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�� �y 2 y 1��

4 Distance of ( x1 , y1 ) from the origin is �x 21 y 2 1

5 Distance between (r1 , θ 1 ) and (r2 , θ2 ) is �r 21 r 22 2 r 1 r 2 cos �θ 2 θ 1�

6 Coordinates of the point which divides the line joining (X1 , Y1 ) and

(X2, Y2 ) internally in the ratio m1 : m2 are :-

�� � � � � � �� � � � � � � � � � � � � � � � � � � � � � � , ( m 1 + 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 :- ��� �'+, - #

��� �,./ - # r is the equation of line which

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 # - 01 # - �'+234.'.2/5 +3 �'+34.'.2/5 +3 � �

(b) X - intercept # - 60

(c) Y- intercept # - 61

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 + Cos

29 # 1; Sin2 9 # 1 - Cos

2 9 ,

Cos2 9 # 1 – Sin

2 9

30. tan θ # =>? @6A= @ ; cot 9 # 6A= @=>? @ ; sec 9 #

�6A= @ ;

Cosec 9 # �=>? @ ; cot 9 # �B0? @

31. 1 + tan2 9 # sec

2 9 ; tan

2 9 # sec

2 9 - 1 ;

Sec2

9 - tan2 9 # 1

32. 1 + cot2 9 # cosec

2 ; cot

2 9 # cosec

2 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 C6

450 EF

600 C3

900

H�

1200 2C3

1350 3C 4

1500

JHK

1800

C

Sin

0

12

�√�

√ �

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%/ O�5%/ P� � 5%/ O 5%/ P

tan (A - B) = 5%/ O � 5%/ P��5%/ O 5%/ P

38. tan QEF AS # ��5%/ O� � 5%/ O

tanQ EF AS # � � 5%/ O� � 5%/ O

39. SinC + SinD = 2 sin TU �V� W cos TU � V� W

SinC - SinD = 2 cos TU � V� W sin TU � V� W

CosC + CosD = 2 cos TU � V� W cos TU � V� W

CosC - CosD = 2 sin TU � V� W sin TV � 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 - sin

2B

Sin (A +B). sin (A – B) = sin2A - sin

2B

42. Sin 2θ = 2 sinθ cosθ = � 5%/ -��5%/8 -

43. Cos2 θ =cos2θ

- sin2-θ

= 2cos2 θ

-1 = 1 – 2 sin2 θ

=

� � 5%/8 -� � 5%/8 - ;

44. 1 + cos 2θ = 2 cos2 θ; 1 – cos 2 θ = 2 sin

2 θ

45. tan 2 θ =� 5%/ -��5%/8 - ;

46. sin 3 9 = 3 sin 9 - 4 sin 39;

cos 3 9 = 4 cos 3

9 - 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#

68�08�18� 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

50. Area of triangle =

�� 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 THF [ \/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 + by

2 +2gx +2fy + c = 0 may represent a pair of straight

line is

abc + 2fgh – af2 –bg

2 - ch

2 = 0

i.e. ab c dc e fd f ga = 0.

5. Ax2 + 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 ax2 + 2hxy + by

2 + 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 + 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

= a2 is the equation of a circle whose centre is

(h, k) and radius is a.

3. X2 + y

2 + 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 y

2put 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 + y

2 = a

2

(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.

(C) Tangent in terms of slope m is

Y = mx [ a √l� 1

4. For the circle x2 + y

2 + 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 n21 2dm 1 2fn 1 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 = at

2, 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 (at21, 2at1) is

yt1 = x + at2

1

(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&8 = 1 (a o b) is x = a cos θ

and y = b sin θ .

4. For the ellipse �8�8 +

�8&8 = 1, ao b, b2 =a

2 (1 =e

2)

And �8%8 +

�8&8 = 1, ap b, a2 = b

2 (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 ex 1s and SP = sex 1 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 u808 - v818 = 1, b

2 = a

2 (e

2 -1) and for

v818 – u808 = 1, a2 = b

2 (e

2 – 1).

4. Parametric equations of hyperbola u808 -

v818 = 1 are

X = a sec 9 , y = b tan 9

5. For the hyperbola u808 -

v818 = 1

(a) Equation of tangent at (x1 , y1 ) are

uu � 08 -

vv �18 = 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 , y

2,

z

2 ) is

��m 2 m 1 �� �n 2 n 1�� �x 2 x 1�� 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 + m

2 + n

2 = 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 n̂ a ı{ | { k~ a 1 b 1 c 1 a 2 b 2 c 2 a

a~ � b~ = - ( b~ � a~ )

4. a~ · b~ � c~ = �a~ b~ c~� = �a 1 b 1 c 1a 2 b 2 c 2a 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~~~~ |

6. Volume of parallelepiped : | a~ b~ c~ |

�b 1 e 1 g 1b 2 e 2 g 2b 3 e 3 g 3� = |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/w�N 0 � p () � 1

2. p ( AUB ) = P (A) + P (B) - P (A�B). IF A and B are mutually

exclusive then P (A�B) = 0 and P (A�B) = P(A) + P(B)

3 P (A) = 1 – P (A) = 1 - P (A)

4. P(A�B) = 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(A�B) + P(A�B)

6. P(B) = P(A�B) + P(A�B)

7. limθ � 0

,./ � -- = 1 ; limx � 0

�+�w��� N� = 1

lim θ � 0

,./ � -- = limθ � 0 ,./ � -� - � m = m

limθ � 0 cos . = 1;

limx � a �� –%���% = nan

8 . limx � 0 (1 + x)

�� = e ;

limx � 0 (1 + kx)�� =

limx � 0 �w1 kxN ����� = e

K.

D I F F E R E N T I A L C A L C U L A S

1. F(x) = limh � 0 3 w� � ^ N �3 w�N^ ; where f ‘ (x) is derivative of

function f (x) with respect to x.

F (a) =

limh � 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 � �� ;

����� = nyn-1

����

4. ��� logx =

�� ; ��� (logu) =

�  � � ��

��� loga x = �� �+� % ;

��� loga u = �  �+� % �

� ��

5. ��� ¢a�£ = a

x log a ;

��� ¢a £ =au log a � � ��

6. ��� ¢e�£ = e

x ;

��� ¢e £ = eu � � ��

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

2u � � ��

10. ��� cot x = - cosec

2x ;

��� cot u = - cosec2u � � ��

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

2x = 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

-1x =

�� � � �8 ; ��� cot

-1 u = ��� �  8 � � ��

18. ��� sec

-1x =

��√�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 =

±3w�N� ��²/�� + c, (n � -1)

5. ­ �� dx = log x + c ;

­ �%��& dx = �% log ¢ax b£ + c ;

­ 3³w�N3w�N 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 ;

­ √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�� EFW + 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, cot

2 x change to

�� (1 – cos2x);

�� (1 – cos2x); �� (1 + cos2x); sec

2x - 1 and cosec

2x – 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. ­ ��√%8��8 = 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 + c

5. ­ √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

-1 T�%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

If the integrand contain Proper substitution to be used

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

�%�& ,./ � , �%�& '+,� ,

1a 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 =

��58��58

tan x = t then dx = �5��58

14

15

16

1a� sin� x b� cos� x

��w¹�º� »N

Expression containing

fractional power of x or (ax

+b)

sin 2t = �5��58 cos 2x =

��58��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 ­ � ��� I � ­ IIdx� 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 = ­ ba f(t) dt =­ ba f(m) dm

3

a­ b f(x) dx = -

a­ b f (x) dx

4 b­ af(x) =

c­ fa (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 0 f(x) dx +

a­ 0 f (2a – x) dx

If f (2a - x ) = f (x) then

2a­ 0 f(x) dx = 2

a­ 0 f (x) dx

e. g.

π­ 0 sinnx dx = 2 ­ π 20́ sin

nx dx as

sinnx = sin

n (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 � n 0 4 �n 1 y 3 y 5 … y n 1� 2�y 2 y 4 y 6 Á y n 2� y n �

Where h = 1�0? , and y0, y1, y2, y3, -------- yn are the

values of y when x = a, a + h, a + 2h, -------, b

In words : ­ 10 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 ­ 10 y dx is approximately given by ­ 10 y dx

= ¾� � �n 0 n Î � 2 �n 1 n 2 n 3 Á n Î 1� �

In words : ­ 10 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) + 5w5��N�! ∆� f (x0)

+ 5w5�� Nw5��N ! ∆ f(x0) + _____

Y =y0 + t y0 + 5w5��N�!

2 y0

+ 5 w5��N w5�� N ! ∆ y0 + _____

Newton’s Backward formula of Interpolation.

t = ���/^

F(xn + th) = f (xn) + t f ( xn ) + 5 w5��N�! � f( xn )

+ 5 w5�� Nw5�� N ! × f(xn) + _____

or y = yn + t yn +

5w5��N�! yn + 5 w5��Nw5��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 � ��� Õ 3�� ���3�� Õ� 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 � � –� Õ 3�� ��� 3�� Õ� 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 - 3w� .N3w� .N , 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

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 w���N __________ w%�&N w%�'N w%��N_________ � f(a)

+ � w��%Nw��'Nw���N_____________ w&�%Nw&�'Nw&��N_____________ � f(b)

+ � w��%Nw��&Nw���N__________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

Yn = (C1 + C2x + C3x2) (m1)

x

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 x́)

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

I is the length of class interval.

(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.

(II) Median - If the variates are arranged in accending or

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

group Cf is the cumulative frequency

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�

= �∑ 3�8ã T∑ 3�ã W�

Where ∑ 4i = N

(b) Method of assumed mean

S.D. = σ = �∑ 4.�.8ã 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 .�8ã 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 = ∑w���N~~~ w�� �N~~~h ∑w���~N8 ∑ w����N8 =

∑ ������∑ ��� �∑ ���

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� ∑w�N8ã � Ø∑ y� ∑w�Nã �Ù

For bi variate frequency table

r = ∑ ��� w∑ ä¨N . w∑ äåNæ

ç∑ è�8 � w∑ ä¨N8æ �∑ 3�8��∑ äå8�æ

= ∑ éê � ∑ é ∑ êë

çì∑ éí � w∑ éNë íî ì∑ êí � T∑ êë Wíî

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

-1 � r ï 1, r lies between -1 & 1

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 = ð�ð�

(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

regression cofficient are equal

(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 ó �ó � joule.

= 2.303 n RT log ô �ô � joule

(5) H = ∑ H P - ∑ 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 = �K a . C mole / dm3

(4) ¢OH£ = a . C = �K b . C mole / dm3

(5) PH

= - log 10 ¢H�£ , POH

= - log10 ¢OH£

(6) PH

+ POH

= 14

(7) Kh = h2

. C = û ¤û % # û ¤û &

(8) Kh = ^8w� �^N = h

2 =

û ¤û % . û &

(9) Molarity = �� ¹2ø ��Xò.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 �üè

(4) C. E. = E. C. E. x 96500

(5) E'2��Õ = E�w+�.NÕ + E�wø2�NÕ = E�w+�.N Õ - E�w+�.NÕ

(6) Equivalent weight = O5.ñ5.ó%�2/'�

(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

ãÕã per unit time

(6) T = Õ.K�

�

P H Y S I C S

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 = ¥8ø = rω�

C.P. force = �¥8ø = m r ω� ; v = hµ r g ; tanθ =

¥8ø�

G R A V I T A T I O N

V = � òø ; V c = � ò

� � ^ = �g h wR hN

T = 2π �w��^NXò = 2π � w� � ^N� ̂ ; T

2 r

3

Ve = ��ò �

= �2 gR ; 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

I = ∑ m r� = ­ r� d m ; I = M K 2

; τ = I α

KE = �� I ω2

; For rolling body, K.E. = �� MV

2 T1 û8ø8 W

Conservation of angular momentum I1 ω1 = I2 ω2

M.I.of ( i ) ring = Mr2 , ( ii ) disc =

òø8� ,

(iii) hollow sphere = � Mr

2 (iv) solid sphere =

�J Mr2

,

(v) thin rod = òß8�� , (vi) rect.bar = M T ß8�� &8��W

Equation of motion, ( i ) ω = ω0 + αt ; (ii ) θ = ω0 t + �� α t

2 ;

(iii) ω� = ω0 2 + 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 = - ω

2 x ;

���5 = ω √a� x� ; x = a sin ( ω t + α )

T = �E�

= 2π ��� = �E h%'',¹2ø  /.5 �.,¹�%'2�2/5

=2π � �%,, 3+ø'2 ¹2ø  /.5 �.,¹�%'2�2/5

K .E. = �� m ω� (a

2 - x

2) ; P.E. =

�� M ω2 x

2 ;

Total Energy = �� m a

2 ω� = 2π� m a

2 n

2

For simple pendulum, T = 2π � �� ; For oscillating magnet, T = 2π � �òP

R = �a21 a22 2a 1 a 2 cos�α 1 – α 2� ; ËbÎ =

0 � =>? � � � 0 � =>? � �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 = � ; Tensile stress =

èO ; Y = ò � �E ø8ß

Volume Strain = � ó ó ; Volume stress =

èO = dP ;

K = - V �ô�ó

Shearing strain = ∆ � ß = ∆ θ ; Shearing stress =

èO ;

n = èO ∆- ; 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 T 5�

��W

In - ve x - direction , y = a sin 2 π T 5�

��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 óó �   ,á when source is approaching towards stationary

listner

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 = ôß . ���

Perpendicular position , N = n = ô�ß ���

For both positions , Tp2 = a constant

Air columns : closed at one end, n = óF 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 , ýO5 = σ T4

Newton’s law , �ý�5 = k �θ θ 0�

Radiation correction ∆ θ # � � wθ θN

KINETIC 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� =

� òó C2 =

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.

W A V E T H E O R Y A N D

I N T E R F E R E N C E O F L I G H T

n = '�'� =

���� ; n =

,./ .,./ ø

Bright 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 = �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 = »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� � � Õ

Energy per unit volume = �� k ε 0 E

2

C = ýó ; For parallel plate condenser, C =

O ü Õ � �

Energy of a charged condenser = �� QV =

�� CV2 =

�� 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 =

óß �

Tangent Galvanometer, I = � ø P

�

µ Õ / tan θ = k tan θ

M A G N E T I S M

M= 2ml; Baxil = � ÕFH

��ÆX ; Beqa = �

ÕFH �ÆX

For any point, B = �

ÕFH �ÆX √3 cos� 9 1 ;

� = tan-1

T�� tan 9W OR tan� = �� tan 9

Vaxial = � ÕFH �Æ8 , Veqn = 0, Any point, V =

� ÕFH � '+, @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 ���/OW α 2

tan θ = � �� �

e = e0 sinωt = 2 π fnAB sin2πnt

I = 2�

= I0 sinωt; erms = 2 Õ√� , Irms =

ß Õ√�

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 ^8E � 28 , En =

� 2�

� � Õ8 /8^8 ,

v~ = �� =

� 2�

� � Õ8 '^X T �¹8 �/8W

µ ��

� ü� Õχ ηX = P

�ã�5 = - λ N = N0e

- λ t

T =

�+� 2 ��

= Õ.K�

� ; λ =

Õ.K� � ; λ =

!¦æ¦ !ã

E L E C T R O N S A N D P H O T O N S

A photon = hv = ^ '�

; w = hv0 = h '�

Õ

� �m V

2 max = h (v - v0) = hc à�

� �

� Õá