XII MATHS FORMULAS.pdf

CBSE 12 Maths Formulae

1. FORMULA LIST
Relations and Functions
(1) Relation: Let andA Bbe two sets. Then a relation Rfrom set Ato set Bis a subset of A B.
(2) Types of Relations:
2a. Empty Relation: A relation R on a set A is said to an empty relation iffRi.e.
, ,a b R a b A   i.e. no element of A will be related to any other element of A with the help of
the relation R.
2b. Universal Relation: A relation R on a set A is said to be a universal relation iff
R A A i.e., , a,b Aa b R   i.e. each element of A is related to every other element of A with
the help of the relation R.
(3) A relation onR Ais:
3a. Reflexive Relation if
, for alla a R a A 

3b. Symmetric Relation if
 , , all ,a b R b a R a b A   

3c. Transitive Relation if
  , , , , all , ,a b R b c R a c R a b c A    

(4) A relation R in a set A is said to be an Equivalence Relation if R is reflexive, symmetric and
transitive.
(5) Equivalence Class
Let Rbe an equivalence relation on a set Aand let a A. Then, we define the equivalence class of
aas    , is related to : ,a b A b a b A b a R    
(6) Function: Let andA Bbe two non-empty sets. Then a function ‘f’ from set Ato set Bis a
rule which associates elements of set Ato elements of set Bsuch that:
(a) All elements of set Aare associated to elements in set .
(b) An element of set Ais associated to a unique element in set B.
(7) Vertical Line Test: A curve in a plane represents the graph of a real function if and only if no
vertical line intersects it more than once.


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2. (8) Types of functions:
8a. One-One Function (or Injective Function): A function y f x is said to be one –
one iff different pre-images have different images or if images are same
then the pre-images are also same.
i.e. 
1 2 1 2
f x f x x x   or 
1 2 1 2
x x f x f x  
8b. Many-One Function: A function in which at least two pre-
images have same image is called as many-one function.


8c. Into Function: A Function is said to be an ‘into’ function if there is at least one element
in the co-domain of the function such that it has no pre-image in domain.
8d. Onto Function (or Surjective Function): A function is said to be onto function if each
element of co-domain has a pre-image in its domain. Alternately, A function ‘f’ will be called an
onto function iff
co-domain
f
R f
8e. Bijective Function: A function which is one – one and onto both is called as bijective
function.
(9) Horizontal Line Test:fis one-one function if no line parallel to x-axis meets the graph in more
than one point.
(10) Let Abe any finite set having nelements. Then,
10a. Number of one-one functions from Ato Aare !n
10b. Number of onto functions from Ato Aare !n
10c. Number of bijective functions from Ato Aare !n
(11) If A and B are finite sets containing mand nelements, then
11a. Total number of relations from the set A to set B is 2
mn
.
11b. Total number of relations on the set A is
2
2
m
.
11c. Total number of functions from the set A to set B is
m
n.
11d. Total number of one-one functions from the set A to set B is
n
m
Pif n m, otherwise 0. Amit Bajaj | amitbajajmaths.com | [email protected] amitbajajmaths.com

3. 11e. Total number of onto functions from set A to set B is
1
( 1) if
n
n r n m
r
r
C r m n


  ,
otherwise 0.
11f. Total number of bijective functions from the set A to set B is ! , fm i m n, otherwise 0.
(12) COMPOSITION OF FUNCTIONS
If fand gare two functions then their composition
12a. fogis defined iff
g f
R D and   fog x f g x
12b. gofis defined iff
f g
R D and   gof x g f x
(13) a. If :f A Band :g B Cthen:gof A C.
b. If :f A Band :g B Athen fogand gofare both defined, where:gof A Aand
:fog B B
(14) Identity Function: A function ‘I’ on a set A is said to be an identity function iff:I A A,
I x x.
(15) Equal Functions: A function fwill be equal to another function giff
(i)
f g
D D and (ii)   
f g
f x g x x D or D  
(16) Invertible Functions: A function :f A B is said to be an invertible function iff there exist
another function :g B Asuch that
B
fog Iand
A
gof I. And we write
1
g f


(17) a. A function is invertible iff it is one-one and onto.
b. In the above definition fand gare both inverse of each other i.e.
1
f g

and
1
g f

.
c.
1
1
f f


.
d.
 
1
1 1
fog g of
  

e.
1 1
fof f of I
 
 
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4. Trigonometry and Inverse Trigonometry
0
(1)180 radians
(2) , is measured in radians
l
r

 


(3) Trigonometric Ratios of Special Angles
145tan
2
1
45sin
2
1
45cos




   
   
   
0
(4)sin 90 cos ; cos 90 sin
tan 90 cot ; cot 90 tan
sec 90 cos ; cosec 90 secec
   
   
   
    
   
   

  (5)cos cos ; sin sin ; tan tan            
1 1 1 sin cos
(6)sec ; cosec ; cot ; tan ; cot
cos sin tan cos sin
 
    
    
    
2 2 2 2 2 2
(7)sin cos 1 ; 1 tan sec ; 1 cot cos ec          
(8) 1 sin 1 ; 1 cos 1 ; tanx x x        
 
 
 
 
 
 
(9)sin sin cos cos sin
(10)sin sin cos cos sin
(11)cos cos cos sin sin
(12)cos cos cos sin sin
tan tan
(13) tan
1 tan tan
tan tan
(14)tan
1 tan tan
A B A B A B
A B A B A B
A B A B A B
A B A B A B
A B
A B
A B
A B
A B
A B
  
  
  
  

 


 


  
   
   
   
(15) 2sin cos sin sin
(16) 2cos sin sin sin
(17)2cos cos cos cos
(18) 2sin sin cos cos
A B A B A B
A B A B A B
A B A B A B
A B A B A B
   
   
   
   

360tan
2
3
60sin
2
1
60cos



3
1
30tan
2
1
30sin
2
3
30cos


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5. (19)sin sin 2sin cos
2 2
(20)sin sin 2cos sin
2 2
(21)cos cos 2cos cos
2 2
(22)cos cos 2sin sin
2 2
A B A B
A B
A B A B
A B
A B A B
A B
A B A B
A B
    
 
   
   
    
 
   
   
    
 
   
   
    
  
   
   

2
2
2 2 2 2
2
2
2 tan
(23)sin 2 2sin cos
1 tan
1 tan
(24)cos 2 cos sin 2cos 1 1 2sin
1 tan
2tan
(25) tan 2
1 tan
A
A A A
A
A
A A A A A
A
A
A
A
 


      




1 cos2
(26)sin
2
1 cos 2
(27)cos
2
A
A
A
A

 

 

  
cot cot 1
(28)cot
cot cot
cot cot 1
(29)cot
cot cot
A B
A B
B A
A B
A B
B A

 


 


3
3
3
2
(30)sin3 3sin 4sin
(31)cos3 4cos 3cos
3tan tan
(32)tan3
1 3tan
A A A
A A A
A A
A
A
 
 




(33) Area of triangle formula:
ab C bc A ac B
1 1 1
sin sin sin
2 2 2
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6.  
 
2 2
2 2
(34) Trigonometric Equation General Solution
(i) sin 0 ,
( ) cos 0 2 1 ,
2
( ) tan 0 ,
( ) sin sin 1 ,
( ) cos cos 2 ,
( ) tan tan ,
sin sin
( ) cos cos
tan
n
n n Z
ii n n Z
iii n n Z
iv n n Z
v n n Z
vi n n Z
vii
  

 
  
    
    
    
 
   
   
  
    
   
   


2 2
,
tan
n n Z  
 


   



 
(35)sin 0 ; cos 1 ; tan 0
n
n n n
  
   
Inverse Trigonometry
 
   
   
 
 
 
1
1
1
1
1
1
(1)
sin 1 , 1 ,
2 2
cos 1 , 1 0 ,
tan ,
2 2
cos 1 , 1 , 0
2 2
sec 1 , 1 0 ,
2
cot 0 ,
R
ec R
R
R
 

 
 









 
 
 
 

 

 
 
 
   
 
 
 
   
 
Inverse Function Domain Range

   
     
   
   
   
   
1 1
1 1
1 1
1 1
1 1
1 1
(2)sin sin for all 1 , 1
(3)cos cos for all 1 , 1
(4)tan tan for all
(5)cos cos for all 1
(6)sec sec for all 1
(7)cot cot for all
x x x
x x x
x x x R
ec x ec x x
x x x
x x x R



 
 
 
 
 
 
    
    
   
   
   
   
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7.  
 
 
 
1 1
1 1
1
1
1
1
(8)sin cos ; for all 1
1
(9)cos sec ; for all 1
cot , for 01
(10) tan
cot ,for 0
ec x x
x
x x
x
x x
x xx

 
 



 
 
 
 
 
 
 
 
  
 
   
 
1 1
1 1
1 1
(11)sin cos ;for all 1 , 1
2
(12)tan cot ;for all
2
(13)sec cosec ;for all 1
2
x x x
x x x R
x x x



 
 
 
   
  
  
 
 
 
1 1 1 2 2 2 2
1 1 1 2 2 2 2
1 1 1 2 2
1 1
(14)sin sin sin 1 1 ; if either 1 or 0, 1, 1
(15)sin sin sin 1 1 ; if either 1 or 0, 1, 1
(16)cos cos cos 1 1 ; if 0, 1, 1
(17)cos cos cos
x y x y y x x y xy x y
x y x y y x x y xy x y
x y xy x y x y x y x y
  
  
  
  
         
         
          
 
1 2 2
1 1 ; if , 1, 1xy x y x y x y     
 
 
1 1 2
1 1 2
1 1
(18)2sin sin 2 1 ; if
2 2
(19)2cos cos 2 1 ; if 0 1
x x x x
x x x
 
 
    
   
1
1 1 1
1
tan , if 1
1
(20)tan tan tan , if 0, 0 and 1
1
tan , if 0, 0 and 1
1
x y
xy
xy
x y
x y x y xy
xy
x y
x y xy
xy



  

  
  
 

 
       
 

 
     
  
1
1 1 1
1
tan , if 1
1
(21)tan tan tan , if 0, 0 and 1
1
tan , if 0, 0 and 1
1
x y
xy
xy
x y
x y x y xy
xy
x y
x y xy
xy



  

  
   
 

 
        
 

 
      
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8.  
 
1 1 3
1 1 3
3
1 1
2
1 1
(22)3sin sin 3 4 ; if
2 2
1
(23)3cos cos 4 3 ; if 1
2
3 1 1
(24)3tan tan ; if
1 3 3 3
x x x x
x x x x
x x
x x
x
 
 
 
    
   
 
    
 

1
2
2
1 1
2
1
2
2
sin , 1 1
1
1
(25)2 tan cos , 0
1
2
tan , 1 1
1
x
x
x
x
x x
x
x
x
x

 

  
    
 

 
    

  
 

   
  

1 1 1
1 1 1
1
(26)cot cot cot
1
(27)cot cot cot
xy
x y
y x
xy
x y
y x
  
  
 
 
 
 
 
 
 
 

(28)Some useful substitutions
2 2
2
2 2
2
2 2
2
a tan or cot
1 tan or cot
sin or cos
1 sin or cos
sec or cosec
1 sec or cosec
;
cos 2
;
x x a a
x x
a x x a a
x x
x a x a a
x x
a x a x
a x a x
x a
a x a x
a x a x
 
 
 
 
 
 

 
 
 
 
 
 
  

 


 

 

Expression Substitution

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9. Ch 3 Matrices

(1) A matrix in which number of rows is equal to number of columns, say n is known as square
matrix of order n.
(2) Properties of Transpose of a Matrix:
  
   
 ( ) ( )
( ) ( )
( )
T T
T T T
T T
T T T
T
T T T
i A A ii A B A B
iii kA kA iv AB B A
v ABC C B A
   
 


(3) A square matrix
ij
A a 
 
is called a symmetric matrix, if for all ,
T
ij ji
a a i j A A   .
(4) A square matrix
ij
A a 
 
is called a skew-symmetric matrix, if
for all ,
T
ij ji
a a i j A A    

(5) All main diagonal elements of a skew-symmetric matrix are zero.
(6) Every square matrix can be uniquely expressed as the sum of symmetric and skew-symmetric
matrix.
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10. Determinants
(1) A square matrix Ais a singular matrix if 0A
(2) For any square matrix A, the |A| satisfy following properties.
2a. |AB| = |A| B|
2b. If we interchange any two rows (or columns), then sign of determinant changes.
2c. If any two rows or any two columns are identical or proportional, then value of
determinant is zero.
2d. If we multiply each element of a row or a column of a determinant by constant k, then
value of determinant is multiplied by k.
2e. Multiplying a determinant by k means multiply elements of only one row (or one
column) by k.
2f. If elements of a row or a column in a determinant can be expressed as sum of two or
more elements, then the given determinant can be expressed as sum of two or more
determinants.
2g. If to each element of a row or a column of a determinant the equimultiples of
corresponding elements of other rows or columns are added, then value of determinant
remains same.
(3) Let
ij
A a 
 
be a square matrix. The Minor
ij
Mof an element
ij
aof Ais the determinant of
the matrix obtained by deleting
th
irow and
th
jcolumn of A. Minor of an element of a square
matrix of order n(2n) is a determinant of order 1n.
(4) The Cofactor
ij
Aof an element
ij
aof a square matrix
ij
A a 
 
is defined as  1
i j
ij ij
A M

  .
(5)A=Sum of product of elements (of any row or column) with their corresponding cofactors.
(6) If elements of a row (or column) are multiplied with cofactors of any other row (or column),
then their sum is zero.
(7) Adjoint of a matrixAis the transpose of a cofactor matrix.
(8) If andA Bare square matrices of the same order n, then:
(a)    
n
A adj A A I adj A A  (b)   adj AB adj B adj A
(c)
   
T T
adj A adj A (d)
1n
adj A A

 Amit Bajaj | amitbajajmaths.com | [email protected] amitbajajmaths.com

11. (e)  
2n
adj adj A A A

 (f)
 
 
2
1n
adj adj A A



(g)  
1
,
n
adj kA k adjA k R

 

 
1 1
( )h adjA adjA
 


(9) A square matrix Aof order nis invertible if there exists a square matrix Bof the same order
such that
n
AB I BA  . We write,
1
A B

.
(10) Properties of inverse of a matrix:
(a) Every invertible matrix possesses a unique inverse.
 

1
1
b A A


 (c)  
1
1 1
AB B A

 
 (d)
11
A
A


 
1
1
T
T
e A A


 (f)  
11
A adj A
A


(11) Let    
1 1 2 2 3 3
, , , and ,A x y B x y C x y be any three points in XY-plane. Then, area of triangle
ABCis given by
1 1
2 2
3 3
1
1
1
2
1
x y
x y
x y
  .
(12) A system AX Bof nlinear equations has a unique solution given by
1
X A B

 , if
0A.
(13) If 0 and 0A adjA B , then the system is consistent and has infinitely many solutions.
But if 0 and 0A adjA B , then the system is inconsistent. Amit Bajaj | amitbajajmaths.com | [email protected] amitbajajmaths.com

12. Continuity and Differentiability
(1) Limits
1
lim
n n
n
x a
x a
na
x a





;
1
sin
lim
0

x
x
x
;

0
tan
lim 1
x
x
x


0
1
lim 1
x
x
e
x


 ;
0
1
lim log
x
e
x
a
a
x


 ;

0
log 1
lim 1
x
x
x



(2) A function ( )f xis continuous at x aif lim ( ) ( )
x a
f x f a

 i.e. lim ( ) lim ( ) ( )
x a x a
f x f x f a
 
 
  .
(3) Following functions are continuous everywhere:
(a) Constant function (b) Identity function (c) Polynomial function
(d) Modulus function (e) Exponential function (f) Sine & Cosine functions
(4) Following functions are continuous in their domains:
(a) Logarithmic function (b) Rational function
(c) Tan, Cot, Sec & Cosec functions (d) all inverse trigonometric functions
(5) If fis continuous function then
1
andf
f
are continuous in their domains.
(6) A function ( )f xis differentiable at x aif
( ) ( )
lim
x a
f x f a
x a



exists finitely i.e.

0 0
( ) ( ) ( ) ( )
lim lim
h h
f a h f a f a h f a
h h
 
   


.
(7) Every differentiable function is continuous but, converse is not true.
(8) Following functions are differentiable everywhere/their defined domain:
(a) Polynomial function (b) exponential function (c) constant function (d) Logarithmic function (e) trigonometric & inverse trigonometric functions (9) The sum, difference, product, quotient and composition of two differentiable functions is
differentiable.
(10) Some Standard Derivatives:
   
     
     
 
1
2 2
1
( ) ( ) log ( )
( ) log ( ) sin cos ( ) cos sin
( ) tan sec ( ) cot cos ( ) sec sec tan
( ) cos cos cot
n n x x
e
x x
e
d d d
i x n x ii x iii e e
dx dx x dx
d d d
iv a a a v x x vii x x
dx dx dx
d d d
viii x x ix x ec x x x x x
dx dx dx
d
xi ecx ecx x
dx

  
   
   
 
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13.      
     
1 1 1
2
2 2
1 1 1
2
2 2
1 1 1
( ) sin ( ) cos ( ) tan
11 1
1 1 1
( ) cot ( ) sec ( ) cosec
1 1 1
d d d
xii x xiii x xiv x
dx dx dx xx x
d d d
xv x xvi x xvii x
dx dx dxx x x x x
  
  
   
 
   
  

(11) Chain rule:
   
If and , then .
dz dz dy
z f y y g x
dx dy dx
  

(12) Product Rule

then
du dv
v u
dx dx
dy
y uv
dx
 

(13) Quotient Rule
2
f then
du dv
u dy dx dx
y
v dx
v u
I
v

 
(14) For a function in the parametric form, say  ,y f t x g t  we have:
2
2
and
dy
dy d y d dydt
dxdx dx dt dx
dt
 
 
  
 
 
   
 
 

(15) Rolle’s Theorem:
Let fbe a real valued function defined on ,a bsuch that:
(a) continuous on ,a b (b) differentiable on ,a b (c) f a f b
then, there exist a real number ,c a b such that 
'
0f c.
(16) Mean Value Theorem:
Let fbe a real valued function defined on ,a bsuch that:
(a) continuous on ,a b (b) differentiable on ,a b
then, there exist a real number ,c a b such that  

'
f b f a
f c
b a



.
Amit Bajaj | amitbajajmaths.com | [email protected] amitbajajmaths.com

14. Ch 6 Applications of Derivatives

(1) Tangents and Normals
If y f x , then
P
dy
dx
 
 
 
is slope of tangent to y f x at a point P.
If y f x , then
1
P
dy
dx

 
 
 
is slope of normal to y f x at a point P.
(2) If tangent is parallel to x-axis, then 0
dy
dx
 ; If tangent is parallel to y-axis, then 0
dx
dy

(3) If  
1 1
,P x yis a point on the curve y f x , then:
Equation of tangent at Pis  
1 1
P
dy
y y x x
dx
 
  
 
 

Equation of normal at Pis  
1 1
1
P
y y x x
dy
dx
   
 
 
 

(4) The angle between the tangents to two given curves at their point of intersection is defined as the
angle of intersection of two curves.

(5) Approximations:
Let y = f (x), xbe a small increment in x and ybe the increment in y corresponding to the
increment in x, i.e., y = f (x + x) – f (x). Then
dy
y x
dx
  
Also,   
'
f x x f x f x x    
Increasing and Decreasing Function
(6) A function fis said to be:
6a. Increasing on ,a bif   
1 2 1 2 1 2
in , for all , ,x x a b f x f x x x a b    .
Alternatively,  
'
0 for each in ,f x x a b
6b. Decreasing on ,a bif   
1 2 1 2 1 2
in , for all , ,x x a b f x f x x x a b    .
Alternatively,  
'
0 for each in ,f x x a b
6c. Strictly increasing on ,a bif   
1 2 1 2 1 2
in , for all , ,x x a b f x f x x x a b    .
Alternatively,  
'
0 for each in ,f x x a b .
6d. Strictly decreasing on ,a bif   
1 2 1 2 1 2
in , for all , ,x x a b f x f x x x a b    .
Alternatively,  
'
0 for each in ,f x x a b Amit Bajaj | amitbajajmaths.com | [email protected] amitbajajmaths.com

15. (7) A function fis monotonic on ,a bif it is strictly increasing or strictly decreasing on,a b.
(8) A point cin the domain of a function fat which either 
'
0f cor fis not differentiable is
called a critical point.
Maxima and Minima
(9) First Derivative Test
Given a curve y = f(x),
(a) For the stationary point at x = a,
(i) if
dx
dy
changes sign from negative to positive as x increases through a, the point S is a
minimum point,
(ii) if
dx
dy
changes sign from positive to negative as x increases through a, the point S is a
maximum point,
(iii) if
dx
dy
does not change sign as x increase through a, the point S is a point of inflexion.
(b) A stationary point is called a turning point if it is either a maximum point or a minimum point.
(10) Second Derivative Test
Given a curve y = f(x),
(a)
dx
dy
= 0 and
2
2
dx
yd
0 at x = aS(a, f(a)) is a turning point.
(i) If
2
2
dx
yd
> 0, then S is a minimum point. (ii) If
2
2
dx
yd
< 0, then S is a maximum point.
(b)
dx
dy
= 0 and
2
2
dx
yd
= 0 at x = a, go back to First Derivative Test.
(11) Working rule for finding absolute maxima and/or absolute minima:
Step1: Find all critical points of fin the given interval.
Step 2: Take end points of the interval. Step 3: At all these points (listed in step 1 and 2), calculate the values of
f.
Step 4: Identify the maximum and minimum values of fout of values calculated in step 3.
Amit Bajaj | amitbajajmaths.com | [email protected] amitbajajmaths.com

16. Integrals
(1) Standard Formulas
1
2
3 2
( ) ; 1
1
1
( ) log
( )
log
1 1
( )
1
( ) 2
2
( )
3
n
n
e
x
x
e
x
i x dx c n
n
ii dx x c
x
a
iii a dx c
a
iv dx c
x x
v dx x c
x
vi x dx x c

   

 
 
  
 
 







2
2
( ) sin cos
( ) cos sin
( ) sec tan
( ) cos cot
( ) sec tan sec
( ) cos cot cos
vii x dx x c
viii x dx x c
ix x dx x c
x ec x dx x c
x x x dx x c
xi ecx x dx ecx c
  
 
 
  
 
  



 


( ) tan log sec ( ) cot log sin
( ) sec log sec tan log tan
4 2
( ) cos log cosec cot log tan
2
xii x dx x c
xiii x dx x c
x
xiv x dx x x c c
x
xv ecx dx x x c c

 
 
 
     
 
 
    




Amit Bajaj | amitbajajmaths.com | [email protected] amitbajajmaths.com

17. 2 2
2 2
1
2 2
2 2
2 2
1
2 2
2 2
2 2
1 1
( ) log
2
1 1
( ) log
2
1 1
( ) tan
1
( ) log
1
( ) sin
1
( ) log
x a
xvi dx c
x a a x a
a x
xvii dx c
a x a a x
x
xviii dx c
x a a a
xix dx x x a c
x a
x
xx dx c
aa x
xxi dx x x a c
x a



 
 

 
 
 
 
 
  
   

 
 
 
 
   








2
2 2 2 2 2 2
2
2 2 2 2 1
2
2 2 2 2 2 2
( ) log
2 2
( ) sin
2 2
( ) log
2 2
x a
xxii x a dx x a x x a c
x a x
xxiii a x dx a x c
a
x a
xxiv x a dx x a x x a c

      
 
    
 
 
      




(2) By Parts (ILATE Rule):
If andu vare two functions of x, then
du
uvdx u vdx vdx dx
dx
 
   
 
   

i.e. (first function) x (integral of second function) – integral of {(derivative of first function) x
(integral of second function)}
We can choose the first function as the function which comes first in the word ILATE, where
I stands for inverse trigonometric functions, L for logarithmic functions, A for algebraic
functions, T for trigonometric functions and E for exponential function.
(3)   
' x x
f x f x e dx e f x c   
 

(4) Integration by Partial fraction of Rational Function of the form

 
P x
Q x
:
If degree of P x degree of Q x, then divide  byP x Q x Amit Bajaj | amitbajajmaths.com | [email protected] amitbajajmaths.com

18.       
     
         
         
    
2 2
2
2
2 2
2
2
22
( )
( )
( )
( )
( ) here, can't befactorised.
px q A B
i
x a x b x a x b
px q A B
ii
x ax a x a
px qx r A B C
iii
x a x b x c x a x b x c
px qx r A B C
iv
x a x bx a x b x a
px qx r A Bx C
v x bx c
x a x bx cx a x bx c


   


 
 
 
     
 
 
   
  
  
    
Form Partial Fraction

(5) For Integrals of the form
2
2
or
dx dx
ax bx c ax bx c   
 
use completing the square
method and then applying formulastoxvi xxi.
(6) For Integrals of the form
 
2
2
2
or or
px q px q
dx dx px q ax bx c dx
ax bx c ax bx c
 
  
   
  
, write
 
2d
px q A ax bx c B
dx
     where A and B are determined by comparing coefficients on
both sides.
(7) For Integrals of the form
 
2 2 2 2
2
1 1 1
or or or
sin cos sin cos
1 1 1
or or
sin 2 cos 2sin cos
dx dx dx
a b x a b x a x b x
dx dx dx
a b x a b xa x b x
  
 
  
  

Algorithm:
Step 1: Divide numerator and denominator by
2
cosx
Step 2: Replace
2
secx, if any, in denominator by
2
1 tanx
Step 3: Put
2
tan so that secx t xdx dt . This will reduce the integral in form
2
1
dx
at bt c 


Step 4: Evaluate the integral now using completing the square method. Amit Bajaj | amitbajajmaths.com | [email protected] amitbajajmaths.com

19. (8) For Integrals of the form
1 1 1
or or
sin cos sin cos
dx dx dx
a b x a b x a x b x  
  

Algorithm:
Put
2
2 2
2tan 2 1 tan 2
sin , cos
1 tan 2 1 tan 2
x x
x x
x x

 
 

(9) For Integrals of the form
sin cos
sin cos
a x b x
dx
c x d x



.
Algorithm:
Put   Numerator Denominator Derivative of Denomina torA B 
(10) For Integrals of the form
2
4 2 4 2
1 1
or or tan or cot
1 1
x
dx dx x dx x dx
x x x x 

   
   
or
4 4
1
sin cos
dx
x x


Algorithm:
Step1: Divide numerator and denominator by
2
x.
Step2: Express the denominator in the form
2
21
x k
x
 
 
 
 

Step3: Introduce
1 1
ord x d x
x x
   
 
   
   
in the numerator.
Step4: Substitute
1 1
orx t x t
x x
    as the case may be.
(11) For Integrals of the form
sin cos
presenceof sin 2
x x
dx
x


,PutDerivativeof Numerator=t
(12) For Integrals of the form
 
21
Putdx cx d t
ax b cx d
 
 


(13) For Integrals of the form
 
2
2
1
Putdx px q t
ax bx c px q
 
  


(14) For Integrals of the form
 
2
1 1
Putdx ax b
tax b px qx r
 
  

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20. (15) For Integrals of the form
   
2 2
2 2 2 2
1 1
Put toobtain and then put
tdt
dx x c dt u
tax b cx d a bt c dt

  
   
 

(16) First fundamental theorem of integral calculus:
Let the area function be defined by    
x
a
A x f x dx
for all x a where the function fis
assumed to be continuous on ,a b. Then 'A x f x for all ,x a b
(17) Second fundamental theorem of integral calculus:
Let fbe a continuous function of x defined on the closed interval [a, b] and let F be another function
such that    
d
F x f x
dx
 for all xin the domain of f, then      
b
a
f x dx F b F a 
.
(18) Properties of Definite Integral




   
   
     
   
   
 
     
   
0 0
0
( )
( )
( ) where
( )
( )
2 if . . is even function
( )
0 if . . is odd function
b b
a a
b a
a b
b c b
a a c
b b
a a
a a
a
a
a
i f x dx f t dt
ii f x dx f x dx
iii f x dx f x dx f x dx a c b
iv f x dx f a b x dx
v f x dx f a x dx
f x dx f x f x i e f
vi f x dx
f x f x i e f


 
   
  
 

 


  

 
 
  
 
 


 
     
   
2
0
0
2 if 2
( )
0 if 2
a
a
f x dx f a x f x
vii f x dx
f a x f x

 


  


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21. (19) Limit as a Sum
         
0
lim 2 ... ( 1 where
b
h
a
f x dx h f a f a h f a h f a n h nh b a

           
 

(20)
 
1
1 2 3 ... 1
2
n n
n

     
(21)
 
 2
2 2 2
1 2 1
1 2 3 ... 1
6
n n n
n
 
     
(22)
0 0
1
lim 1 lim
1
x
x
x x
e x
x e
 

 


(23)
2 1 1
...
1
n
n r
a ar ar ar a
r
  
     

 


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22. Application of Integration
Area between a Curve and an Axis
In general, if A(x) is an area function under the curve y = f(x), then area under the curve y = f(x)
from x = a to x = b is given by
 
b
a
A f x dx
where f(x)0 for axb.
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23. Differential Equations
(1) Order of a differential equation: is the order of the highest order derivative appearing in the
equation.
(2) Degree of a differential equation: is the degree of the highest order derivative, when
differential coefficients are made free from radicals and fractions.
(3) Solution of a differential equation:
3a. Variable separable form:
An equation whose variables are separable can be put into the form
 
1 2
0f x dx f y dy  . Integrating, the general solution is given by
 
1 2
f x dx f y dy C  
, where Cis an arbitrary constant.
3b. Homogeneous differential equation:
A differential equation of the form

 
,
,
f x ydy
dx g x y
 where  , and ,f x y g x y are both
homogeneous function of the same degree in andx yi.e. an equation of the form
dy y
F
dx x
 

 
 
.
To solve, puty vx, then
dy dv
v x
dx dx
  and on substituting these values in the given
equation it will be reducible to variable separable.
Another form:
dx x
G
dy y
 

 
 
, can be solved by substitution x vy
3c. Linear Differential Equation:
Equation of the form .
dy
P y Q
dx
  where both andP Q are functions of xonly.
To solve, find Integrating Factor (I.F.) =
P dx
e

and the solution is given by
y.( . .) .( . .)I F Q I F dx c 
.
Another Form: Equation of form .
dx
P x Q
dy
  where both andP Q are functions of yonly.
To solve, find Integrating Factor (I.F.) =
P dx
e

and the solution is given by
.( . .) .( . .)x I F Q I F dy c 
.
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24. Vectors
(1) Position vector of a point  , ,P x y zis given by
ˆ
OP r x i y j zk   



and its magnitude is
given by
2 2 2
x y z  .
(2) If
1 2 3
ˆˆ ˆa a i a j a k  

, then
1 2 3
, anda a a are direction ratios of a

.
(3) Triangle law of vector addition: AB BC AC 
  

(4) The vector sum of the three sides of a triangle taken in order is 0

.
(5) If

ˆ
r a i b j c k  


, then , ,a b care proportional to its direction ratios and its direction cosine
are , ,l m ngiven by
2 2 2 2 2 2 2 2 2
, ,
a b c
a b c a b c a b c
  
     
. Here
2 2 2
1l m n  
(6) Two vectors anda b


are collinear if and only if there exists a nonzero scalar such that
b a


.
(7) If
1 2 3 1 2 3
ˆ ˆˆ ˆ ˆ ˆanda a i a j a k b bi b j b k     

then the two vectors are collinear if and only
if :
31 2
1 2 3
aa a
b b b
 
(8) The vector sum of two co- initial vectors is given by the diagonal of the parallelogram whose
adjacent sides are the given vectors.
(9) For a given vectora

, the vector ˆ
a
a
a


gives the unit vector in the direction of a

.
(10) If    
1 1 1 2 2 2
, , and , ,P x y z Q x y z are any two points, then the vector joining andP Q is
given by    
2 1 2 1 2 1
ˆˆ ˆPQ x x i y y j z z k     
(11) Section Formula: The position vector of a point R dividing a line segment joining the
points P and Q whose position vectors are anda b


respectively, in the ratio m :n
(i) Internally, is given by
mb na
m n




(ii) externally, is given by
mb na
m n





(12) The Scalar (dot) Product of two given vectors anda b


having angle  between them is
. cosa b a b 
  

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25. (13) If
1 2 3 1 2 3
ˆ ˆˆ ˆ ˆ ˆanda a i a j a k b bi b j b k     

, then:
1 1 2 2 3 3
.a b a b a b a b  

.
(14)
2
.a a a
  

(15) Projection of vector
.
on
a b
a b
b




and Projection of vector
.
on
a b
b a
a



.
(16) If a b


, then . 0a b



(17) For mutually perpendicular unit vectors
ˆˆ ˆ, andi j k, we have:
ˆ ˆˆ ˆ ˆ. . . 1i i j j k k  

and
ˆ ˆˆ ˆ ˆ ˆ. . . 0i j j k k i  
(18) The Vector (cross) Product of two given vectors anda b


having angle  between them is
ˆsina b a b n 
  
, where ˆn is a unit vector perpendicular to the plane containing anda b


.
(19) If
1 2 3 1 2 3
ˆ ˆˆ ˆ ˆ ˆanda a i a j a k b bi b j b k     

, then:
1 2 3
1 2 3
i j k
a b a a a
b b b
 
 


(20) If||a b


, then 0a b 



(21) For mutually perpendicular unit vectors
ˆˆ ˆ, andi j k, we have:
ˆ ˆˆ ˆ ˆ 0i i j j k k     

and
ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ, ,i j k j k i k i j     
(22) Area of a triangle: If anda b


represent the adjacent sides of a triangle then its area is
given as
1
2
a b


.
Also, Area of
1 1 1
2 2 2
ABC AB AC BC BA CB CA     
     

(23) If , ,a b c

 
are p.v. of the vertices , , ofA B C ABC , then Area of
1
2
ABC a b b c c a     
 
   

(24) Area of a parallelogram having two adjacent sides as anda b


= a b



(25) Unit vectors perpendicular to the plane of anda b


are
a b
a b







.
(26) . .a b b a
 
 
and  a b b a   
  
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26. (27) For any two vectors anda b


, we have 
22 2
2
.a b a b a b  
  
  
.
(28) For any two vectors anda b


, we have  
2 2 2
2
2a b a b a b    
  
  

(29) Scalar Triple Product
Let , ,a b c

 
be three vectors. Then the scalar .a b c
 
is known as scalar triple product and is
denoted by a b c 
 
 
.
(30) Geometrically, a b c 
 
 
represents the volume of the parallelepiped whose coterminous
edges are , anda b c

 
.
(31) Properties of Scalar Triple Product:
(a)  . . .a b c b c a c a b    
       

(b) a b c b a c c b a a c b            
       
          

(c)  . .a b c a b c  
    

(d) a b c 
 
 
= 0 , if or ora b b c c a  
 
   
i.e. , ,a b c

 
are coplanar.
(e) a b c a b c    
   
    

(f)   a b c a b a c     
     

(g) If
1 2 3 1 2 3 1 2 3
ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ, anda a i a j a k b bi b j b k c c i c j c k        
 
, then:
1 2 3
1 2 3
1 2 3
a a a
a b c b b b
c c c
  
 
 

(h) Four points , , andA B C D are coplanar if 0AB AC AD 
 
  

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27. Three Dimensional Geometry
LINE
(1) Equation of a line through a given point and parallel to given vector


Vector Form: Equation of a line passing through point with position vectora

and parallel to the
vector b

is given by: r a b
 
 
,
where r

is position vector of any point on the line.
Cartesian Form: Cartesian equation of a line passing through the point
 
1 1 1
, ,x y zand parallel to
the line having D-ratios
 , ,a b c is given by:
1 1 1
x x y y z z
a b c
  
 

(2) Equation of a line passing through two points
Vector Form: Equation of a line passing through two points with position vector a

and b

is
given by
  ,r a b a R    
  

Cartesian Form: Cartesian equation of a line passing through the points with
 
1 1 1
, ,x y zand
 
2 2 2
, ,x y z

is given by:
1 1 1
2 1 2 1 2 1
x x y y z z
x x y y z z
  
 
  

(3) Angle between two lines: Angle between the lines with d.r.’s
 
1 1 1
, ,a b cand  
2 2 2
, ,a b cis
given by:
1 2 1 2 1 2
2 2 2 2 2 2
1 1 1 2 2 2
cos
a a b b c c
a b c a b c

 

   

(4) Two lines with d.r.’s
 
1 1 1
, ,a b cand  
2 2 2
, ,a b c
4a. will be perpendicular to each other iff
1 2 1 2 1 2
0a a b b c c  
4b. will be parallel to each other iff
1 1 1
2 2 2
a b c
a b c
 
(5) Angle between the two lines with direction cosines
 
1 1 1
, ,l m nand  
2 2 2
, ,l m nis given by
1 2 1 2 1 2
cosl l m m n n
   Amit Bajaj | amitbajajmaths.com | [email protected] amitbajajmaths.com

28. (6) Skew Lines: Two lines in the space which are neither parallel and nor intersecting.
(7) If two lines are coplanar then they are either parallel or intersecting. Also, two lines are
neither parallel nor intersecting iff they are non-coplanar.
(8) Shortest Distance (SD) between two lines:
Vector Form: If
1 1
r a b 
 
and
2 2
r a b 
 
are the vector equations of two lines then the
shortest distance between them is the length of the line of the line segment perpendicular to both
of them, it is denoted by ‘d’.
  
1 2 2 1
1 2
.b b a a
d
b b
 


 
 
 
Cartesian Form: If
1 1 1
1 1 1
x x y y z z
a b c
  
  and
2 2 2
2 2 2
x x y y z z
a b c
  
  are the equations of two
skew lines then the shortest distance between them ‘d’ is given by:
     
2 1 2 1
1 1 1
2 2 2
2 2 2
1 2 2 1 1 2 2 1 1 2 2 1
x x y y z z
a b c
a b c
d
a b a b b c b c c a c a
  

    

(9) Two skew lines
1 1
r a b 
 
and
2 2
r a b 
 
will be intersecting each other iff the shortest
distance between them is zero iff  
1 2 2 1
. 0b b a a  
 
iff the lines are coplanar.
(10) Two skew lines
1 1 1
1 1 1
x x y y z z
a b c
  
  and
2 2 2
2 2 2
x x y y z z
a b c
  
  will be intersecting
each other iff
2 1 2 1
1 1 1
2 2 2
0
x x y y z z
a b c
a b c
  
.
(11) Distance between two parallel lines: Distance between two parallel lines
1
r a b 
 
and
2
r a b 
 
and is given by:
 
2 1
b a a
d
b
 


 
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29. PLANE
(12) Normal form of the Plane:

Vector Form: Equation of a plane having unit vector normal to the plane ˆnand at a distance of
‘d’ from the origin is ˆ.r n d

,
where,
r

=Position vector of any point on the plane, ˆn= Unit vector normal to the plane
d= Distance of plane from the origin to the plane
Cartesian Form: Equation of the plane with
 , ,l m nas d-cosine of the normal to the plane is
lx my nz d  
Where, d= Distance of plane from the origin to the plane.
(13) General form of the equation of the plane:
Vector Form: Equation of plane having normal vector to it as n

is given by .r n p
 

Note: 1. Here ‘p’ is not the distance from the origin to the plane
2. In order to convert the equation .r n p
 
to unit vector normal form we divide the
equation by
n

i.e. the unit vector normal form becomes .
n p
r
n n



 
and then ,d= Distance of plane from the origin to the plane =
p
n

Cartesian Form: Equation of plane with d.r.’s of normal vector to it as
 , ,a b cis
0ax by cz p   
Note: 1. Here ‘p’ is not the distance from the origin to the plane
2. In order to convert the equation 0ax by cz p    to normal form we divide the
equation by
2 2 2
a b c  i.e. the normal form becomes
2 2 2 2 2 2 2 2 2 2 2 2
0
a b c p
x y z
a b c a b c a b c a b c
   
       
Amit Bajaj | amitbajajmaths.com | [email protected] amitbajajmaths.com

30. and then ,d= Distance of plane from the origin to the plane =
2 2 2
p
a b c 
.
(14) Equation of a plane passing through a given point and with given normal vector:
Vector Form: Equation of a plane passing through the point a

and having normal to the plane as
vector n

is given by
 . 0r a n 
  


Cartesian Form:: Equation of a plane passing through the point
 
1 1 1
, ,x y z and having  , ,a b c
as the d.r. of the normal to the plane is
     
1 1 1
0a x x b y y c z z     

(15) Equation of a plane passing through three non collinear points:
Vector Form: Equation of plane passing through three points
    , andA a B b C c
 
, where
, anda b c

 
are the position vectors of three points, andA B C , is given by:
     . 0r a b a c a     
 

    

Cartesian Form: Equation of the plane passing through the three points
     
1 1 1 2 2 2 3 3 3
, , , , , and , ,A x y z B x y z C x y z is:
1 1 1
2 1 2 1 2 1
3 1 3 1 3 1
0
x x y y z z
x x y y z z
x x y y z z
  
   
  

(16) Intercept form of the plane: Equation of a plane having , andx y z intercept as
, anda b crespectively is given by 1
x y z
a b c
  
(17) Equation of a plane passing through the intersection of two given planes:
Vector Form: Equation of a plane passing through the intersection of two planes
1 1 2 2
. and .r n d r n d 
   
is given by
 
1 2 1 2
. , wherer n n d d R
     
  
Amit Bajaj | amitbajajmaths.com | [email protected] amitbajajmaths.com

31. Cartesian Form: Equation of plane passing through the intersection of the two planes
1 1 1 1
0a x b y c z d    and
2 2 2 2
0a x b y c z d    is given by
 
1 1 1 1 2 2 2 2
0a x b y c z d a x b y c z d        , whereR
(18) Coplanarity of two lines:
Vector Form: Two lines
1 1
r a b 
 
and
2 2
r a b 
 
will be coplanar iff they lie in a common
plane iff  
1 2 2 1
. 0b b a a  
   
.
Cartesian Form: Two lines
1 1 1
1 1 1
x x y y z z
a b c
  
  and
2 2 2
2 2 2
x x y y z z
a b c
  
  will be coplanar
iff
2 1 2 1 2 1
1 1 1
2 2 2
0
x x y y z z
a b c
a b c
  


And equation of the plane containing these two lines is:
1 1 1
1 1 1
2 2 2
0
x x y y z z
a b c
a b c
  

(19) Angle between two planes:
Vector Form: Angle between two intersecting planes is the angle between the normal to the
planes i.e. Angle ‘’ between the planes
1 1 2 2
. and .r n d r n d 
   
is given by
1 2
1 2
.
cos
n n
n n

 
 
Cartesian Form: Angle ‘’ between planes
1 1 1 1
0a x b y c z d    and
2 2 2 2
0a x b y c z d    is
given by
1 2 1 2 1 2
2 2 2 2 2 2
1 1 1 2 2 2
cos
a a b b c c
a b c a b c

 

   


(20) Angle between a line and a plane:
Vector Form: Angle ‘’between a line r a b 
 
and a plane.r n d
 
is given by
.
sin
.
b n
b n





Amit Bajaj | amitbajajmaths.com | [email protected] amitbajajmaths.com

32. Cartesian Form: Angle ‘’between a line
1 1 1
1 1 1
x x y y z z
a b c
  
  and a plane
0ax by cz d    is given by
1 1 1
2 2 2 2 2 2
1 1 1
sin
aa bb cc
a b c a b c

 

   

(21) Distance of a point from a line:
Vector Form: Distance of a point a

(not on the plane) from a plane .r n p
 
is given by
.r n p
D
n


 

Cartesian Form: Distance of a point  
1 1 1
, ,x y z(not on the plane) from a plane
0ax by cz d    is given by
1 1 1
2 2 2
ax by cz d
D
a b c
  

 
Amit Bajaj | amitbajajmaths.com | [email protected] amitbajajmaths.com

33. Linear Programming
(1) Theorem 1 Let R be the feasible region (convex polygon) for a linear programming problem
and let Z = ax + by be the objective function. When Z has an optimal value (maximum or
minimum), where the variables x and y are subject to constraints described by linear inequalities,
this optimal value must occur at a corner point (vertex) of the feasible region.
(2) Theorem 2 Let R be the feasible region for a linear programming problem, and let
Z = ax + by be the objective function. If R is bounded, then the objective function Z has both a
maximum and a minimum value on R and each of these occurs at a corner point (vertex) of R.
(3) If R is unbounded, then a maximum or a minimum value of the objective function may not
exist. However, if it exists, it must occur at a corner point of R. (4) If two corner points of the feasible region are both optimal solutions of the same type, i.e.,
both produce the same maximum or minimum, then any point on the line segment joining these two points is also an optimal solution of the same type. (5) Corner Point Method:
Step 1: Find the feasible region of the linear programming problem and determine its
corner points (vertices) either by inspection or by solving the two equations of the lines intersecting at that point.
Step 2: Evaluate the objective function Z = ax + by at each corner point. Let M and m,
respectively denote the largest and smallest values of these points.
Step 3a: When the feasible region is bounded, M and m are the maximum and minimum
values of Z. Step 3b: In case, the feasible region is unbounded, we have:
(a) M is the maximum value of Z, if the open half plane determined by ax + by > M has
no point in common with the feasible region. Otherwise, Z has no maximum value.
(b) Similarly, m is the minimum value of Z, if the open half plane determined by ax + by
<m has no point in common with the feasible region. Otherwise, Z has no minimum value. Amit Bajaj | amitbajajmaths.com | [email protected] amitbajajmaths.com

34. Probability
Number of favorable outcomes in ( )
(1) ( )
Number of possible outcomes in ( )
E n E
P E
S n S
 
   (2) 1P A P A 
       (3)P A B P A P B P A B    
(4) If andA B are mutually exclusive events, then
  0P A B  .      (5)P A B P B P A B   
(6) Probability of occurrence of exactly one of andA B=
         2P A P B P A B P A B P A B      

(7) If , andA B C are three events, then
               P A B C P A P B P C P A B P B C P A C P A B C             

(9)

(10) Conditional Probability:
 |P A BProbability of occurrence of Awhen Bhas already
occured =
 
 
P A B
P B


   (11) | 1 |P A B P A B 
(12) Multiplication Theorem on Probability:
         . | . |P A B P A P B A P B P A B  
(13) Independent events: If andA B are independent events, then
     P A B P A P B  
(14) If andA B are independent events, then
   
  at least oneof or 1P A B P A B P A P B   

(15) If andA B are independent events, then and ; and ; andA B A B A B are all independent.
(16) Three events A, B and C are said to be mutually independent, if:
(a)
     P A B P A P B   (b)      P B C P B P C   Amit Bajaj | amitbajajmaths.com | [email protected] amitbajajmaths.com

35. (c)  P A C P A P C   (d)  P A B C P A P B P C    
(17) Bayes’ Theorem:
If
1 2 3
, , ,...
n
E E E E are n non empty events which constitute a partition of sample space S i.e.
1 2 3
, , ,...
n
E E E E are pair- wise disjoint and
1 2 3
...
n
E E E E S     andA is any event, then
 

   
1
|
| ; where 1,2,3,...,
|
i i
i n
i i
i
P E P A E
P E A i n
P E P A E

 


For Example, for two events
1 2
andE E , we have
 

       
 
 
       
1 1 2 2
1 2
1 1 2 2 1 1 2 2
| |
| and |
| | | |
P E P A E P E P A E
P E A P E A
P E P A E P E P A E P E P A E P E P A E
 
 

(18) Probability Distribution: If a random variable Xtakes values
1 2 3, , ,...,
nx x x x with respective
probabilities
1 2 3, , ,...,
np p p p , then
 
1 2 3
1 2 3
: .....
: .....
n
n
X x x x x
P X p p p p
is known as probability distribution of X. Here,
1
1
n
i
i
p


(19) Mean (Expectation) / Expected Value of Random Variable:
 
1
n
i i
i
E X x p

 

(20) Variance of Random Variable:
 
2
2 2
1 1
Var
n n
x i i i i
i i
X x p x p

 
 
    
 
 
(21) Standard Deviation of a Random Variable: Var X
x
 Amit Bajaj | amitbajajmaths.com | [email protected] amitbajajmaths.com