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MATHEMATICS FORMULAE EXPLORER
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This book is dedicated to my Parents – Mrs. S. Geethabai
Copies can be obtained from No. 9, New No. 29, First Street Bank Colony, MMC Chennai – 600 051, Tamilnadu
Phone : 044 – 2555 9594
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MATHEMATICS FORMULAE EXPLORER CONTENTS S. No.
Topics
Page No.
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34.
Algebra Analytical Geometry 3D- Analytical Geometry Boundary Value Problems Coordinate Geometry Commercial Arithmetic Complex Numbers Data Analysis Determinants Differential Calculus Differential Equations Discrete Mathematics Fourier Series Fourier Transform Graphs Integral Calculus Laplace Transform Matrices Measurement Mensuration Multiple Integrals Number Work Numbers and Operations Ordinary Differential Equations Partial Differential Equations Probability Pure Arithmetics Sets Statistics Tables Theoretical Geometry Trignometry Vector Algebra Z-Transform
004 007 020 023 028 031 035 040 044 048 051 057 062 068 070 071 074 076 079 087 092 093 094 099 102 108 117 118 120 122 123 131 139 143
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1. Algebra Expansion and Factorisation
H.C.F x L.C.M of two expression =Product of the two expressions
Equation Two expression connected by a sign of equality is
is consistent equation, if the equality holds for some value of the variable/unknown
an inconsistent equation, if the equality holds for no value of the variable/unknown
an identical equation, if the equality holds for any value of the variable/unknown Mathematics Formulae Explorer - Page 4 of 146
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Simultaneous Equation
Simultaneous Equation of the type
is consistent and has only one set of solution if
is consistent and has no solution if
have infinite number of solutions if
Laws of Indices • • • • • • • • • •
… . ! "!#, %# &#%!%'( %!()(
*
√
, ( '-.( "
, ,
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Logarithms • • • • • • • • • •
/ -)
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0 0, 0 0 & 3, 4
-) / !%-) 0 0, 0 0 & 3, 4
-) -) -) -) -) -)
-) -)
-)
-)
-) -) -) -) -)
-)
[ in all cases from third formulae, a > 0, b > 0 & a, b , 0 0, 5 0 0 ]
Some standard forms of the Binomial Expansion
…. ! ! …. ! !
…. ! ! …. ! !
….
….
…. ….
. . . … .
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2. Analytical Geometry
Introduction ‘Geometry’ is the study of Points, Lines, Curves, Surfaces, etc and their properties. In the 17th century AD, the methods of Algebra were applied in the study of Geometry and thereby ‘Analytical Geometry’ emerged out. The renowned French philosopher and Mathematician Rane Descartes (1596-1650) showed how the methods of Algebra could be applied to the study of Geometry. Locus The path traced by a point when it moves according to specified geometrical conditions is called the Locus of the point. Straight Lines A straight line is the simplest geometrical curve. Every straight line is associated with an equation. •
Slope-Intercept Form : y = mx + c
•
Point –Slope Form : y-y1 = m(x – x1)
• Two Point Form : • • •
Intercept Form : , where ‘a’ and ‘b’ are x and y intercepts.
Normal Form : # 7 #% 7 &
General Form : ax + by + c = 0
Length of the Perpendicular •
The length of the perpendicular from the point (x1, y1) to the line ax+by+c=0 is 8 8 9
•
The length of the perpendicular from the Origin to the line ax+by+c=0 is
8
9
8
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Slope of an equation ax + by + c = 0 For ax + by + c = 0, Slope m =
:(""%%(! " :(""%%(! "
Angle between two straight line
If ; is the angle between the two straight lines, then ;
! <
<
Condition for Parallel and Perpendicular •
If the two straight lines are Parallel, then their slopes are equal. i.e., m1=m2
•
If the two straight lines are Perpendicular, then the product of their slopes is -1. i.e., m1 x m2= -1
The condition for three straight lines to be concurrent is = if Condition for Concurrent
= ,
Equation of the Straight line passing through the intersection of the two lines •
>
represents a straight line passing through the intersection of the straight lines and .
Pair of Straight Lines •
Combined
•
Pair of straight lines passing through the origin is ax +2hxy+by =0
•
The Straight line is ( i ) Real and Distinct if h2 > ab ( ii ) Coincident if h2 = ab ( iii ) Imaginary if h2 < ab
equation
of
the
ax2+2hxy+by2+2gx+2fy+c=0,
pair of straight lines is where a, b, c, f, g, h are constants. 2
2
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Slopes of pair of straight line
?
•
Sum of the slopes of pair of straight lines, m1+m2 =
•
Product of the slopes of pair of straight lines, m1m2 =
Angle between the pair of Straight line •
• •
Angle between the pair of straight lines passing through the origin is ; ! 8
8
9?
If the straight lines are parallel, then h2 = ab If the straight lines are perpendicular, then c(""%%(! " (""%%(! "
Condition to represent a pair of straight line •
The
general second degree equation ax +2hxy+by +2gx+2fy+c= 0 represent a pair of straight lines is 2
condition
for
a
2
abc+2fgh-af2-bg2-ch2 = 0.
Circle Definition A circle is the locus of a point which moves in such a way that its distance from a fixed point is always constant. The fixed point is called the Centre of the Circle and the constant distance is called the Radius of the circle. •
The equation of circle when the centre is (h, k) and radius ‘r’ is (x – h)2 + (y – k)2 = r2
•
If the centre is origin, equation of circle is x2+y2 = r2
•
The equation of circle, if the end points of a diameter are given by (x – x1) (x – x2) + (y – y1) ( y – y2) = 0
•
The General equation of the circle is x2 + y2 + 2gx + 2fy + c = 0 with centre is (-g, -f) and radius is 9) " Mathematics Formulae Explorer - Page 9 of 146
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Tangent to the Circle •
Equation of the tangent to a circle at a point (x1, y1) is xx1 + yy1 + g(x + x1) + f(y + y1) + c = 0
•
Length of the tangent to the circle from a point (x1, y1) is
•
If PT2 = 0, then the point is on the Circle.
•
If PT2 > 0, then the point is outside the Circle.
•
If PT2 < 0, then the point is inside the Circle.
•
Condition for the line y = mx + c to be a tangent to the circle x2 + y2 = a2 is c2 = a2 (1 + m2)
•
Point of contact of the tangent y = mx + c to be a tangent to the circle , D x2 + y2 = a2 is C 9 9
@A B ) "
•
Equation of any tangent to a circle if of the form E √
•
Two tangent can be drawn from a point to a circle is
m2(x2 – a2) – 2mxy +(y2 – a2) = 0. This is a Quadratic equation in ‘m’. Thus ‘m’ has two values. But ‘m’ is the slope of the tangent. Thus, two tangents can be drawn from a point to a circle.
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Family of Circles Concentric Circles Two (or) more circles having the same centre are called Concentric Circle. Circles Touching Internally or Externally Two circles may touch each other either internally or externally. Let C1, C2 be the centres of the circles and r1, r2 be their radii and P, the point of contact. •
Two circle touch externally, if C1C2 = r1 + r2
•
Two circle touch internally, if C1C2 = r1 - r2
Orthogonal Circles Two circles are said to be Orthogonal if the tangent at their point of intersection are at right angles. Condition for Orthogonal •
Condition for two circles to cut orthogonal is 2g1g2 + 2f1f2 = c1+c2
Conic Definition A conic is the locus of a point which moves in a plane, so that its distance from a fixed point bears a constant ratio to its distance from a fixed straight line. The fixed point is called focus, the fixed straight line is called directrix and the constant ratio is called eccentricity, which is denoted by ‘e’. Classification with respect to the General Equation of a Conic The equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 represents either a (nondegenerate) conic or a degenerate conic. If it is a conic, then it is •
a Parabola if B2- 4AC = 0
•
an Ellipse
•
a Parabola if B2- 4AC > 0
if B2- 4AC < 0
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Parabola ( y2 = 4ax ) Definition The locus of a point whose distance from a fixed point is equal to the distance from a fixed line is called a Parabola. i.e., Parabola is a conic whose eccentricity is 1. Definitions •
The fixed point used to draw the parabola is called the Focus F. Here, the focus is F(a,o).
•
The fixed line used to draw a parabola is called the Directrix of the parabola. Here, the equation of the directrix is x = - a
•
The axis of the parabola is the axis of symmetry. The curve y2 = 4ax is symmetrical about x-axis and hence x-axis or y = 0 is the axis of the parabola y2 = 4ax. Note that the axis of the parabola passes through the focus and perpendicular to the directrix.
•
The point of intersection of the parabola and its axis is called its Vertex. Here, the vertex is V(0,0).
•
The Focal Distance is the distance between a point on the parabola and its focus.
•
A chord which passes through the focus of the parabola is called the Focal Chord of the parabola
•
Latus Rectum is a focal chord perpendicular to the axis of the parabola. Here, the equation of the latus rectum is x = a.
•
End points of Latus Rectum is L (a, 2a) and L/(a, -2a)
•
Length of Latus Rectum = 4a. Length of Semi-Latus Rectum is 2a.
General form of the standard equation of a Parabola The General form of the standard equation of the parabola is • F ? (open rightwards) •
• •
F ? ( open leftwards)
? F (open upwards )
? F (open downwads) Mathematics Formulae Explorer - Page 12 of 146
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Ellipse
Definition
The locus of a point in a plane whose distance from a fixed point bears a constant ratio, less than one to its distance from a fixed line is called Ellipse. Definitions Focus : The fixed point is called focus, denoted as F(ae,0) Directrix : The fixed line is called directrix l of the ellipse and its equation is (
Major axis : The line segment AA/ is called the major axis and the length of the major axis is 2a. The equation of the major axis is y = 0. Minor axis : The line segment BB/ is called the minor axis and the length of the minor axis is 2b. The equation of the minor axis is x = 0. Centre : The point of intersection of the major axis and minor axis of the ellipse is called the Centre of the Ellipse. Vertices : The points of intersection of the ellipse and its major axis are called its vertices. Focal Distance : The focal distance with respect to any point P on the ellise is the distance of P from the referred focus. Focal Chord : A chord which passes through the focus of the ellipse is called the focal chord of the ellipse. Latus Rectum : It is the focal distance perpendicular to the major axis of the Ellipse. The equation of the latus rectum are x = + ae, x = - ae. Eccentricity : ( B
End Points of Latus Rectum are G(, E and other latus rectum are G (, E .
Length of the Latus Rectum are
Special Property : Thanks to the symmetry about the origin, it permits the second Focus F2(-ae,0) and the second directrix x = (
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General forms of Standard Ellipses The General forms of Standard Equation of Ellipses, if the centre C(h,k) is
? F ?
F
a>b
Focal Property of an Ellipse The sum of the focal distances of any point on an ellipse is constant and is equal to the length of the major axis.
Hyperbola Definition
The locus of a point in a plane whose distance from a fixed point bears a constant ratio, greater than one to its distance from a fixed line is called Hyperbola.
Definitions Focus : The fixed point is called focus, denoted as F(ae,0) Directrix : The fixed line is called directrix l of the hyperbola and its equation is (
Transverse axis : The line segment AA/ joining the vertices is called the transverse axis and the length of the transverse axis is 2a. The equation of the transverse axis is y = 0. Conjugate axis : The line segment joining the points B(0, b) and B/ (0, -b) is called the conjugate axis and the length of the conjugate axis is 2b. The equation of the conjugate axis is x = 0.
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Centre : The point of intersection of the transverse and conjugate axes of the hyperbola is called the Centre of the Hyperbola.
Vertices : The points of intersection of the hyperbola and its transverse axis are called its vertices. Latus Rectum : It is the focal chord perpendicular to the transverse axis of the Hyperbola. The equation of the latus rectum are x = + ae, x = - ae. Eccentricity : ( B
End Points of Latus Rectum are G(, E and other latus rectum are G (, E .
Length of the Latus Rectum are
The other form of the Hyperbola If the transverse axis is along y-axis and the conjugate axis is along x-axis, then the equation of the hyperbola is of the form
For this type of hyperbola, we have the following points. •
Center is C(0,0)
•
Vertices A(0, a) and A/(0, -a)
•
Foci are F(0, ae) and F(0, -ae)
•
Equation of transverse axis is x = 0
•
Equation of conjugate axis is y = 0
•
Equations of Latus rectum is E(
• • •
End points of conjugate axis is (b, 0) and (-b, 0) Equations of directrices is E
End points of Latus rectum is E
(
, ( , E
, (
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Parametric form of Conics Conic
Parametric equations
Parabola
x = at2 y = 2at
Ellipse
Hyperbola
! !
. ! !
x = a sec ;, y = b tan ;
Range of parameter
Any point on the conic
∞ I J I ∞
t
x = a cos ;, y = b sin ;
Parameter
;
‘t’ or (at2, 2at) M;M or
I ; I 2L
∞ I J I ∞
t
;
I ; I 2L
(acos ;, #% ;
‘t’ or
! . ! , ! !
M;M or
(a sec;, ! ;
Equation of Chord Conic
Parabola
Ellipse
Hyperbola
Equation of Chord joining (x1, y1) and (x2, y2)
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Equation of Tangent and Normal Conic Parabola
Ellipse
Hyperbola
Equation of Tangents at (x1, y1)
Equation of Normal at (x1, y1)
Equation of Chord and Tangent at Parametric Form Conic
Parabola
Ellipse
Hyperbola
Equation of Chord at Parametric Form
Equation of Tangents at Parametric Form
Chord joining the points ! & ! at ‘t’ is is ! ! ! !
yt = x + at2
Chord joining the points ; & ; at ‘;’ is is ; ; ; ; #; #%; # #% ; ; #
Chord joining the points ; & ; at ‘;’ is is ; ; ; ; #(; !; #
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Results connected with Conics Conic
Parabola
Ellipse
Condition that y=mx+c may be a tangent to the conic
Point of Contact
C , D
N , O
E 9
N , O
E 9
where
Hyperbola
Equation of any tangent is of the form
where
Asymptotes Definition An asymptote to a curve is the tangent to the curve such that the point of contact is at infinity. In particular, the asymptote touches the curve at ∞ ∞. Results regarding Asymptotes •
•
The equations of the asymptotes to the hyperbola
is
The combined equation of asymptotes is i.e.,
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• •
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The asymptotes pass through the centre C(0,0) of the hyperbola.
The slopes of asymptotes are
.
i.e.,
the transverse axis and
conjugate axis bisect angles between the asymptotes.
•
• •
If 7 is the angle between the asymptotes then the slope of
! 7 .
Angle between the asymptotes is 7 !
is
Angle between the asymptotes is 7 #( (
Rectangular Hyperbola ( xy = c2 where
Definition
)
A hyperbola is said to be a rectangular hyperbola if its asymptotes are at right angles. Results • • • • •
• • • • • •
Eccentricity of the Rectangular Hyperbola is ( √ and also b2 = a2(e2-1) The Vertices of the rectangular hyperbola are , and
√ √
√
,
√
The foci are (a, a) and (-a, -a) The equation of the transverse axis is y = x and the conjugate axis is y = - x. If the centre of the rectangular hyperbola is (h, k) then (x – h) ( y – k) = c2 The parametric equation of the rectangular hyperbola xy = c2 is x = ct, y= !
Equation of the tangent at (x1, y1) to the rectangular hyperbola xy = c2 is xy1+yx1 = 2c2 Equation of the tangent at ‘t’ is x + yt2 = 2ct Equation of normal at (x1, y1) to the rectangular hyperbola xy = c2 is xx1- yy1 = x12- y12 Equation of normal at ‘t’ is y - xt2 = ct3 !
Two tangents and four normals can be drawn from a point to a rectangular hyperbola.
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3. Three Dimensional Analytical Geometry The equation of the Sphere whose centre is (a, b, c) and radius ‘r’ is P
The equation of the Sphere has the following three characteristics. • It is of second degree equation in x, y, z • The coefficients of x2, y2, z2 are equal • The terms xy, yz and zx are absent If the coefficients of x2, y2, z2 are each unity, then the coordinates of the centre of the Sphere are
C (""%%(! " , (""%%(! " , (""%%(! " PD and square of the radius is equal to the sum of the squares of the coordinates of the centre minus the constant term.
The equation of a Sphere whose centre is (x1, y1, z1) is P P P Equation of a Sphere with the extremities of diameter at the points (x1, y1, z1) and (x2, y2, z2) is P P P P Two Spheres S1 and S2 whose radii are r1 and r2 touch externally if the distance between their centres is equal to the sum of their radii. d = r1 + r2 The point of contact is the point which divides internally the line joining the centres in the ratio of the radii.
Two Spheres S1 and S2 whose radii are r1 and r2 touch internally if the distance between their centres is equal to the difference of their radii. d = r1 ~ r2
The point of contact is the point which divides externally the line joining the centres in the ratio of the radii. Mathematics Formulae Explorer - Page 20 of 146
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To find the condition that the plane Ax + By + Cz + D = 0 may touch the Sphere P . ' RP is S. T' :R U S T : . ' R Condition for the two Spheres to cut Orthogonally
P . ' RP and P . ' R P is
. . ' ' R R . ' R . ' R
The General Equation of a Right Circular Cylinder If the axis of the required cylinder is
7 -
V
PW
and radius is ‘r’ then the
equation of a circular cylinder is - - 7 V P W 7 V P W – - The equation of the Cylinder whose generators intersect the curve ? ) " , P
and parallel to the line
-
P
is
-P ? -P P P ) -P " P
The equation of the cylinder whose generators touch the sphere
and are parallel to the line
-
P
P
is
- P - P
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The equation of the Cone whose vertex is at the point 7, V, W and whose generators intersect the conic is ) " , P
The equation of the Right Circular Cone whose vertex is at 7, V, W and its axis
at the line
7 -
V
PW
and whose semi-vertical angle ; is
- 7 V P W - 7 V P W # ;
The equations of the enveloping cone whose vertex is at 7, V, W and whose generators touch the sphere P is
7 V W 7 V P W 7 7 V V WP W
The equation of the tangent plane at the point (x1, y1, z1) to the cone P )P "P ? is
? )P ? "P P) " P
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4. Boundary Value Problems VIBRATION OF STRING Equation and Conditions
Y Y Y! Y
Initial velocity zero.
Y Y Y! Y Initial velocity is given
Boundary Conditions , ! , Z! -, ! , Z!
Y, , [ Y! [, "
, ! , Z! -, ! , Z! ,
Y, ) Y!
Correct Solution
Most General Solution
, ! : # & : #% & : # &! : #% &!
, !
, ! : # & : #% & : # &! : #% &!
, !
]
\ : #% ^
]
\ : #% ^
_ _! # -
_ _! #% -
ONE DIMENSIONAL HEAT FLOW EQUATION Equation and Conditions
Boundary Conditions
Y. Y . 7 Y! Y
., ! , Z!
Y. Y . 7 Y! Y
., ! , Z!
Beginning point ‘A’ and Ending point ‘B’ are at zero temperature
Beginning point ‘A’ is at zero temperature and Ending point ‘B’ is at non-zero temperature k.
.-, ! , Z! ., "
.-, ! F, Z! ., "
Correct Solution
Most General Solution
., ! S # & T #% & (7 & !
., !
., ! S # & T #% & (7 & !
., !
]
_ 7 \ T #% ( ^
]
_ 7 \ T #% ( ^
_ !
-
_ !
-
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SQUARE PLATE - Condition I Conditions of Square Plate
• .,
f(x) y=a 0o x=0
Boundary Conditions
0o x=a
• .,
• .,
• ., "
y=0 0o
Correct Solution ., : # & : #% & : (& : (&
Most General Solution ]
., \ : #% ^
_ _ #% ?
SQUARE PLATE - Condition II Conditions of Square Plate
• .,
0o y=a 0o x=0
Boundary Conditions
0o x=a y=0
f(x)
Correct Solution ., : # & : #% & : (& : (&
• .,
• .,
• ., " Most General Solution ]
., \ : #% ^
_ _ #% ?
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SQUARE PLATE - Condition III Conditions of Square Plate
• .,
0o y=a 0o x=0
Boundary Conditions
f(x) x=a
• .,
• .,
• ., "
y=0
0o
Correct Solution ., : # & : #% & : (& : (&
Most General Solution ]
., \ : #% ^
_ _ #% ?
RECTANGULAR PLATE - Condition I Conditions of Rectangular Plate
• .,
f(x) y=b 0o x=0
Boundary Conditions
0o x=a y=0
0o
Correct Solution ., : # & : #% & : (& : (&
• .,
• .,
• ., " Most General Solution ]
., \ : #% ^
_ _ #% ?
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RECTANGULAR PLATE - Condition II Conditions of Rectangular Plate
Boundary Conditions
• .,
0o
• .,
y=b 0o x=0
• .,
0o x=a
• ., "
y=0
f(x)
Correct Solution ., : # & : #% & : (& : (&
Most General Solution ]
., \ : #% ^
(
_ _ _ ( (
_
RECTANGULAR PLATE - Condition III Conditions of Rectangular Plate
• .,
0o y=b 0o x=0
Boundary Conditions
f(y) x=a y=0
0o
Correct Solution ., : # & : #% & : (& : (&
• .,
• .,
• ., " Most General Solution ]
., \ : #% ^
_ _ #% ?
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RECTANGULAR PLATE – Infinite Plate - Condition I Conditions of Rectangular Plate - Infinite Plate ∞
Boundary Conditions
• .,
0o x=0
• .-,
• ., ∞
0o x=l
• ., "
y=0
f(x)
Correct Solution
Most General Solution
., : # & : #% & : (& : (&
]
., \ : #% ^
_ _ ( -
RECTANGULAR PLATE – Infinite Plate - Condition II Conditions of Rectangular Plate - Infinite Plate 0o y=l f(y) y=0
∞
0o
Correct Solution ., : # & : #% & : (& : (&
Boundary Conditions
• .,
• ., -
• .∞,
• ., " Most General Solution ]
., \ : #% ^
_ _ ( -
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5. Co ordinate Geometry Introduction The Modern terms Co-ordinates, abscissa and ordinate were contributed by German Mathematician Gottfried Wilhelm Von Neibliz in 1692. Rene Descartes invented co-ordinate geometry. Distance Formula between two points A(x1, y1) and B(x2, y2)
U%#!(,
ST 9
Mid-Point Formula between two points A(x1, y1) and B(x2, y2)
`% @%! " ST C , D
Centroid Formula between three points A(x1, y1), B(x2, y2) and C(x3, y3)
:(!% C
,
D
Area of the Triangle from the given three points A(x1, y1), B(x2, y2) and C(x3, y3) S(
Condition for the three points A(x1, y1), B(x2, y2) and C(x3, y3) to be Collinear Area of the Parallelogram from the given four points A(x1, y1), B(x2, y2), C(x3, y3) and D(x4, y4) S(
Slope (or) Gradient of the Line
If ; is the angle of inclination, then Slope, m = tan ;
Slope of the line joining two points A(x1, y1) and B(x2, y2) Slope, m =
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Slope of the line ax+by+c = 0
Slope =
:(""%%(! "
:(""%%(! "
Equation of Straight Line with Slope m and y-intercept c Equation of straight line is y = m x + c
Equation of Straight Line with Slope m and point A(x1, y1) Equation of straight line is y – y1 = m (x – x1) Equation of Straight Line with Slope m and joining two points A(x1, y1) and B(x2, y2) Equation of straight line is
Equation of Straight Line with x intercept a and y intercept b Equation of straight line is
Condition for two lines to be Parallel
Two lines are Parallel, then their slopes are equal. i.e., m1 = m2 Condition for two lines to be Perpendicular Two lines are Perpendicular, then their product of their slopes gives -1 i.e., m1 x m2 = -1 Equation of Straight Lines with different cases •
Any line parallel to ax + by + c = 0 is ax + by + k = 0 (differ only by constant)
•
Any line parallel to x-axis is y=k ( k is constant)
•
Any line parallel to y-axis is x = c ( c is constant)
•
The line which is perpendicular to the line ax + by + c = 0 is of the form bx – ay + k = 0
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Circumcentre, Centroid and Orthocentre Circumcentre : The perpendicular bisector of the sides of a triangle are concurrent. The point of concurrence is called circumcentre. Centroid of a triangle : The medians of a triangle meet at a point. This point is known as centroid. Orthocentre of a triangle : The altitudes of a triangle meet at a point. This point is called Orthocentre. Slope of both axes •
The Slope of x-axis = 0
•
The Slope of y-axis = not defined
Concurrency of Three Lines
Condition that the lines , and may be concurrent if,
Intersection of Two Straight Lines
The two lines if not parallel in a plane intersect in a unique point.
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6. Commercial Arithmetic Basic Definitions
Percentage
%
%
Mb # &((!)( "
Profit and Loss
Profit = Selling Price (S.P) – Cost Price (C.P)
Loss = Cost Price (C.P) – Selling Price (S.P)
Selling Price = Cost Price + Profit
Selling Price = Cost Price - Loss
Cost Price = Selling Price – Profit
Cost Price = Selling Price + Loss
Profit (in percent) =
Loss (in percent) =
Selling Price = Cost Price + x% of Cost Price,
,
if Profit is x%.
Cost Price = Selling Price x d e, if Profit is x%.
Selling Price = Cost Price - x% of Cost Price,
,
if Loss is x%.
A!- @"%! :#! @%(
A!- G##
:#! @%(
x 100 =
x 100 =
c.@:.@
:#! @%(
:.@c.@
:#! @%(
x100
x100
Cost Price = Selling Price x d e, if Loss is x%.
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Discount and Market Price
Discount = Marked Price – Actual Selling Price
Discount in Percent = Marked Price –
Actual Selling Price = Marked Price – Discount = Marked Price -
Marked Price = d
U%#.! %
• • • • • •
`F( @%(
%U%#.!
x 100
x Marked Price
e c(--%) @%(
Successive (2nd) discount is calculated on the balance after deduction of the first discount from the marked price and so on.
Simple Interest •
S!.- c(--%) @%(
Simple Interest (S.I) =
@fg
= PNi, where P is the Principal, N is the Period
in years and R% is the rate of interest for 1 year. % unit principal for one year
@
h
g
h
f
g
= interest for
fg
h @g
@f
Amount (A) = Principal + Interest
@
S
fg
Interest = Amount - Principal
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Compound Interest (C.I) f
•
Compound Interest (C.I) = @ d e @, where P is the Principal, N is the Period in years and R% is the rate percent annually.
•
Amount, A = @ d
g
g
e
f
•
Principal = Amount – Compound Interest
•
Difference between C.I and S.I for 2 years = @ d
•
e g
Difference between C.I and S.I for 3 years = d e @g
g
Recurring Deposit (R.D)
Recurring Deposit is a special type of deposit in which a person deposits a fixed sum every month over a period of years and receives a large sum at the end of the specified number of years. Since the deposit is made month after month, it is called Recurring Deposit. Recurring Deposits are also known as Cumulative Term Deposits. The amount deposited every month is called the Monthly Deposit.
Total Interest =
@fg
, where N =
,
P be the Monthly Instalments, R % be the rate of Interest and ‘n’ be the number of monthly instalments.
Amount Due = Amount Deposited + Total Interest
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Fixed Deposit Fixed Deposit are deposits for a fixed period of time and the depositor can withdraw his money only after the expiry of the fixed period. It is also known as Term Deposits. However, in the case of necessity, the depositor can get his fixed deposit terminated earlier to get a loan from the bank under terms laid down by the bank. There are two types of fixed deposits, namely
Short Term Deposits Long Term Deposits
Short Term Fixed Deposits are accepted by the banks for a short period ranging from 46 days to one year. The interest paid on this deposit is Simple Interest. Long Term Fixed Deposits are accepted by the banks for a period of one year or more. The interest paid on this type of deposit is Compound Interest.
Quarterly Interest =
@g
Half Yearly Interest =
@g
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7. Complex Numbers The Complex Number System •
A Complex number is of the form a+ib, where ‘a’ and ‘b’ are real numbers and ‘I’ is called the imaginary unit, having the property i2 = -1.
•
If z = a+ib then ‘a’ is called the real part of z, denoted by Re(z) and ‘b’ is called the imaginary part of z and is denoted by Im(z).
•
If z = a+ib is a complex number then the negative of z is denoted by –z and it is defined as –z = -a + i (-b).
•
Basic Algebraic Operations with Complex Numbers (a + ib) + (c + id) = (a + c) + i (b + d) (a + ib) - (c + id) = (a - c) + i (b - d) (a + ib) (c + id) = (ac - bd) + i (ad + bc)
•
If z = a + i b, then the conjugate of z is denoted by Pi and
•
Properties of Complex Numbers
is defined by Pi %.
PPi % % :j.)!( " Pi P. %. (. , Pi P
Z is real / the imaginary part is zero
g(P
h P
PPi
PPi %
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Conjugate of the sum is the sum of their conjugates iiiiiiiiii i Pi P P P
Conjugate of the product of two complex numbers is the product of their conjugates
iiiiii i Pi P P P
The conjugate of the quotient of two complex numbers is the quotient of their conjugates.
P iii i P
•
iiiii P P iii P
P iii
The Modulus (or) Absolute value of z = a+ib is denoted by |P| is defined by √
• The Amplitude (or) Argument of z = a+ib is denoted by arg z or arg z is defined by ; !
•
It is obvious that |Pi| |P|. Also, |P| √PPi
•
g(P [ |P| and h P [ |P|
•
The Modulus of a product of two complex numbers is equal to the
•
The above result can be extended to any finite number of complex
•
The Modulus of a quotient of two complex numbers is equal to the
product of their moduli. |P P | |P ||P |
numbers. i.e., |P P … . P | |P ||P ||P | … |P | quotient of their moduli. < < P P
|P |
|P |
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Triangle Inequality The Modules of sum of two complex numbers is always less than or equal to the sum of their moduli.
|P P | [ |P | |P |
|P P | [ |P | |P |
|P P l … P | [ |P | |P | l |P | •
The Modulus of the difference of two complex numbers is always greater
•
Polar form of a Complex Number P % # ; % #% ;
•
than or equal to the difference of their moduli. |P P | m |P | |P |
For any two complex numbers P P
|P P | |P ||P |
)P P ) P ) P ) ) P ) P P P
•
The above result can be extended to any finite number of complex numbers .
|P P … … . P | |P ||P | … … . . |P |
)P P … … … . P ) P ) P l … … . . ) P
•
The Exponential form of a Complex Number (%; is known as Euler’s
Formula and is defined by (%; # ; % #% ;
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General Rule for determining the argument ;
Let z = a + ib where a, b n R. In First Quadrant,
; 7
In Third Quadrant,
; _ 7
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Take 7 ! || ||
In Second Quadrant, ; _ 7
In Fourth Quadrant, ; 7
Both cos ; and sin ; are positive.
Z lies in the first quadrant.
Sin ; is positive and cos ; is negative. Z lies in the second quadrant.
Both cos ; and sin ; are negative.
Z lies in the third quadrant.
Sin ; is negative and cos ; is positive. Z lies in the fourth quadrant.
; 7
; _ 7
; _ 7
; 7
Theorem For any polynomial equation P(x) = 0 with real coefficients, imaginary (complex) roots occur in conjugate pairs. De Moivre’s Theorem For any rational number n, opq rs t qtr rs is the value or one of the values of # ; % #% ; Mathematics Formulae Explorer - Page 38 of 146
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Roots of a Complex Number Working Rule to find the nth roots of a Complex Number •
Write the given number in polar form
•
Add 2k_ to the argument
•
Apply De Moivre’s theorem
•
Put k=0, 1, 2,3, …upto (n-1)
nth roots of unity • •
R
Sum of the roots is zero.
•
The roots are in Geometric Progression with common ratio w.
•
The arguments are in Arithmetic Progression with common difference
•
Product of the roots =
_
Cube Roots of Unity •
If R
%√
, then R
%√
•
The sum of the cube roots of unity is zero.
•
w3 = 1
1+w+w2 = 0.
Fourth Roots of Unity •
1+w+w2 +w 3= 0.
•
w4 = 1
•
The values of w used in cube roots of unity and in fourth roots of unity are different. Mathematics Formulae Explorer - Page 39 of 146
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8. Data Analysis Statistics is the study of the methods of collecting, organizing and analyzing quantitative data, and drawing conclusions. The data are collected on samples from various populations of people, animals and things by different methods such as observations, interviews, etc. Statistics is used in almost every field such as business, education, science, psychology, research, etc. The word ‘data’ is the plural form of datum, which means facts and figures. Data Data represent factual information (in the form of measurements or statistics) which is used as a basis for reasoning, discussion or calculation. Data are classified as either Primary or Secondary. Primary Data Primary data are the data which are collected directly for a specific purpose for the first time and they are original in character. Examples : Questionnaires, Interviews, etc., Secondary Data Secondary data are data already collected, analyzed and presented in written form ready for people to use. Examples : Government reports, books, articles, maps, etc., Types of Data Data can be qualitative or quantitative. Names of persons, marital status, etc., are examples of qualitative data. Quantitative Data Quantitative data are measurements expressed in terms of numbers. Income of individuals, production of a car company, exports in units of a garment company, marks of students, etc., are all quantitative data. Quantitative data can further be classified as continuous data and discrete data. Continuous Data : Takes numerical values within a certain range. Example : Height of a person. Discontinuous (or) Discrete Data : Takes only whole-number values. Example : The number of boys in each class can be expressed only in whole numbers.
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Displaying Data Tables, Charts and Graphs are examples of visual representation of data. Graphs or Charts show the relationship between changing things and are used to make facts clearer and more understandable. Line Graph A Line Graph is used to show continuous data. The dependent data is plotted along the y-axis and the independent data along the x-axis. Multiple-Line Graph A multiple-line graph can effectively compare similar data over the same period of time. Pie Chart A pie chart is a circular chart divided into segments. Each segment illustrates relative magnitudes or frequencies. It shows the component parts of a whole. A pie chart uses percentages to compare information since they are the easiest way to represent a whole (100%). In a Pie chart, the arc length, central angle and area of each segment is proportional to the quantity it represents. Exploded Pie Chart A chart with one or more segments separated from the rest of the disc is called an exploded pie chart. Formation of Frequency Tables Classification and Tabulation Collection of data in the form of numbers alone will not help us to make decisions or form conclusions. Since just a huge collection of numbers does not have any meaning, it is necessary to classify the numbers as values and pictures before presentation. Classification is the process of grouping data according to their common characteristics. Tabulation is the process of arranging the classified data in tabular form. Notes •
The number of times a particular observation or a variable ‘x’ occurs in a data set is called its frequency which is denoted by ‘f’. Mathematics Formulae Explorer - Page 41 of 146
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•
Frequency distributions show the actual number of observations falling in each range of observations.
•
In a continuous distribution the data are obtained by measurement.
•
The vertical bar ‘|’ which represents each occurrence of a variable ‘x’ or observation is called a tally mark.
•
The mid-value of a class interval is called its class mark.
•
Class boundaries are actual or true limits of a class interval in a grouped distribution table and are continuous.
Measures of Central Tendency The classification and tabulation of statistical data is a process of condensing the entire data. The graphs / charts give a visual presentation and make the comparisons easier. But for analysis of given numerical data, some description of the given data is needed. The statistical average is a numerical value around which the greatest proportion of the data concentrates. For example, if we say in a class of 40 students, the mathematics marks vary from 40 to 95, but most of them secured 70 marks then 70 is the statistical average marks of the class. Such values are called measures of central tendency. The three important measures of central tendency are • • •
Arithmetic mean (or) Average Median Mode
Arithmetic Mean (A.M) The Arithmetic Mean of a collection of data is a measure of central tendency and it helps in interpreting the data. The arithmetic mean (or) AM is commonly known as the mean or the average of a given set of data. Arithmetic Mean (A.M) of Ungrouped Data
The formula used is S. `,
Median of Ungrouped Data
u
c. " v#('!%#
f. ( " v#('!%#
∑
Median is the middle value or the mean of the middle two values, when a set of observed data is arranged in numerical order. Median divides the distribution into two equal halves such that there are as many observations less than it as there are greater than it. Mathematics Formulae Explorer - Page 42 of 146
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In a set of N observations, when N is odd, the arranged data in the numerical order is the median.
!?
f !?
In a set of N observations, when N is even, the average of
f
median.
and
observation of
f !?
observation
observation of the arranged data in numerical order is the
Mode of Ungrouped Data Mode is the data which occurs most frequently in the given set of observations (data). It is possible to have more than one mode. Range of Ungrouped Data The difference between the highest and lowest values of the observed data is called the Range.
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9. Determinants Singular / Non Singular
A Square Matrix A is said to be Singular if |S| . Otherwise it is said to be Non-Singular. Adjoint of A Let A = [ aij ] be a square matrix of order n. Let Aij be the cofactor of aij. The adjoint of A is nothing but the transpose of the cofactor matrix [Aij ] of A. Theorem If A is a Square matrix of order n, then
A (Adjoint A) = |S| In = (adjoint A) A
where In is the identity matrix of order n.
Theorem If a matrix A possesses an inverse then it must be unique. Theorem If A is a non singular matrix, there exists an inverse which is given by
S
|S|
j%! S
Reversal Law for Inverses If A, B are any two non-singular matrices of the same order, then AB is also non-singular and ST T S
Reversal Law for Transposes
If A and B are matrices conformable to multiplication, then STA TA SA Inverses and Transposes
For any non-singular matrix A, SA S A Matrix Inversion Method
For a system of n linear non-homogeneous equations in ‘n’ unknowns is represented by AX = B, then its unique solution is given by X = A-1B. Mathematics Formulae Explorer - Page 44 of 146
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Properties of Determinants •
The Value of a determinant is unaltered by interchanging its rows and columns
•
If any two rows (columns) of a determinant are interchanged the determinant changes its sign but its numerical value is unaltered.
•
If two rows (columns) of a determinant are identical then the value of the determinant is zero.
•
If every element in a row (or column) of a determinant is multiplied by a constant “K” then the value of the determinant is multiplied by K.
•
If every element in any row (column) can be expressed as the sum of two quantities then given determinant can be expressed as the sum of two determinants of the same order with the elements of the remaining rows (columns) of both being the same.
•
A determinant is unaltered when to each element of any row (column) is added to those of several other rows (columns) multiplied respectively by constant factors.
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Rank of a Matrix The matrix A is said to be of rank r, if •
A has atleast one minor of order r which does not vanish
•
Every minor of A of order (r+1) and higher order vanishes
In other words, the rank of a matrix is the order of any highest order non vanishing minor of the matrix. The rank of A is denoted by xS.
The rank of an m x n matrix A cannot exceed the minimum of m and n. i.e., xS [ % , . Elementary Transformation on a Matrix Let A be an mxn matrix. An elementary row (column) operation on A is of any one of the following three types. • • •
The interchange of any two Ith and jth rows (columns). i.e., g% y gj
Multiplication of a Ith row (column) by a non zero constant C. i.e., g% z : g% Addition of any multiple of one row (column) with any other row (column). i.e., g% y g% Fgj
Echelon Form A matrix A (of order m x n) is said to be in Echelon form (Triangular form) if • Every row of A which has all its entries 0 occurs below every row which has a non-zero entry. •
The first non zero entry in each non zero row is 1.
•
The Number of zeros before the first non zero element in a row is less than the number of such zeros in the next row.
Note : • Any matrix can be brought to Echelon matrix form. • The Rank of a matrix in Echelon form is equal to number of non zero rows of the matrix. Mathematics Formulae Explorer - Page 46 of 146
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Cramer’s Rule Method ( Determinant Rule) For a system of non-homogeneous equation with 3 unknowns, the system is Consistent and has Unique Solution, if %" ∆ . Solution is
∆ ∆
,
∆ ∆
P
∆P ∆
.
Consistency for a given System of Equations by using Rank Method • • •
xS xS, T , !?( (|.!%# ( %#%#!(! ?# #-.!%.
xS xS, T f. ( " .FR# , !?( (|.!%# ( #%#!(! ?'( %"%%!( . ( " #-.!%#. xS xS, T f. ( " .FR# , !?( (|.!%# ( #%#!(! ?'( .%|.( #-.!%.
Consistency for a System of Homogeneous Equation A System of Homogeneous equations is always consistent. • •
xS xS, T f. ( " .FR# , !?( !%'%- #-.!% %# !?( .%|.( #-.!% xS xS, T f. ( " .FR# , !?( ##!( ?# !%'%- #-.!%
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10. Differential Calculus Derivatives of Standard Functions :
#!! ( ( . . . -) #% }. # # ~. #% ! . #( ! . #( #( . #( ! #( . #( ! -) . , -) -) ( . , #% . √ .
.
}.
#
√
!
!
~. 16. a. .'R .'R/ .'/ R ./ 'R
Seven Indeterminant Forms , , 0x ∞, ∞∞, 1] , ∞
, 0
] ]
#( . √ .
.
.
.
.
.
.
}.
~.
#(
√
#% ? # ? # ? #% ? ! ? #( ? ! ? #( ? #( ? #( ? ! ? #( ? #( ? ! ? #% ? √
# ?
√
! ? ! ?
.
#( ? . √
.
#( ?
. √ . .' .'/ './
. './ .'/ . ' '
Maclaurin’s Series "/ "// "/// " " l ! ! ! Mathematics Formulae Explorer - Page 48 of 146
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Curvature of Curve The rate of bending of a curve in any interval is called the Curvature of the curve in that interval. Cartesian Curve y = f(x) Polar Curve r = f(ϴ ϴ) 1
1
Sin Ψ =
cos Ψ =
1
sin =
tan =
cos =
tan Ψ =
Radius of Curvature
p= r sin
Parametric Form
Implicit Form
The reciprocal of the Let x=f(t) and y=g(t) be the Let f(x,y)=o be the implicit Curvature of a curve at parametric equations of form of the given curve. any point is called the the given curve. Radius of Curvature at the point and is denoted by x
/
Polar Form
Let r = f(ϴ ϴ) be the given curve in polar coordinates. / x
x
d"/ )/ e
/
x
"/ )// "// )/
" "
/
" " " " " " "
Centre of Curvature in the Cartesian Form
Circle of Curvature
u, u ,
The equation of the circle of curvature is
where ,
u x u
Local Maxima and Minima for functions of one variable
Given y=f(x), (i) if f/(c)=0 and f//(c)>0, then f has a local minimum at c. (ii) if f/(c)=0 and f//(c) 0 and A0 (or B>0) iii) f(a,b) is not an extremum if AC-B2 < 0 and iv) If AC-B2 > 0, the test is inconclusive.
where λ is called Lagrange Multiplier which is independent of x,y,z, The necessary condition for a maximum or minimum are Y" , Y
Y" , Y
Y" YP
Solving the above equations for four unknowns λ, x, y, z, we obtain the point (x,y,z). The point may be a maxima, A function f(x,y) at (a,b) or f(a,b) is said minima or neither which is decided by the to be a Stationary Value of f(x,y) if physical consideration. fx(a,b)=0 and fy(a,b)=0.
Stationary Value
Jacobians
Properties of Jacobian
If u1, u2, u3, …….un are functions of n 1. If u and v are the functions of x and y, Y.,' Y, variables x1, x2, x3, …xn, then the then 1. Y, Y.,' Jacobian of the transformation from x1, x2, x3, …xn to u1, u2, u3, …….un is 2. If u,v are the functions of x,y and x,y defined by are themselves functions of r,s then Y. ¢ ¡ Y ¡ £ ¡Y. Y
l ¤
l
Y. § Y ¦ Y. , . , … … . ¨. ,. ,……. £ ¦ Y , , … . . Y.. ¦ Y ¥
then
Y.,' Y,
Y, Y,#
Y.,' Y,#
.
3. If u,v,w are functionally dependent function of three independent variables x,y,z then
Y.,',R Y,,P
0
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11. Differential Equations Definition An equation involving one dependent variable and its derivatives with respect to one or more independent variables is called a Differential Equation.
Differential Equation are of two types namely • •
Ordinary Differential Equations Partial Differential Equations
Definition An Ordinary Differential Equation is a differential equation in which a single independent variable enters either explicitly or implicitly.
Order and Degree of a Differential Equation
Definition The Order of a differential equation is the order of the highest order derivative occurring in it. The degree of the differential equation is the degree of the highest order derivative which occurs in it, after the differential equation has made free from radicals and fractions as far as the derivatives are concerned.
Differential Equations of First Order and First Degree
For the solutions of first order and first degree equations, we shall consider only certain special types of equations of the first order and first degree. They are • • •
Variable Separable Homogeneous Linear
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Variable Separable
Variables of a differential equation are to be rearranged in the form " ) " )
The above equation can be rewritten as " ) " )
,
) "
) "
The solution is ©
) " © ) " Homogeneous Equations
Definition
"
" ,
A differential equation of first order and first degree is said to be homogeneous if it can be put in the form
" ,
Solving this, by putting y = vx, we get the solution.
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Linear Differential Equation Definition
A first order differential equation is said to be linear in y, if the power of the terms
and y are unity.
@ ª, where P and Q are function of x only.
A differential equation of order one satisfying the above condition can always be put in the form
@ ª where P and Q are functions of y only.
Similarly a first order linear differential equation in x will be of the form
The solution of the equation which is linear in y is given as
(« @ « ª (« @ where (« @ is known as an integrating factor and it is denoted by I.F.
Similarly, the solution of the equation which is linear in x is given as
(« @ « ª (« @ where (« @ is known as an integrating factor and it is denoted by I.F.
We frequently use the following properties of Logarithmic and Exponential functions • • •
(-) S S
( -) S S
( -) S S
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SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS A General Second Order non-homogeneous linear differential equation with constant coefficients is of the form
// / ¬
where a0, a1, a2 are constants a0 0, and X is a function of x. The equation
// / ,
is known as a homogeneous linear second order differential equation with constant coefficients. : If > is a root of & & , then (> is a solution of
// / .
Theorem
Definition : The equation & & is called the characteristic equation of // / , .
General Solution : The General Solution of a linear equation of second order with constant co-efficient consists of two parts namely the Complementary Function (C. F) and the Particular Integral (P.I). Method of finding Complementary Function (C.F)
Let > , > be the two roots of & & then the solution of // / , is
S(> T(> , %" > > ( (- %#!%! ® S T(> , %" > > (- ( S # T #% , %" > % > %
where A and B are arbitrary constants.
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Method of finding Particular Integral (P. I )
•
Suppose X is of the form ( , where ‘a’ is a constant.
Formula 1 P. I =
"U
Formula 2 P. I =
"U
(
"U
( U;U (
Formula 3 P. I =
•
"U
( , R?( "U
(
(
;
(
%" "U , U %# "! " "U
%" ; , !?( U %# (&(!( ! " "U
When X is of the form sin ax (or) cos ax Formula 1 P. I =
# #% U # #% "U # #%
Formula 2
Sometimes we cannot form U . Then we shall try to get U, U . Multiplying and Dividing by the conjugate of the demoninator and get the solution. Formula 3
If , P. I If , P. I
U
U
#
U
#
#% U #%
#%
#
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Suppose X is of the form
Working Rule : Take the Particular Integral as
, %" " ® @. h , %" "
Since Particular Integral is also a solution of (aD2+bD+c) y = f(x), take according as f(x) = x or x2. By Substituting y value and comparing the like terms, one can find c0, c1 and c2.
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12. Discrete Mathematics Discrete Mathematics deals with several selected topics in Mathematics that are essential to the study of many Computer Science areas. Among many topics, only two topics, namely ‘Mathematical Logic’ and ‘Groups’ have been introduced here. These topics will be very much helpful to the students in certain practical applications related to Computer Science.
Mathematical Logic Logic deals with all types of reasoning’s. These reasoning’s may be legal arguments or mathematical proofs or conclusions in a scientific theory. Logic is widely used in many branches of sciences and social sciences. It is the theoretical basis for many areas of Computer Science such as Digital Logic, Circuit Design, Automata Theory and Artificial Intelligence. Logic Statement (or) Proposition A statement or a proposition is a sentence which is either true or false but not both. Truth Value of a Statement The truth or falsity of a statement is called its truth value. If a statement is true, we say that its truth value is TRUE or T and if it is false, we say that its truth value is FALSE or F. Simple Statements A Statement is said to be Simple if it cannot be broken into two or more statements. Compound Statements If a statement is the combination of two or more simple statements, then it is said to be a Compound Statement. Basis Logical Connectives Three basis connectives are Conjunction which corresponds to the English word ‘and’ denoted by the symbol ¯, Disjunction which corresponds to the word ‘or’ denoted by the symbol ° and Negation which corresponds to the word ‘not’ denoted by the symbol ±
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Truth Tables A table that shows the relation between the truth values of a compound statement and the truth values of its sub-statements is called the truth table. A truth table consists of rows and columns. The initial columns are filled with the possible truth values of the sub-statements and the last column is filled with the truth values of the compound statement on the basis of the truth values of the sub-statements written in the initial columns. If the compound statement is made up of n sub-statements, then its truth table will contain 2n rows. Logical Equivalence Two compound statements A and B are said to be logically equivalent or simply equivalent, if they have identical last columns in their truth tables. Negation of a Negation Negation of a Negation of a statement is the statement itself. Equivalently we write ±(± ±p) ²p.
Conditional and Bi-Conditional Statements
In Mathematics, we frequently come across statements of the form “If p then q”. Such statements are called Conditional statements or implications and denoted by & z | and read as ‘p implies q’. If p and q are two statements, then the compound statement & z | | z & is called a Bi-Conditional statement and is denoted by & y |. p T T F F
q T F T F
TRUTH TABLE p¯q p°| T T F T F T F F
&z| T F T T
&y| T F F T
Tautologies and Contradiction A statement is said to be a Tautology, if the last column of its truth table contains only T. In other word, it is true for all logical possibilities. A statement is said to be a Contradiction, if the last column of its truth table contains only F. In other word, it is false for all logical possibilities.
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Groups Binary Operation A binary operation * on a non-empty set S is a rule, which associates to each ordered pair (a, b) of elements a, b in S an element a*b in S. Multiplication table for a Binary Operation Any binary operation * on a finite set S = {a1, a2, a3, …., an} can be described by means of multiplication table. This table consists of ‘n’ rows and ‘n’ columns. Place each element of S at the head of one row and one column, usually taking them in the same order for columns as for rows. The operator * is placed at the left hand top corner. The nxn=n2 spaces can be filled by writing ai * aj in the space common to the ith row and the jth column of the table.
List of Symbols used
Z - for every
n - belongs to
³ - there exists ´ - such that , - implies Definition A non-empty set G, together with an operation * i.e,, (G, *) is said to be a Group if it satisfies the following axioms. • • • •
Closure axiom : Z a,b n G, , a * b n G
Associative axiom : Z a, b, c n G, (a * b) * c = a * (b * c)
Identity axiom : Z a n G, ³ e n G, such that a * e = e * a = a
Inverse axiom : Z a, e n G, ³ a-1 n G, such that a * a-1 = a-1 * a = e
Here e is called the identity element of G and a-1 is called the inverse of a in G.
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Definition (Commutative Property) A binary operation * on a set S is said to be commutative, if a * b = b * a, Z a, b n S. Definition If a Group satisfies the Commutative property then it is called an Abelian Group (or) a Commutative Group, otherwise it is called a non-abelian group.
Order of a Group The Order of a Group is defined as the number of distinct elements in the underlying set. If the number of elements is finite, then the group is called a finite group and if the number of elements is infinite then the group is called an infinite group. The Order of a group G is denoted by o(G).
Definition ( Semi-Group ) A non-empty set S, together with an operation * i.e,, (S, *) is said to be a SemiGroup if it satisfies the following axioms. • •
Closure axiom : Z a,b n S, , a * b n S
Associative axiom : Z a, b, c n S, (a * b) * c = a * (b * c)
Definition ( Monoid ) A non-empty set M, together with an operation * i.e,, (M, *) is said to be a Monoid if it satisfies the following axioms. • • •
Closure axiom : Z a,b n M, , a * b n M
Associative axiom : Z a, b, c n M, (a * b) * c = a * (b * c)
Identity axiom : Z a n M, ³ e n M, such that a * e = e * a = a Mathematics Formulae Explorer - Page 60 of 146
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Properties of Groups •
The identity element of a group is unique.
•
The inverse of each element of a group is unique.
•
Let G be a group. Then for all a, b, c n G
o a * b = a * c , b = c G("! :(--!% GR
o b * a = c * a , b = c g%)?! :(--!% GR
• •
In a group G, ^ , for every a n G
Reversal Law : Let G be a group. a, b n G. Then µ ^
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13. Fourier Series Results :
Fourier Series of f(x) Interval
(-l, l)
(0, 2l)
_, _
, _
f(x)
a0
_ _ \ d # #% e
© " -
"
_ _ \ d # #% e -
© " -
"
_ _ \ d # #% e -
© " _
_ _ \ d # #% e -
© " _
"
"
]
^
]
^
]
^
]
^
-
-
, "%# -
_
_
, "%# _
an -
_ © "# -
, "%# -
_ © "#
_
_ © "# _ _
, "%# _
_ © " # _
bn -
_ © "#% -
, "%# ('( -
_ © "#%
_
_ © "#% _ _
, "%# ('( _
_ © " #% _
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Half Range Fourier Sine Series of f(x) Interval
(0, l)
(0, _)
]
f(x)
_ " \ #% ^ ]
" \ #% ^
a0
an
0
0
bn -
0
_ © "#% -
0
© "#% _
_
Half Range Fourier Cosine Series of f(x) Interval
(0, l)
(0, _)
f(x) ]
_ \ # " ^
]
" \ # ^
a0 -
© "
_
© " _
an -
_ © "#
_
© " # _
bn
0
0
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COMPLEX FORM OF FOURIER SERIES Interval
(-l, l)
(0, 2l)
Complex form of Fourier Series ]
" \
^] ]
%_ : ( -
%_ -
" \ : (
_, _ "
, _ "
^] ]
\ : (%
^] ]
\ : (
^]
%
Fourier Coefficients -
%_ : © " ( - -
-
%_ : © " ( -
_
: © " (% _ _
_
: © " (% _
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Parsevals Identity Interval
(-l, l)
(0, 2l)
_, _
, _
Parsevals Identity « " - -
« " -
_ « " _ _
_ « " _
∑] ^
∑] ^
a0
∑] ^
∑] ^
-
© " -
-
© "
_
© " _ _
_
© " _
an -
_ © "# -
-
_ © "#
_
_ © "# _ _
_
_ © " # _
bn -
_ © "#% -
-
_ © "#%
_
_ © "#% _ _
_
_ © " #% _
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Parsevals Identity for Half Range Fourier Sine Series Interval
(0, l)
(0, _)
Parsevals Identity for Half Range Fourier Sine Series
a0
an
« " ∑] ^ -
0
0
" ∑] « ^ _
0
-
bn
_
-
_ © "#% -
_
_ © "#% _
0
Parsevals Identity for Half Range Fourier Cosine Series Interval
(0, l)
(0, _)
Parsevals Identity for Half Range Fourier Cosine Series
" « -
« " _
_
a0
an
_ © "# -
∑^
© " -
∑^
© " _
]
]
_
_
_
bn
-
0
_
0
_ © " # _
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HARMONIC ANALYSIS Interval
(0, 2l)
(0, 2_)
Fourier Series
"
_ _ \ d # #% e -
"
_ _ \ d # #% e -
]
^
]
^
a0
an
bn
`( '-.( " ¶ · % , -
`( '-.( _ ¸ " # - ¹ % , -
`( '-.( _ º " #% »
`( '-.( " ¶ · % , _
`( '-.( ¶ " # · % , _
`( '-.( ¶ " #% · % , _
-
% , -
Harmonic Analysis for Half Range Fourier Sine Series Interval
(0, l)
(0, _)
Harmonic Analysis for Half Range Fourier Cosine Series ]
" \ #% ^ ]
_ -
" \ #% ^
a0
an
0
0
0
0
bn `( '-.( _ º " #% » % , -
-
`( '-.( ¶ " #% · % , _
Harmonic Analysis for Half Range Fourier Cosine Series of f(x) Interval
(0, l)
(0, _)
Harmonic Analysis for Half Range Fourier Sine Series ]
_ " \ # ^ ]
" \ # ^
a0 `( '-.( " ¶ · % , - `( '-.( " ¶ · % , _
an
bn
`( '-.( _ º" # » % , -
-
`( '-.( ¶ " # · % , _
0
0
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14. Fourier Transforms Fourier Transform ¼ " "
√ _
√ _
]
]
© " (
]
© ¼ " (
]
Fourier Sine Transform %#
%#
#
¼ " ) ¼# ½# ¼ " (%# ¼#
¼ (%# " ¼# ¼ "
# ¼
¼ " % ¼¾
¼ #
" ¿ %# ¼#
¼ iiiiiiiii " iiiiiiiii ¼ #
iiiiiiÁ ¼ # iiiiiiii ¼ À " ¼ À iiiiiiii " Á iiiiii ¼# ¼ "#
¼# ¼ #
]
¼# " © " #% # _ ]
" © ¼# " #% # # _
¼# " ) ¼# # ½# # ¼# "
¼ " #
Fourier Cosine Transform ]
¼ " © " # # _ ]
" © ¼ " # # # _
¼ " ) ¼ # ½ # ¼ "
---
¼# " ¼# "
# ¼ #
¼ " #
¼# À "/ Á #¼ # ]
© ¼# ". ½# ) #
]
© ")
¼ # # ¼# #
¼# "#
¼ " # # ---
¼ " ¼ "
# ¼
¼ " # #
¼ À "/ Á " # ¼# # _ ]
© ¼ ". ½ ) #
]
© ")
¼ # ¼ #
¼ "#
---
---
---
---
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Convolution of two functions "µ)
√ _
]
© "!) ! ! ]
Convolution theorem for Fourier Transforms ¼ " µ ) ¼#. ½# Parsevals Identity
«]|"| «]|¼#| #, where ¼# ¼ " ]
]
Parsevals Identity for Fourier Sine Transforms
«]|"| «]|¼# #| #, where ¼# # ¼# " ]
]
Parsevals Identity for Fourier Cosine Transforms
«]|"| «]|¼ #| #, where ¼ # ¼ " ]
]
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15. Graphs Quadratic Graph To draw a straight line, two points are sufficient but to graph of a quadratic, more numbers of point required. For each value of x, the equation y = ax2 + bx + c gives the corresponding value of y. The set of all such ordered pairs (x,y) which defines the graph is called quadratic graph. Quadratic Polynomials A polynomial with degree 2 is called quadratic polynomial. The general form of a quadratic polynomial is y = ax2 + bx + c, where a, b, c are real numbers such that a 0, and x is a variable. Value of a Quadratic Polynomial
Let y = ax2 + bx + c be a quadratic polynomial and let a be a real number. Then y = ax2 + bx + c is known as the value of the quadratic polynomial y = f(x) and it is denoted by y = f(Â). i.e.,
f(Â) = aÂ2 + b + c
Solving Quadratic Equation by Graphical Method
In Algebra, we have solved the quadratic equation by algebraic method. Now we are going to solve this quadratic equation by Graphical Method. Type – I First draw the graph of the equation y = ax2 + bx + c Here y = 0 is the equation of x=axis Get the points of intersection of the curve y = ax2 + bx + c with x-axis. The x- coordinates of the intersecting points will give the roots of the given equation. Type – II Split the quadratic equation into two equations representing a parabola and a straight line. Draw their graphs. The x- coordinates of the points of intersection of the parabola and the straight line will give the roots of the given quadratic equation. Mathematics Formulae Explorer - Page 70 of 146
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16. Integral Calculus Integrals of Standard Functions : . © . ©
,
-) ,
. © ( ( . ©
, -)
}. © #% # ~. © # #%
! -) #( -) # ! -) #% . © -) #( #( -) #( ! _ _ . © -) ! #( -) #( ! _ . © -) ! . ©
. © #( !
. © #( !
. © #( ! #( . © #( ! #( }. © #% ? # ? ~. © # ? #% ? . ©
! ? -) # ?
. © ! ? -) #% ?
. © #( ? ! ? #( ? . © #( ? ! ?
. © #( ? ! ? #( ? . © #( ? ! ?
! . « -) , (x>a) }. © #% √ √ ~. © 9 #% . © #% √ √ -) . ©
√ -) √ √ √ . © 9 -) √ . © 9 #% √ √ . © 9
-) . © -) C D , I 3 . © #% √
. © # √ . © !
}. © ! ~. © #( √
. © #( √ . © #% ? √ . © # ? √ . © ! ?
. © ! ?
. © #( ? √
. © #( ? √ ( . © ( #%
#% # ( }. © ( #
# #% ~. © √ √ ] . © ( # . ©
# ?
]
. © ( #%
. © -) -) _
_
}. © #% © #
}…… _ ………..
, if n is even.
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} . « #% # _
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………..
© " , %" "%# ('( ".!%® } . © " Ã
,
}. © . ' .' ./' .//' ./// '
%" " " ('( -%F( ".!%®
}~. © . ' .' © ' .
, %" " %# ".!%
© " , }. © " Ã
}}. © " © "
only if both m and n are even.
} . © " © " © "
}…… }…… _
l . T(.--% " .-
%" " " -%F( ".!%
Double Integral in Cartesian Coordinates Properties of the Definte Integrals Double Integral over region R may be . © " © " evaluated by two successive integrations. If A is described as ÄÅ [ [ Ä . © " © " Å [ [ 35 Å [ [ , Ê Ê
JÆÇ5 È Ä, É È Ä, Ì
I = ÎÌ Ä, , Ï Ï
.! %#
Volume of Solids of Revolution
Ð-. , Ð © _ " © _
. © "
~. © " © " © "
2) ÎÌ Ï represents the Volume of the Region R. Area of Bounded Regions
S( © " ©
}. © " © " © "
Triple Integration in Cartesian Coordinates
Ë Ë
1)
. © " © "
Double Integral in Polar Coordinates
. © " © "
Ë Ë
I = Í Ê Ê Ä,
.! %#
Ã
© " , %" " " ®
,
. © "
%" " "
© " , %" " %# ('( ".!% ® Ã
,
%" " % ".!%
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Length of a Curve
© C D © B"/ )/
.! %#
c."( S(, S
Surface Area of a Solid
(Parametric Form)
_ © C D © 9"/ ! )/ ! !
.! %#
(Parametric Form)
Gamma Function
Ñ « ( , 0 0 ]
Ñ Ñ
Ñ ! , %" %# %!()(
Ñ / √_
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17. Laplace Transform Basic Definitions: Definition of Laplace Transform ]
Ò# G "! © (#! "! !,
! 0 0.
"! G Ò#
L { t. f(t) } = Ò/(s)
Differentiation of Transform
Inverse Laplace Transform
G
G "! )! G "! G )!
Convolution
Linearity
G ( "! Ò#
G (
!
G Ò# (! "!
G
G
(#
(
!
G "!
"!
G "! .! (# ¼#
Scaling
Differentiation of Function GÀ"/ !Á #G "! "
G% !z "! G% #z] # ¼#
G " ! # G "! # " … "
G "! #
G% !z] "! G% #z # ¼# Final Value
Integration of the Function
#
G (!
#
G !
!
Table of Laplace Transform G G !
#
G ! G¾
√_!
G ¸
¿ !
√_
!
√#
¹
#
#/
G ! (! ( n=1,2,3…)
G ! (
!
(
¼#, 0 0
Initial Value
GÀ"// !Á # G "! #" "/
& © (#! "! ! (&#
G "!
¼# "! .!
!
f periodic with period p
t - shifting Second Shifting Theorem
G © "! !
© )! . "! . .
"! Ò#
Ò#
"! µ )! © "! . )! .
First Shifting Theorem !
] "! © G "! # ! #
Integration of Transform
G #% R!
#
!
#
#
G .!
( #
G # R! ( n=1,2,3…)
#
R R
# # R
G #%? R!
G #? R!
#
R
R
# # R
G Ó! (#
Mathematics Formulae Explorer - Page 74 of 146
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GÀ(! (! Á
# #
G (! #% R!
R # R
# GÀ(! (! Á # # G (! # R!
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# # R
R G # R! ## R
G R! #% R!
R ## R
G Ó!, .%! #!(& ".!%
G # ! # !
^ #
# #
G #% R! R! # R!
G #% R! R! # R! G ! #% R!
G ! # R!
G .!
#
R# R
(a2≠b2)
R # R R# # R
# R # R (# #
G Ó! (#
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18. Matrices Basic Definitions: Row Matrix
Transpose of Matrix
One row and any number of columns.
The matrix obtained by interchanging rows and columns of the matrix A is called the Transpose of A and is denoted by A/ (or) AT (read as A Transpose).
Eg:- [ 1 2 3 4 ] Column Matrix 1 Eg:-¶2· 3 Square Matrix
Only one column and any number of rows.
1 4 2 3 × ÉÞ ¶2 5· 5 6 3 6 Triangular Matrix
1 Eg ÝÄ É Õ 4
A Square matrix in which all the entries Number of rows is equal to number of above the main diagonal are zero, is called Lower Triangular Matrix. If all the columns. entries below the main diagonal are zero, 1 2 3 it is called Upper Triangular Matrix. 1 2 Eg:- Õ × ¶4 5 6· 3 4 Eg:7 8 9 1 2 3 Zero Matrix (or) Null Matrix Upper Traingular Matrix A ¶0 1 5· 0 0 0 ¶0 0 0· 0 0 0
0
All elements are zeros. 0 0 Eg:- 0 Õ × 0 0 Diagonal Matrix
Lower Traingular Matrix A
1 ¶2 1
0 1
0 0 3 0· 2 3
Symmetric Matrix A Square matrix all of whose elements except those in the leading diagonal are A Square matrix A = {aij} is said to be Symmetric when aij=aji for all i and j. zero, is called a diagonal matrix 3 0 Eg:- Õ × 0 5
Scalar Matrix
2 ¶0 0
0 0 3 0· 0 4
(i.e., A =AT) Skew Symmetric Matrix
A Square matrix A = {aij} is said to be It is a diagonal matrix whose elements in Skew Symmetric when aij= - aji for all i diagonal are same. and j. 2 0 0 (i.e., A = - AT) 3 0 Eg:- Õ × ¶0 2 0· 0 3 Singular Matrix 0 0 2 A Square matrix A is said to be Singular if Identity Matrix the determinant value of A is zero. A Square matrix in which diagonal elements are ‘1’ and all other elements ‘0’ Inverse of a Matrix is called Identity Matrix. S j%! S 1 0 0
S 1 0 Eg:- Õ × ¶0 1 0· 0 1 0 0 1 Mathematics Formulae Explorer - Page 76 of 146
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Multiplication of Matrices : Two matrices A and B are conformable for the product AB only, if the number of Column in A (Pre-Multiplier) is the same as the number of rows in B (PostMultiplier). A [mxn]
B [nxp]
=
AB [mxp]
Matrix Multiplication is always associative : If A,B,C are mxn, nxp, pxq matrices, then (AB)C = A(BC). Multiplication of a matrix by a Unit Matrix : If A is a Square Matrix of order n and I is the Unit Matrix of same order n, then A.I = I.A = A. Note : AB=0 (NULL) does not necessarily imply A=0 (or) B=0 (or) both A,B=0.
Properties of Matrix Addition : • Matrix Addition is commutative if both are same order. A+B = B+A. • It is also associative. A+(B+C) = (A+B)+C. • Additive Identity : A Null matrix of same order is the identity matrix. • Additive Inverse : For matrix [A], additive inverse is [-A]. [A] + [-A] = 0. Properties of Matrix Multiplication : • Matrix Multiplication is not commutative. • Matrix Multiplication is distributive over matrix addition. • A,B,C is of order mxn, nxp and nxp, then A(B+C) = AB + AC. • A,B,C is of order mxn, mxn, and nxp, then (A+B)C = AC + BC. Characteristic Equation : The equation ßS >hà is said to be Characteristic Equation of the transformation or the Characteristic Equation of the matrix A. Eigen Values : To solve the characteristic equation, we get characteristic roots. They are called Eigen Values. Eigen Vectors : To find the eigen vectors, solve (A->I)=0 for the different values of á.
Cayley-Hamilton Theorem : Every Square Matrix satisfies its own characteristic equation.
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Properties of Eigen values and Eigen Vectors: • • • • • • • • • • • • •
Sum of the eigen values is equal to the sum of the main diagonal elements. Product of the eigen values is equal to its determinant value. The eigen values of A and AT are the same. The characteristic roots of a triangular matrix are just the diagonal elements of the matrix. If > is a eigen value of a matrix A, then 1/> is the eigen value of A-1. If > is an eigen value of an orthogonal matrix, 1/> is also its eigen value. If > , > , > , … > are eigen values of a matrix A, then Am has the eigen values > , > , > , … > . The eigen values of a real symmetric matrix are real numbers. The eigen values corresponding to distinct eigen values of a real symmetric matrix are orthogonal. The similar matrices have same eigen values. If a real symmetric matrix of order Z has equal eigen values then the matrix is a scalar matrix. The eigen vector X of a matrix A is not unique. If A and B are nxn matrices and B is a non-singular matrix then A and B-1AB have same eigen values.
Diagonalisation of a Matrix : If a Square matrix A of order n has ‘n’ linearly independent eigen vectors, then a matrix P can be found such that P-1AP is a diagonal matrix. Fundamental theorem on Quadratic Form : Any Quadratic form may be reduced to Canonical form by means of a nonsingular transformation. Quadratic Form : A homogeneous polynomial of the second degree in any number of variables called a Quadratic Form. The matrix corresponding to the Quadratic form in three variables is
:(""%%(! "
ä S ã :(""%%(! "
â
:(""%%(! "
:(""%%(! " :(""%%(! "
:(""%%(! "
:(""%%(! "
ç :(""%%(! " æ :(""%%(! "
å
Nature of Quadratic Form: Let ¬/ S¬ be the given quadratic form in the variables x1,x2,x3,….xn. i.e., ¬/ S¬ l … . . Let the rank of A be r, then ¬/ S¬ contains only ‘r’ terms. The number of positive terms in the above equation of ¬/ S¬ is called the index of the quadratic form and it is denoted by ‘s’. The difference between the number of positive terms and the negative term is called the Signature of the quadratic form. Signature = 2s-r, where ‘s’ is equal to the number of positive terms and ‘r’ is equal to the rank of A. Mathematics Formulae Explorer - Page 78 of 146
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19. Measurements Denominate Number A Denominate Number is one that refers to a unit of measurement which has been established by law or by general usage. Examples are 2 inches, 8 pounds, 3 seconds, etc.,
Compound Denominate Number A compound Denominate Number is one that consists of two or more units of the same kind. Examples are 4 foot 3 inches, 3 hours 15 minutes, 1 pound 14 ounces, 5 rupees 30 paise etc.,
Denominate numbers are used to express measurements of many kinds, such as • Linear (Length) •
Square (Area)
•
Cubic (Volume)
•
Weight (pounds)
•
Time (Seconds)
•
Angular (degrees)
This classification is by no means complete. Systems of currency (dollars and cents, pounds sterling and pence, etc.,) would, for instance, be considered denominate numbers, and the various foreign systems of weights and measures would, of course, come under the same head, though they are beyond the scope of this webpage.
To gain facility in working out arithmetic problems involving denominate numbers it is necessary to know the most common tables of measures, such as are given here for reference. Note the abbreviation used, since these are in accordance with the manner in which the values are usually written.
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TABLE OF MEASUREMENTS
1. Length or Linear Measure
Linear units are used to measure distances along straight lines.
U.S. (or) English System Nautical Measure
12 inches (in. or “)
= 1 foot (ft. or ‘)
6080 feet
= 1 English nautical mile
3 feet or 36 inches
= 1 yard (yd.)
1.15 land miles
= 1 English nautical mile
60 nautical mile
= 1 degree of arc
5 ½ yards or 16 ½ feet = 1 rod (rd.) 220 yards or ½ mile
(at the equator)
= 1 furlong (fur.)
320 rods or 8 furlongs = 1 mile (mi.)
360 degrees of arc = circumference of earth at Equator
1,760 yards
= 1 mile
1 fathom
= 6 ft. (of depth)
5, 280 feet
= 1 mile
1 hand
= 4 in.
Metric System
Unit
Metres
1 millimetre (mm.)
= 0.001
= 0.03937 in.
10 millimetres
=
1 centimetre (cm.)
= 0.01
= 0.3937 in.
10 centimetres
=
1 decimetre (dm.)
= 0.1
= 3.937 in.
10 decimetres
=
1 Metre (M.)
= 1
= 39.3707 in.
10 metres
=
1 dekametre (Dm.)
= 10
= 32.809 ft.
10 dekametres
=
1 hectometre (Hm.)
= 100
= 328.09 ft.
10 hectometres
=
1 kilometre (Km.)
= 1000
= 0.52137 mile
10 kilometres
=
1 myriametre (Mm.)
= 10000
= 6.2137 miles
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2. Square Measure This is used to measure the area of a surface. It involves two dimensions, length and width.
SQUARE OR AREA MEASURE
144 square inches
= 1 square root (sq. ft.)
9 square feet
= 1 square yard (sq. yd.)
30 ¼ square yards
= 1 square rod (sq. rd.)
160 square rods
= 1 Acre (A.)
640 acres
= 1 square mile
SURVEYORS’ SQUARE MEASURE
625 square links (sq. 1.) = 1 square rod (sq. rd.) ( 1 linear link = 7.92”) 16 square rods = 1 square chain (sq. ch.) 10 square chains = 1 acre (A.) 640 acres = 1 square mile
METRIC SQUARE MEASURE
100 square millimetres(sq. mm.) = ¼ square centimetre (sq. cm.) 100 square centimetres
= 1 square decimetre (sq. dm.)
100 square decimetres
= 1 square metre (sq. m.)
100 square metres
= 1 square hectometre (sq. Hm.) or hectare (H)
100 square hectares
= 1 square kilometre (sq. km.)
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3. Cubic Measure This is used to measure the volume or amount of space within the boundaries of three-dimensional figures. It is sometimes referred to as Capacity. CUBIC OR VOLUME MEASURE 1728 cubic inches (cu.in.) = 1 cubic foot (cu.ft.) 27 cubic feet = 1 cubic yard (cu.yd.) 1 cubic yard = 1 load of sand or dirt 128 cubic feet = 1 chord of wood (cd.) 24 ¾ cubic feet = 1 perch of stone (pch.) LIQUID MEASURE OF CAPACITY 4 gills (gi.) 2 pints 4 quarts
= 1 pint (pt.) = 1 quart (qt.) = 1 gallon (gal.)
The imperial gallon is used in the United Kingdom. 1 Imperial gallon = 1.20094 U.S. gallon APOTHECARIES LIQUID MEASURE 60 drops or minims = 1 fluid drachm 8 fluid drachms = 1 fluid ounce DRY MEASURE OF CAPACITY 2 pints (pt.) = 1 quart (qt.) 8 quarts = 1 peck (pk.) 4 pecks = 1 bushel (bu.)
METRIC MEASURE OF CAPACITY 1000 cubic millimetres (cu.mm) = 1 cubic centimetre (c.c) 1000 cubic centimetres = 1 cubic decimetre (cu. dm.) 1000 cubic decimeters = 1 cubic metre (cu. m.) 10 centilitres (cl.) = 1 decilitre (dl.) 10 decilitres = 1 litre (l.) = 1 cubic metre 10 cubic litres = 1 dekalitre (Dt.) 10 dekalitres = 1 hectolitre (Hl.)
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4. Measures of Weight These are used to determine the quantity of matter a body contains. Four scales of weight are used. •
Tray – for weighting gold, silver and other precious metals.
•
Apothecaries – used by chemists for weighting chemicals.
•
Avoirdupois - used for all general purposes.
•
Metric – used in scientific work.
AVOIRDUPOIS WEIGHT 16 drams (dr. ) = 1 ounce (oz.) 16 ounces = 1 pound (lb.) 7000 grains (gr.) = 1 pound 14 lb. = 1 stone (st.) 2 st. = 1 quarter (qtr.) 112 lb. = 1 cwt. 2240 lb. = 1 ton TROY WEIGHT 24 grains (gr.) = 1 pennyweight (dwt.) 20 pennyweights = 1 ounce (oz.) 12 ounces = 1 pound (lb.) 5760 grains = 1 pound = 1 carat (kt.) 3 ~ grains The carat, as defined in the table, is used to weigh diamonds. The same term is used to indicate the purity of gold. In this case, a carat means a twenty-fourth part. Thus, 14 kt. Gold means that 14 parts are pure gold and that 10 parts are of other metals. APOTHECARIES WEIGHT 20 grains (gr.) = 1 scruple (sc.) 3 scruples = 1 drachm (dr.) 8 drams = 1 ounce (oz. ) 12 ounces = 1 pound (lb. ) 5760 grains = 1 pound The above table is now obsolete but is given for historical interest.
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METRIC WEIGHT 10 milligrams (mg. ) = 1 centigram (cg. ) 10 centigrams = 1 decigram (dg. ) 10 decigrams = 1 gram (g. ) 10 grams = 1 dekagram (Dg.) 10 dekagrams = 1 hectogram (Hg.) 10 hectograms = 1 kilogram (Kg. ) 10 kilograms = 1 myriagram (Mg.) 10 myriagrams = 1 quintal (Q. ) 10 quintals = 1 tonne (T.)
5. MEASURE OF TIME 60 seconds (sec. ) = 1 minute (min. or ‘) 60 minutes = 1 hour (hr.) 24 hours = 1 day (da. ) 7 days = 1 weel (wk. ) 2 weeks = 1 fortnight 365 days = 1 common year 366 days = 1 leap year 12 calendar months = 1 year 10 years = 1 decade 100 years = 1 century (C.)
6. ANGULAR OR CIRCULAR MEASURE Angular (L) or Circular (0) Measure 60 seconds ( ‘’ ) = 1 minute ( ‘ ) 60 minutes = 1 degree(0) 90 degrees = 1 right angle (L) or 1 Quadrant 360 angle degrees = 4 right angles 360 arc degrees = 1 Circumference (0) 7. MONEY
UNITED STATES MONEY 10 mills (m. ) = 1 cent (c., or ct. ) 10 cents = 1 dime (d.) 10 dimes = 1 dollar ($) 10 dollars = 1 eagle (E. )
ENGLISH MONEY 2 half pennies = 1 penny (d. ) 12 pence = 1 shilling (s. ) 20 shillings = 1 pound (₤) 21 shillings = 1 guinea
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THE METRIC SYSTEM Since the metric system is based on decimal values, all ordinary arithmetical operations may be performed by simply moving the decimal point. The metric system is a system of related weights and measures. The metre is the basis from which all other units are derived. The unit of capacity, the litre, is the volume of 1 Kg. (1,000 g.) of water, and thus is represented by a 1,000 c.c. The unit of capacity, the litre and its derivates are used for both dry and liquid measure. In the following tables whenever the metric equivalents of standard measures are given, metric equivalents of other denominations may be found by simply moving the decimal point to the right or the left as may be necessary.
EQUIVALENT VALUES
LINEAR MEASURE 1 inch = 2.5400 centimetres 1 foot
= 0.3048 metre
1 yard = 0.9144 metre 1 rod
= 5.0292 metres
1 mile = 1.6093 kilometres
1 centimetre = 0.3937 inch 1 decimetre = 3.9379 inches 1 decimetre = 0.3281 foot 1 metre
= 39.3700 inches
1 metre
= 3.2808 feet
1 metre
= 1.0936 yards
1 kilometre = 3280.83 feet 1 kilometre = 1093.611 yards 1 kilometre = 198.838 rods
SQUARE MEASURE 1 sq. inch = 6.4516 sq. centimeters 1 sq. foot = 0.0929 sq. metre 1 sq. yard = 0.8361 sq. metre 1 sq. rod = 25.2930 sq. metres 1 acre = 4046.8730 sq. metres 1 acre = 0.404687 hectare 1 sq. mile = 258.9998 hectares 1 sq. mile = 2.5900 kilometres 1 sq. centimetre = 0.15550 sq. inch 1 sq. decimetre = 15.5000 sq. inches 1 sq. metre = 15550.0000 sq. inches 1 sq. metre = 10.7640 sq. feet 1 sq. metre = 1.1960 sq. yards 1 hectare = 2.4710 acres 1 hectare = 395.3670 sq. rods 1 hectare = 24.7104 sq. chains 1 sq. kilometre = 247.1040 acres 1 sq. kilometer = 0.3861 sq. mile The hectare is the unit of land measure.
1 kilometre = 0.62137 mile
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CUBIC MEASURE 1 cu. inch 1 cu. foot 1 cu. yard 1 cord
= = = =
16.3872 cu. centimetres 28.3170 cu. decimeters 0.7645 cu. metre 3.624 cu. metres
1 cu. centimetre = 1 cu. decimeter = 1 cu. metre = 1 cu. metre =
0.0610 cu. inch 0.0353 cu. foot 1.3079 cu. yards 0.2759 cord
The cubic metre when used for measuring wood is called a ster.
CAPACITY 1 gallon U.S. = 3.7853 litres 1 gallon U.K. = 4.546 litres
WEIGHT 1 grain 1 ounce troy 1 pound troy 1 ounce avoirdupois 1 pound avoirdupois 1 ton 1 gram 1 gram 1 gram 1 kilogram 1 kilogram 1 tonne 1 tonne
= 0.0648 gram = 31.103 grams = 0.3732 kilogram = 28.350 grams = 0.4536 kilogram = 1.0160 tonne = 15.4324 grains = 0.0322 ounce troy = 0.0353 ounce avoirdupois = 2.6792 pounds troy = 2.2046 pounds avoirdupois = 0.9842 ton = 2,204.6223 pounds avoirdupois
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20. Mensuration Square If one side of a square is ‘a’, then its •
Area = a2
•
Perimeter = 4a
•
Diagonal = √
Rectangle If the length and breadth of a rectangle are ‘a’ and ‘b’ respectively, then •
Area = a x b
•
Perimeter = 2(a+b)
•
Diagonal = √
Parallelogram If one side = ‘a’ and height = ‘h’, then •
Area = a x h
Rhombus If d1, d2 be the diagonals of a rhombus, then
• Area =
B
•
Side, a =
•
Perimeter =
Trapezium
Area = #. " &--(- #% (#U%#!( (!R(( &--(- #% (#
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Triangle
• Area = #( -!%!. ( 9## # # , where a,b,c are the sides and #
•
Area of the equilateral triangle of side ‘a’ = B , its Perimeter = 3a, its altitude =
√
.
Circle If r is the radius of a Circle, then
• Area = _ •
•
Perimeter = 2 _r
Diameter =
Cube If ‘a’ is side of a Cube, then
• Volume = •
•
Total Surface = ~ Diagonal = √
Cuboid If length, breadth and height of a Cuboid are a, b, c respectively, then
• Volume = •
•
Total Surface Area =
Diagonal = √
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Hollow Cylinder A Solid bounded by two coaxial cylinders of the same height and different radii is called a Hollow Cylinder. If ‘R’ is external and ‘r’ is internal radii of a hollow cylinder of height h, then
•
Area of each end (Top and Bottom) =
•
Curved Surface Area (Lateral Surface Area) =
•
Total Surface Area =
•
Volume of the Cylinder =
square units
square units
square units
cubic units
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Right Circular Cone
•
Slant Height,
•
Volume of Cone =
cubic units
•
Curved Surface Area =
square units
•
Total Surface Area =
square units
Hollow Cone made from a sector of radius ‘r’ and central angle ,
• Radius of Cone = •
Radius of Sector = Slant height of Cone
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Sphere
•
Volume of Sphere =
•
Surface Area of Sphere =
cubic Units square units
Hemisphere
•
Volume of Hemi Sphere =
•
Curved Surface Area of Hemi Sphere =
•
Total Surface Area = Cursed Surface Area + Area of Circular Base = + =
cubic Units square units
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21. Multiple Integrals 1. Change of Order of Integration
If the limits of the inner integral is a function of x ( or function of y), then the first integration should be with respect to y (or with respect to x). Draw the region of integration by using the given limits. If the integration is first with respect to x keeping y as a constant, then consider the horizontal strip and find the new limits accordingly. If the integration is first with respect to y keeping x as a constant, then consider the vertical strip and find the new limits accordingly. After finding the new limits, evaluate the inner integral first and then the outer integral.
•
• •
•
•
In evaluating double integrals by changing Cartesian to polar coordinates, put x = r cos ;, y = r sin ; and dxdy = r. dr. d; in the given integral and then find the new limits for ; and r and then evaluate.
To change the three dimensional Cartesians to Cylindrical Coordinates, we have to put x = r cos ;, y = r sin ;, z = z.
To change the three dimensional Cartesians to spherical polar coordinates, we have to put x = r sin ;. Cos , y = r sin ; sin , z = r cos ;.
h
_/ « #
; ;
}
}
…… ,
_
……
,
%" %#
%" %# ('(
®
The area included between the curves y = f1(x) and y = f2(x) and the ordinates x = a and x = b is given by
"
S( © © "
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22. Number Work Arithmetic Progression (A. P) •
The first term of an Arithmetic Progression = a
•
Common Difference = d
•
nth term, tn = a + (n-1)d
• Number of terms of an Arithmetic Progression, •
•
Sum of ‘n’ terms, c
-
c -, if first and last term are known
Geometric Progression (G. P) •
The first term of an Geometric Progression = a
•
Common Ratio = r
• nth term, tn = a r n-1 •
Sum of ‘n’ terms c
• Sum to infinity terms, c] Special Series
, %" 0 1
® , %" I 1
• ∑ l … … .
• ∑ l … … .
• ∑ l … … . d
~
e
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23. Number & Operations The Number System of Algebra ELEMENTARY MATHEMATICS ELEMENTARY MATHEMATICS is concerned mainly with certain elements called Numbers and with certain operations defined on them. The unending set of symbols 1,2,3,4,5,6,7,8,9,10,11,12,13,…. used in counting are called Natural numbers. In adding two of these numbers, say, 5 and 7, we begin with 5 ( or with 7) and count to the right seven (or five) numbers to get 12. The sum of the two natural numbers is a natural number, i.e., the sum of the tow members of the above set is a member of the set. In subtracting 5 from 7, we begin with 7 and count to the left five numbers to 2. It is clear, however, that 7 cannot be subtracted from 5, since they are only four numbers to the left of 5. INTEGERS INTEGERS : In order that subtraction be always possible, it is necessary to increase our set of numbers. We prefix each natural number with a + sign (in practice, it is more convenient not to write the sign) to form the positive integers, we prefix each natural number with a – sign ( the sign must always be written ) to form the negative integers, and we create a new symbol 0, read zero. On the set of integers … -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, ….. the operations of addition and subtraction are possible without exception. To add two integers such as +7 and -5, we begin with +7 and count to the left five numbers to +2, or we begin with -5 and count to the right (indicated by the sign +7) seven numbers to +2. To subtract +7 and -5, we begin with -5 and count to the left (opposite to the direction indicated by +7) seven numbers to -12. To subtract -5 from +7, we begin with +7 and count to the right (opposite to the direction indicated by -5) five numbers to +12. If one is to operate easily with integers, it is necessary to avoid the process of counting. To do this, we memorize an addition table and establish certain rules of procedure. Now, we may state Rule 1 : To add two numbers having like signs, add their numerical values and prefix their common sign. Rule 2 : To add two numbers having unlike signs, subtract the smaller numerical value from the larger and prefix the sign of the number having the larger numerical value. Mathematics Formulae Explorer - Page 94 of 146
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Rule 3 : To subtract a number, change its sign and add. Rule 4 : To multiply or divide two numbers (never divide by 0!), multiply or divide the numerical values, prefixing + sign if the two numbers have like signs and a – sign if the two numbers have unlike signs. Every positive integer m is divisible by E E . A positive integer m > 1 is called a Prime if its only factors or divisors are E E . Otherwise, m is called Composite. For example 2, 7, 19 are primes, while 6 = 2.3, 18 = 2.3.3 and 30 = 2.3.5 are composites. In these examples, the composite numbers have been expressed as products of prime factors, i.e., factors which are prime numbers. Clearly, if m = r.s.t is such a factorization of m, then –m = (-1).r.s.t is such a factorization of m.
THE RATIONAL NUMBERS
, where m and n are integers. Thus, the rational numbers include the integers and common fractions. Every rational number has an infinitude of representations, for example, the integer 1 may be represented by , , , … and the fraction ~ may be represented by , , , … A fraction is said to be expressed in ~ lowest terms by the representation , where m and n have no common prime factor. The most useful rule concerning rational numbers is, therefore The set of rational numbers consists of all numbers of the form
Rule 5 : The value of a rational number is unchanged if both the numerator and denominator are multiplied or divided by the same nonzero number.
# If two rational numbers have representations and , where n is a positive # # integer, then > if r > s and < if r < s. Thus, in comparing two rational numbers it is necessary to express them with the same denominator. Of the many denominators (positive integers) there is always a least one, called the least common denominator. For the fractions and , the least common } denominator is 15. We conclude that < since = < = . } } } }
Rule 6 : The sum (difference) of two rational numbers expressed with the same denominator is a rational number whose denominator is the common denominator and whose numerators is the sum (difference) of the numerators.
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Rule 7 : The product of two or more rational numbers is a rational number whose numerator is the product of the numerators and whose denominator is the product of the denominators of the several factors.
Rule 8 : The quotient of two rational numbers can be evaluated by the use of Rule 5 with the least common denominator of the two numbers as the multiplier.
DECIMALS In writing numbers, we use a positional system, that is, the value given any particular digit depends upon its position in the sequence. For example, in 423 the positional value of the digit 4 is 4(100). While in 234 the positional value of the digit 4 is 4(1). Since the positional value of a digit involves the number 10, this system of notation is called the decimal system. In this system, 4238.75 means 4(1000) + 2(100) + 3(10) + 8(1) + 7(
) + 5(
)
PERCENTAGE }
The symbol %, read percent, means per hundred. Thus 5% is equivalent to
or 0.05. •
Any number, when expressed in decimal notation, can be written as a percent by multiplying by 100 and adding the symbol %.
•
Conversely, any percentage may be expressed in decimal form by dropping the symbol % and dividing by 100.
•
When using percentages, express the percent as a decimal and , when possible, as a simple fraction.
THE IRRATIONAL NUMBERS The existence of numbers other than the rational numbers may be inferred from either of the following considerations. We may conceive of a non repeating decimal constructed in endless time by setting down a succession of digits chosen at random The length of the diagonal of a square of side 1 is not a rational number. i.e., there exists no rational number a such that a2 = 2. Numbers such as Mathematics Formulae Explorer - Page 96 of 146
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√ , √ , √ _ .! ! √ √} are called irrational numbers. The first three of these are called radicals. The radical √ is said to be of order n, n is called the index, and a is called the radicand.
}
THE REAL NUMBERS The set of real numbers consists of the rational and irrational numbers. The real numbers may be ordered by comparing their decimal representations. •
We assume that the totality of real numbers may be placed in one-to-one correspondence with the totality of points on a straight line.
•
The number associated with a point on the line, called the coordinate of the point, gives its distance and direction from the point (called the origin) associated with the number 0. If a point A has coordinate a, we shall speak of it as the point A(a).
•
The directed distance from point A(a) to point B(b) on the real number scale is given by AB = b – a. the midpoint of the segment AB has coordinate . THE COMPLEX NUMBERS
be such a number, say, √ , then by definition √
In the set of real numbers, there is no number whose square is -1. If there is to Note carefully that √ √ √ 9 √ is incorrect. In
order to avoid this error, the symbol i with the following properties is used.
If a > 0, √ %√,
%
Then √ √ √ %√ %√ % . Also, √ √ %√ %√ % . √~ √~
Numbers of the form a + bi, where a and b are real numbers, are called Complex Numbers. In the complex number a + bi, a is called the real part and bi is called as the imaginary part. Numbers of the form ci, where c is real, are called imaginary numbers or sometimes pure imaginary numbers. The complex number a + bi is a real number when b = 0 and a pure imaginary number when a = 0. Mathematics Formulae Explorer - Page 97 of 146
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Only the following operations will be considered here. •
To add (subtract) two complex numbers, add (subtract) the real parts and add (subtract) the pure imaginary parts, i.e., (a + ib) + (c + id) = ( a + c) + (b+d)i
•
To multiply two complex numbers, form the product treating I as an ordinary number and then replace i2 by -1. i.e., (a + ib) (c + id) = ( ac - bd) + (bc+ad)i
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24. Ordinary Differential Equations General form of a Linear Differential Equation of the nth order with constant coefficients is
è è l … … . . ¬, where , , , . . are constants. è
è
The Solution of above equation consists of • Complementary Function (C.F) • Particular Integral (P.I) Also,
¬ © ¬ U
¬ ( © ¬ ( U
¬ ( © ¬ ( U
Complementary Function
An auxillary equation is given by l … . Solving this, we get , , , … … , # ′r′ éppêq.
Nature of the Roots
Roots of an Auxillary Equation Roots are Real and Distinct.
Roots are m1, m2 where m1 m2. Roots are Real and Equal.
Roots are m1, m2 where m1 m2. Roots are imaginary.
7EV
Complementary Function (C.F)
S( T( S T( S T(
(7 S # V T #% V Mathematics Formulae Explorer - Page 99 of 146
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To find Particular Integral Particular Integral P. I =
"U
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¬
X
Particular Integral
P. I = Pf1 + Qf2 X, where X is a function of x.
(
where P = «
Q=«
P. I =
"U
=
Sin 7x (or) cos 7x
( ë #% #
P. I =
(
"
, %" "
( , %" " , "/
"//
"U
" " " " "
(
"/
=
" " " " "
( , %" "/ , "//
"U
By expanding "U , we get a solution. P. I =
"U
# 7 #% 7
Replacing D2 by – 7 P. I =
P. I =
"U
"U
(
(
"U
ë
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Homogeneous Equation of Euler Type
(P ,
z = log x
xD = D/
x2D2 = D/ (D/ - 1) x3D3 = D/ (D/ - 1) (D/ - 2) x4D4 = D/ (D/ - 1) (D/ - 2) (D/ - 3)
Some Standard Binomial Expansion
l … …. l … ….
l … ….
l … ….
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25. Partial Differential Equation Let us assume that z will always represent a function of x and y. i.e., z = f(x,y) where x and y are two independent variables and z is a dependent variable.
Notations
&
, | Y YP
, Y YP
, # Y Y P
, ! Y Y Y P
Y P
Y
•
If the number of constants to be eliminated is equal to the number of independent variables then the required Partial Differential Equation will be of First Order.
•
If the number of constants to be eliminated is more than the number of independent variables then the required Partial Differential Equation will be of Second Order or higher order.
•
If the number of functions to be eliminated is one, then the required Partial Differential Equation will be of first order otherwise it will be of second order or higher order.
•
Eliminating ф from ф., ' gives a Partial Differential Equation
. 8'
. ' 8
To Solve f(p, q) = 0 Let z = ax + by + c be the solution. Then p = a, q = b, we get f(a,b)=0 Solving, we get ф
The Complete Integral is z •
= ax + ф y + c. There is no singular integral.
The complete integral of Partial Differential Equation of the type Z = px + qy + f(p, q) is z= ax + by + f(a,b)
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To Solve f(z, p, q) = 0 Let z=f(x + ay) be the solution. Put u = x + ay, then z = f(u).
&
P , .
Solving " P,
|
P .
, . which is an ordinary differential equation, we get . P
P
the required solution.
To Solve f1(x, p) = f2(y, q) Let f1(x, p) = f2(y, q) = k p = F1(x, k) q = F2(y, k)
Then P « ¼ , F « ¼ , F To Solve F(xmp, ynq) = 0 and F(z, xmp, ynq) = 0 If , , !?(
Xmp = (1 – m)P
Ynq = (1 – n)Q, where @ Solution is F ( P, Q ) = 0 F (z, P, Q) = 0
YP
Y¬
, ª
YP
Yí
If , , !?(
put log x = X and log y = Y xp = P and yq = Q Solution is F ( P, Q ) = 0 F (z, P, Q) = 0
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Lagranges Linear Equation The standard form is Pp and z.
+ Qq = R
where P, Q and R are functions of x, y
The subsidiary equation is
P ª g @
Choose any three multiplier l, m, n such that
P - P , R?(( -@ ª g @ ª g -@ ª g
i.e., - P = 0
Solving, we get u(x, y, z ) = c1
Similarly choose another set of three multipliers l/, m/, n/ such that
P -/ / / P , R?(( -/ @ / ª / g @ ª g -/ @ / ª / g Solving, we get v(x, y, z ) = c2
The Solution is given by ф(u, v) = 0
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HOMOGENEOUS LINEAR PARTIAL DIFFERENTIAL EQUATION
The general form of Linear Partial Differential Equation is
Y P Y P Y P l … ¼ , Y Y Y Y
The Solution is z = Complementary Function (C.F) + Particular Integral (P.I)
To find Complementary Function (C.F)
Auxillary Equation is … …
This equation has n roots say , , , … . . Case (i)
If the roots are real ( or imaginary) and different say l . . , then the C.F is
P " " " l … " Case (ii)
If any two roots are equal say l …then the C.F is
P " " " " … … " Case (ii)
If any three roots are equal say } l …then the C.F is
P " " " " … … "
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To find Particular Integral ( P. I ) If F(x, y) = eax+by, then
Rule 1
@. h
( ( &'% ( ф, / ф, фU, U
If ф, , !?( ("( ! #(%'. Rule 2
If F(x, y) = sin (mx+ny) or cos(mx+ny), then
@. h
#% # фU, U/
Replace D2 by –m2, D/2 by –n2 and DD/ by –mn in ф, provided the denominator is not equal to zero. If the denominator is zero, then refer to case (iv). Rule 3
If F(x, y) = xmyn then
@. h
фU, U/ фU, U/
Expand фU, U/ by using Binomial theorem and then operate on xmyn. Note 1
U
", means integrate f(x,y) with respect to ‘x’ one time assuming ‘y’ as a
constant. U/ ", means integrate f(x, y) with respect to ‘y’ one time assuming ‘x’ as a constant.
Note 2
In xmyn, if m < n, then try to write фU, U/ as фU/ and if n < m, write фU, U/
as ф U U/
U
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Rule 4
If F(x, y) is any other function, resolve фU, U/ into linear factors say U U/ U U/ … . . (!, !?( !?(
@. h Now,
U
U U
U/ U
U/
¼,
¼, « ¼, R?((
Note : If the denominator is zero in Rule 1 and Rule 2, then apply rule 4 to find Particular Integral.
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26. Probability Random Experiment Any experiment whose outcomes cannot be predicted in advance or determine in advance is a random experiment. Trial Each performance of the random experiment is called a trial. Sample Space The set of all possible outcomes of a random experiment is called a sample space and is denoted by S. Sample Point Each element of the sample space is called a sample point. Event An event is a subset of a sample space. Equally Likely Events Two or more events are said to be equally likely if each one of them has an equal chance of occurring. In tossing a coin, getting a head and getting a tail are equally likely events. Mutually Exclusive Events Two events A and B are said to be mutually exclusive events if the occurrence of any one of them excludes the occurrence of the other event. i.e., they cannot occur simultaneously. Favourable Events or Cases The number of outcomes of cases which entail the occurrence of the event in an experiment are called favourable events or favourable cases. Probability Let A be any event and the number of outcomes of an experiment favourable to the occurrence of A be ‘m’ and let ‘n’ be the total number of outcomes which are all equally likely. Then the probability of occurrence denoted by P(A) is defined as f. ( " "'.-( .! (# " S @S A!- . ( " .! (# Mathematics Formulae Explorer - Page 108 of 146
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Definition (Conditional Probability)
The conditional Probability of an event B, assuming that the event A has already happened and is denoted by P(B/A) and defined as
@ S T
@SîT @S
, provided P(A) 0
Similarly
@ T S
@SîT @T
, provided P(B) 0
Theorem (Multiplication theorem on probability) The Probability of the simultaneous happening of two events A and B is given by
T @S î T @S. @ C D S Definition
Two events A and B are independent if
@S î T @S. @T
Baye’s Theorem
Suppose A1, A2, A3, …. An are mutually exclusive and exhaustive events such that (Ai) > 0 for i = 1,2,3,4,…,n. Let B be any event with P(B) > 0, then
T @S% @ S% S% @C D T T T T @S @ S @S @ S l . @S @ S
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Points to remember • •
[ @S [
Number of outcomes which are not favourable to the event A = n – m. Probability of non-occurrence of A denoted by A/ is given by
P(A/) + P(A) = 1 •
If P(A) = 0, then A is an impossible event. i.e., Probability of an impossible event is zero. That is P(A) = 0.
•
Probability of the sure event is 1. That is P(S) = 1. S is called sure event.
Addition Theorem on Probability •
If A and B are any two events then P(AUB) = P(A) + P(B) – P (AîB)
•
If A and B are mutually exclusive events, then P(AUB) = P(A) + P(B)
Definition - Random Variable If S is a sample space with a probability measure and X is a real valued function defined over the elements of S, then X is called a Random Variable. Types of Random Variable •
Discrete Random Variable
•
Continuous Random Variable
Discrete Random Variable
Continuous Random Variable
Definition : Discrete Random Variable Definition : Continuous Random Variable If a random variable takes only a finite or a countable number of values, it is called a Discrete Random Variable. Example Number of Aces when ten cards are drawn from a well shuffled pack of 52 cards.
A Random variable X is said to be continuous if it can take all possible values between certain given limits. i.e., X is said to be continuous if its values cannot be put in 1-1 correspondence with N, the set of Natural Numbers. Example The Life length in hours of a certain light bulb. Mathematics Formulae Explorer - Page 110 of 146
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Discrete Random Variable
Continuous Random Variable
Probability Density Function (p.d.f)
Probability Mass Function (p.m.f) • •
p(x) is non-negative for all real x.
∑ &% , where pi probability at X = xi
is
the
Also, @¬ m @¬ I 3 @¬ [ @¬ 0 3 Distribution Function (Cumulative Distribution Function) The distribution function of a random variable X is defined as
¼ [ @¬ [
\ &% ∞ I I ∞ %ï
Properties of Distribution Function • • • • •
F(x) is a non-decreasing function of x.
[ ¼ [ , ∞ I I ∞
¼ ∞ ðtñz] ¼ ¼∞ ðtñz] ¼
@¬ ¼ ¼
•
@ [ [ « " > 0 for all real X.
• «] " ]
Cumulative Distribution Function If X is a continuous random variable, the function given by ∞
¼ [ @¬ [ © "! !
∞
∞ I I ∞ where f(t) is the value of the probability density function of X at t is called the distribution function or cumulative distribution of X.
Properties of Distribution Function • • • • • •
F(x) is a non-decreasing function of x.
[ ¼ [ , ∞ I I ∞
¼ ∞ ðtñz] «] " ]
¼∞ ðtñz] «] " ]
@ [ [ ¼ ¼ F/(x) = f(x)
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Discrete Random Variable
Continuous Random Variable
Mathematical Expectation
Mathematical Expectation
Expectation of a Discrete Random Expectation of a Continuous Random Variable Variable Definition
Definition
If X denotes a discrete random variable which can assume the values x1, x2, ….,xn with respective probabilities p1,, p2, ….pn then the mathematical expectation of X, denoted by E(X) is defined by
Let X be a continuous random variable with probability density function f(x). Then the mathematical expectation of X is defined as
ò¬ & & l &
\ &% % R?(( \ &%
]
ò¬ © " ]
Note
If is function such that ¬ is a random variable and ò ¬ exists Hence the mathematical expectation then E(X) of a random variable is simply the arithmetic mean. ] ò © " ] Result %^
%^
]
If ¬ is a function of the random ò © " variable X, then ]
ò \ @¬ Properties
Ð%( " ¬ ò ò
Properties
•
E (Constant) = Constant
•
E (Constant) = Constant
•
E (cX) = cE(X)
•
E (cX) = cE(X)
•
E (aX+b) = aE(X)+b
•
E (aX+b) = aE(X)+b
•
Var (XEc) = Var(X)
•
Var (XEc) = Var(X)
•
Var (aX) = a2Var(X)
•
Var (aX) = a2Var(X)
•
Var (Constant) = o
•
Var (Constant) = o
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Moments
Expected values of a function of a random variable X is used for calculating the moments. The two types of moments are • •
Moments about the origin Moments about the mean which are called Central Moments.
Moments about the origin If X is a discrete random variable for each positive integer r ( r = 1,2,3…) the rth moment
μ ò¬ \ &% % /
/ First Moment : μ ò¬ ∑ &% %
/ Second Moment : μ ò¬ ∑ &% %
Moments about the Mean : (Central Moments) For each positive integer n, (n=1,2,3,…) the nth central moment of the discrete random variable is
u \% u &% μ ò¬ ¬
The algebraic sum of the deviations about the arithmetic mean is always zero. i.e., μ Second moment about the mean is called the variance of the random variable X. i.e., μ ' ¬ ò¬ ò¬
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Theoretical Distributions
The values of random variables may be distributed according to some definite probability distribution is called theoretical distribution. Theoretical distribution are based on expectations on the basis of previous experience.
Discrete Distributions
Definition of Binomial Distribution
A random variable X is said to follow Binomial distribution if its probability mass function is given by
@¬ & & |è Constants of Binomial Distribution •
Mean = np
•
Variance = npq
• •
, , , , … . .
Standard Deviation = √Ð%( = &| p+q = 1
¬~T. & denotes that the random variable X follows Binomial Distribution with parameters n and p. In Binomial Distribution, mean is always greater than the variance.
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Poisson Distribution
Poisson Distribution is a limiting case of Binomial Distribution under the following conditions. • • •
nth number of trials is indefinitely large. i.e., z ∞
p the constant probability of success in each trial is very small i.e., & z
np = >, is finite where > is a positive real number. When an event occurs rarely, the distribution of such an event may be assumed to follow a Poisson Distribution.
Definition A Random Variable X is said to have a Poisson Distribution if the probability mass function of X is
(> > @¬ & , , , , , … . . " # ( > 0 0 ! The mean of the Poisson Distribution is >, and the Variance is also The parameter of the Poisson Distribution is
>.
>.
Examples of Poisson Distribution •
The number of printing errors at each page of a book by a good publication.
•
The number of telephone calls received at a telephone exchange in a given time interval.
•
The number of defective articles in a packet of 100, produced by a good industry.
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Continuous Distribution Function Normal Distribution Definition A continuous random variable X is said to follow a normal distribution with parameters µ and ô, if the distribution function is
"
ô√ _
(
õ ô , ∞
I I ∞, ∞ I ö I ∞ 35 ÷ 0 0.
¬~fõ, ô denotes that the random variable X follows Normal Distribution with mean õ and standard deviation ô. The Normal Distribution is also called Gaussian Distribution. Constants of Normal Distribution • Mean = õ • Variance = ô2 • Standard Deviation = ô
Properties of Normal Distribution • The normal curve is bell shaped. • It is symmetrical about the line X= õ. i.e., about the mean line. • Mean = Median = Mode = õ • The height of the normal curve is maximum at X = õ and ô√ _ is the maximum height. • It has only one mode at X= õ. The normal curve is unimodal. • The normal curve is asymptotic to the base line. • The points of inflection are at X= õ ô. • Since the curve is symmetrical about X= õ, the skewness is zero. • A normal distribution is a close approximation to the binomial distribution when n, the number of trials is very large and p the probability of success is close to ½. i.e., neither p nor q is so small. • It is also a limiting form of Poisson Distribution i.e., > z ∞, Poisson Distribution tends to normal distribution. • Area Property : @õ ô I ø I ö ÷ . ~ ~ @õ ô I ø I ö 2ô . } @õ ô I ø I ö 3ô .
Standard Normal Distribution
A random variable X is called a Standard Normal Variate if its mean is zero and its standard deviation is unity. i.e., N(0,1). The formula that enables to change ¬õ from the x-scale to the z-scale and vice versa is ù ô . ]
]
]
© " © P P © P P © P P . } ]
]
]
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27. Pure Arithmetic Basic Definitions
Uniform Speed, Time and Distance
c&(( A% (
U%#!( A% (
U%#!( c&((
U%#!( c&(( A% ( Relative Speed
Relative Speed = Sum of the speeds of two bodies when they are moving along straight path in the OPPOSITE DIRECTION.
Relative Speed = Difference of the speeds of two bodies when they are moving along straight path in the SAME DIRECTION.
Average Speed Average Speed =
A!- U%#!( :'(( A!- A%( AF(
[not average of speeds]
Resultant Speed •
Resultant [ or effective ] speed of a boat = Speed of the boat in still water – Speed of the stream, when the boat is moving up stream
•
Resultant speed of the boat = Speed of the boat in still water + Speed of the stream, when the boat is moving down stream.
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28. Sets Symbols N : Set of Natural Numbers = { 1, 2, 3, 4, …} W
: Set of Whole Numbers = { 0, 1, 2, 3, 4, ….}
I+ or Z+ : Set of Positive integers = { 1, 2, 3, 4, …} I- or Z- : Set of Negative integers = { -1, -2, -3, -4, …} I or Z Q
: Set of integers = { -3, -2, -1, 0, 1, 2, 3, …}
: Set of Rational Numbers = { x : x = | when p n I, qn I, q 0} &
Q/
: Set of Irrational Numbers
R
: Set of Real Numbers
n
: Belongs to
û
: Subset A û B means A is a proper subset of B
ú
: Does not belongs to
⊇
: Subset A ⊇ B means A contains B
þ
: Superset A þ B means A is a superset of B. i.e., A Properly contains B
ý ±
: Proper Subset A ý B means A is a proper subset of B
: Difference : A ± B means a set containing all elements of A which are not elements of B
: Universal Set /
(or) ‘ (or) - : Complement A/ or A’ or A- means complement set of A, is the set of elements of U which do not belong to A
î
: Union (or) join : A B means a set of elements which belong either to A or to V or to both
: Intersection (or) meet : A î B means a set of elements which belong to both A and B
N(A)
: Cardinal No. of A : means the number of elements in the set A
P(A)
: Power set of A
AxB
: A x B = Cartesian product of A and B Mathematics Formulae Explorer - Page 118 of 146
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Operation on Sets
Union of Sets : A B = / n pé n pé n r Intersection of Sets : A î B = / n r n
u / n r ú Complement of a Set : A/ or Ac or S
Set Difference : A – B = / n r ú Properties of Union Set Union is Commutative. i.e., Set Union is Associative.
i.e.,
AB=B A
A (B C) = (A B) C
Properties of Intersection
AîB=BîA
Set Intersection is Commutative. i.e., Set Intersection is Associative.
A î (B î C) = (A î B) î C
i.e.,
Properties of Set Difference Set Difference is not Commutative. i.e., Set Difference is not Associative.
i.e.,
Distributive Property
A B B A
A (B C) (A B) C
Union is distributed over intersection.
i.e.,
Intersection is distributed over union.
i.e.,
A (B î C) = (A B) î (A C)
A î (B C) = (A î B) (A î C)
Other Laws
S T S T S î T
S T : S T : nA î B nB î C nA î C nA î B î C
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29. Statistics Class Boundaries •
Lower Class Boundary = Lower Class Limit -
•
Upper Class Boundary = Upper Class Limit +
where d is the common difference between the upper class limit of a class and the lower class limit of the next class.
Class Mark Class Mark = =
GR( :-## G% %! &&( :-## G% %!
GR( :-## T. &&( :-## T.
Width or Size of a Class Width or Size of a Class = Upper Class Boundary – Lower Class Boundary ¼(|.( " :-## A!- ¼(|.(
•
Relative Frequency =
•
Percentage Frequency =
¼(|.( " :-## A!- ¼(|.(
Range Range = Difference between the maximum values and minimum values of a set of observations.
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Arithmetic Mean
u
∑ %
u and
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∑ " ∑"
Arithmetic Mean = S
: ∑ " , where C is common interval. ∑"
Rang e= Maximum Value – Minimum Value
Standard Deviation = ô B
∑
iii%# !?( ( , u
∑ ∑ =ô B
, S S %# !?( ##. ( (
Standard Deviation for Disordered Series
ô
∑ " ∑"
iiii%# !?( ( , u
∑ " ∑ " ô
N O ∑" ∑"
Variance = ô
Standard Deviation of first n natural number = B
•
The Standard Deviation of a series remain unchanged when each value is added (or) subtracted by the same quantity.
•
The Standard Deviation of a series gets multiplied (or) divided by the same quantity k, if each value is multiplied (or) divided by k.
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30. Tables Addition Tables 0+1 =1 0+2 =2 0+3 =3 0+4 =4 0+5 =5 0+6 =6 0+7 =7 0+8 =8 0+9 =9 0 + 10 = 10
1+1 =2 1+2 =3 1+ 3 = 4 1+4 =5 1+5 =6 1+6 =7 1+7 =8 1+8 =1 1 + 9 = 10 1 + 10 = 11
2+1 = 3 2+2 = 4 2+3 = 5 2+4 = 6 2+5 = 7 2+6 = 8 2+7 = 9 2 + 8 = 10 2 + 9 = 11 2 + 10 = 12
3+1 = 4 3+2 = 5 3+3 = 6 3+4 = 7 3+5 = 8 3+6 = 9 3 + 7 = 10 3 + 8 = 11 3 + 9 = 12 3 + 10 = 13
4+1 = 5 4+2 = 6 4+3 = 7 4+4 = 8 4+5 = 9 4 + 6 = 10 4 + 7 = 11 4 + 8 = 12 4 + 9 = 13 4 + 10 = 14
5+1 = 6 5+2 = 7 5+3 = 8 5+4 = 9 5 + 5 = 10 5 + 6 = 11 5 + 7 = 12 5 + 8 = 13 5 + 9 = 14 5 + 10 = 15
6+1 = 7 6+2 = 8 6+3 = 9 6 + 4 = 10 6 + 5 = 11 6 + 6 = 12 6 + 7 = 13 6 + 8 = 14 6 + 9 = 15 6 + 10 = 16
7+1 = 8 7+2 = 9 7 + 3 = 10 7 + 4 = 11 7 + 5 = 12 7 + 6 = 13 7 + 7 = 14 7 + 8 = 15 7 + 9 = 16 7 + 10 = 17
8+1 = 9 8 + 2 = 10 8 + 3 = 11 8 + 4 = 12 8 + 5 = 13 8 + 6 = 14 8 + 7 = 15 8 + 8 = 16 8 + 9 = 17 8 + 10 = 18
9 + 1 = 10 9 + 2 = 11 9 + 3 = 12 9 + 4 = 13 9 + 5 = 14 9 + 6 = 15 9 + 7 = 16 9 + 8 = 17 9 + 9 = 18 9 + 10 = 19
10 + 1 = 10 10 + 2 = 12 10 + 3 = 13 10 + 4 = 14 10 + 5 = 15 10 + 6 = 16 10 + 7 = 17 10 + 8 = 18 10 + 9 = 19 10 + 10 = 20
11 + 1 = 12 11 + 2 = 13 11 + 3 = 14 11 + 4 = 15 11 + 5 = 16 11 + 6 = 17 11 + 7 = 18 11 + 8 = 19 11 + 9 = 20 11 + 10 = 21
Multiplication Tables 0x1 =0 0x2 =0 0x3 =0 0x4 =0 0x5 =0 0x6 =0 0x7 =0 0x8 =0 0x9 =0 0 x 10 = 0
1x1 =1 1x2 =1 1x3 =1 1x4 =1 1x5 =1 1x6 =1 1x7 =1 1x8 =1 1x9 =1 1 x 10 = 1
2x1 = 2 2x2 = 4 2x3 = 6 2x4 = 8 2 x 5 = 10 2 x 6 = 12 2 x 7 = 14 2 x 8 = 16 2 x 9 = 18 2 x 10 = 20
3x1 = 3 3x2 = 6 3x3 = 9 3 x 4 = 12 3 x 5 = 15 3 x 6 = 18 3 x 7 = 21 3 x 8 = 24 3 x 9 = 27 3 x 10 = 30
4x1 = 4 4x2 = 8 4 x 3 = 12 4 x 4 = 16 4 x 5 = 20 4 x 6 = 24 4 x 7 = 28 4 x 8 = 32 4 x 9 = 36 4 x 10 = 40
5x1 = 5 5 x 2 = 10 5 x 3 = 15 5 x 4 = 20 5 x 5 = 25 5 x 6 = 30 5 x 7 = 35 5 x 8 = 40 5 x 9 = 45 5 x 10 = 50
6x1 = 6 6 x 2 = 12 6 x 3 = 18 6 x 4 = 24 6 x 5 = 30 6 x 6 = 36 6 x 7 = 42 6 x 8 = 48 6 x 9 = 54 6 x 10 = 60
7x1 = 7 7 x 2 = 14 7 x 3 = 21 7 x 4 = 28 7 x 5 = 35 7 x 6 = 42 7 x 7 = 49 7 x 8 = 56 7 x 9 = 63 7 x 10 = 70
8x1 = 8 8 x 2 = 16 8 x 3 = 24 8 x 4 = 32 8 x 5 = 40 8 x 6 = 48 8 x 7 = 56 8 x 8 = 64 8 x 9 = 72 8 x 10 = 80
9x1 = 9 9 x 2 = 18 9 x 3 = 27 9 x 4 = 36 9 x 5 = 45 9 x 6 = 54 9 x 7 = 63 9 x 8 = 72 9 x 9 = 81 9 x 10 = 90
10 x 1 = 10 10 x 2 = 20 10 x 3 = 30 10 x 4 = 40 10 x 5 = 50 10 x 6 = 60 10 x 7 = 70 10 x 8 = 80 10 x 9 = 90 10 x 10 = 100
11 x 1 = 11 11 x 2 = 22 11 x 3 = 33 11 x 4 = 44 11 x 5 = 55 11 x 6 = 66 11 x 7 = 77 11 x 8 = 88 11 x 9 = 99 11 x 10 = 110
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31. Theoretical Geometry Definition The Geometry which deals with properties and characters of various geometrical shapes with axioms / theorems without accurate measurements is known as Theoretical Geometry. History Geometry was developed by the Egyptians more than 1000 years before Christ to help them mark out of their fields after the floods from the Nile, but was abstracted by the Greeks into logical system of proofs many centuries later. For measurements, the length of line and sizes of angles were needed. For logical system of proofs, basic postulates or axioms were necessary. Now the study of Geometry is useful in our daily life in many ways. Axioms (or) Postulates Some Geometrical statements are accepted and they are without any proof. Such statements are called Axioms. An axiom is a self-evident truth. Let us learn some important axioms. Axiom – 1 Given any two distinct points in a plane, there exists one and only one line passing through them. Axiom – 2 Two distinct lines cannot have more than one point in common. Axiom – 3 Given a line and a point not on the line, there is one and only one line which passes through the given point and is parallel to the given line. Complementary Angles Two angles are complementary if their sum is 90o. Supplementary Angles Two angles are said to be supplementary if their sum is 180o. Adjacent Angles Two angles are adjacent angles if both angles have a common vertex.
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Linear pair Two adjacent angles form a linear pair if the two non-common arms are in a straight line. Theorems A theorem is a geometrical statement which needs to be proved. To prove a theorem, the following five important steps are followed. • • • • •
Draw the figure Write all the data Write what is to be proved, using letters of figures Write the construction, if necessary which will help to prove the theorem Write the proof with statements and reasons
Theorem If a ray stands on a line, then the sum of the two adjacent angles so formed is 180o. Theorem If the sum of two adjacent angles is 180o, then their outer arms are in the same straight line. Corollary : Corollary is also a Geometrical Statement which can be proved from the theorem. Corollary If two straight line intersect each other, the sum of the four angles so formed is equal to 360o (or) 4 right angles. Corollary If any number of straight lines meet at a point, the sum of all the angles so formed is equal to 360o (or) 4 right angles. Corollary If from a given point on a line, any number of rays are drawn on the same side of it, the sum of all the angles so formed is equal to two right angles (180o) Theorem If two lines intersect, the vertically opposite angles so formed are equal.
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Parallel Lines Two (or) more lines are said to be parallel to each other, if they are in the same plane and do not intersect when produced on either side. i.e., distance between them remains same. Transversal A straight line which intersect two or more lines at distinct points is called a transversal. When a transversal intersect two lines, four pairs of angles are formed. Playfair’s Axiom Lines which are parallel to the same line are parallel to each other. Theorem If a transversal intersects two parallel lines then the pair of corresponding angles are equal. Converse of the above theorem If a transversal intersects two straight lines such that a pair of corresponding angles are equal, then the two lines are parallel. Theorem If a transversal intersects two parallel line then • Each pair of alternate angles are equal • The interior angles on the same side of the transversal are supplementary Theorem The sum of three angles of a triangle is 180o. Theorem If a side of a triangle is produced, the exterior angle so formed is equal to the sum of the two interior opposite angles. Important Notes • In a triangle, the sum of any sides is always greater than the third side. • Every triangle should have atleast two acute angles. • The sum of the angles of a triangle is 180o or 2 right angles. • The sum of the angles of an n-sided polygon is (2n-4) right angles. • In any right angled triangle, the square on the hypotenuse is equal to the sum of the squares in the other two sides (Pythagorus theorem) Mathematics Formulae Explorer - Page 125 of 146
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Angles • • • • • •
g%)?! )-(
S.!( )-( I
v!.#( )-( 0 .! I
g("-( )-( 0 .! I ~
S)-(# S T ( : &-( (!, %" S T
S)-(# S T ( c.&&-( (! , %" S T
Triangle In a triangle ABC, A, B, C are the vertical angles and sides BC, CA, AB are denoted by a, b, c respectively. Then • • • •
A+B+C =
a + b > c, a + c > b, b + c > a
~ I , 4 ~ I 3, ~ 3 I 4
Exterior angle = Sum of two opposite interior angles
Polygon • •
Sum of interior angles of a polygon of n sides = (2n – 4) right angles = (2n -4) x Sum of exterior angles of a convex polygon = ~
•
Each angle of a regular polygon of n sides =
•
Number of sides of a regular polygon each
having internal angle = %#
~
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Theorem If two chord of a circle intersect inside the circle (or outside) when produced the rectangle formed by the two segments of one chord is equal in area to the rectangle formed by the two segments of the other chord, then PA x PB = PC x PD
Secant Theorem If PAB is a secant to a circle intersecting it at A and B and PT is a tangent at T, then PA x PB = PT2
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Basic Proportionality Theorem (or) Tales Theorem If the straight line is drawn parallel to one side of a triangle it cuts the other two sides proportionally.
Angle Bisector Theorem If the vertical angle of a triangle is bisected internally (or) externally, the bisector divides the base internally (or) externally in to two segments which have the same ratio as the order of two sides of the triangle.
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Theorem [AAA – Similarity] If two triangles are equiangular to one another, then the two triangles are similar. Theorem [SAS – Similarity] If two triangles have one angle of the one equal to one angle of the other and the sides about the equal angles proportional, then two triangles are similar. Theorem [SSS – Similarity] If two triangles have their corresponding sides proportional then the two triangles are similar. Areas of Similar Triangle Similar triangles are to one another as the squares on their corresponding sides (or) the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides. Similar Triangles In
•
• •
If D, E are midpoints of AB, AC in triangle ABC,
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Right angled triangle ABC, Right angled at A
If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, the triangle on each side of the perpendicular are similar tto the whole (original) triangle and to each other.
•
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32. Trigonometry Trigonometry is that branch of mathematics which deals with the study of the relationship between the sides and angles of triangle. Basic Definitions:
#% ;
v&&#%!( c% ( #( ; &!(.#(
! ;
! ;
Inter Relations
# ; #( ; ! ;
!;
#( ; # ;
S j(! c% ( &!(.#(
v&&#%!( c% ( S j(! c% (
S j(! c% ( v&&#%!( c% (
&!(.#( S j(! c% (
&!(.#( #( ; #% ; v&&#%!( c% ( #% ; # ; # ; ! ; #% ; !
Identities
#% ; # ; #% ; # ; # ; #% ;
! ; #( ; #( ; ! ;
#( ; ! ;
! ; #( ; #( ; ! ; #( ; ! ;
_ g %# Radian Measure
_ g %# g % ()((# _
Trigonometric Ratios standard angles
for
certain
;
}
~
cos
√
sin
tan
sec
cosec
cot
∞ ∞
Also, sin ;
cos ; tan ;
√
√
√
√ √
√
√
∞
√
√
#% ;
√
√
∞
# ;
! ;
cosec ; #( ; sec ; cot ;
#( ;
! ;
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Allied Angles
Trigonometrical ratio’s of 90E;, 180E;, 270E;, 360E; in terms of those of ; can be found easily by the following rule known as A-S-T-C rule. When the angle is 90E; (or) 270E;, the trigonometrical ratio changes from sine to cosine, tan to cot, sec to cosec and vice versa.
When the angle is 180E; (or) 360E;, the trigonometrical ratio remains the same. i.e., sin; --> sin ;, cos ; --> cos ;, etc.,
In each case the sign (+) or (-) is premultiplied by the A-S-T-C quadrant rule. S A II (90 – 180) I (0 – 90 ) T C III (180 – 270) IV (270 – 360 )
A : all ratio’s are positive in the I Quadrant S : sine is positive in the II Quadrant T : tan is positive in the III Quadrant C : cos is positive in the IV Quadrant
Compound Angle Formulae
#% S T #% S # T # S #% T #% S T #% S # T # S #% T # S T # S # T #%S #% T # S T # S # T #% S #% T ! S T ! S T
! S ! T ! S ! T ! S ! T ! S ! T
To convert product in to sum or difference formulae #% S # T
#% S T #% S T
# S # T
# S T # S T
# S #% T
#% S #% T
#% S T #% S T
# S T # S T
Formulae for A, 2A & 3A angles
#% S #% S # S
# S #% S # S # S ® #% S #% S
# S ! S
! S ! S
! S ! S
! S ! S
#% S #% S #% S
# S # S # S
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#% S #%
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S S #
Hyperbolic Functions
If -)( , ( is called as the Exponential Function.
S S # #% S # S # ® S
#%
Hyperbolic Functions are defined in terms of exponential function as below :
S ! #% S S !
( (
! ?
#% ? # ?
# ?
S ! # S S !
! ?
#( ?
S ! ! S S !
Formula to convert a sum or difference in to product :
:U : U #% : #% U #% C D # C D #% : #% U # C
#% ?
:U : U D #% C D
:U : U # : # U # C D # C D # : # U #% C
:U : U D #%
( (
# ? #% ? ! ? # ?
#( ?
#% ?
Hyperbolic Identities
# ? #% ?
! ? #( ?
! ? #( ?
# ? #% ? # ? #% ? # ? #% ?
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Relationship between Trigonometric and Hyperbolic Functions :
#% #
(% (% %
(è (
# %
(è
= %
%
=
( (è
%
(
G% C D , z
%# !%- . (
(% (%
#% %
Standard Results
( (è %
G% #% D C z ]
= % #% ? # ?
G% ! C D z
G% / ( z G% C D ( z∞
G% ( z∞ G% / z∞
General Solutions of #% ; #% 7
; _ 7,
R?(( n ù
General Solutions of # ; # 7
; _ 7,
R?(( n ù
General Solutions of ! ; ! 7
; _ 7,
R?(( n ù
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Properties of Triangles
Consider a triangle ABC. It has three angles A, B and C. The sides opposite to the angles A, B, C are denoted by the corresponding small letters a, b, c respectively. Thus a = BC, b = CA, c = AB. We can establish number of formulae connecting these three angles and sides.
Sine Formula
c% T #% : g, #% S
In any triangle ABC,
R?(( g %# !?( %.# " !?( %. %-( " !?( !%)-( ST:.
Napier’s Formulae In any triangle ABC,
• !
ST
!
:S
• ! •
T:
!
!
!
:
S
T
Cosine Formulae In any triangle ABC, the following results are true with usual notation.
• # S
• # T
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Projection Formulae
In any triangle ABC,
• # : # T • # S # :
• # T # S Sub – Multiple (half) angle formulae
In any triangle ABC, the following results are true.
#%
# # S
#
S ##
!
# # S ##
#%
# # T
#
T ##
!
# # T ##
#%
# # :
#
: ##
!
# # : ##
where s =
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Area Formulae ( ∆ (!(# ( " !%)-(
In any triangle ABC,
• ∆
#% :
• ∆
#% S
• ∆
#% T
• ∆
g
• ∆ g #% S #% T #% :
• ∆ 9## # # Are true with the usual notations and these are called Area formulae.
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Inverse Trigonometrical functions (Inverse circular function)
Properties of Principal Inverse Trigonometric Functions
#% #%
# #
! !
#% C D #(
# C D #(
! C D !
#( #(
#( #(
#( C D #%
#( C D #
#% #%
#( #(
#% #
•
! !
# #
! C D !
#( _ #(
_ _ ! !
If xy < 1, then !
! ! ! !
#( #(
! !
_
• #% #% #% 9 √
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33. Vector Algebra Basic Definitions Dot and Cross Product
be any two vectors and Let subtending an angle ;, between them. represent the unit vector Also, let perpendicular to the plane containing . (i.e.,) the vectors and and . Then, Perpendicular to both we have | # ; c- ª.!%! . |
Properties
. . and $ $ ! . !̂ #̂ . #̂ F . F $ ! !̂ #̂ #̂ F$ F $ $ ! #̂ F ; # F !̂ ; F$ !̂ #̂ $ F$ . !̂ ! . #̂ #̂ . F
Analytic Expressions for the Dot and Cross Product
and ! # F If
| #% ; Ð(! ª.!%! | , , forms a right handled ! # F then where system. and . and Angle Between the vectors ! # F % % If ; is the angle between them, then . # ; | | Scalar Trible Product (or) Box Further, the vectors are Perpendicular, Product _ _ if ; # ; # . . denoted by [ ] is called as the scalar triple product (or) , . the box product of the vectors , , . , %# @(&( %.- Also, to %"
.
Angle Between the vectors using Cross Product
and
If ; is the angle between them, then #% ; | |
Further, the vectors are Parallel, if ; _ , #% ; . ,
%# @--(- to , Also, %"
Properties
. = .
. = . = .
] is equal to the value [ of the coefficient determinant , , of the vectors . If any two vectors are identical in a box product then the value is equal to zero. In such a case, we say that the vectors are Coplanar.
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Vector Triple Product
Equation of a Straight Line
. . =
Equation of straight line passing through a point and parallel to is .
. = .
= ! .
Scalar Product of Four Vectors
. 8 .
. . = 8 .
Equation of a Plane Equation of plane passing through and parallel to . and ' is a point
Vector Product of Four Vectors
= # . + t '
= [ ] - [ ]
and ' are scalars. where .
] ] = [ - [ Gradient
& = !
Y Y
#
Y Y
F
Divergence ¼
Y YP
Directional Derivative =
Unit Tangent Vector =
& . | |
! < ! <
Normal Derivative = |&| Unit Normal Vector
& |&|
= & . ¼ Divergence ¼ =
Y" Y
Y" Y
= 0 , ¼ is Solenoidal. & .¼ Curl ¼
! = ' Y Curl ¼ = & ¼ Y ¼
Y" YP
# F Y Y ' Y YP ¼ ¼
Angle between the Surfaces is given = 0 , ¼ is Irrotational. & ¼ by, cos ; =
& .& |& ||& |
Laplace Operator
Y Y Y & Y Y YP
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Line Integral Statement The Line Integral along the curve C is denoted by
. . ( ¼ , « ¼
if c is a closed curve.
Surface Integral Statement The Surface Integral of ¼ is defined to be
. # È È ¼ c
g
. ¼ . F
Volume Integral Statement Volume Integral of F(x,y,z) over a region enclosing a volume V is given by
) ¼ , , P c
) ¼, , P P Ð
Green’s Theorem in a Plane Statement
If u, v, Y , Y are continuous and one-valued functions in the Region R enclosed Y. Y'
by the curve C, then
Y Y. © . ' È Y Y g
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Gauss Divergence Theorem Statement The Surface Integral of the normal component of a vector function F over a closed surface S enclosing volume V is equal to the volume Integral of the divergence of F taken throughout the volume V is represented by,
È ¼ . # ) & . ¼ ' c
Ð
Stoke’s Theorem Statement The Surface Integral of the normal component of the Curl of a vector function F over an open surface S is equal to the line integral of the tangential component of F around the closed curve C bounding S is given by ,
. Íc & ¼ . # « ¼
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34. Z - Transforms PROPERTIES OF UNILATERAL Z-TRANSFORM
Property
Linearity
Frequency shifting
Time shifting mm0
Scaling in Z-domain (or) Multiplication by Differentiation In Z-domain
Time reversal (Bilateral)
Convolution
Initial Value Theorem
Final Value Theorem
Discrete Sequence
" ) (! "!
Z-Transform
" ) P(A
"
P ¼P \ "%P , m
"
¼PPzP
"F
%^
P ¼P
"
P
¼P P
¼C D P
" F " µ "
" ðtñ ¼P
¼ P µ ¼ P
Pz]
"∞ ðtñP ¼P Pz
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TABLE OF Z-TRANSFORMS "
ù "
F
P
P
P P
F F
P
(
P(è
(
F
P P
P( P
P
F
P P
FF
P P
! F!
(P
F
F (%!
PP P
P P
P-) P
P(%A
P(%A
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#% ! #!
# #%
. d
_
www.MathsHomeWork123.com P#% A
P P # A PP# A
P P # A P
P
_
, m ® , !?(R%#(
P
P
P P
(! "!
¼PPzP(A
Ó
1 Ï *
Ó
Solving Simultaneous Equations with given initial conditions
uP
P P uP
P P uP P P
P P P
uP P
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Inverse Z-Transform
• P
• PÀF Á • P • P F
, P
P
P P
P
P P
, P
P P
• P • PÀFF Á
P
F
, P P P
, P P F
P P
P
, P P P
, P P FF
P P
• PÀFF Á
P P
• P
• P P
P
P
P P
• P
P
P
, P P P
, P P FF P
P
, P P P
, P P
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