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Written By

Md MoinMechanical Engineer

This book contains all necessary formulas and theoremsrequired by ICSE boards students

For ICSE class X

Price : Rs 49/-

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A product by

Flash Education

Based on 2019-20 syllabus

A chapterwise formula book

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Flash Education

Price : Rs 49/-

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INDEX

CHAPTERS NAME PAGE NO.

1. BASIC ALGEBRAIC FORMULA …………………………………………………………… 1

2. BANKING ………………………………………………………………………………………………… 2

3. SHARES AND DIVIDEND ……………………………………………………………………… 2

4. QUADRATIC EQUATION ……………………………………………………………………… 2

5. FACTORISATON ……………………………………………………………………………………… 3

6. PROPORTION ………………………………………………………………………………………….… 3

7. MATRICES ……………………………………………………………………………………………….… 4

8. ARITHMETIC AND GEOMETRIC PROGRESSION ……………….……….… 4

9. REFLECTION ………………………………………………………………………………………………. 5

10. SECTION FORMULA …………………………………………………………………………… 5

11. EQUATION OF THE STRAIGHT LINE …………………………………………….… 6

12. SIMIARITY ……………………………………………………………………………………………….. 7

13. LOCUS …………………………………………………………………………………………………………. 10

14. CIRCLE ………………………………………………………………………………………………………… 13

15. MENSURATION ……………………………………………………………………………………….… 17

16. TRIGONOMETRY ……………………………………………………………………………………… 19

17. STATISTICS …………………………………………………………………………………………….. 21

18. PROBABILITY …………………………………………………………………………………………… 23

Flash Education

1. EXPANSION

(i) (a + b)2 = a2 + 2ab + b2.

= (a - b)2 + 4ab

(ii) (a - b)2 = a2 - 2ab + b2.

= (a + b)2 - 4ab.

(iii) a2 - b2 = (a+b)(a-b).

(iv) (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)

(v) (a + b)3 = a3 + b3 + 3ab(a + b)

= a3 + 3a2b + 3ab2 + b3.

(vi) (a - b)3 = a3 - b3 - 3ab(a - b)

= a3 - 3a2b + 3ab2 - b3.

(vii) a3 + b3 = (a + b)( a2 - ab + b2).

(viii) a3 - b3 = (a - b)( a2 + ab + b2).

(ix) a3 + b3 + c3 + 3abc = (a + b + c)(a2 + b2 + c2 - ab - bc - ca).

2. RULES OF INDICES

If ‘a’ and ‘b’ are any two real numbers and ‘m’ and ‘n’ are any two rational

numbers, then

(i) am.an = am+n

(ii) nmn

m

aaa .

(iii) (am)n = amxn

(iv) (ab)m = am.bm (v) m

mm

ba

ba

.

(vi) a-n = na1

(vii) n1

n aa

(viii) a0 = 1

Flash Education

1. Simple Interest (I) =100r

122)1n(nP

2. Maturity Value (MV) = P x n + I

also, MV = P x n +100r

122)1n(nP

Where ‘P’ is the money deposited per month, ‘n’ is the number of months and‘r’ is the rate of interest.

Page 2

1. Money invested = Number of shares x market value of one share.

2. Annual income = Number of shares x rate of dividend x face value of one

share.

3. Return percentage = %100xinvestment

incomeannual

.

4. Number of share purchased =shareoneonincome

incomeAnnualorshareoneofvaluemarket

investment .

1. Sridhara Acharyya’s formula

The roots of the quadratic equation ax2 + bx + c = 0, a ≠ 0 are given by

a2ac4bb,

a2ac4bbx

22

2. Nature of Roots

A quadratic equation ax2 + bx + c = 0, a ≠ 0 has

(i) Two real and unequal roots if b2 - 4ac > 0(ii) Two real and equal roots if b2 - 4ac = 0(iii) Two unreal and unequal roots if b2 - 4ac < 0(iv) Two real roots if b2 - 4ac ≥ 0

Proportion : An equality of two ratios is called a proportion.

Four (non-zero) quantities a, b, c, d are said to be in proportion iff

a : b = c : d i.e.dc

ba . We write it as a : b : : c : d.

Continued proportion

The non-zero quantities of the same kind, a, b, c, d, e, …, are said to

be in continued proportion iff .....ed

dc

cb

ba

1. Remainder theorem

If a polynomial f(x) is divided by (x - α) then remainder = f(α)

2. Factor theorem

x- α is a factor of the polynomial f(x) if and only if f(α) = 0

Page 3

Some properties of proportion

1. Invertendo

If a : b : : c : ddc

ba then

cd

ab

2. Alternendo


ba then

db

ca

3. Componendo


ba then

ddc

bba

4. Divedendo


ba then

ddc

bba

5. Componendo and dividendo


ba then

dcdc

baba

6. Convertendo


ba then

dcc

baa

7. Iffe

dc

ba

, then each ratio = .sconsequentofSumsantecedentofSum

fdbeca

Flash Education

Let A =

dcba

and B =

srqp

then

1. Addition of Matrices :

A + B =

sdrcqbpa

2. Subtraction of Matrices

(i) A - B =

sdrcqbpa

dscrbqap

AB)ii(

3. Multiplication of Matrices

(i) kA =

kdkckbka

dscqdrcpbsaqbrap

B.A)ii(

Note : (i) For addition or subtraction of two or more matrices, order of the

matrices must be same.

(ii) For multiplication of two matrices, number of column in first matrix and number

of row in second matrix must be same.

(iii) A.B B.A (iv) A - B B - A

1. Arithmetic progression

Let the first term be ‘a’ and common difference be ‘d’ and ‘n’ be the nth

term of an AP series, then

(i) Genera term (or last term)

an = a + (n-1)d or l = a + (n-1)d

(ii) Sum of n term

(a) ]d)1n(a2[2nSn (b) )la(

2nSn [ when last term (l) is known ]

Flash Education

( Where k is aconstant number )

4. Reflection of P (x, y) in a line parallel to x - axis is P’ (x, -y + 2a).

5. Reflection of P (x, y) in a line parallel to y - axis is P’ (-x + 2a, y).

2. Geometric progression

Let the first term and the common ratio of a GP series be ‘a’ and ‘r’

respectively, then

(i) General term (or last term)

an = arn-1 or l = arn-1

(ii) Sum of n terms 1rif,r1r1.aSn

n

(iii) Sum of n terms 1ror1rif,1r1r.aS

n

n

(iv) Sum of n terms Sn = na, if r = 1

P (x1, y1) Q (x2,y2)

Page 5

1. Reflection of P (x, y) in the x-axis is P’ (x, -y).

2. Reflection of P (x, y) in the y-axis is P’ (-x, y).

3. Reflection of P (x, y) in the origin is P’ (-x, -y).

1. Distance Formula

212

212 )yy()xx(PQ

Flash Education

2. Section formula

21

1221

mmxmxmx

and21

1221

mmymymy

The co-ordinate of R which divides the line segment joining the points

P (x1, y1) and Q (x2, y2) in the ration m1 : m2 are

21

1221

21

1221

mmymym,

mmxmxm

.

3. Mid-point formula

2xxx 12 and

2yyy 12

The co-ordinate of the mid-point of the line segment joining the points

P (x1, y1) and Q (x2, y2) are

2yy,

2xx 1212

4. Centroid formula

The co-ordinate of the centroid (G) of a triangle whose vertices are

A (x1, y1), B (x2, y2) and C (x3, y3) are

3yyy,

3xxx 321321

1. Slope of a straight line

(i) Slope (m) = tan θ (where θ is inclination) (ii) Slope (m) =12

12

xxyy

.

2. Equation of straight lines

(i) Equation of x - axis is y = 0.

(ii) Equation of y - axis is x = 0.

Page 6

P (x1, y1) R (x, y) Q (x2, y2)

1 1

B (x2,y2) C (x3,y3)

P (x1, y1) R (x, y) Q (x2, y2)

m1 m2

Flash Education

(iii) Equation of a line parallel to x - axis is y = b.

(iv) Equation of a line parallel to y - axis is x = a.

(v) Equation of straight line in slope - intercept form

y = mx + c ( where m = slope and c = y - intercept )

(vi) Equation of straight line in point-slope form

y - y1 = m (x - x1)

(vii) Equation of straight line in two point form

)xx(xxyyyy 112

121

3. Condition for Parallelism and perpendicularity

Two lines with slope m1 and m2 are

(i) Parallel if and only if m1 = m2

(ii) Perpendicular if and only if m1m2 = -1

Page 7

1. Similarity of triangles

Two triangles are called similar if and only if they have the same shape,

but not necessarily the same size.

In two similar triangles, corresponding angles are equal and corresponding

side are proportional.

If two triangles ABC and PQR are similar,

then we shall find that

(i) ,RC,QB,PA

(ii)RPCA

QRBC

PQAB

Flash Education

Flash Education

2. Axioms of similarity of triangles

(i) A.A (Angle-Angle) axiom of similarity

If two angles of a triangle are equal to two angles of another triangle,

then the two triangles are similar.

In the adjoining diagram,

∆s ABC and PQR are such that

A = P and B = Q

∆ ABC ~ ∆ PQR

(ii) S.A.S (Side-Angle-Side) axiom of similarity

If one angle of a triangle is equal to one angle of another triangle and

the sides including these angles are proportion, then the two triangle are

similar.

In the adjoining diagram, ∆s ABC and

PQR are such that

A = P andPRAC

PQAB

∆ ABC ~ ∆ PQR

(iii) S.S.S (Side-Side-Side) axiom of similarity

If the three sides of one triangle are proportional to the three

side of another triangle,then the two triangles are similar.

In the adjoining diagram, ∆s ABC and

PQR are such that

RPCA

PRAC

PQAB

,

∆ ABC ~ ∆ PQR

3. Basic theorem of proportionality

If a line is drawn parallel to one side of a triangle intersecting

the other two sides, then the other two sides are divided in the same ratio.

Conversely : If a straight line divides any two sides of a triangle in the

same ratio, then the straight line is parallel to the third side of the

triangle.

4. Relation between areas of similar triangles

The ratio of the areas of two similar triangle is equal to the

ratio of the squares of any two corresponding sides.

2

2

2

2

2

2

RPCA

QRBC

PQAB

PQRΔofAreaABCΔofArea

Corollary 1: The Ratio of the areas of two similar triangles is equal to the

ratio of the squares of any two corresponding altitudes.

2

2

PNAD


Corollary 2 : The Ratio of the areas of two similar triangles is equal to the

ratio of the squares of any two corresponding median.

Flash Education

2

2

2

2

2

2

RZCF

QYBE

PXAD


Locus : The locus of a point is the path traced out by the point moving

under given geometrical condition(s).

Theorem on locus

Theorem 1 : The locus of a point, which is equidistant from two fixedpoints, is the perpendicular bisector of the line segment joining the twofixed points. (Fig. 1)

Conversely : Any point on the perpendicular bisector of a line segmentjoining two fixed points is equidistant from the fixed points. (Fig. 2)

Theorem 2 : The locus of a point, which is equidistant from twointersecting straight lines, consists of a pair of straight lines which bisectthe angles between the two given lines.

Conversely : Any point on the bisector of an angle is equidistant from thearms of the angle.

Locus in some standard cases

(ii) The locus of a point, which is equidistant from

two fixed points, is the perpendicular bisector

of the line segment joining the two fixed point.

(i) The locus of a point, which is equidistant

From two intersecting straight lines, consists

of a pair of straight lines which bisect the

angles between the two given lines.

Page 10

Flash Education

(iii) The locus of a point, which is

ekmquidistant from two parallel straight

lines, is a straight line parallel to the given

lines and midway between them.

(iv) The locus of a point, which is at a given

distance from a given straight line parallel to

the given line and at given distance from it.

(v) The locus of the centre of a wheel, which

moves on a straight horizontal road, is a

straight line parallel to the road and at a

distance equal to the wheel radius of the

wheel.

(vi) The locus of a point, which is inside a

circle and is equidistant from two points on

the circle, is the diameter of the circle which

is perpendicular to the chord of the circle

joining the given points.

(vii) The locus of the mid-points of all parallel

chords of a circle is the diameter of the

circle which is perpendicular to the given

parallel chords.

(viii) The locus of a point (in a plane), which is

at a given distance r from a fixed point (in a

plane), is a circle with the fixed point as its

centre and radius r.

Flash Education

(ix) The locus of a point which is equidistant

from two given concentric circles of radii r1 and

r2 is the circle of radius2rr 21 concentric with

the given circles. It lies midway between them.

(x) The locus of a point which is equidistant

from a given circle consists of a pair of circles

concentric with the given circle.

(xi) If A, B are fixed points, then the locus of a

point P such that APB = 90o is the circle with

AB as diameter.

(xii) The locus of the mid-points of all equal

chords of a circle is the circle concentric with

the given circle and of radius equal to the

distance of equal chords from the centre of

the given circle.

(xiii) The locus of centre of circles touching a

given line PQ at a given point T on it is the

straight line perpendicular to PQ at T.

Flash Education

Chord properties of circles

Theorem 1 : The straight line drawn from the centre

of a circle to bisect a chord, which is not a diameter,

is perpendicular to the chord.

i. e If a chord of a circle with centre C, and CM

bisects the chord AB, then

CM AB

Theorem 2 : The perpendicular to a chord from the

centre of the circle bisects the chord.

i. e If a chord AB of a circle with centre C, and CM

is perpendicular to the chord AB, then

AM = MB

Theorem 3 : One and only one circle can be drawn

passing through three non-collinear points.

Theorem 4 : Equal chord of a circle are equidistant

from the centre.

i. e If AB and CD are equal chords of a circle with

centre O, then

OM = ON

Theorem 5 : Chords of a circle that are equidistant

from the centre of the circle are equal.

i.e If AB and CD are chords of a circle with centre

O ; OM AB; ON CD, then AB = CD

CIRCLE

Page 13

Chord properties of circles

Theorem 1 : The straight line drawn from the centre

of a circle to bisect a chord, which is not a diameter,

is perpendicular to the chord.

i.e. If AB is a chord of a circle with centre C, and CM

bisects the chord AB, then

CM AB

Theorem 2 : The perpendicular to a chord from the

centre of the circle bisects the chord.

i.e. If AB is a chord of a circle with centre C, and CM

is perpendicular to the chord AB, then

AM = MB

Theorem 3 : One and only one circle can be drawn

passing through three non-collinear points.

Theorem 4 : Equal chord of a circle are equidistant

from the centre.

i.e. If AB and CD are two equal chords of a circle

with centre O, then

OM = ON

Theorem 5 : Chords of a circle that are equidistant

from the centre of the circle are equal.

i.e. If AB and CD are chords of a circle with centre

O ; OM AB; ON CD, then AB = CD

Flash Education

Angle properties of a circle

Theorem 6 : The angle subtended by an arc of a circle at the centre is

doubled the angle subtended by it at any point on the remaining part of the

circle.

i.e. In the figures AOB = 2ACB

Theorem 7 : Angles in the same segment of a circle are equal.

i.e. In the figure ACB = ADB

Theorem 8 : The angle in a semicircle is a right angle.

i.e. In the figure ACB = 90o

Flash Education

Theorem 9 : The opposite angle of a cyclic quadrilateral aresupplementary. (Converse of this theorem is also true)

i.e. In the figure A + C = 180o and B + D= 180o

Tangent properties of circles

Theorem 10 : The tangent at any point of a circle

and the radius through the point are perpendicular

to each other.

i.e. In the figure, CPT’PT

Theorem 11 : If two tangents are drawn from an external point to a

circle, then

(i) The tangent are equal in length.

(ii) The tangent subtend equal at the centre of the circle.

(iii) The tangents are equally inclined to the line joining the point and

the centre of the circle.

i.e. In the figure, (i) PA = PB (ii) BCPACP (iii) CPBAPC .

Flash Education

Theorem 12 : If two circles touch, the point of contact lies on the

straight line through their centres.

Theorem 13 : If two chords of a circle intersect internally or

externally, then the products of the lengths of segments are equal.

i.e. In the figures PA.PB = PC.PD

Theorem 14 : If a line touches a circle and from

the point of contact, a chord is drawn, the angles

between the tangent and the chord are

respectively equal to the angles in the

corresponding alternate segments.

i.e. In the figure (i) RPTRSP (ii) PR'TRQP

Theorem 15 : If a chord and a tangent

intersect externally, then the product

of the lengths of the segments of the

chord is equal to the squares of the length

of the tangent from the point of contact

to point of intersection. i.e. PA.PB = PT2

Flash Education

Flash Education

Circumference and area of a circle

1. Circle

Let the radius of circle be ‘r’, then

(i) Circumference = 2 r

(ii) Area = r2

2. Circular ring

Let the internal and external radii be ‘r’

and ‘R’, then

(i) Thickness = R - r

(ii) Area of ring (shaded portion) = (R2 - r2)

3. Semicircle

Let the radius of the semi-circle be ‘r’, then

(i) Perimeter = r + 2r

= (+ 2)r

(ii) Area = 21 r2

Page 17

Cylinder

1. Solid cylinder

Let ‘r’ be the radius and ‘h’ be the height

of a solid cylinder

(i) Curved (lateral) surface area = 2rh

(ii) Total surface area = 2r(h + r)

(iii) Volume = r2h.

2. Hollow cylinder

Let ‘R’ and ‘r’ be the external and internal radii of a

hollow cylinder, and ‘h’ be its height, then

(i) Thickness = R - r.

(ii) Area of cross-section = (R2 - r2).

(iii) External curved surface area = 2Rh.

(iv) Internal curved surface area = 2 rh.

(v) Total curved surface area = 2 (Rh + rh + R2 - r2).

(vi) Volume of material = (R2 - r2)h.

Cone

Let ‘r’ be the radius of the cone, ‘h’ be its height and

‘l’ be the slant height, then

(i) Slant height (l) = 22 hr .

(ii) Curved or lateral surface area = rl.

(iii) Total surface area = r(l + r).

(iv) Volume = hr31 2 .

Sphere

1. Solid Sphere

Let ‘r’ be the radius of solid sphere, then

(i) Surface area = 4r2

(ii) Volume = 3r34

2. Hemisphere

Let ‘r’ be the radius of solid hemisphere, then

(i) Curved surface area = 2r2

Page 18

Flash Education

(ii) Total surface area = 3r2

(iii) Volume = 3r32

3. Spherical shell

Let ‘R’ and ‘r’ be the radii of the outer and the inner

sphere, then

(i) Thickness = R - r.

(ii) Volume = )rR(34 33

4. Hemispherical shell

Let ‘R’ and ‘r’ be the radii of the outer and the inner

hemisphere, then

(i) Thickness = R - r

(ii) Base area = )rR( 22

(iii) External curved surface area = 2R2

(iv) Internal curved surface area = 2r2

(v) Total surface area = )rR3( 22

(vi) Volume of material = )rR(32 33

1. Trigonometrical Ratios :

(i) sin θ = p/h (vi) cosec θ = h/p

(ii) cos θ = b/h

(iii) tan θ = p/b

(iv) cot θ = b/p

(v) sec θ = h/b

Flash Education

Page 19

2. Quotient relation

(i)

cossintan (ii)

sincoscot

(iii)

sin1eccos (iv)

cos1sec

3. Square relation

(i) sin2θ + cos2θ = 1 (ii) sec2θ - tan2θ = 1 (iii) cosec2θ - cot2θ = 1

4. Trigonometrical ratios with standard angles

StandardAngle→

T.Ratio ↓

0o 30o 45o 60o 90o

sin 021

21

23 1

cos 123

21

21 0

tan 031 1 3 ∞

cot ∞ 3 131 0

sec 132 2 2 ∞

cosec ∞ 2 232 1

5. Trigonometrical Ratios of complementary angles :

(i ) Sin (90o - θ) = cos θ (ii) cos (90o - θ) = sin θ

(iii) tan (90o - θ) = cot θ (iv) cot (90o - θ) = tan θ

(v ) sec (90o - θ) = cosec θ (vi) cosec (90o - θ) = sec θ

Page 20

Flash Education

1. Mean

(i) For ungrouped data

Mean (x ) =n

x...xxx n321

Or, (x ) =nx

Where x1, x2, x3, … , xn are the variates and ‘n’ is the total number of

observations.

(ii) For grouped data

Mean (x ) =n321

nn332211

f...fffxf...xfxfxf

Or, (x ) =

ffx

Where x1, x2, x3, … , xn are the different variates with frequencies f1, f2,

f3, … , fn respectively.

(iii) For continuous distribution

Mean (x ) =n321

nn332211

f...fffyf...yfyfyf

Or, (x ) =

ffy

(Direct method)

Let A be the assumed mean, then

Mean (x ) = A +

ffd

, where d = y - A (Short cut method)

Mean (x ) = A + cffu

, where

cayu

(Step deviation method)

Where y1, y2, y3, … , yn are the class mark with frequencies f1, f2, f3, … , fn

respectively of n classes.

Flash Education

2. Median

(i) For ungrouped data

Let ‘n’ be the total number of observations, then

evenisnif,2

obsevationth21nnobservatioth

2n

oddisnif,nobservatioth21n

Median

(ii) For grouped data

evenisfif,2

obsevationth2

1fnobservatioth

2f

oddisfif,nobservatioth2

1f

Median

3. Quartiles

evenisfif,4f


1f

)Q(quartielower)i( 1

evenisfif,4f3


)1f(3

)Q(quartielower)ii( 3

13 QQrangequartileInter)iii(

)iv( Semi inter-quartile range =2QQ 13

Flash Education

Let A be the event associated with a random experiment, then

Probability of occurring A : P (A) =outcomespossibleofnumberTotal

Atofavourableoutcomesofnumber

i.e P(A) =)s(n)A(n

Some important sample space

(i) Sample space when one coin is tossed

S ={H,T} and n(S) = 2

(ii) Sample space when two coin is tossed or one coin is tossed twice

S = {HH, HT, TH, TT} and n(S) = 4

(iii) Sample space when three coin is tossed

S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}

and n(S) =8

(iv) Sample space when one dice is thrown

S ={1, 2, 3, 4, 5, 6} and n(S) = 6

(v) Sample space when two dice is thrown

S = {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6)

(2,1), (2,2), (2,3), (2,4), (2,5), (2,6)

(3,1), (3,2), (,3), (3,4), (3,5), (3,6)

(4,1), (4,2), (4,3), (4,4), (4,5), (4,6)

(5,1), (5,2), (5,3), (5,4), (5,5), (5,6)

(6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}

and n(S) = 36

Note : (i) 1)A(P0

(ii) P (A) + P(not A) = 1

(iii) P (Impossible event) = 0

(iv) P (Sure event) = 1

Flash Education

Flash Education

About this book

F - Book is the chapterwise formula book which contains all the

necessary formulas and theorems needed in board exam.

What is the full of F - Book ?

F - Book stands for Formula Book.

What is special about this book ?

The most important thing about this book is that it contains the

chapterwise formula that will help the students to find the formula very easily.

Is this book important ?

Yes, this book will help the students to remind or find the formula in

less time. It is very effective during the exam time.

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