TNPSC – MATHS IMPORTANT FORMULAS

S.No Name of the figure (ngah;)

Figure (glk;)

Area (Sq. Units) gug;gsT (r.myF)

Perimetre (P) Rw;wsT

1. Triangle Kf;Nfhzk;

1

2 x b x h AB + BC + CA

2. Right triangle Nrq;Nfhz Kf;Nfhzk;

1

2 x b x h

(base + height + hypotenuse)

3.

Equilateral triangle rkgf;f

Kf;Nfhzk;

3

4 a2 where

( 3 = 1.732)

AB + BC + CA= 3A; Altitude ,

h = 3

2a units

4.

Isosceles triangle

,U rkgf;f Kf;Nfhzk;

h x 2 2a h 2a + 2 2 2a h

5.

Scalene triangle

(mrkgf;f Kf;Nfhzk;)

( )( )( )s s a s b s c

Where a = 2

a b c

AB + BC + CA = (a + b + c)

6. Quadrilateral

(ehw;fuk;)

1

2 x d x (h1 + h2) AB + BC + CD + DA

7. Parallelogram (,izfuk;)

b x h 2 x (a + b)

8. Rectangle (nrt;tfk;)

l x b 2 x (l + b)

9. Trapezium

(rhptfk;)

1

2 x h x (a + b) AB + BC + CD + DA

10. Rhombus (rha;rJuk;)

1

2 x d1 x d2 where

d1, d2 are diagonals

(%iytpl;lk;)

4a

11. Square

a2 4a

Square (%iytpl;lk;) (d) (Diagonal length) = 2a

No.

Name of the figure

figure Surface

area tisgug;G

Total surface area

Volume

1. Right circular

cylinder(Neh;tl;l jpz;k cUis)

2πrh 2πr(h + r) πr2h

2.

Right circular hollow

cylinder(Neh;tl;l cs;sPlw;w cUis)

2πh(R + r) 2π(R + r) (R – r + h)

πR2h – πr2h =

πh(R2 – r2) = πh(R + r)(R – r)

3. Right circular cone (Neh;tl;l jpzkf; $k;G)

πrl πr(l + r) 21

3r h

4. Frustum of a

cone (,ilf;fz;lk;)

---------

-------- 2 21 ( )

3h R r Rr

5. Solid sphere

(jpz;kf;Nfhsk;)

4πr2 ______ 34

3r

6. Hollow Sphere

(cs;sPlw;w Nfhsk;)

-------

______ 3 34 ( )

3R r

7.

Solid hemisphere

(jpz;k miuf;Nfhsk;)

2πr2 3πr2

32

3r

8.

Hollow Hemisphere (cs;sPlw;w

miuf;Nfhsk;)

2π(R2 + r2) 2π(R2 + r2) + π(R2 - r2) =

π(3R2 + r2)

gad;gLj;jg;gl;l cNyhfj;jpd;

fd msT = 3 32 (

3R r )

9. tl;lf;Nfhzg;gFjp $k;ghf khw;wg;gLfpwJ

tisgug;G = tl;lf;Nfhzg;gFjpapd; gug;G

2

360rl r

tpy;ypd; ePsk;( l ) = $k;gpd; mbr;Rw;wsT(2πr)

10. Volume of water flows out through a pipe = (Cross section area x Speed x Time) Foha; topNa ghAk; jz;zPh; fd msT = (FWf;F ntl;Lg; gug;G X Ntfk; X Neuk;)

11.

No. of new solids obtained by recasting = Volumeof thesolid whichismelted

Volumeof onesolid whichismade

cUf;fp jahhpf;fg;gLk; Gjpa fd cUtq;fspd; vz;zpf;if

= ; ; ; ; ;

; ; ; ; ;

cUffggll fd cUtjjpd fdmsT

cUthffggll xU fd cUtjjpd fdmsT

12. Conversions: 1 m3 = 1000 litres, 1 d.m3 = 1 litre, 1000 cm3 = 1 litre, 1000 litres = 1 kl 1 kP3 = 1000yp> 1nlrp kP3 = 1yp> 1000 nr.kP3 = 1yp> 1000yp = 1 fp.yp

CUBE (fdr;rJuq;fs;)

CUBIOD (fdr; nrt;tfk;)

tl;lf;Nfhzg;gFjpapd; tpy;ypd; ePsk; xU tl;lf;Nfhzg;gFjpapd; ikaf;Nfhzk; kw;Wk; Muk; r vdpy;>

tpy;ypd; ePsk; 2360

l r

myFfshFk;.

tl;lf;Nfhzg;gFjpapd; gug;gsT xU tl;lf;Nfhzg;gFjpapd; ikaf;Nfhzk; kw;Wk; Muk; r vdpy;>

tl;lf;Nfhzg;gFjpapd; gug;gsT 2360

r

rJu myFfshFk;.

tl;lf;Nfhzg;gFjpapd; gug;gsT = 2

lr rJu myFfs;

tl;lf;Nfhzg;gFjpapd; Rw;wsT tpy;ypd; ePsk; l, tl;lf;Nfhzg;gFjpapd; Muk; r vdpy;> mjd; Rw;wsT

P=l + 2r myFfs;.

Length of Arc If is the central angle and r is the radius of a sector, then its arc length

is given by 2360

l r

units.

Area of a Sector If is the central angle and r is the radius of a sector, then the area of

the sector is 2360

r

square units.

Area of sector = 2

lr square units.

Key concept:

Volume of a cube (fd msT )= 3a cubic units The Total surface area of a cube (T.S.A) (nkhj;jg;gug;G)

= 26a square units. The Lateral surface area of a cube (L.S.A) (gf;fg;gug;G)

= 24a square units. diagonal = 3a

Volume of a cuboid (fd msT )= l b h cubic units

The Total surface area of a cuboid (nkhj;jg; gug;G) = 2( )lb bh lh sq.units.

The Lateral surface area of a cuboid (gf;fg;gug;G) = 2( )l b h sq.units.

2 2 ( )( )a b a b a b 2 2 2( ) 2a b a b ab

2 2 2( ) 2a b a b ab 2 2( ) ( ) 4a b a b ab

3 3 2 2( ) ( )( )a b a b a ab b 3 3 2 2( ) ( )( )a b a b a ab b

3 3 3 2 2 3 3( ) 3 3 3 ( )a b a b a b ab a b ab a b 3 3 3 2 2 3 3( ) 3 3 3 ( )a b a b a b ab a b ab a b

2 2 2 2( ) 2( )a b c a b c ab bc ca 3 3 3 2 2 23 ( )( ) a b c abc a b c a b c ab bc ca

If a + b + c = 0 then 3 3 3 3 a b c abc

2( )( ) ( )x a x b x a b x ab 3 2( )( )( ) ( ) ( )x a x b x c x a b c x ab bc ca x abc

n(n+1)1 + 2 + 3 + ...... + n =

2

2

2 Last Number+11 3 5 ........ (2 1)2

n n

2 2 2 2 ( 1)(2 1)1 2 3 ......6

n n nn

2 2 2 2 (2 1)(2 1)1 3 5 ...... (2 1)3

n n nn

2

3 3 3 31

1 2 3 .......2

n nn

2 2 2 2 -n(n+1)1 2 3 4 ......... up to 'n' terms =2

2 1

2 4 6 ....... 2 2 1 2 3 ..... 12

n nn n n n

Logarithms Rule i) Product rule:ngUf;fy; tpjp: loga (mn) = loga m + loga n

ii) Quotient rule:tFj;jy; tpjp: loga m

n

= loga m - loga n.

iii) Power rule: gb tpjp: loga mn = nloga m. iv) loga 1 = 0. & v) loga a = 1

Perimeter of a Sector If l is the arc length and r is the radius of a sector, then its perimeter P

is given by the formula P=l + 2r units.

APPRECIATION AND DEPRECIATION [cah;T kw;Wk; tPo;r;rp] kf;fs; njhif cah;T> thfdq;fspd; kjpg;gPL Nghd;w gioa nghUl;fspd; tpiy

eph;zaj;jpid fPNo Fwpg;gpl;l tpjpiag; gad;gLj;jpf; fhzyhk;.

The growth or Appreciation (A): After ‘n’ years

kf;fs; njhif tsh;r;rp (m) tpiyNaw;wk; n Mz;LfSf;F gpwF 1100

nr

A p

After ‘n’ years Depreciation tPo;r;rp 1100

nr

D p

The growth or Appreciation (A): Before ‘n’ years or ‘n’ years ago

kf;fs; njhif tsh;r;rp (m) tpiyNaw;wk; n Mz;LfSf;F Kd;G

1100

n

pA

r

Before ‘n’ years or ‘n’ years ago Depreciation tPo;r;rp n Mz;LfSf;F Kd;

1100

n

pD

r

Arithmetic Progression (A.P.) [$l;Lj;njhlh; ] General form of an A.P. is a, a + d, a + 2d …with first term a, and

Common Difference = d

The general term or the nth term of an A.P. is tn = a + (n – 1) d →

1l a

nd

Sum to n terms of an A.P / Sn = 2 ( 1)2

na n d

Sn = 2

n [a + a + (n-1)d] or Sn = 2

n [a + l]

Where l = tn = a + (n – 1) d = last term

GEOMETRIC PROGRESSION[ngUf;Fj;njhlh; thpir]

ngUf;Fj;njhlh; thpirapd; nghJ tbtk; a, ar, ar2, ar3, … a ≠ 0 ngUf;Fj;njhlh; thpirapd; n - Mk; cWg;G tn = arn-1 Sum of n terms of a G.P[xU ngUf;Fj;njhlhpd; n cWg;Gfspd; $Ljy;]

Sn = ( 1)

1

na r

r

The sum of infinite geometric series[Kbtwh ngUf;Fj;njhlh; njhFg;gpd; $Ljy;]

S= 1

a

r

MATHS FORMULAS

For the system of equations

1 1 1

2 2 2

0

0

a x b y C

a x b y c

where 2 2 2 21 1 2 20, 0a b a b

If 1 2 1 2 0a b b a or 1 1

2 2

a b

a b then the system of equations has a unique solution

If 1 1 12 2 2

a b c

a b c then the system of equations has infinitely many solutions

If 1 1 12 2 2

a b c

a b c then the system of equations has no solution

1 1 1

2 2 2

0

0

a x b y C

a x b y c

,q;F 2 2 2 21 1 2 20, 0a b a b Mfpa rkd;ghLfspd; njhFg;gpw;F

1 2 1 2 0a b b a mjhtJ 1 1

2 2

a b

a b vdpy; xNunahU jPh;T (unique solution) cz;L

1 1 1

2 2 2

a b c

a b c vdpy; Kbtpyp vz;zpf;ifapy; jPh;Tfs; (infinitely many solutions)

cz;L

1 1 1

2 2 2

a b c

a b c vdpy; jPh;T VJkpy;iy (no solution)

The Basic relationship between the zeros and the coefficients of 2( )p x ax bx c

sum of zeros: 2

b coefficient of xa

a coefficient of x

product of zeros 2

tanc cons t terma

a coefficient of x

2( )p x ax bx c -d; nfOf;fSf;Fk;> g+r;rpaq;fSf;Fk; ,ilNaahd mbg;gilj; njhFg;G

G+r;rpaq;fspd; $Ljy;> 2

b xa

a x

d; nfO

d; nfO

G+r;rpaq;fspd; ngUf;fw;gyd;

2

ca

a x

khwpyp cWg; G

d; nfO

Nature of roots of a quadratic equation

The roots of the equation 2 0ax bx c are given by 2 4

2

b b acx

a

If 2 4 0b ac we get two distinct real roots 2 4

2

b b acx

a

; &

2 4

2

b b acx

a

If 2 4 0b ac , then the equation has two equal roots 2

bx

a

If 2 4 0b ac , then 2 4b ac is not a real number. Therefore there is no real root

for the given quadratic equation.

,Ugbr; rkd;ghl;bd; %yq;fspd; jd;ik

2 0ax bx c vDk; ,Ugbr; rkd;ghl;bd; %yq;fs; 2 4

2

b b acx

a

vd mwpNthk;

2 4 0b ac vdpy; ,U ntt;Ntwhd nka;naz; %yq;fs; cs;sd. mitfs;>

2 4

2

b b acx

a

kw;Wk;

2 4

2

b b acx

a

2 4 0b ac vdpy; rkd;ghl;bw;F ,U rkkhd nka;naz; %yq;fs; cs;sd. rk

%yk; 2

bx

a

MFk;

2 4 0b ac vdpy; 2 4b ac xU nka;naz; my;y. Mifahy;> ,Ugbr; rkd;ghl;bw;F nka;naz; %yq;fs; ,y;iy

Therefore, if ,a are the roots of

2 0ax bx c then

the sum of the roots b

aa

the product of roots, c

aa

2 0ax bx c -d; %yq;fs; ,a vdpy;

%yq;fspd; $Ljy;> b

aa

%yq;fspd; ngUf;fw;gyd;> c

aa

Area of a quadrilateral ehw;fuj;jpd; gug;G

1 2 2 3 3 4 4 1 2 1 3 2 4 3 1 4

1{( ) ( )}

2x y x y x y x y x y x y x y x y sq.units / r. myFfs;

The distance between 1 1 2 2( , ), ( , )P x y Q x y is 2 2

2 1 2 1( ) ( )x x y y

1 1 2 2( , ), ( , )P x y Q x y Mfpa Gs;spfSf;F ,ilNaAs;s njhiyT 2 2

2 1 2 1( ) ( )x x y y

Area of Triangle Kf;Nfhzj;jpy; gug;G If 1 1 2 2( , ), ( , )A x y B x y and 3 3( , )C x y are the vertices of a ABC then the area of the

ABC is 1 2 3 2 3 1 3 1 2

1{ ( ) ( ) ( )}

2x y y x y y x y y sq.units

1 1 2 2( , ), ( , )A x y B x y kw;Wk; 3 3( , )C x y Mfpatw;iw Kidfshff; nfhz;l ABC -d;

gug;G 1 2 3 2 3 1 3 1 2

1{ ( ) ( ) ( )}

2x y y x y y x y y r. myFfs;

Equation of straight lines (Neh;f;Nfhl;bd; rkd;ghLfs;) Straight Line Equation

x – axis y = 0

y – axis x = 0

Parallel to x - axis y = k

Parallel to y – axis x = k

Parallel to ax + by + c = 0 ax + by + k = 0

Perpendicular to ax + by + c = 0 bx – ay + k = 0

Given Equation

Passing through the origin y mx

Slope m, y-intercept c y mx c

Slope m a point 1 1( , )x y 1 1( )y y m x x

Passing through two points 1 1 2 2( , ), ( , )x y x y 1 12 1 2 1

y y x x

y y x x

x- intercept a and y-intercept b 1

x y

a b

Neh;f;Nfhl;bd; rkd;ghLfs;

Neh;f;NfhL rkd;ghL x – mr;R y = 0 y – mr;R x = 0

x – mr;rpw;F ,iz y = k

y – mr;rpw;F ,iz x = k ax + by + c = 0 f;F ,iz ax + by + k = 0 ax + by + c = 0 f;F nrq;Fj;J bx – ay + k = 0

nfhLf;fg;gl;lit rkd;ghL Mjp topr; nry;Yk; Neh;f;NfhL y mx

rha;T m kw;Wk; y-ntl;Lj;Jz;L c y mx c

rha;T m kw;Wk; xUGs;sp 1 1( , )x y 1 1( )y y m x x

1 1 2 2( , ), ( , )x y x y Mfpa ,U Gs;spfs; topr; nry;Yk; Neh;f;NfhL

1 1

2 1 2 1

y y x x

y y x x

x- ntl;Lj;Jz;L a kw;Wk; y-ntl;Lj;Jz;L b 1

x y

a b

,g;nghOJ (x1, y1) kw;Wk; (x2, y2) Mfpa ,UGs;spfis ,izf;Fk; Nfhl;Lj;Jz;bid m : n vd;w nfhLf;fg;gl;l tpfpjj;jpy; cl;Gwkhfg; gphpf;Fk; Gs;spapd; Maj;njhiyTj; J}uq;fisf; fhz;Nghk;.

To find the coordinates of the point which divides internally the line segment joining two given points (x1, y1) and (x2, y2) in the given ratio m : n

2 1 2 1,mx nx my ny

m n m n

AB I ntspg;Gwkhf m : n (m > n) vd;fpw tpfpjj;jpy; gphpf;Fk; Gs;sp D ,d; Maj;njhiyTj;J}uk;

Hence the point which divides AB externally in the ratio m : n (m > n) is given by

2 1 2 1,mx nx my ny

m n m n

Middle Point Formula (or) Mid – Point Formula ikag;Gs;sp #j;jpuk; my;yJ eLg;Gs;sp #j;jpuk;

1 2 1 2,2 2

x x y y

1 1 2 2( , ),( , )x y x y kw;Wk; 3 3( , )x y Mfpa cr;rpg; Gs;spfisf; nfhz;l Kf;Nfhzj;jpd; eLf;Nfhl;L ikak; fhz;Nghk;.

xU Kf;Nfhzj;jpw;F %d;W eLf;NfhLfs; cz;L. mit G vd;fpw Gs;spapy; re;jpf;Fk;. me;jg;Gs;sp, xU Kf;Nfhzj;jpd; eLf;Nfhl;L ikak; (Centroid) vdg;gLk;.

We are now able to find the coordinates of the centroid of the triangle whose vertices are the given points 1 1 2 2( , ),( , )x y x y and 3 3( , )x y .

There are three medians of a triangle and they are concurrent at a point G, called the centroid of the triangle.

1 2 3 1 2 3,3 3

x x x y y y

LCM & HCF

LCM of fractions = min

LCM of numerotors

HCF of deno ators

njhFjpapYs;s vz; fspd; kP.rp.k

gpd; d vz; fspd; k.P rp.kgFjpapYs;s vz; fspd; kP.ng.t

=

HCF of fractions = min

HCF of numerotors

LCMof deno ators

njhFjpapYs;s vz; fspd; kP.ng.tgpd; d vz; fspd; k.P ng.t

gFjpapYs;s vz; fspd; kP.rp.k=

First number second number = LCM x HCF ,uz;L vz;fspd; ngUf;fw;gyd; = LCM x HCF

STATISTICS Range and Coefficient of Range

Range (tPr;R) = L – S

Coefficient of Range (tPr;R nfO) = L SL S

Quartiles deviation (Q.D.)and Coefficient of Quartile deviation

Quartiles deviation (fhy;khd tpyf;fk;) = 3 12

Q Q

Coefficient of Quartile deviation (fhy;khd tpyf;ff;nfO) 3 13 1

Q Q

Q Q

Mean deviation and Coefficient of Mean deviation

Mean deviation (ruhrhp tpyf;fk;) = Dn

Coefficient (ruhrhp tpyf;ff;nfO) = Median or Mode

Mean deviation

Mean or

RATIO AND PROPORTION In a Proportion: Product of Extremes = Product of Means (tpfpjrkj;jpy;

Standard deviation (jpl;ltpyf;fk;)

n

2d= d = x - x

Variance (khWghL) = 2

Remarks: It is quite interesting to note the following: The S.D. of any n successive terms of an A.P. with common difference d is,

2 1

12

nd

Thus,

S.D. i, i+1, i+2, ….., i+n is 2 1

12

n

, i

S.D. of any n consecutive even integers, is given by 2 1

2 ,12

nn

S.D. of any n consecutive odd integers, is given by 2 1

2 ,12

nn

Mutually exclusive events: (xd;iwnahd;W tpyf;Fk; epfo;r;rpfs;) ( ) ( ) ( )P A B P A P B

Not Mutually exclusive events: (xd;iwnahd;W tpyf;fh epfo;r;rpfs;) ( ) ( ) ( ) ( )P A B P A P B P A B

Dimensional Diagrams [tpsf;fg; glq;fspd; tiffs;;] One-dimensional diagrams [xU ghpkhd tpsf;fg; glq;fs;] Line Diagram [Nfhl;L tpsf;fg;glk;] Simple Diagram [rhjhuz gl;il tpsf;fg;glk;] Multiple Bar Diagram [gy; mq;fg; gl;il tpsf;fg;glk;] Sub-divided Bar Diagram [$W gl;il tpsf;fg;glk; (gFjp gl;il)] Percentage Bar Diagram [rjtPj gl;il tpsf;fg; glk;]

Two-dimensional diagrams [,U ghpkhd tpsf;fg;g glq;fs;] Rectangles [nrt;tfq;fs;] Squares [rJuq;fs;] Pie-diagrams [tl;l tpsf;fg; glq;fs;]

Three-dimensional diagrams [Kg;ghpkhd tpsf;fg; glq;fs;] Cubes, Cylinders, Spheres, Prisms, Pyramids, etc…,

Graphs: [tiuglq;fs;] Histogram [guty; nrt;tfg; glk;] – To find Mode Frequency Polygon [epfo;ntz; gy Nfhz tbtk;] Frequency Curve [epfo;ntz; tisNfhL] Ogive [tsh; epfo; tistiufs; (xift;)] - To find Median Lorenz Curve [yhud;]; tistiu] – Study about dispersion(Variability)