UNIT – 1 (Numbers, Quantification and Numerical Applications)
1. Modulo Arithmetic – 𝑎 𝑚𝑜𝑑 𝑏 = 𝑟, where 𝑟 is the remainder when 𝑎 is divided by 𝑏
2. Congruence Modulo – If 𝑎 and 𝑏 are any two integers and 𝑛 is a positive integer such
that 𝑛/(𝑎 − 𝑏) ie., (𝑎 − 𝑏) is divisible by 𝑛, then 𝑎 is said to be congruent to 𝑏 modulo
𝑛 and written as
𝑎 ≡ 𝑏 (𝑚𝑜𝑑 𝑛) or 𝑎 − 𝑏 ≡ (𝑚𝑜𝑑 𝑛) or 𝑎 − 𝑏 = 𝑘𝑛 for some integer 𝑛
3. Properties of Congruences:
If 𝑎, 𝑏, 𝑐 are any integers and 𝑛 is a positive integer, then
(a) 𝑎 ≡ 𝑏 (𝑚𝑜𝑑 𝑛) ⟺ 𝑎 = 𝑏 + 𝑘𝑛 for some integer 𝑘
(b) 𝑎 ≡ 𝑏 (𝑚𝑜𝑑 𝑛) ⟺ 𝑎 and 𝑏 leave the same remainder when divided by 𝑛
(c) An integer is congruent to its remainder i.e, 𝑎 ≡ 𝑟 (𝑚𝑜𝑑 𝑛)
(d) Every integer is congruent to itself i.e, 𝑎 ≡ 𝑎 (𝑚𝑜𝑑 𝑛)
(e) If 𝑎 ≡ 𝑏 (𝑚𝑜𝑑 𝑛) then 𝑏 ≡ 𝑎 (𝑚𝑜𝑑 𝑛)
(f) If 𝑎 ≡ 𝑏 (𝑚𝑜𝑑 𝑛) then −𝑎 ≡ −𝑏 (𝑚𝑜𝑑 𝑛)
(g) If 𝑎 ≡ 𝑏 (𝑚𝑜𝑑 𝑛) and 𝑏 ≡ 𝑐 (𝑚𝑜𝑑 𝑛), then 𝑎 ≡ 𝑐 (𝑚𝑜𝑑 𝑛)
(h) (𝑎 + 𝑏)(𝑚𝑜𝑑 𝑛) ≡ 𝑎 (𝑚𝑜𝑑 𝑛) + 𝑏 (𝑚𝑜𝑑 𝑛)
(i) If 𝑎 ≡ 𝑏 (𝑚𝑜𝑑 𝑛), then (𝑎 + 𝑘) ≡ (𝑏 + 𝑘) (𝑚𝑜𝑑 𝑛), for some integer 𝑘
(j) If 𝑎 ≡ 𝑏 (𝑚𝑜𝑑 𝑛) and 𝑐 ≡ 𝑑 (𝑚𝑜𝑑 𝑛), then (𝑎 + 𝑐) ≡ (𝑏 + 𝑑) (𝑚𝑜𝑑 𝑛)
(k) If 𝑎 ≡ 𝑏 (𝑚𝑜𝑑 𝑛) and 𝑐 ≡ 𝑑 (𝑚𝑜𝑑 𝑛), then (𝑎 − 𝑐) ≡ (𝑏 − 𝑑) (𝑚𝑜𝑑 𝑛)
(l) 𝑎. 𝑏 (𝑚𝑜𝑑 𝑛) = 𝑎(𝑚𝑜𝑑 𝑛). 𝑏(𝑚𝑜𝑑 𝑛)
(m) If 𝑎 ≡ 𝑏 (𝑚𝑜𝑑 𝑛) and 𝑐 ≡ 𝑑 (𝑚𝑜𝑑 𝑛), then 𝑎𝑐 ≡ 𝑏𝑑 (𝑚𝑜𝑑 𝑛)
(n) If 𝑎 ≡ 𝑏 (𝑚𝑜𝑑 𝑛), then 𝑘𝑎 ≡ 𝑘𝑏 (𝑚𝑜𝑑 𝑛), for some integer 𝑘
(o) If 𝑎 ≡ 𝑏 (𝑚𝑜𝑑 𝑛), then 𝑎𝑘 ≡ 𝑏𝑘 (𝑚𝑜𝑑 𝑛), for some integer 𝑘
4. Alligation & Mixture:
If two ingredients are mixed together, then the rule of allegation is
𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑜𝑓 𝑐ℎ𝑒𝑝𝑎𝑒𝑟 𝑖𝑛𝑔𝑟𝑒𝑑𝑖𝑒𝑛𝑡 𝑐𝑜𝑠𝑡 𝑝𝑟𝑖𝑐𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑑𝑒𝑎𝑟𝑒𝑟 𝑖𝑛𝑔𝑟𝑒𝑑𝑖𝑒𝑛𝑡−𝑀𝑒𝑎𝑛 𝑃𝑟𝑖𝑐𝑒
=
𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑜𝑓 𝑑𝑒𝑎𝑟𝑒𝑟 𝑖𝑛𝑔𝑟𝑒𝑑𝑖𝑒𝑛𝑡 𝑀𝑒𝑎𝑛 𝑃𝑟𝑖𝑐𝑒−𝑐𝑜𝑠𝑡 𝑝𝑟𝑖𝑐𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑐ℎ𝑒𝑎𝑝𝑒𝑟 𝑖𝑛𝑔𝑟𝑒𝑑𝑖𝑒𝑛𝑡
𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑜𝑓 𝑐ℎ𝑒𝑝𝑎𝑒𝑟 𝑖𝑛𝑔𝑟𝑒𝑑𝑖𝑒𝑛𝑡 𝑑−𝑚
Or =
𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑜𝑓 𝑑𝑒𝑎𝑟𝑒𝑟 𝑖𝑛𝑔𝑟𝑒𝑑𝑖𝑒𝑛𝑡 𝑚−𝑐
5. Boats and Streams:
Upstream – If the boat rows/sails along the direction of the stream then it is said to move
downstream
Downstream – If the boat rows/sails in the direction opposite to that of the stream then
it is said to move upstream
If 𝑥 km/h is the speed of boat in still water and 𝑦 km/h is the speed of the stream, then
Speed of boat in upstream = (𝑥 − 𝑦) km/h
Speed of boat in downstream = (𝑥 + 𝑦) km/h
𝑑𝑜𝑤𝑛𝑠𝑡𝑟𝑒𝑎𝑚 𝑠𝑝𝑒𝑒𝑑 × 𝑢𝑝𝑠𝑡𝑟𝑒𝑎𝑚 𝑠𝑝𝑒𝑒𝑑
Average speed of boat =
𝑠𝑝𝑒𝑒𝑑 𝑖𝑛 𝑠𝑡𝑖𝑙𝑙 𝑤𝑎𝑡𝑒𝑟
6. Pipes and Cisterns:
(a) If two inlet pipes can fill a tank in 𝑥 and 𝑦 hours respectively, then time taken to fill
1 𝑥𝑦
the whole tank = 1 1 =
( + ) 𝑥+𝑦
𝑥 𝑦
(b) If three inlet pipes can fill a tank in 𝑥, 𝑦 and 𝑧 hours respectively, then time taken to
1 𝑥𝑦𝑧
fill the whole tank = 1 1 1 =
( 𝑥+ 𝑦 + 𝑧 ) 𝑥𝑦+𝑦𝑧+𝑧𝑥
, (c) If an inlet pipe can fill a tank in 𝑥 hours and an outlet pipe can empty a tank in 𝑦
1 𝑥𝑦
hours (𝑦 > 𝑥), then time taken to fill the whole tank = 1 1 =
( 𝑥− 𝑦 ) 𝑦−𝑥
(d) If an inlet pipe can fill a tank in 𝑥 hours and an outlet pipe can empty a tank in 𝑦
1 𝑦𝑥
hours (𝑥 > 𝑦), then time taken to empty the whole tank = 1 1 =
( 𝑦 − 𝑥) 𝑥−𝑦
(e) If two inlet pipes take 𝑥 and 𝑦 hours respectively to fill the tank and the third outlet
pipe takes 𝑧 hours to empty the tank and all three of them are opened together, then
1 𝑥𝑦𝑧
time taken to fill the whole tank = 1 1 1 =
(𝑥+𝑦−𝑧) 𝑦𝑧+𝑧𝑥−𝑥𝑦
7. Races and Games:
S. No. Statements Mathematical Interpretations
1 A beats B by 𝑡 seconds A finishes the race 𝑡 seconds before B
finishes
2 A beats B by 𝑥 metres A reaches the finishing point and B is 𝑥
metres behind A
3 A beats B by 25 m or 7 seconds B covers a distance of 25 m in 7 seconds
4 A gives B a start of 𝑡 seconds A starts 𝑡 seconds after B starts from the
same starting point
5 A gives B a start of 𝑥 metres While A starts from the starting point, B
starts 𝑥 metres ahead of the same starting
point at the same time
6 Game of 100 A game of 100 means that the person
among the participants who scores 100
points first is the winner
7 A beats B by 20 points A scores 100 points and B scores 80 points
8 In a game of 100, A can give B While A need to score 100 points, B needs
20 points to score only (100-20) = 80 points
8. Numerical Inequalities:
If 𝑎, 𝑏, 𝑐 are real numbers, then
(a) If 𝑎 > 𝑏 and 𝑏 > 𝑐 then 𝑎 > 𝑐
(b) If 𝑎 < 𝑏 and 𝑏 < 𝑐 then 𝑎 < 𝑐
(c) If 𝑎 > 𝑏 then 𝑎 + 𝑐 > 𝑏 + 𝑐
(d) If 𝑎 > 𝑏 then 𝑎 − 𝑐 > 𝑏 − 𝑐
(e) If 𝑎 > 𝑏 and 𝑝 > 0 then 𝑝𝑎 > 𝑝𝑏
𝑎 𝑏
(f) If 𝑎 > 𝑏 and 𝑝 > 0 then >
𝑝 𝑝
1 1
(g) If 𝑎 > 𝑏 then <
𝑎 𝑏
(h) If 𝑎, 𝑏 are positive real numbers such that 𝑎 < 𝑏 and if 𝑛 is any positive rational
1 1
number then (i) 𝑎𝑛 < 𝑏 𝑛 (ii) 𝑎−𝑛 > 𝑏 −𝑛 (iii) 𝑎𝑛 < 𝑏𝑛
(i) For any real number 𝑎, −|𝑎| ≤ 𝑎 ≤ |𝑎|
(j) Types of Intervals
➢ Closed Interval: 𝑎 ≤ 𝑥 ≤ 𝑏 written as [𝑎, 𝑏]
➢ Open Interval: 𝑎 < 𝑥 < 𝑏 written as (𝑎, 𝑏)
, ➢ Semi-closed or Semi-open Interval: 𝑎 < 𝑥 ≤ 𝑏 written as (𝑎, 𝑏] or 𝑎 ≤ 𝑥 < 𝑏
written as [𝑎, 𝑏)
➢ Set of real numbers 𝑥 satisfying 𝑥 < 𝑎 and 𝑥 > 𝑎 are written as (−∞, 𝑎) and
(𝑎, ∞) respectively.
➢ Set of real numbers 𝑥 satisfying 𝑥 ≤ 𝑎 and 𝑥 ≥ 𝑎 are written as (−∞, 𝑎] and
[𝑎, ∞) respectively.
➢ |𝑥| ≤ 𝑎 ⟹ −𝑎 ≤ 𝑥 ≤ 𝑎
➢ |𝑥| ≥ 𝑎 ⟹ 𝑥 ≤ −𝑎, 𝑥 ≥ 𝑎
UNIT – 2 (Algebra)
1. Types of Matrices
(a) Column Matrix: A matrix is said to be a column matrix if it has only one column.
(b) Row Matrix: A matrix is said to be a row matrix if it has only one row.
(c) Square Matrix: A matrix in which the number of rows are equal to the number of
columns, is said to be a square matrix.
(d) Diagonal Matrix: A square matrix is said to be a diagonal matrix if all its non
diagonal elements are zero.
(e) Scalar Matrix: A diagonal matrix is said to be a scalar matrix if its diagonal
elements are equal.
(f) Identity Matrix: A square matrix in which elements in the diagonal are all 1 and
rest are all zero.
(g) Zero Matrix: A matrix is said to be zero matrix or null matrix if all its elements are
zero.
2. Properties of Matrix addition:
(a) Commutative Law: If A and B are two matrices of same order, then A+B=B+A
(b) Associative Law: If A, B and C are matrices of same order, then
A+(B+C) = (A+B)+C
(c) Existence of additive identity: If A is any matrix and O be a zero matrix, then A +
O = O + A, O is called the additive identity
(d) Existence of additive inverse: For any matrix A, A + (-A) = (-A) + A = O. -A is
called the additive inverse of A.
3. Properties of scalar multiplication: If A and B are two matrices of the same order and
𝑘 and 𝑙 are scalars, then
(a) 𝑘 (𝐴 + 𝐵 ) = 𝑘𝐴 + 𝑘𝐵
(b) (𝑘 + 𝑙)𝐴 = 𝑘𝐴 + 𝑙𝐴
4. Properties of Matrix multiplication:
(a) Commutative Law: 𝐴𝐵 ≠ 𝐵𝐴 for any two matrices A and B
(b) Associative Law: For any three matrices A, B and C (𝐴𝐵 )𝐶 = 𝐴(𝐵𝐶)
(c) Distributive law: For any three matrices A, B and C (i) 𝐴(𝐵 + 𝐶 ) = 𝐴𝐵 + 𝐴𝐶 and
(ii) (𝐴 + 𝐵 )𝐶 = 𝐴𝐶 + 𝐵𝐶
(d) Existence of multiplicative identity: For any square matrix A, there exist an
identity matrix of same order such that 𝐴𝐼 = 𝐼𝐴 = 𝐴
5. Properties of transpose of the matrices: For any matrices A and B of suitable orders,
(a) (𝐴′ )′ = 𝐴
(b) (𝑘𝐴)′ = 𝑘𝐴′
(c) (−𝐴)′ = −𝐴′
(d) (𝐴 + 𝐵 )′ = 𝐴′ + 𝐵′
1. Modulo Arithmetic – 𝑎 𝑚𝑜𝑑 𝑏 = 𝑟, where 𝑟 is the remainder when 𝑎 is divided by 𝑏
2. Congruence Modulo – If 𝑎 and 𝑏 are any two integers and 𝑛 is a positive integer such
that 𝑛/(𝑎 − 𝑏) ie., (𝑎 − 𝑏) is divisible by 𝑛, then 𝑎 is said to be congruent to 𝑏 modulo
𝑛 and written as
𝑎 ≡ 𝑏 (𝑚𝑜𝑑 𝑛) or 𝑎 − 𝑏 ≡ (𝑚𝑜𝑑 𝑛) or 𝑎 − 𝑏 = 𝑘𝑛 for some integer 𝑛
3. Properties of Congruences:
If 𝑎, 𝑏, 𝑐 are any integers and 𝑛 is a positive integer, then
(a) 𝑎 ≡ 𝑏 (𝑚𝑜𝑑 𝑛) ⟺ 𝑎 = 𝑏 + 𝑘𝑛 for some integer 𝑘
(b) 𝑎 ≡ 𝑏 (𝑚𝑜𝑑 𝑛) ⟺ 𝑎 and 𝑏 leave the same remainder when divided by 𝑛
(c) An integer is congruent to its remainder i.e, 𝑎 ≡ 𝑟 (𝑚𝑜𝑑 𝑛)
(d) Every integer is congruent to itself i.e, 𝑎 ≡ 𝑎 (𝑚𝑜𝑑 𝑛)
(e) If 𝑎 ≡ 𝑏 (𝑚𝑜𝑑 𝑛) then 𝑏 ≡ 𝑎 (𝑚𝑜𝑑 𝑛)
(f) If 𝑎 ≡ 𝑏 (𝑚𝑜𝑑 𝑛) then −𝑎 ≡ −𝑏 (𝑚𝑜𝑑 𝑛)
(g) If 𝑎 ≡ 𝑏 (𝑚𝑜𝑑 𝑛) and 𝑏 ≡ 𝑐 (𝑚𝑜𝑑 𝑛), then 𝑎 ≡ 𝑐 (𝑚𝑜𝑑 𝑛)
(h) (𝑎 + 𝑏)(𝑚𝑜𝑑 𝑛) ≡ 𝑎 (𝑚𝑜𝑑 𝑛) + 𝑏 (𝑚𝑜𝑑 𝑛)
(i) If 𝑎 ≡ 𝑏 (𝑚𝑜𝑑 𝑛), then (𝑎 + 𝑘) ≡ (𝑏 + 𝑘) (𝑚𝑜𝑑 𝑛), for some integer 𝑘
(j) If 𝑎 ≡ 𝑏 (𝑚𝑜𝑑 𝑛) and 𝑐 ≡ 𝑑 (𝑚𝑜𝑑 𝑛), then (𝑎 + 𝑐) ≡ (𝑏 + 𝑑) (𝑚𝑜𝑑 𝑛)
(k) If 𝑎 ≡ 𝑏 (𝑚𝑜𝑑 𝑛) and 𝑐 ≡ 𝑑 (𝑚𝑜𝑑 𝑛), then (𝑎 − 𝑐) ≡ (𝑏 − 𝑑) (𝑚𝑜𝑑 𝑛)
(l) 𝑎. 𝑏 (𝑚𝑜𝑑 𝑛) = 𝑎(𝑚𝑜𝑑 𝑛). 𝑏(𝑚𝑜𝑑 𝑛)
(m) If 𝑎 ≡ 𝑏 (𝑚𝑜𝑑 𝑛) and 𝑐 ≡ 𝑑 (𝑚𝑜𝑑 𝑛), then 𝑎𝑐 ≡ 𝑏𝑑 (𝑚𝑜𝑑 𝑛)
(n) If 𝑎 ≡ 𝑏 (𝑚𝑜𝑑 𝑛), then 𝑘𝑎 ≡ 𝑘𝑏 (𝑚𝑜𝑑 𝑛), for some integer 𝑘
(o) If 𝑎 ≡ 𝑏 (𝑚𝑜𝑑 𝑛), then 𝑎𝑘 ≡ 𝑏𝑘 (𝑚𝑜𝑑 𝑛), for some integer 𝑘
4. Alligation & Mixture:
If two ingredients are mixed together, then the rule of allegation is
𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑜𝑓 𝑐ℎ𝑒𝑝𝑎𝑒𝑟 𝑖𝑛𝑔𝑟𝑒𝑑𝑖𝑒𝑛𝑡 𝑐𝑜𝑠𝑡 𝑝𝑟𝑖𝑐𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑑𝑒𝑎𝑟𝑒𝑟 𝑖𝑛𝑔𝑟𝑒𝑑𝑖𝑒𝑛𝑡−𝑀𝑒𝑎𝑛 𝑃𝑟𝑖𝑐𝑒
=
𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑜𝑓 𝑑𝑒𝑎𝑟𝑒𝑟 𝑖𝑛𝑔𝑟𝑒𝑑𝑖𝑒𝑛𝑡 𝑀𝑒𝑎𝑛 𝑃𝑟𝑖𝑐𝑒−𝑐𝑜𝑠𝑡 𝑝𝑟𝑖𝑐𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑐ℎ𝑒𝑎𝑝𝑒𝑟 𝑖𝑛𝑔𝑟𝑒𝑑𝑖𝑒𝑛𝑡
𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑜𝑓 𝑐ℎ𝑒𝑝𝑎𝑒𝑟 𝑖𝑛𝑔𝑟𝑒𝑑𝑖𝑒𝑛𝑡 𝑑−𝑚
Or =
𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑜𝑓 𝑑𝑒𝑎𝑟𝑒𝑟 𝑖𝑛𝑔𝑟𝑒𝑑𝑖𝑒𝑛𝑡 𝑚−𝑐
5. Boats and Streams:
Upstream – If the boat rows/sails along the direction of the stream then it is said to move
downstream
Downstream – If the boat rows/sails in the direction opposite to that of the stream then
it is said to move upstream
If 𝑥 km/h is the speed of boat in still water and 𝑦 km/h is the speed of the stream, then
Speed of boat in upstream = (𝑥 − 𝑦) km/h
Speed of boat in downstream = (𝑥 + 𝑦) km/h
𝑑𝑜𝑤𝑛𝑠𝑡𝑟𝑒𝑎𝑚 𝑠𝑝𝑒𝑒𝑑 × 𝑢𝑝𝑠𝑡𝑟𝑒𝑎𝑚 𝑠𝑝𝑒𝑒𝑑
Average speed of boat =
𝑠𝑝𝑒𝑒𝑑 𝑖𝑛 𝑠𝑡𝑖𝑙𝑙 𝑤𝑎𝑡𝑒𝑟
6. Pipes and Cisterns:
(a) If two inlet pipes can fill a tank in 𝑥 and 𝑦 hours respectively, then time taken to fill
1 𝑥𝑦
the whole tank = 1 1 =
( + ) 𝑥+𝑦
𝑥 𝑦
(b) If three inlet pipes can fill a tank in 𝑥, 𝑦 and 𝑧 hours respectively, then time taken to
1 𝑥𝑦𝑧
fill the whole tank = 1 1 1 =
( 𝑥+ 𝑦 + 𝑧 ) 𝑥𝑦+𝑦𝑧+𝑧𝑥
, (c) If an inlet pipe can fill a tank in 𝑥 hours and an outlet pipe can empty a tank in 𝑦
1 𝑥𝑦
hours (𝑦 > 𝑥), then time taken to fill the whole tank = 1 1 =
( 𝑥− 𝑦 ) 𝑦−𝑥
(d) If an inlet pipe can fill a tank in 𝑥 hours and an outlet pipe can empty a tank in 𝑦
1 𝑦𝑥
hours (𝑥 > 𝑦), then time taken to empty the whole tank = 1 1 =
( 𝑦 − 𝑥) 𝑥−𝑦
(e) If two inlet pipes take 𝑥 and 𝑦 hours respectively to fill the tank and the third outlet
pipe takes 𝑧 hours to empty the tank and all three of them are opened together, then
1 𝑥𝑦𝑧
time taken to fill the whole tank = 1 1 1 =
(𝑥+𝑦−𝑧) 𝑦𝑧+𝑧𝑥−𝑥𝑦
7. Races and Games:
S. No. Statements Mathematical Interpretations
1 A beats B by 𝑡 seconds A finishes the race 𝑡 seconds before B
finishes
2 A beats B by 𝑥 metres A reaches the finishing point and B is 𝑥
metres behind A
3 A beats B by 25 m or 7 seconds B covers a distance of 25 m in 7 seconds
4 A gives B a start of 𝑡 seconds A starts 𝑡 seconds after B starts from the
same starting point
5 A gives B a start of 𝑥 metres While A starts from the starting point, B
starts 𝑥 metres ahead of the same starting
point at the same time
6 Game of 100 A game of 100 means that the person
among the participants who scores 100
points first is the winner
7 A beats B by 20 points A scores 100 points and B scores 80 points
8 In a game of 100, A can give B While A need to score 100 points, B needs
20 points to score only (100-20) = 80 points
8. Numerical Inequalities:
If 𝑎, 𝑏, 𝑐 are real numbers, then
(a) If 𝑎 > 𝑏 and 𝑏 > 𝑐 then 𝑎 > 𝑐
(b) If 𝑎 < 𝑏 and 𝑏 < 𝑐 then 𝑎 < 𝑐
(c) If 𝑎 > 𝑏 then 𝑎 + 𝑐 > 𝑏 + 𝑐
(d) If 𝑎 > 𝑏 then 𝑎 − 𝑐 > 𝑏 − 𝑐
(e) If 𝑎 > 𝑏 and 𝑝 > 0 then 𝑝𝑎 > 𝑝𝑏
𝑎 𝑏
(f) If 𝑎 > 𝑏 and 𝑝 > 0 then >
𝑝 𝑝
1 1
(g) If 𝑎 > 𝑏 then <
𝑎 𝑏
(h) If 𝑎, 𝑏 are positive real numbers such that 𝑎 < 𝑏 and if 𝑛 is any positive rational
1 1
number then (i) 𝑎𝑛 < 𝑏 𝑛 (ii) 𝑎−𝑛 > 𝑏 −𝑛 (iii) 𝑎𝑛 < 𝑏𝑛
(i) For any real number 𝑎, −|𝑎| ≤ 𝑎 ≤ |𝑎|
(j) Types of Intervals
➢ Closed Interval: 𝑎 ≤ 𝑥 ≤ 𝑏 written as [𝑎, 𝑏]
➢ Open Interval: 𝑎 < 𝑥 < 𝑏 written as (𝑎, 𝑏)
, ➢ Semi-closed or Semi-open Interval: 𝑎 < 𝑥 ≤ 𝑏 written as (𝑎, 𝑏] or 𝑎 ≤ 𝑥 < 𝑏
written as [𝑎, 𝑏)
➢ Set of real numbers 𝑥 satisfying 𝑥 < 𝑎 and 𝑥 > 𝑎 are written as (−∞, 𝑎) and
(𝑎, ∞) respectively.
➢ Set of real numbers 𝑥 satisfying 𝑥 ≤ 𝑎 and 𝑥 ≥ 𝑎 are written as (−∞, 𝑎] and
[𝑎, ∞) respectively.
➢ |𝑥| ≤ 𝑎 ⟹ −𝑎 ≤ 𝑥 ≤ 𝑎
➢ |𝑥| ≥ 𝑎 ⟹ 𝑥 ≤ −𝑎, 𝑥 ≥ 𝑎
UNIT – 2 (Algebra)
1. Types of Matrices
(a) Column Matrix: A matrix is said to be a column matrix if it has only one column.
(b) Row Matrix: A matrix is said to be a row matrix if it has only one row.
(c) Square Matrix: A matrix in which the number of rows are equal to the number of
columns, is said to be a square matrix.
(d) Diagonal Matrix: A square matrix is said to be a diagonal matrix if all its non
diagonal elements are zero.
(e) Scalar Matrix: A diagonal matrix is said to be a scalar matrix if its diagonal
elements are equal.
(f) Identity Matrix: A square matrix in which elements in the diagonal are all 1 and
rest are all zero.
(g) Zero Matrix: A matrix is said to be zero matrix or null matrix if all its elements are
zero.
2. Properties of Matrix addition:
(a) Commutative Law: If A and B are two matrices of same order, then A+B=B+A
(b) Associative Law: If A, B and C are matrices of same order, then
A+(B+C) = (A+B)+C
(c) Existence of additive identity: If A is any matrix and O be a zero matrix, then A +
O = O + A, O is called the additive identity
(d) Existence of additive inverse: For any matrix A, A + (-A) = (-A) + A = O. -A is
called the additive inverse of A.
3. Properties of scalar multiplication: If A and B are two matrices of the same order and
𝑘 and 𝑙 are scalars, then
(a) 𝑘 (𝐴 + 𝐵 ) = 𝑘𝐴 + 𝑘𝐵
(b) (𝑘 + 𝑙)𝐴 = 𝑘𝐴 + 𝑙𝐴
4. Properties of Matrix multiplication:
(a) Commutative Law: 𝐴𝐵 ≠ 𝐵𝐴 for any two matrices A and B
(b) Associative Law: For any three matrices A, B and C (𝐴𝐵 )𝐶 = 𝐴(𝐵𝐶)
(c) Distributive law: For any three matrices A, B and C (i) 𝐴(𝐵 + 𝐶 ) = 𝐴𝐵 + 𝐴𝐶 and
(ii) (𝐴 + 𝐵 )𝐶 = 𝐴𝐶 + 𝐵𝐶
(d) Existence of multiplicative identity: For any square matrix A, there exist an
identity matrix of same order such that 𝐴𝐼 = 𝐼𝐴 = 𝐴
5. Properties of transpose of the matrices: For any matrices A and B of suitable orders,
(a) (𝐴′ )′ = 𝐴
(b) (𝑘𝐴)′ = 𝑘𝐴′
(c) (−𝐴)′ = −𝐴′
(d) (𝐴 + 𝐵 )′ = 𝐴′ + 𝐵′
