Compound Interest (3 ~ 6 marks)

A Amount P Principle n Number of years R Rate % per year

(

)

(

)

(

) (

) (

)

(

) (

)

(

)

*(

)

+

The Compound Interest and Simple Interest for the first year is always same C.I.1 = S.I.1

( )

(

) ( )

Shares and Dividend (3 ~ 6 marks)

Nominal Value = Face Value = Par value

Dividend = Income

Investment = No. of shares x Market value per share

Total Nominal value = No. of shares x Face value per share

( )

Sales Proceeds = No. of shares x selling price of each share (Market Value of each

share)

( )

For better investment, compare the Y% both shares.

Discount shares means Market Value < Face Value.

Premium share means Market Value > Face Value.

Share at par means Market Value. = Face Value.

Sales Tax + VAT (2 ~ 5 marks)

The tax imposed by state govt. on sale of goods within their respective states or by

central government is called sales tax.

If an article is to be sold for Rs.100 and 5% sales tax is applicable so the purchaser will

have to pay Rs.105.

i.e. sales tax (given as percentage) is to be added to the selling price of any article.

Usually Selling Price Rs.100

Selling Price. + tax Rs.105

Vat to be paid = x % of ( selling price cost price ) = x % of value addition

Banking (4~ 8 marks)

Savings account:

Calculate the principle qualifying for interest for every month. (Max. value till the tenth

and then if it decreases then the smaller value) Time is always 1/12. Calculate the

interest using Simple Interest formulae = PRT/100

Recurring account:

Fixed amount of money is to be kept in the bank and then after a stipulated period

interest and amount is calculated.

( )

;

S

n no. of months

Maturity value = P/month x n + S.I.

OR ( )

Inequations

(3 ~ 4 marks)

Natural numbers (N) = {1, 2, 3, 4, 5, }

Whole numbers (W) = {0, 1, 2, 3, 4, 5, } (W0N)

Integers (I, Z) = {-, .-3, -2, -1, 0, 1, 2, 3, 4, 5, }

Real numbers (R) = {-, .0 } (all decimals and fractions in between are

included)

Quadratic Equations (4 ~ 5 marks)

Let the Quadratic equation be ax2 + bx + c = 0 (a 0 )

The expression ( b2 4ac ) is called discriminant

= ( b2 4ac )

If = 0, roots are equal and real.

If > 0 and is a perfect square, roots are real, unequal and rational.

If > 0 and is not a perfect square, roots are real, unequal and irrational.

If < 0, roots are imaginary. (i.e. not real)

Sum of the roots of the equation =

& product of the roots =

A quadratic equation will have 2 solutions.

Eg.

x2 25 = 0

x = + 5

x2 6x + 9 = 0

( x 3) 2 = 0

x = 3, 3

Remainder & Factor Theorem (3 ~ 4 marks)

Remainder Theorem :

If f(x) is a polynomial, an expression with variable x and is divided by (x a); the

remainder is the value of f(x) at x = a. i.e. the remainder is f(a).

eg. f(x) = x 3 + 2 x 2 + 4; (x 3); f(3) = 33 + 2 x 32 + 4 = 49.

Factor Theorem :

When a polynomial f(x) is divided by x a, the remainder = f (a). And, if remainder f(a)

= 0;

x - a is a factor of polynomial f(x). OR if (x a) is a factor, the remainder is 0.

Ratio & Proportion (3 ~ 5 marks)

If a, b, c & d are in proportion

, i.e. ad = bc (product of means = product of

extremes) also called inproportion

If a, b & c are in proportion

, i.e. b2 = ac(where b is the mean proportion and c is

the third proportion.) also continued proportion.

If a, b, c, d are in continued proportion

, then c = dk, b = dk2, a = dk3.

If a, b, c & d are in proportion, then using properties of ratio.

By Alternendo

By Invertendo

By Componendo

By Dividendo

By Comp & Div

Property of ratio

Matrices (3 ~ 5 marks)

Size of Matrix = No. of rows x No. of columns. (Rh Rows horizontal)

*

+ Only same order matrix can be added or subtracted

*

+

Multiplication of matrix

Matrix A m x n can only be multiplied with matrix B p x q if n = p and the resultant matrix

will have m rows and q cols i.e. R m x q

Matrix multiplication is not commutative. (i.e. AB BA)

Matrix multiplication is associative.

A (BC) = (AB) C

A x I = I x A = A where I is a unit matrix of suitable order.

A ( B + C ) = AB + AC (distributive property)

If AB = AC then B C, A may or may not be zero.

Coordinate Geometry (12 ~ 16 marks)

Distance formulae : ( ) ( )

Mid Point Formulae :

Section formulae :

Centroid Formulae :

Slope

is the angle the line makes with the x - axis, +ve if it is

anticlockwise ve if it is clockwise.

For parallel lines m1= m2 Slopes are equal.

For perpendicular lines m1 x m2 = 1 Slopes are negative reciprocal each other.

Equation of a Line

Equation of x - axis : y = 0 Equation of line parallel to x- axis : y = a

Equation of y - axis : x = 0 Equation of line parallel to y- axis : x = b

Slope of x-axis = tan 0 = 0 Slope of line parallel to x- axis = 0

Slope of y - axis = tan 90 = undefined(1/0) Slope of line || to y- axis : undefined (1/0)

A point on x axis is taken as (a , 0 ) and on y axis is taken as ( 0 , b )

x coordinate is called abscissa and y is ordinate.

Circumcentre is a point equidistant from the vertices of a . It is the centre of the circle

drawn about a . It is the point if intersection of the bisector of sides of a .

For a right angled triangle it is midpoint of the hypotenuse.

Incentre is a point equidistant from the sides of a triangle. It is the centre of a circle

drawn inside a triangle. It is a point of intersection of the bisectors of the angles of a

triangle.

Centroid is the point of intersection of the medians of a triangle. Centroid divides the

median in the ratio 2 : 1

Orthocentre is the point of intersection of the altitudes of a triangle.

For a parallelogram x1 + x3 = x2 + x4 and y1 + y3 = y2 + y4 where x and y are coordinates

of the parallelogram.

For any points

P( a ,b ) Mx P(a , b) Reflected in x axis Invariant point is a point which

P( a ,b ) My P(a , b) Reflected in y axis does not change under reflection.

P( a ,b ) Mo P(a , b) Reflected in origin eg. P( 2 , 0 ) Mx P(2 , 0)

Symmetry (3 ~ 5 marks)

If a figure is divided into two congruent parts with respect to a line, then the figure is

said to be symmetric about the line and the line is called the axis of symmetry. Any

figure may have line and point symmetry.

Slope Intercept form {m , c = y intercept} y = mx + c

Slope point form {P(x1 , y1) & m} y y1 = m (x x1)

Two point form {P(x1 , y1) & Q(x2 , y2 )}

Similarity and Size Transformation (6 ~ 8 marks)

ABC ~ PQR

By A. A. test or S.A.S test or by S.S.S. test C. P. S. T.

[ ]

Areas of ~ s are proportional to squares of their corresponding sides.

A1 = A2 A median divides a into 2 s with equal area.

Areas of s meeting at common vertex and base through the same st line is proportional to their bases.

Loci (5 ~ 8 marks)

It is a path traced by a moving particle which fulfils a given condition. Theorem 1 : Locus of points equidistant from two fixed points is the perpendicular

bisector of the line joining the two fixed points.

Theorem 2 : Locus of points equidistant from two fixed lines is the angle bisector of the

angle formed where the two lines intersect.

Theorem 3 : Locus of point B such that AB2 + BC2 = AC2 is the circle with AC as

diameter.

Circles (4~ 8 marks)

Chord Properties: (all theorems are applied to same circle or equal circles i.e. circles with

equal radius)

A straight line drawn from centre of a circle bisecting the chord is to it.

A straight line drawn from centre of a circle bisects the chord.

Only one circle can be drawn passing through three non collinear points.

Equal chords are equidistant from the centre of the circle.

Equidistant chords are equal.

Angle Properties: The angle which an arc of a circle subtends at the centre is double that which it

subtend at any point on the remaining part of the circumference.

Angles in the same segment are equal.

The angle subtended by a semi circle is a right angle.

Opposite angles of a cyclic quadrilateral are supplementary.

Exterior angle of a cyclic quadrilateral is equal to its interior opposite angle.

Arc Properties: If two arcs subtend equal angles at the center they are equal.

If two arcs are equal they subtend equal angles at the center.

If two chords are equal they cut off equal arcs.

If two arcs are equal they cut off equal chords.

Tangent Properties: The tangent to any circle and radius through the point of contact are to each other. Two tangents drawn to a circle from an external point are

i) equal in length,

ii) equally inclined to each other through the central line,

iii) equally inclined at the centre.

If two circles touch each other the point of contact lies on the straight line drawn

through the centres.

If two chords intersect each other then the product of their lengths of their segments is

equal.

If a tangent and a chord intersect then the product of the lengths of the segments of the

chord is equal to square of the length of the tangent from the point of contact to point

of intersection.

The angle between a tangent and a chord through the point of contact is equal to the

angle in the alternate segment. (Alternate segment theorem T. C. A. S. A.)

Mensuration (12 ~ 14 marks)

Area Circumference Perimeter

Circle r 2 = D2/4 2 r = D 2 r = D

Semi Circle

r + 2 r

Ring ( R2 r 2 ) = ( R r) (R + r) 2 ( R + r ) 2 ( R + r )

Sector

Distance travelled by a wheel = No. of rev. x circumference of wheel

Volume Curved Surface Area Total Surface Area

Cylinder r2h 2 rh 2r ( h + r )

Hollow Cylinder / Pipe (R2 r2) h 2h (R + r) 2 (Rh + rh + R2 r2)

Cone

rL r (L + r)

Sphere

4r2 4r2

Hemisphere

2r2 3r2

Spherical Shell

( )

4 ( R2 + r2 ) 4 ( R2 + r2 )

Hemispherical Shell

( )

2 ( R2 + r2 ) ( 3 R2 + r2 )

Volume of Big Sphere = No. of lead shots x Volume of each lead shot.

Volume = Area of cross section x length ( height )

1m3 = 1000 ltrs = 106 cm3

Trigonometry (10 ~ 16 marks)

oSh aCh oTa

Trigometric Identities

Complementary Angle

Sin = Cos ( 90 )

Cos = Sin ( 90 )

Tan A = Cot ( 90 )

Cosec = Sec (90 )

Sec = Cosec (90 )

Cot = Tan (90 )

Standard Angles

0o 30o 45o 60o 90o

Sin 0

Cos 1

Tan 0

1

Statistics (8 ~ 16 marks)

Mean ( ) : The average value of the given data is mean

Ungrouped

Grouped

a = assumed mean

Short cut method d = x a and

Step Deviation method

and Mean =

( i = class width)

Median (Q2) The exact middle value of the given data.

Raw data Odd ,

- Even

(

)

Ungrouped frequency distribution Median (Q2)

{

}

,

-

Lower Quratile (Q1) ,

- ,

-

Upper Quartile (Q3) { ( )

}

{

}

Grouped frequency distribution (all values from OGIVE) Median (Q2)

{

}

,

-

Lower Quartile (Q1) {

}

,

-

Upper Quartile (Q3) { ( )

}

{

}

Inter Quartile Range = Q3 Q1

Semi Inter Quartile Range =

Mode : (z) The most occurring value in a ungrouped data and the class mark with highest frequency in grouped data.

Raw Data = the most occurring number

Ungrouped Mean = the number with the highest frequency.

Grouped draw a histogram and the highest bar gives the mode.

Probability (3 ~ 5 marks)

Probability of an impossible event = 0

All possible outcomes added = 1

Sample space denotes all possible outcomes.

Complementary event P (A) + P (A) = 1 or P(A) = 1 P(A)