ; Stefan’s law , ý O5 = σ T 4 Newton’s law , ý 5 = k θ θ0 Radiation correction ∆ θ # wθ θNKINETIC THEORY
Regnault’s method: mocp Tθ – - - W = wm wN (θ1 - θ2)Cp - Cv = , cp - cv = ò , '¹ '¥ = ¹ ¥ = γ L = Li + Le , Le = ô ó c = ∑ '/ , c = ∑ '8 / , R.M.S. vel, C = hc = ∑ '8 / P = ρ C = ò ó C 2 = ß / 8 ó
K.E. per unit vol. =
p ; K.E. per mole = RT
C =
ò ; K.E. PER MOLECULE =
ã = Kt
T H E R M O D Y N A M I C S
Van der Waals’ equation, TP %
ó8W (V - b) = RT
covolume, b = 4 actual volume occupied by molecules.
I N T E R F E R E N C E O F L I G H T
n = ' ' = ; n = ,./ . ,./ øBright Point :- Path Difference = n λ ; xn = Þ n λ
Dark Point :- Path Difference = (2n – 1)
, xn = Þ (2n - 1 ) X = Þ λ ; λ = Þ X ; d = d1 d2
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 = »
F E Õ ø8
(ii) charged cylinder = »
E Õ ø = % ð Õ ø
(iii) any charged conductor at the point near it = ð
Õ
Mech. Force per unit area of charged conductor = ð8
Energy per unit volume =
k ε 0 E
2
C = ý
ó ; For parallel plate condenser, C =
O üÕ
Energy of a charged condenser =
QV = CV 2 = ý8 In series, # … … … … . / In parallel, C = C1 + C2 + C3 + ………….+ Cn
C U R R E N T E L E C T R I C I T Y
Wheatstone’s Net Work,
= F Meter Bridge, # Potentiometer, ü ü #
While assistin & opposing, ü
ü #
Internal resistance of a cell, r = T
W R
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 =
/OP θ AMMETER, s = ß ß ß ; voltmeter, R = ó ß Tangent Galvanometer, I = ø P µÕ / tan θ = k tan θ
M A G N E T I S M
M= 2ml; Baxil = Õ FH ÆX ; Beqa = Õ FH ÆXFor any point, B = Õ
FH ÆX √3 cos 9 1 ; = tan-1 T tan 9W OR tan = tan 9 Vaxial = Õ FH
Æ8 , Veqn = 0, Any point, V =
Õ FH '+, @ Æ8
E L E C T R O M A G N E T I C I N D U C T I O N
e = -
5 ; charge induced =
Straight conductor, e = B l V
Earth Coil BH = T /O W α 1 , Bv = T /O W α 2
tan θ =
e = e0 sinωt = 2 π fnAB sin2πnt
I = 2 = I0 sinωt; erms = 2Õ √ , Irms = ßÕ √ XL = ω L = 2 π f L Xc = = E 3 Z = R Tω L W
A T O M S, M O L E C U L E S A N D N U C L E I
rn = Õ /8 ^8 E 28 , En = 2 Õ8 /8^8 , v~ = = 2 Õ8 '^X T ¹8 /8W µ ü Õχ ηX = Pã 5 = - λ N = N0e- λ t T = +2 = Õ.K ; λ = Õ.K ; λ = !¦æ¦ ! ã
