CONTENTS
1. Chapter wise formula sheet
2. Important Questions:
i. Numbers, Quantification & Numerical Applications ii.
Numerical inequalities
iii. Matrices
iv. Determinants
v. Differentiation
vi. Applications of Derivatives
vii. Integrals
viii. Differential equations
ix. Probability
x. Inferential statistics
xi. Time-based data
xii. Perpetuity Sinking finds & EMI
xiii. Returns, Growth & Depreciation
xiv. Linear Programming
4. Previous year question paper (2023-24)
5. Previous year question Paper (2022-23)
,MATHEMATICS IS A GREAT MOTIVATOR FOR ALL HUMANS....
BECAUSE ITS CAREER STARTS WITH ZERO AND ITS NEVER ENDS (
INFINITY )
Important basic Concepts and Formulas
1.Numbers Quantification and numerical application
Modulo arithmetic:
When we divide integer a by integer b (≠ 0), we have a = b q + r, where q , r ∈
1, 0≤ r < b. Here a is dividend, b is divisor, q is quotient and r is remainder.
Congruence modulo :
a≡ b (mod n) means 1-b is divisible by n , where n > 1
Alligation or Mixture:
M= d-m
M= m-c
( Repeated Dilution )= After n number of repeated dilutions the quantity of
pure liquid = x( 1-y/x)n
Boats and streams:
Let speed of boat in still water be x km/h and speed of stream be y km/h, time taken by
downstream be t1 and time taken upstream be t2 , downstream speed be u and
upstream speed be v
Downstream speed (u) = (x + y) km/h , Upstream speed (v) = (x - y) km/h
Speed of the boat in still water(x)= u+v/2 Speed of the stream (y)= u-v/2
When the distance covered by the boat in downstream is same as the distance covered
by the boat upstream, then,t2\t1 =x+y/x-y
Average speed downstream speed x upstream speed/ speed in still water=
(x+y)(x-y)/x=x2-y2/x
,If a boat takes t hours to row to a certain place d km apart and returns back, then
distance between two places
(d) = t(x2-y2)/2x
If a boat takes t hours more in going upstream than downstream for covering the same
distance d then distance between two places (d)=t(x2-y2)/2y
2.NUMERICAL INEQUALITY
Relation between AM and GM between two numbers a and b
𝑎+𝑏
AM= and GM = √𝑎𝑏 , AM -GN>0 OR AM>GM
2
Inequalities involving modulus are define as: if I is any positive real
number and x ∈ R, then
i. |X| <| iff-|<x<|i.e. iff x ∈ (-|,|)
ii. |X|< | iff-| <x<| i.e. iff x ∈ [-|,|]
iii. |X| >| iff either x <-| or x>| i.e. iff x=(- ∞ ,-|) U ( I,∞ )
iv. |X| > | iff either x <-| or X >|i.e. iff x = (∞ ,-|] U [|,∞)
3.MATRICES
Types of matrices
(i) Row Matrix (ii) column matrix (iii) Null or zero matrix or void matrix
(iv) Square matrix (v) Diagonal matrix (vi) Scalar matrix (vii) identity matrix
Matrix polynomial
If f (x)=x2-3x+2 is a polynomial and A is a square matrix then F(A)=A2-3A+2| is a
matrix polynomial
, Properties of transpose
I. (A’)’ = A
II. (A ± B)’= A’ ± B’
III. (kA’) = kA’
IV. (AB)’ = B’A’
V. (ABC)’ = C’B’A’
Symmetric and skew-symmetric matrices
I. symmetric matrix ; A square matrix A is called symmetric matrix if
A’ = A.
II. skew symmetric matrices; A square matrix A is called skew -
symmetric matrix If A’=-A
III. Every element in the principal diagonal of a skew symmetric matrix
is always zero
IV. If A is square matrix then A+A’ is symmetric and A-A' is skew
symmetric matrix
V. Every matrix A can be uniquely expressed as the sum of
symmetric and skew symmetric matrix i.e. A=½ (A+A’) + ½ (A-A')
4. Determinants
To every square matrix A=[aij] of order n, aij ∈ R, we can associate a unique real
number called determinant of matrix A and denoted by det A=|A|
Properties of determinants:
1. If each element in a row (or column) of a determinant is zero, then
the value of determinant is
zero.
1. Chapter wise formula sheet
2. Important Questions:
i. Numbers, Quantification & Numerical Applications ii.
Numerical inequalities
iii. Matrices
iv. Determinants
v. Differentiation
vi. Applications of Derivatives
vii. Integrals
viii. Differential equations
ix. Probability
x. Inferential statistics
xi. Time-based data
xii. Perpetuity Sinking finds & EMI
xiii. Returns, Growth & Depreciation
xiv. Linear Programming
4. Previous year question paper (2023-24)
5. Previous year question Paper (2022-23)
,MATHEMATICS IS A GREAT MOTIVATOR FOR ALL HUMANS....
BECAUSE ITS CAREER STARTS WITH ZERO AND ITS NEVER ENDS (
INFINITY )
Important basic Concepts and Formulas
1.Numbers Quantification and numerical application
Modulo arithmetic:
When we divide integer a by integer b (≠ 0), we have a = b q + r, where q , r ∈
1, 0≤ r < b. Here a is dividend, b is divisor, q is quotient and r is remainder.
Congruence modulo :
a≡ b (mod n) means 1-b is divisible by n , where n > 1
Alligation or Mixture:
M= d-m
M= m-c
( Repeated Dilution )= After n number of repeated dilutions the quantity of
pure liquid = x( 1-y/x)n
Boats and streams:
Let speed of boat in still water be x km/h and speed of stream be y km/h, time taken by
downstream be t1 and time taken upstream be t2 , downstream speed be u and
upstream speed be v
Downstream speed (u) = (x + y) km/h , Upstream speed (v) = (x - y) km/h
Speed of the boat in still water(x)= u+v/2 Speed of the stream (y)= u-v/2
When the distance covered by the boat in downstream is same as the distance covered
by the boat upstream, then,t2\t1 =x+y/x-y
Average speed downstream speed x upstream speed/ speed in still water=
(x+y)(x-y)/x=x2-y2/x
,If a boat takes t hours to row to a certain place d km apart and returns back, then
distance between two places
(d) = t(x2-y2)/2x
If a boat takes t hours more in going upstream than downstream for covering the same
distance d then distance between two places (d)=t(x2-y2)/2y
2.NUMERICAL INEQUALITY
Relation between AM and GM between two numbers a and b
𝑎+𝑏
AM= and GM = √𝑎𝑏 , AM -GN>0 OR AM>GM
2
Inequalities involving modulus are define as: if I is any positive real
number and x ∈ R, then
i. |X| <| iff-|<x<|i.e. iff x ∈ (-|,|)
ii. |X|< | iff-| <x<| i.e. iff x ∈ [-|,|]
iii. |X| >| iff either x <-| or x>| i.e. iff x=(- ∞ ,-|) U ( I,∞ )
iv. |X| > | iff either x <-| or X >|i.e. iff x = (∞ ,-|] U [|,∞)
3.MATRICES
Types of matrices
(i) Row Matrix (ii) column matrix (iii) Null or zero matrix or void matrix
(iv) Square matrix (v) Diagonal matrix (vi) Scalar matrix (vii) identity matrix
Matrix polynomial
If f (x)=x2-3x+2 is a polynomial and A is a square matrix then F(A)=A2-3A+2| is a
matrix polynomial
, Properties of transpose
I. (A’)’ = A
II. (A ± B)’= A’ ± B’
III. (kA’) = kA’
IV. (AB)’ = B’A’
V. (ABC)’ = C’B’A’
Symmetric and skew-symmetric matrices
I. symmetric matrix ; A square matrix A is called symmetric matrix if
A’ = A.
II. skew symmetric matrices; A square matrix A is called skew -
symmetric matrix If A’=-A
III. Every element in the principal diagonal of a skew symmetric matrix
is always zero
IV. If A is square matrix then A+A’ is symmetric and A-A' is skew
symmetric matrix
V. Every matrix A can be uniquely expressed as the sum of
symmetric and skew symmetric matrix i.e. A=½ (A+A’) + ½ (A-A')
4. Determinants
To every square matrix A=[aij] of order n, aij ∈ R, we can associate a unique real
number called determinant of matrix A and denoted by det A=|A|
Properties of determinants:
1. If each element in a row (or column) of a determinant is zero, then
the value of determinant is
zero.
