Docsity
Docsity

Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity


Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan


Guidelines and tips
Guidelines and tips

Basic algebra mathematics formula sheet, Cheat Sheet of Algebra

Formula sheet in which given mathematic formula, basic algebra formula, indices and surds, logarithms, progression, permutations and combinations.

Typology: Cheat Sheet

2021/2022

Uploaded on 02/07/2022

rajeshi
rajeshi 🇺🇸

4.1

(9)

237 documents

1 / 10

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
SOME IMPORTANT MATHEMATICAL FORMULAE
Circle : Area =
π
r2; Circumference = 2
π
r.
Square : Area = x2 ; Perimeter = 4x.
Rectangle: Area = xy ; Perimeter = 2(x+y).
Triangle : Area =
1
2
(base)(height) ; Perimeter = a+b+c.
Area of equilateral triangle =
3
4
a2 .
Sphere : Surface Area = 4
π
r2 ; Volume =
4
3
π
r3.
Cube : Surface Area = 6a2 ; Volume = a3.
Cone : Curved Surface Area =
π
rl ; Volume =
1
3
π
r2 h
Total surface area = .
π
r l +
π
r2
Cuboid : Total surface area = 2 (ab + bh + lh); Volume = lbh.
Cylinder : Curved surface area = 2
π
rh; Volume =
π
r2 h
Total surface area (open) = 2
π
rh;
Total surface area (closed) = 2
π
rh+2
π
r2 .
SOME BASIC ALGEBRAIC FORMULAE:
1.(a + b)2 = a2 + 2ab+ b2 . 2. (a - b)2 = a2 - 2ab+ b2 .
3.(a + b)3 = a3 + b3 + 3ab(a + b). 4. (a - b)3 = a3 - b3 - 3ab(a - b).
5.(a + b + c)2 = a2 + b2 + c2 +2ab+2bc +2ca.
6.(a + b + c)3 = a3 + b3 + c3+3a2b+3a2c + 3b2c +3b2a +3c2a +3c2a+6abc.
7.a2 - b2 = (a + b)(a – b ) .
8.a3 – b3 = (a – b) (a2 + ab + b2 ).
9.a3 + b3 = (a + b) (a2 - ab + b2 ).
10.(a + b)2 + (a - b)2 = 4ab.
11.(a + b)2 - (a - b)2 = 2(a2 + b2 ).
12.If a + b +c =0, then a3 + b3 + c3 = 3 abc .
INDICES AND SURDS
1. am an = am + n 2.
m
am n
a
n
a
=
. 3.
m n mn
(a ) a=
. 4.
m m m
(ab) a b=
.
5.
. 6.
0
a 1, a 0=
. 7.
1
m
am
a
=
. 8.
y
x
a a x y= =
9.
x x
a b a b= =
10.
a 2 b x y± = ±
, where x + y = a and xy = b.
S B SATHYANARAYANA
M. Sc., M.I.E ., M Phil .
9481477536
1
pf3
pf4
pf5
pf8
pf9
pfa

Partial preview of the text

Download Basic algebra mathematics formula sheet and more Cheat Sheet Algebra in PDF only on Docsity!

SOME IMPORTANT MATHEMATICAL FORMULAE

Circle : Area =

π

r

2

;

Circumference = 2

π

r.

Square : Area = x

2

; Perimeter = 4x.

Rectangle: Area = xy ; Perimeter = 2(x+y).

Triangle : Area =

(base)(height) ; Perimeter = a+b+c.

Area of equilateral triangle =

a

2

.

Sphere : Surface Area = 4

π

r

2

; Volume =

π

r

3

.

Cube : Surface Area = 6a

2

; Volume = a

3

.

Cone : Curved Surface Area =

π

rl ; Volume =

π

r

2

h

Total surface area =.

π r l +

π r

2

Cuboid : Total surface area = 2 (ab + bh + lh); Volume = lbh.

Cylinder : Curved surface area = 2

π

rh; Volume =

π

r

2

h

Total surface area (open) = 2

π

rh;

Total surface area (closed) = 2

π

rh+

π

r

2

SOME BASIC ALGEBRAIC FORMULAE:

1.(a + b)

2

= a

2

  • 2ab+ b

2

. 2. (a - b)

2

= a

2

  • 2ab+ b

2

3.(a + b)

3

= a

3

  • b

3

  • 3ab(a + b). 4. (a - b)

3

= a

3

  • b

3

  • 3ab(a - b).

5.(a + b + c)

2

= a

2

  • b

2

  • c

2

+2ab+2bc +2ca.

6.(a + b + c)

3

= a

3

  • b

3

  • c

3

+3a

2

b+3a

2

c + 3b

2

c +3b

2

a +3c

2

a +3c

2

a+6abc.

7.a

2

  • b

2

= (a + b)(a – b ).

8.a

3

  • b

3

= (a – b) (a

2

  • ab + b

2

9.a

3

  • b

3

= (a + b) (a

2

  • ab + b

2

10.(a + b)

2

  • (a - b)

2

= 4ab.

11.(a + b)

2

  • (a - b)

2

= 2(a

2

  • b

2

12.If a + b +c =0, then a

3

  • b

3

  • c

3

= 3 abc.

INDICES AND SURDS

  1. a

m

a

n

=

a

m + n

m

a

m n

a

n

a

m n mn

(a ) = a

. 4.

m m m

(ab) = a b

.

m m

a a

m

b

b

a = 1, a ≠ 0

m

a

m

a

y x

a = a ⇒ x =y

x x

a = b ⇒ a = b

a ± 2 b = x ± y

, where x + y = a and xy = b.

S B SATHYANARAYANA

M. Sc., M.I.E ., M Phil.

LOGARITHMS

x

a m log m x

a

(a > 0 and a ≠ 1)

  1. log a

mn = logm + logn.

  1. log a

m

n

= logm – logn.

  1. log a

m

n

= n logm.

  1. log b

a =

log a

log b

  1. log a

a = 1.

  1. log a
  1. log b

a =

a

log b

  1. log a
  1. log (m +n) ≠ logm +logn.
  2. e

logx

= x.

  1. log a

a

x

= x.

PROGRESSIONS

ARITHMETIC PROGRESSION

a, a + d, a+2d,-----------------------------are in A.P.

n

th

term, T n

= a + (n-1)d.

Sum to n terms, S n

= [ ]

n

2a (n 1)d

If a, b, c are in A.P, then 2b = a + c.

GEOMETRIC PROGRESSION

a, ar, ar

2

,--------------------------- are in G.P.

Sum to n terms, S n

n

a(1 r )

1 r

if r < 1 and S n

n

a(r 1)

r 1

if r > 1.

Sum to infinite terms of G.P,

a

S

1 r

If a, b, c are in A.P, then b

2

= ac.

HARMONIC PROGRESSION

Reciprocals of the terms of A.P are in H.P

a a + d a + 2d

----------------- are in H.P

If a, b, c are in H.P, then b =

2ac

a + c

MATHEMATICAL INDUCTION

1 + 2 + 3 + -----------------+n =

n(n 1)

n

2

2

2

  • -----------------+n

2

2

n(n 1)(2n 1)

n

S B SATHYANARAYANA

M. Sc., M.I.E ., M Phil.

  1. Mid point formula

1 2 1 2

x x y y

  1. Centriod formula

1 2 3 1 2 3

x x x y y y

  1. Area of triangle when their vertices are given,

[ ]

1 2 3

1 2 3 2 3 1 3 1 2

x (y y )

x (y y ) x (y y ) x (y y )

STRAIGHT LINE

Slope (or Gradient) of a line = tangent of an inclination = tanθ.

Slope of a X- axis = 0

Slope of a line parallel to X-axis = 0

Slope of a Y- axis = ∞

Slope of a line parallel to Y-axis = ∞

Slope of a line joining (x 1

, x 2

) and (y 1

, y 2

2 1

2 1

y y

x x

If two lines are parallel, then their slopes are equal (m 1

= m 2

If two lines are perpendicular, then their product of slopes is -1 (m 1

m 2

EQUATIONS OF STRAIGHT LINE

  1. y = mx + c (slope-intercept form)

y - y 1

= m(x-x 1

) (point-slope form)

2 1

1 1

2 1

y y

y y (x x )

x x

(two point form)

x y

a b

(intercept form)

x cosα +y sinα = P (normal form)

Equation of a straight line in the general form is ax

2

+ bx + c = 0

Slope of ax

2

  • bx + c = 0 is –

a

b

  1. Angle between two straight lines is given by, tanθ =

1 2

1 2

m m

1 m m

Length of the perpendicular from a point (x 1

,x 2

) and the straight line ax

2

  • bx + c

= 0 is

1 1

2 2

ax by c

a b

S B SATHYANARAYANA

M. Sc., M.I.E ., M Phil.

Equation of a straight line passing through intersection of two lines a 1

x

2

  • b 1

x + c 1

= 0 and a 2

x

2

  • b 2

x + c 2

= 0 is a 1

x

2

  • b 1

x + c 1

  • K(a 2

x

2

  • b 2

x + c 2

) = 0, where K is

any constant.

Two lines meeting a point are called intersecting lines.

More than two lines meeting a point are called concurrent lines.

Equation of bisector of angle between the lines a 1

x + b 1

y+ c 1

= 0 and

a 2

x + b 2

y + c 2

= 0 is

1 1 1 2 2 2 2

2 2 2 2

1 1 2 2

a x b y c a x b y c

a b a b

PAIR OF STRAIGHT LINES

  1. An equation ax

2

+2hxy +by

2

= 0, represents a pair of lines passing through origin

generally called as homogeneous equation of degree2 in x and y and

angle between these is given by tanθ =

2

2 h ab

a b

ax

2

+2hxy +by

2

= 0, represents a pair of coincident lines, if h

2

= ab and the same

represents a pair of perpendicular lines, if a + b = 0.

If m 1

and m 2

are the slopes of the lines ax

2

+2hxy +by

2

= 0,then m 1

  • m 2

2h

b

and m 1

m 2

a

b

  1. An equation ax

2

+2hxy +by

2

+2gx +2fy +c = 0 is called second general second

order equation represents a pair of lines if it satisfies the the condition

abc + 2fgh –af

2

  • bg

2

  • ch

2

The angle between the lines ax

2

+2hxy +by

2

+2gx +2fy +c = 0 is given by

tanθ =

2

2 h ab

a b

ax

2

+2hxy +by

2

+2gx +2fy +c = 0, represents a pair of parallel lines, if h

2

= ab and

af

2

= bg

2

and the distance between the parallel lines is

2

2 g ac

a(a b)

ax

2

+2hxy +by

2

+2gx +2fy +c = 0, represents a pair of perpendicular lines

,if a + b = 0.

S B SATHYANARAYANA

M. Sc., M.I.E ., M Phil.

where as the transformation begins at 180

0

or 360

0

, the same trigonometric functions

will be retained, however the signs (+ or -) of the functions decides ASTC rule.

COMPOUND ANGLES

Sin(A+B)=sinAcosB+cosAsinB.

Sin(A-B)= sinAcosB-cosAsinB.

Cos(A+B)=cosAcosB-sinAsinB.

Cos(A-B)=cosAcosB+sinAsinB.

tan(A+B)=

tan A tan B

1 tan A tan B

tan(A-B)=

tan A tan B

1 tan A tan B

tan A

π  

1 tan A

1 tan A

tan A

π  

1 tan A

1 tan A

tan(A+B+C)=

tan A tan B tan C tan A tan B tan C

1 (tan A tan B tan B tan C tan C tan A)

sin(A+B) sin(A-B)=

2 2 2 2

sin A − sin B = cos B −cos A

cos(A+B) cos(A-B)=

2 2

cos A −sin B

MULTIPLE ANGLES

1.sin 2A=2 sinA cosA. 2. sin 2A=

2

2 tan A

1 + tan A

3.cos 2A =

2 2

cos A −sin A

2

sin A

2

cos A − 1

2

2

1 tan A

1 tan A

  1. tan 2A=

2

2 tan A

1 − tan A

, 5. 1+cos 2A=

2

2 cos A , 6.

2

cos A =

(1 cos 2A)

  1. 1-cos 2A=

2

2sin A , 8.

2

sin A =

(1 cos 2A)

, 9.1+sin 2A=

2

(sin A + cos A) ,

  1. 1-sin 2A=

2

(cos A − sin A) =

2

(sin A − cos A), 11.cos 3A=

3

4 cos A − 3cos A

  1. sin 3A=

3

3sin A − 4sin A, 13.tan 3A=

3

2

3 tan A tan A

1 3 tan A

S B SATHYANARAYANA

M. Sc., M.I.E ., M Phil.

HALF ANGLE FORMULAE

  1. sin θ =

2sin cos

θ θ

. 2) sin θ =

2

2 tan

1 tan

θ  

θ  

. 3) cos θ =

2 2

cos sin

θ θ

2

cos 1 2sin

θ

θ = −

. 5)

2

cos 2 cos 1

θ

θ = −

. 6)

2

2

1 tan

cos

1 tan

θ  

θ =

θ  

2

2 tan

tan

1 tan

θ  

θ =

θ  

2

1 cos 2 cos

θ

  • θ =

. 9)

2

1 cos 2sin

θ

− θ = .

PRODUCT TO SUM

2 sinA cosB = sin(A+B) + sin(A-B).

2 cosA sinB = sin(A+B) – sin(A-B).

2 cosA cosB = cos(A+B) + cos(A-B).

2 sinA sinB = cos(A+B) – cos(A-B).

SUM TO PRODUCT

Sin C + sin D =

C D

2sin

C D

cos

Sin C –sin D =

C D C D

2 cos sin

Cos C + cos D =

C D C D

2 cos cos

Cos C- cos D =

C D C D

2sin sin

OR

Cos C- cos D =

D C D C

2sin sin

PROPERTIES AND SOLUTIONS OF TRIANGLE

Sine Rule:

a b c

2R

sin A sin B sin C

= = = , where R is the circum radius of the

triangle.

Cosine Rule: a

2

= b

2

  • c

2

-2bc cosA or cosA =

2 2 2

b c a

2bc

S B SATHYANARAYANA

M. Sc., M.I.E ., M Phil.

1

0

p

n

Lt

n

→∞

=

, if

p > 0

and

p

n

Lt n

→∞

= ∞

if

p > 0

0 0 0 0

sin tan

in radians 1

sin tan

x x x x

x x x x

Lt Lt x Lt Lt

x x x x

→ → → →

= = = =

π

→ →

= =

0 0

0 0

sin tan

180

x x

x x

Lt Lt

x x

π

π →

=

2

sin 2

x

x

Lt

x

1 1

0 0

sin tan

lim 1 lim

x x

x x

x x

− −

→ →

= =

=

1

lim

n n

n

x a

x a

na

x a

, where n is an integer or a fraction.

0 0

1 1

lim log , lim log 1

x x

x x

a e

a e

x x

→ →

− −

= = =

1

0

1

lim 1 , lim 1

n

n

x x

e n e

n

→∞ →

 

  • = + =

 

 

→ →

=  

 

lim lim

x a x a

kf x k f x

lim lim lim

x a x a x a

f x g x f x g x

→ → →

± = ±  

 

lim ( ). ( ) lim ( ) .lim ( )

x a x a x a

f x g x f x g x

→ → →

=

→ →

 

= ≠  

 

lim

lim lim ( ) 0

lim

x a

x a x a

x a

f x

f x

provided g x

g x g x

  1. A function

f ( x ) is said to be continuous at the point

x = a

if

(i)

lim ( )

x a

f x

exists (ii) ( )

f a is defined (iii)

lim ( ) ( )

x a

f x f a

=

  1. A function

f ( x )

is said to be discontinuous or not continuous at

x = a

if

(i) ( )

f x is not defined at

x = a (ii)

lim ( )

x a

f x

does not exist at

x = a

(iii)

0 0

lim lim

x a x a

f x f x f a

→ + → −

≠ ≠

  1. If two functions

f x

and

g x

are continuous then

f x + g x

is continuous

S B SATHYANARAYANA

M. Sc., M.I.E ., M Phil.