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8/4/2019 MATHS GENERALLI

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Mathematicia Generalli by Er.OP GuptaList of formulae e-id: [email protected]

MATHEMATICIA GENERALLI

Logarithmic Relationsa) ba

b

log plog p

log ab)

1a

p

log plog a

c) n

a alog m nlog m d) . a a alog m n log m log n

e)

a a a

mlog log m log n

nf) 1alog a ,

p

a alog a plog a p

Exponential Relationsa)

22

1 . . ... . ...2! !

n

nx

e e e

x xa x log a log a log a

n

b)

2 3

1 ... ...2! 3! !

n

x x x xe xn

for allx.

Logarithmic Seriesa)

2 3 4

1 ... , 1 12 3 4

ex x x

log x x x

b) 2 3 4

1 ... , 1 12 3 4

ex x x

log x x x .

Binomial Expansionsa) 0 0 1 1 2 20 1 2 ... ...

n n n n n n n n n r r n n n n

r na b C a b C a b C a b C a b C a b ifn Z .

b) 21 21 1 ... ... n n n n r n n

r nx C x C x C x C x ifn is apositiveinteger.

c) 2 31 1 21 1 ...

2! 3!

n n n n n nx nx x x such that 1 1 x and n Z or n Q .

d)1 2 2 3 1. . ...

n nn n n nx a x a x a x a

x a.

Trigonometric Seriesa)

3 5 7

sin ...3! 5! 7!

x x x

x x

b)

2 4 6

cos 1 ...

2! 4! 6!

x x x

x

c)

352tan ...

3 15

xx x x

Sum of Special Sequencesa) Sum of first n natural numbers:

1

11 2 3 ...

2

n

r

n nn r

b) Sum of squares of first n natural numbers: 2 2 2 2 2

1

1 2 11 2 3 ...

6

n

r

n n nn r

c) Sum of cubes of first n natural numbers:

2

3 3 3 3 3

1

11 2 3 ...

2

n

r

n nn r

d) Sum of any constant kto n times: 1

... to times

n

r

k k k k n k nk .

(By OP Gupta


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MATHEMATICIA\ CLASS 12\ OP GUPTA\ MG -


III

List of formulae of Mensuration for using in the problems of Maxima and Minima

Followings are also ofimportance, though questions on them are rarely found in maxima and minima:

01. Circle:Perimeter i.e. circumference 2 r

Area2r

02. Equilateral triangle:Perimeter 3a

Area23

4a

03. Rectangle:

Perimeter 2 l b Area lb

04. Square:Perimeter 4a

Area2a

05. Cuboid:

Lateral surface area 2 . l b h

Total surface area 2 lb bh hl Volume lbh

06. Cube:

Lateral surface area24 a

Total surface area26a

Volume3a

07. Sphere:

Surface area2

4 r

Volume34

3r

08. Hemisphere:

Curved surface area2

2 r Total surface area

23 r

Volume32

3r

09. Cylinder:Curved surface area 2 rh

Total surface area22 2r rh

Volume2r h

10. Cone:

Curved surface area2 2 2, rl where l r h

Total surface area2r rl

Volume21

3r h

11. Frustum of a cone:

Curved surface area 22

,l R r where l h R r

Total surface area 22 2 2,l R r R r where l h R r

Volume 2 21

3h R r r R

12. Sector and segment of Circle:

Area of the sector of angle2

360r

Length of the arc of a sector of angle 2360

r

Area of the segment of a circle Area of the corresponding sector Area of corresponding triangle

2 2 sin cos360 2 2

r r 2 2

1sin

360 2

r r

Circumference i.e. arc length of semicircle of radius r r

Perimeter of semicircle of radius r 2 r r

13. Area of a trapezium1

2 (Sum ofparallelsides) (Distance betweenparallelsides)

14. Area of a rhombus

1

2 (Product ofdiagonals)


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IV

Algebraic Identitiesa)

2 2 22 a b a ab b

b) 3 3 2 2 33 3 a b a a b ab b

c) 3 3 2 2 33 3 a b a a b ab b

d) 2 2 a b a b a b e) 3 3 2 2 a b a b a ab b

f) 3 3 2 2 a b a b a ab b

g) 2 2 2 2 2 2 2 a b c a b c ab bc ca

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

Concept of InfinityWe consider the existence oftwo symbols and outside the set of real numbers R and would call them minusinfinity andplus infinity respectively with the fact x for every Rx .Thus and are not real numbers but just the symbols (like we usex, y etc.)When we write x , we mean that:

a) x is larger than any real number however large. b) is not a fixed number.

Also

, 1

0, 0 1

1, 1

if c

c if c

if c

.

Symbols and their meaningsS. No. Symbol Meaning

01. N Set of natural numbers02. I Zor Set of integers

03. Q Set of rational numbers

04. R Set of real numbers

05. C Set of complex numbers

06. is an element of (or belongs to)07. is not an element of (or does not belong to)08. S Uor or Universal set

09. : /or Such that

10. Empty set orNull set

11. is subset of12. is superset of13. is proper subset of14. is proper superset of15. Union16. Intersection17. For all18.

Implies

19. if and only if


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VI

General Solutionsa) sin sin 1 ,y x n y where n Z

n .

b) cos cos 2 ,y x n y where n Z .

c) tan tan ,x y x n y where n Z .

Relation in Degree & Radian Measures

Angles in Degree 0 30 45 60 90 180 270 360

Angles in Radian 0c 6

c

4

c

3

c

2

c

c

3

2

c

2c

In actual practice, we omit the exponent c and instead of writingc we simply write and similarly for others.Trigonometric Ratio of Standard Angles

Degree /Radian 0 30 45 60 90

T Ratios 0 6

4

3

2

sin 0 12

1

2

3

2 1

cos 1 3

2

1

2

1

2 0

tan 0 1

3 1 3

cosec 2 2 2

3 1

sec 12

3 2 2

cot 3 1 1

3 0

Trigonometric Ratios of Allied Angles

Angles

2

2

3

2

3

2

2

OR 2

T- Ratios

sin cos cos sin sin cos cos sin sin cos sin sin cos cos sin sin cos cos tan cot cot tan tan cot

cot tan tan

cot tan tan cot cot tan tan

cot cot sec cosec

cosec

sec sec

cosec

cosec sec sec

cosec sec sec cosec cosec sec sec cosec cosec

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