Download Level -1 - Kendriya Vidyalaya Raigarh

KENDRIYA VIDYALAYA
SANGATHAN RAIPUR REGION
MODULES
for
CLASS-XI
MATHEMATICS
Session -2016-17
1
Class XI MLL’s
INDEX
Module No.
Chapter
Marks (Weightage)
Page No.
1.
Sets
9
3
2.
Relations and Functions
5
6
3.
Principle of Mathematical
induction
6
8
4.
Complex Numbers
4
9
5.
Linear Inequalities
6
10
6.
Binomial Theorem
6
11
7.
Conic Section
4
12
8.
Introduction to 3dimentional Geometry
4
15
9.
Limits and Derivatives
6
16
10.
Mathematical Reasoning
3
18
11.
Statistics
6
20
12.
Probability
4
23
13.
Blue Print Half Yearly
Examination
100
25
14.
Blue Print Session Ending
Examination
100
26
15.
Things To Remember
27
2
Module – 1
SETS
Set:
Collection of well-defined and distinct objects is called as Set.
Sets are denoted by capital letters of English alphabets.
Remarks:
(i) Here well-defined means, the statement which contains no adjective.
(ii) Small letter represents the elements or members of a set.
e.g.
(i) Collection of best singers in India. It is not a set as it contains adjective best.
(ii) Collection of singers in India. It is a set.
S.No.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
Symbol
∈
∉
∃
∀
∪
∩
−
∆
′
𝜙
⊂ or ⊆
ℕ
𝑊
ℤ
ℤ+
ℤ−
ℚ
Meaning
Belongs to
Does not belongs to
There exists
For all
Union
Intersection
Difference
Symmetric difference
Complement
Empty Set
Subset
Set of natural numbers
Set of whole numbers
Set of integers
Set of all positive integers
Set of all negative integers
Set of rational numbers
18.
𝑇 or ℚ∗ or ℚ′
Irrational Numbers
Example
3𝜖{1, 2, 3, 4, 5, 6}
3 ∉ { 2, 4, 6, 8}
𝐴 ∪ 𝐵 = {𝑥: 𝑥 ∈ 𝐴 𝑜𝑟 𝑥 ∈ 𝐵}
𝐴 ∩ 𝐵 = {𝑥: 𝑥 ∈ 𝐴 𝑎𝑛𝑑 𝑥 ∈ 𝐵}
𝐴 − 𝐵 = {𝑥: 𝑥 ∈ 𝐴 𝑎𝑛𝑑 𝑥 ∉ 𝐵}
𝐴 △ 𝐵 = (𝐴 − 𝐵) ∪ (𝐵 − 𝐴)
𝐴′ = {𝑥: 𝑥 ∉ 𝐴}
{1, 2, 3, …}
{0, 1, 2, 3, …}
{0, ±1, ±2, ±3, … }
𝑝
{ ∶ 𝑝, 𝑞 𝜖 𝑍, 𝑞 ≠ 0}
𝑞
Numbers which cannot be written as
𝑝
∶ 𝑝, 𝑞 𝜖 𝑍, 𝑞 ≠ 0,
𝑞
3
19.
20.
ℝ
ℝ+
21.
ℝ−
22.
{}
23.
n(A)
3
e.g √5, √−3, √64
All rational and irrational numbers
Set of real numbers
Set of all positive real
numbers
Set of all negative real
numbers
Brackets used to enclose the
elements of a set
Cardinal Number
Number of elements in a set.
𝐴 = {𝑎, 𝑏, 𝑐} ⟹ 𝑛(𝐴) = 3
Representation of Set
Roster / Tabular Form
Set Builder Form
In this representation, elements are listed in curly In this representation common property of elements
braces separated by commas.
is defined.
For example,
For Example, 𝐴 = {1,2,3,4,5}
A = Collection of natural numbers less than 6.
or 𝐴 = {𝑥: 𝑥 ∈ 𝑁, 𝑥 < 6}
3
Types of Set:
S.
Type of Set
No.
1
Empty Set, Null Set,
Void Set
𝜙 𝑜𝑟 {}
2
Singleton Set
3
Subset ⊂ 𝑜𝑟 ⊆
Meaning
Example
𝐴 = {𝑥|𝑥 ∈ 𝑁, 3 < 𝑥 < 4}
A set which has no element.
A set containing only one element.
Set A is said to be subset of set B if all the
elements of A are in B.
𝐴 ⊂ 𝐵 𝑜𝑟 𝐴 ⊆ 𝐵
Set B is said to be subset of set A if B
contains all the elements of A
𝐵 ⊃ 𝐴 𝑜𝑟 𝐵 ⊇ 𝐴
Set which contains countable number of
elements.
Set which contains uncountable number
of elements.
Two or more sets which have same
elements.
A= set of vowels in the word ‘SET’ = {E}
Set of vowels is subset of set of Alphabets.
3
Super Set ⊃ 𝑜𝑟 ⊇
4
Finite Set
5
Infinite Set
6
Equal Sets
7
Equivalent Sets
Two sets A and B are said to be
equivalent if n(A) = n(B)
A= {1, 2, 3} and B= {a, b, c}.
Clearly n(A) = n(B)=3
8
Universal Set, U
Set which is Super Set of all sets under
consideration.
Let A={1,2,3}, B={2,3,5}and C={1,4,6}
Then U={1,2,3,4,5,6} is universal set for A, B
and C.
A ={1,2} then P(A)={ ∅,{1},{1, 2},{2} }
9
Set of alphabets is super set of set of vowels.
𝐴 = {𝑎, 𝑒, 𝑖, 𝑜, 𝑢}
Set of Natural Numbers.
A ={𝑎, 𝑒, 𝑖, 𝑜, 𝑢},
B ={𝑥 |𝑥 𝑖𝑠 𝑣𝑜𝑤𝑒𝑙 𝑖𝑛 𝑡ℎ𝑒 𝑤𝑜𝑑 𝐸𝐷𝑈𝐶𝐴𝑇𝐼𝑂𝑁}
Power Set ,P
Set of all possible subsets of a given set.
Remark:
1. Empty set is subset of itself.
2. Every set is subset of itself.
3. If A ⊂ 𝐵 and 𝑥 ∈ 𝐴, then 𝑥 ∈ 𝐵.
4. If A ⊂ 𝐵 and 𝑥 ∉ 𝐵, then 𝑥 ∉ 𝐴.
5. If A ⊂ 𝐵 and 𝐵 ⊂ 𝐴 then 𝐴 = 𝐵.
6. If A ⊂ 𝐵 and 𝐴 ≠ 𝐵, Then A is called as proper subset of B.
7. If 𝑛(𝐴) = 𝑝 then total number of subsets of A = 2𝑝 and number of proper subsets is 2𝑝 -1.
8. If XϵP(A) ,then 𝑋 ⊂ 𝐴
9. n[P(A)]=2 n(A)
Properties of Sets:
S. No. Property
Meaning
1
Commutative Law
𝐴 ∪𝐵 = 𝐵∪𝐴
𝐴∩𝐵 = 𝐵∩𝐴
2
Associative Law
(𝐴 ∪ 𝐵) ∪ 𝐶 = 𝐴 ∪ (𝐵 ∪ 𝐶)
(𝐴 ∩ 𝐵) ∩ 𝐶 = 𝐴 ∩ (𝐵 ∩ 𝐶)
3
Law of ∅
𝐴 ∪∅=𝐴
∅ ∩𝐴 = ∅
4
Law of U
𝑈∪𝐴 =𝑈
𝑈∩𝐴 =𝐴
5
Idempotent Law
𝐴∪𝐴 =𝐴
𝐴∩𝐴 =𝐴
6
Distributive Law
𝐴 ∩ (𝐵 ∪ 𝐶) = (𝐴 ∩ 𝐵) ∪ (𝐴 ∩ 𝐶)
7
Complement Laws
𝐴 ∪ 𝐴′ = 𝑈
𝐴 ∩ 𝐴′ = ∅
8
De Morgan’s Law
9
10
Law of Double Complementation
Law of Empty Set and Universal Set
(𝐴 ∪ 𝐵)′ = 𝐴′ ∩ 𝐵′
(𝐴 ∩ 𝐵)′ = 𝐴′ ∪ 𝐵′
(𝐴′ )′ = 𝐴
∅′ = 𝑈
𝑈′ = ∅
4
Questions for Practice:
1.
2.
3.
𝐴 = {𝑎, 𝑏, 𝑐, {𝑎, 𝑏}, 𝑑}. Fill in the blanks by putting appropriate symbols.
i) 𝑎_______𝐴 ii) {𝑎, 𝑏}_________𝐴 iii) {{𝑎, 𝑏}}_________𝐴
iv) 𝑑_______𝐴
Write in roster form
i) 𝐴 = {𝑥|𝑥 ∈ 𝑍, 𝑥 2 − 5𝑥 + 6 = 0}
ii) The set of letters of the word “PANIC”
Write the followings in set builder form.
1 2 3
ii) 𝐵 = {0,1,2,3,4}
i) 𝐴 = { , , , … }
2 3 4
4. Represent the following in Roster form
(𝑖) { 2 , 4 , 6 , 8 ,10}
(ii) { 1 , 4 , 8 , 9 , 16 , 25 … … … }
(iii) {
2
3
3
4
5
4
5
6
, , , ,…………….}
(iv) {14 , 21 , 28 , 35 , 42 , … … … … ,98 }
(v){−1 , 1}
5. Represent in roster form
(i) Set of odd numbers.
(ii) {x𝜖N : x2 < 25}
(iii)Set of integers from -5 to 5
6. Classify the following as finite or infinite set.
(i) Set of integers greater than 100.
(ii) Set of concentric circles
(iii) {x∈ 𝑅: 0 < 𝑥 < 1}
(iv) The set of prime numbers less than 100
(v) {x∈ 𝑁, 𝑥 𝑖𝑠 𝑜𝑑𝑑}
7 .State True or False
Let A= {1,2,{3,4},5} state as True or False for the following statements
(i) {3,4}⊂ A (ii) {1,2,5}⊂ A (iii) {∅}⊂ A
(iv) 1𝜖A
(v) {3,4}𝜖 A
Interval’s
Q1. write the below interval’s in set builder form.
(i) [ a,b]
[ii] [a,b)
(iii)[a,b]
(iv) (a,b]
(vi) (-1,3)
(vii) [3,10]
(viii) (-4,0)
Q2. write the below sets in interval form.
(i) {x: 0< x ≪ 10}
(ii) ) {x: - 4 < x < 2}
Q.3.Find the intersection of given two intervals.
(i) [-1,3] and [0,5]
(ii) (2,6) and(-1,2]
Level –I
1.
2.
3.
4.
5.
6.
(v) (-1,3)
(ix) [2,9)
(x) (-1,3]
(iii){ x: 2≪x≪7}
(iii) [1,4) and (2,5)
1, 2, 2,1,3, 4
Let A= 3,6,9,12,15,18, 21 and B= 4,8,12,16, 20 .
Find the cardinal number of the set
Find
A B and A B .
In a school there are 20 teachers who teach Mathematics or Physics. Out of these, 12 teach Mathematics and 4 teach
Physics and Mathematics. How many teach Physics?
Let A, B and C be three sets such that A  B  A  C and A  B  A  C Show that B=C
If 𝑈 = {1,2,3,4,5,6,7,8,9}, 𝐴 = {2,4,6,8}𝑎𝑛𝑑 𝐵 = {2,3,5,7} then verify that
(𝐴 ∪ 𝐵)′ = 𝐴′ ∩ 𝐵′
i)
ii) (𝐴 ∩ 𝐵)′ = 𝐴′ ∪ 𝐵′
Draw appropriate Venn diagram for
i) 𝐴′ ∩ 𝐵′
ii)(𝐴 ∩ 𝐵)′
Level-II
1. If A= {1,2,3,4,5,6,7,8,9}, B={3,4,5,6,7}, C={4,5,6}, D={1,3,5,7}, find:
(i) 𝐴 ∪ (𝐵 ∩ 𝐶) (ii)
B
C'
(iii)
CD
(iv)
C
D
2. Draw appropriate Venn diagram for each of the following:
(i)
A
B '
(ii)
A' B '
Level III
1. In a survey of 25 students, it was found that 15 had taken Mathematics, 12 had taken Physics and 11 had taken
Chemistry, 5 had taken Mathematics and Chemistry, 9 had taken Mathematics and Physics , 4 had taken Physics and
Chemistry and 3 had taken all the three subjects. Find the number of students that had:
i) Only Chemistry ii) at least one of the three subjects
iii) None of the subjects
5
Module-2 : Relations and Functions
S. No.
Concept
1
Cartesian Product
2
Relation
3
Domain
4
Range of a
Relation
Function
5
6
Range of a
function
7
8
Real Valued
Function
Real Function
9
10
Identity Function
Constant Function
11
Modulus Function
12
Signum Function
13
Greatest Integer
Function
Note:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
Meaning
A×B = {(𝑥, 𝑦): 𝑥𝜖𝐴, 𝑦𝜖𝐵}
Example
A={a,b,c} and B={e,f},
then A×B ={(a,e),(a,f),(b,e),(b,f),(c,e),(c,f)}
A subset of cartesian product If A={1,2,3,4,5,6}, B={2,3,5,7,9}
and R is a relation from A to B such that
A×B
R={(a,b): a𝜖A, b𝜖B and a is the multiple of
b},
then
R={(2,2),(3,3),(4,2),(5,5),(6,2),(6,3)}.
Set of all the first elements of the Domain of R = {𝑎: (𝑎, 𝑏)𝜖 𝑅}
ordered pairs of R
In the above example,
domain of R={2,3,4,5,6}
Set of all the second elements of Range of R = {𝑏: (𝑎, 𝑏)𝜖 𝑅}
the ordered pairs of R
In the above example, range of R={2,3,5}
A relation 𝑓 from a set A to a set
B is called a function from A to
B (denoted by 𝑓: 𝐴 → 𝐵) if every
element of A has one and only
one image in B
Set of all the elements of the
codomain of the function
𝑓 which have pre-image in
domain under 𝑓
A function whose range is the
subset of R
A function whose domain and
range are the subsets of R
𝑓(𝑥) = 𝑥, for each 𝑥𝜖𝑅
𝑓(𝑥) = 𝑐, for each 𝑥𝜖𝑅, where c
is a constant
|−5| = 5
|7| = 7
𝑓(𝑥) = |𝑥|, for each 𝑥𝜖𝑅, i.e.
3
3
−𝑥,
𝑥<0
|0| = 0
|− 2| = 2
𝑓(𝑥) = {
𝑥,
𝑥≥0
−1, 𝑥 < 0
𝑓(𝑥) = { 0, 𝑥 = 0
1, 𝑥 > 0
𝑓(𝑥) = [𝑥] , which assumes the
value of the greatest integer, less
than or equal to x, for each 𝑥𝜖𝑅
[𝑥] = −1, − 1 ≤ 𝑥 < 0
[𝑥] = 2,
2≤𝑥<3
[𝑥] = 0,
0≤𝑥<1
[𝑥] = −2, − 2 ≤ 𝑥 < −1
If any of the sets A and B is empty then A×B is empty.
If any of the sets A and B is infinite then A×B is infinite.
If both the sets A and B are finite then A×B is finite.
Set A is called domain of 𝑓 and the set B is called codomain of f.
If 𝑓: 𝐴 → 𝐵 and 𝑓(𝑥) = 𝑦, then 𝑥𝜖𝐴 𝑎𝑛𝑑 𝑦𝜖𝐵.
If 𝑛(𝐴) = 𝑚 and 𝑛(𝐵) = 𝑛, then 𝑛(𝐴 × 𝐵) = 𝑚𝑛
If 𝑛(𝐴) = 𝑚 and 𝑛(𝐵) = 𝑛, then the number of subsets of A×B = 2𝑚𝑛
If 𝑛(𝐴) = 𝑚 and 𝑛(𝐵) = 𝑛, then the number of relations from A to B = 2𝑚𝑛
6
Exercises for Practice
Level-I
1. If A={1,2,3} and B={4,5}, then find A×B and B×A.
2. If A={𝑢, 𝑣} and B={𝑥, 𝑦}, then find A×B and B×A.
𝑦
3. If (𝑥 + 1, 2 ) = (3,7), then find 𝑥 and 𝑦.
4. If A= {𝑎, 𝑏, 𝑐} and B={𝑥, 𝑦}, then find the number of subsets of A×B.
5. A×B ={(𝑎, 𝑥), (𝑏, 𝑦), (𝑐, 𝑦)}, then find A and B.
6. Which of the following relations are functions, justify your answer:
(i){(2,1), (5,2), (5,3)}
(ii) {(2,1), (3,2), (4,1), (5,1)}
(iii) {(2,1), (3,1), (4,3)}
(iv) {(1,1), (2,2), (3,3), (4,4)}
3
2
3
2
7. If 𝑓(𝑥) = 2𝑥 − 7, then find 𝑓(0), 𝑓(3), 𝑓(−2), 𝑓 ( ) and 𝑓(− ).
𝑓
8. If 𝑓(𝑥) = 7𝑥 − 8 and If 𝑔(𝑥) = 𝑥 3 , then find If 𝑓 + 𝑔, 𝑓 − 𝑔, 𝑓. 𝑔 𝑎𝑛𝑑 𝑔.
Level-II
1.
2.
3.
4.
If A = {−2,2}, then find A×A ×A.
If (𝑥, 1), (𝑦, 2), (𝑧, 2) are in A×B, then find A and B, where 𝑛(𝐴) = 3 and 𝑛(𝐵) = 2.
If A= {1,2,5}, B= {1,2,6,7} and C= {3,5,6}, then find C× (A∪B) and (A×B)∩(A×C).
Write the relation R = {(𝑥, 𝑥 3 ): 𝑥 𝑖𝑠 𝑎 𝑝𝑟𝑖𝑚𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 10} in roster form.
5. If 𝑓(𝑥) = 𝑥 2 , then find
𝑓(1.1)−𝑓(1)
.
1.1−1
6. If 𝑓(𝑥) = 3𝑥 3 − 5𝑥 2 + 9, then find 𝑓(𝑥 − 1).
7. Find the range of the functions
(i)
𝑓(𝑥) = 2𝑥 + 4,
𝑥 > 0, 𝑥 𝜖 𝑅
(ii)
𝑓(𝑥) = 𝑥 2 + 3,
𝑥𝜖𝑅
8. Find the domain of the functions
1
𝑥
(i)
𝑓(𝑥) =
(ii)
𝑓(𝑥) = √2𝑥 + 4
Level-III
1. Find domain of the function𝑓(𝑥) =
𝑥 2 +2𝑥+1
𝑥 2 −8𝑥+12
.
𝑥2
2. Let 𝑓 = {(𝑥, 1+𝑥2 ) : 𝑥 ∈ 𝑅} be a function from R into R. Determine the range of 𝑓.
4−𝑥
3. Find domain and range of the real valued function𝑓(𝑥) = 𝑥−4 .
4. Find domain and range of the real valued function𝑓(𝑥) = −|𝑥 | .
5. Find domain and range of the real valued function𝑓(𝑥) = |2𝑥 − 9 | .
2−𝑥
3𝑥−6
𝑥 2.
6. Find domain and range of the real valued function𝑓(𝑥) = |
|.
7. Find domain and range of the real function𝑓(𝑥) = √16 −
8. If 𝑓 = {(1,1), (2,3), (0, −1), (−1, −3)} be a function from Z to Z defined by 𝑓(𝑥) = 𝑎𝑥 + 𝑏, for some
integers 𝑎 , 𝑏. Determine 𝑎 and 𝑏.
9. If 𝑓(𝑥) = 𝑒 𝑥 and 𝑔(𝑥) = log 𝑒 𝑥, then find (𝑓 + 𝑔) (1) and (𝑓. 𝑔) (1).
10. Let 𝑓 be the subset of Z × Z given by {(𝑎𝑏, 𝑎 + 𝑏): 𝑎, 𝑏𝜖 𝑍} . Is 𝑓 a function from Z to Z. Give
justification.
11. Let 𝐴 = {9,10,11,12,13} and 𝑓: 𝐴 → 𝑁given by 𝑓(𝑛) = highest prime factor of 𝑛. Find the range of 𝑓.
12. Draw the graph of the function 𝑓(𝑥) = |𝑥 − 4 |
7
Module 3 : Principles of Mathematical Induction
Introduction: The principle of mathematical induction is a technique to verify the validity of a
pre-derived formula or statement in terms of natural number n.
Principles of Mathematical Induction: Suppose there is a given statement P(n), where n is a
natural number, such that
(i)
P(n) is true for n=1, i.e. P(1) is true, and
(ii)
If P(n) is true for n = k (where k is some positive integer), i.e. if P(k) is true, then P(n) is also
true for n= k+1, i.e. the truth of P(k) implies the truth of P(k+1), then the statement P(n) is
true for all n𝜖N.
Exercises for Practice
Level-I
1. Prove the following by using mathematical induction, for all 𝑛 ∈ 𝑁
(i)
2𝑛 > 𝑛
(ii)
1 + 3 + 5 + ⋯ … … . . +(2𝑛 − 1) = 𝑛2
𝑛(𝑛+1) 2
]
2
𝑛(3𝑛−1)
(iii)
13 + 23 + 33 + ⋯ … … … + 𝑛3 = [
(iv)
1 + 4 + 7 + ⋯ … … . . +(3𝑛 − 2) =
(v)
2
3
𝑛−1
1 + 3 + 3 + 3 + ⋯ … . . +3
=
2
3𝑛 −1
2
Level-II
1. Prove the following by using mathematical induction, for all 𝑛 ∈ 𝑁
(i)
1
1
1
1
𝑛
+ 5.8 + 8.11 + ⋯ … … + (3𝑛−1)(3𝑛+2) = 6𝑛+4
2.5
1
1
1
1
𝑛(𝑛+3)
(ii)
+ 2.3.4 + 3.4.5 + ⋯ … … + 𝑛(𝑛+1)(𝑛+2) = 4((𝑛+1)(𝑛+2)
1.2.3
(iii)
1 + 2 + 3 + ⋯ … … . . +𝑛 < 8 (2𝑛 + 1)2
(iv)
12 + 32 + 52 + ⋯ … … … + (2𝑛 − 1)2 =
(v)
1
2𝑛 + 7 < (𝑛 + 3)
𝑛(2𝑛−1)(2𝑛+1)
3
2
Level-III
1. Prove the following by using mathematical induction, for all 𝑛 ∈ 𝑁
(i)
102𝑛−1 + 1 is divisible by 11.
(ii)
32𝑛+2 − 8𝑛 − 9 is divisible by 8.
(iii) (𝑎𝑏)𝑛 = 𝑎𝑛 𝑏 𝑛
(iv)
𝑥 2𝑛 − 𝑦 2𝑛 is divisible by 𝑥 + 𝑦.
(v)
7𝑛 − 3𝑛 is divisible by 4.
(vi)
(1 + 𝑥)𝑛 ≥ (1 + 𝑛𝑥), where 𝑥 > −1.
8
Module -4 (Complex Numbers)
1. A complex number z = x + iy, where x, y are real numbers and 𝑖 2 = −1.
2. |𝑥| = |𝑥 + 𝑖𝑦| = √𝑥 2 + 𝑦 2
3. If 𝑧̅ = 𝑥 − 𝑖𝑦 is called conjugate of z
𝑦
4. Arg of z = x + iy , 𝜃 = tan−1 𝑥 .
5. Polar form of 𝑧 = 𝑥 + 𝑖𝑦 is 𝑧 = 𝑟 cos 𝜃 + 𝑖 sin 𝜃
𝑦
where 𝑟 = √𝑥 2 + 𝑦 2 and 𝜃 = tan−1 𝑥 .
2.
3.
4.
LEVEL 1
Express √−25 in combination of real and imaginary number.
Express √−36 + √25 in 𝑎 + 𝑖𝑏 form, where ′𝑎′ and ′𝑏′ are real numbers.
Find the value of 𝑖 −63 .
Compute: 𝑖 643 + 𝑖 644 + 𝑖 645 + 𝑖 646 .
5.
Find the conjugate of 𝑧 =
6.
Find the multiplicative inverse of 𝑧 = 5+12𝑖.
7.
If arg z = 3 and |z|=2, then find z in standard form.
1.
7+2𝑖
4−3𝑖
.
2+3𝑖
𝜋
LEVEL 2
1.
Find the square root of the followings
i) 12 − 5𝑖
2.
ii) i
iii)
8−15𝑖
iv) 7 − 30√2𝑖
24+7𝑖
Convert the following into polar form.
i) 3√3 + 3𝑖
ii)
1−𝑖
1+𝑖
iii) −𝑖
𝑥−𝑖
𝑏
2𝑥
3.
If 𝑎 + 𝑖𝑏 = 𝑥+𝑖, where 𝑥 is a real number, prove that 𝑎2 + 𝑏 2 = 1, 𝑎 = 𝑥 2 −1
4.
Solve the following quadratic equations:
i) 𝑥 2 + 3𝑥 + 9 = 0 ii) √3𝑥 2 + √2𝑥 + 3√3 = 0 iii) 𝑥 2 +
𝑥
√2
+1=0
iv) 𝑥 2 − 𝑥 + 2 = 0
1.
LEVEL 3
𝑣
If (𝑥 + 𝑖𝑦)3 = 𝑢 + 𝑖𝑣, 𝑡ℎ𝑒𝑛 show that 𝑥 + 𝑦 = 4(𝑥 2 − 𝑦 2 )
2.
If α and β are the different complex numbers. With | β |=1, then find the value of |1 − α̅β|.
3.
Find the number of non zero integral solutions of the equation |1 − 𝑖|𝑥 = 2𝑥
4.
If (1−𝑖) = 1, find the least positive integral value of 𝑛.
5.
Convert the complex number 𝑧 =
𝑢
β−α
1+𝑖 𝑛
𝑖−1
𝜋
3
𝜋
3
cos + 𝑖𝑠𝑖𝑛
9
in the polar form.
Module 5 (Linear Inequalities)
LINEAR INEQUALITIES
Rules for solving Linear Inequalities
The following rules can be applied to any inequality
 Add or subtract the same n umber or expressions to both the sides.
 Multiply or divide both the sides by same positive number
 The inequality reverses if it is multiplied or divide by a negative number
 a<b⇔b>a
 a < b ⇔ 1/a > 1/b
 x2 ≤ a2 ⇔ x ≤ a and x ≥ -a
Points to remember
 x = 0 is y-axis
 y = 0 is x-axis
 x = k is a line parallel to y-axis passing through (k, 0) of x-axis.
 y = k is a line parallel to x-axis passing through (0, k) of y-axis.
Method to find Graphical Solution
1. Draw lines corresponding to each equation treating it as equality.
2. Find the feasible region: intersection of all the inequalities.
LEVEL – I
1. Solve 24x < 100 when x is a natural number.
2. Solve the inequalities and draw the graph of the solution on number line: 3x – 2 < 2x + 1.
3. Solve the following system of inequalities: 2x + y ≥ 8, x + 2y ≥ 10.
4. Solve the following system of inequalities graphically: x ≥ 3, y ≥ 2.
5. Solve the following system of inequalities graphically: x + y ≤ 9, y>x, x≥0.
6. Find all pairs of consecutive odd positive integers both of which are smaller than 10 such that there sum is
more than 11.
LEVEL – II
1. Solve the inequalities and draw the graph of the solution on number line:
5(2x – 7) – 3(2x + 3) ≤ 0.
2. Solve the following system of inequalities graphically: x + y ≤ 400, 2x + y ≤ 600.
3. Solve the following system of inequalities graphically: 2x + 19 ≤ 6x + 47.
4. The longest side of a triangle is 3 times the shortest side and the third side is 2 cm shorter than the longest
side. If the perimeter of the triangle is at least 61 cm, find the minimum length of the shortest side.
LEVEL – III
1. Solve the following system of inequalities: x ≥ 3, y ≥ 2.
2. Solve the following system of inequalities: 2x + y ≥ 3, 3x + 5y ≥ 2.
3. Solve the following system of inequalities: 3x – 7 < 5 + x, 11 -5x ≤ 1.
4. How many litres of water will have to be added to 1125 litres of 45% solution of acid so that the resulting
mixture will contain more than 25% but less than 30% of acid content?
LEVEL – IV
1. Solve the following system of inequalities: 3( 1 – x) < 2(x + 4)
2. Solve the following system of inequalities: x + y ≥ 1, x ≤ 5, y≤ 4
3. Solve the following system of inequalities:
5𝑥−2
3
−
7𝑥 −3
3
𝑥
>4
4. A manufacturer has 600 litres of a 12% solution of acid. How many litres of a 30% acid solution must be
added to it so that acid content in the resulting mixture will be more than 15% but less than 18%?
10
Module 6: Binomial Theorem
Basic concepts:
Factorial :
Binomial expansions:
n Factorial is denoted by n!
n! = n (n-1)(n-2) …3.2.1
Ex:
5! = 5.4.3.2.1=120
6! = 6.5.4.3.2.1= 720
(a + b)0 = 1
(a + b)1 = a + b
(a + b)2 = a2 + 2ab + b2
(a + b)3 = a3 + 3a2 b +3a b2 +b3
(a + b)4 =a4 +4a3 b +6a2 b2 + 4a b3 + b4
Pascal’s Triangle
1
n=0
n=1
n=2
n=3
n=4
n=5
1
1
`
1 2 1
1 3 3 1
1 4 6 4 1
1
5 10 10 5 1
The formal expression of the Binomial Theorem is as follows:
(a + b)n = nC0 +nC1 an-1 b +nC2 an-2 b2 + nC3 an-3 b3 + … +nCnbn, where n is a positive integer.
General term in the expansion of (a + b)n,
Tr+1= nCr an-r br, where 0 ≤ r ≤n and nCr =
𝑛!
𝑟!(𝑛−𝑟)!
Middle Terms:
In a expansion of (1+x)n where n is a positive integer
(i)
When n is even, there are (n+1) terms i.e. odd number of terms in the expansion and hence only
one middle term
i.e. T(n/2) +1
(ii)
When n is odd, there are (n+1) terms i.e even number of terms in the expansion and hence there
are two middle terms
i.e. T(n+1)/2 and T(n+3)/2.
Practice Questions
LEVEL 1
2
𝑥 5
1. Expand (a) (1 − 2x)5 (𝑏) (𝑥 − 2)
2. Compute 985 using binomial theorem.
3. Find the coefficient of x 5 in the expansion of (x + 3)8 .
4. Write the general term in the expansion of (2x − 3𝑦 2 )6
LEVEL 2
1. Using binomial theorem, indicate which number is larger:(1.1)10000 or 1000.
𝑥3 7
)
6
3
expansion of(3 𝑥 2
2
2. Find the middle term of the expansion of (3 −
1
3. Find the term independent of x in the
− 3𝑥)6.
4. Find a positive value of m for which the coefficient 𝑥 of in the expansion (1 + x)𝑚 is 6.
LEVEL 3
1.3.5.7…..(2𝑛−1) 𝑛 𝑛
1. Show that the middle term in expansion of (1 + x)2𝑛 is
2 𝑥 , where n is a positive
𝑛!
integer.
2. Find a, b and n in the expansion of (a + b)𝑛 if the first three terms of the expansion are 729, 7290 and
30375 respectively.
3. Find the coefficient of 𝑥 5 in the product (1 + 2x)6 (1 − x)7 using binomial theorem.
4. The coefficient of the (r − 1)𝑡ℎ , 𝑟 𝑡ℎ 𝑎𝑛𝑑 (r + 1)𝑡ℎ terms in the expansion of (x + 1)𝑛 are in the ratio
1:3:5, find n and r.
11
Module 7: Conic Sections
Conic Sections – Introduction
 A conic is a shape generated by intersecting two lines at a point and rotating one line around the
other while keeping the angle between the lines constant.
 The resulting collection of points is called a right circular cone. The two parts of the cone
intersecting at the vertex are called nappes.
 A “conic” or conic section is the intersection of a plane with the cone.
 The plane can intersect the cone at the vertex resulting in a point.

The plane can intersect the cone perpendicular to the axis resulting in a circle.
𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑎 𝐶𝑖𝑟𝑐𝑙𝑒 ∶ ( 𝑥 − ℎ)2 + ( 𝑦 − 𝑘)2 = 𝑟 2

The plane can intersect one nappe of the cone at an angle to the axis resulting in an ellipse
Standard Equation of an Ellipse:
Horizontal ellipse
Vertical ellipse
𝑥2
𝑎2
𝑥2
𝑏2
+
+
𝑦2
𝑏2
𝑦2
𝑎2
= 1 (𝑎2 >𝑏 2 )
= 1 (𝑏 2 < 𝑎2 )
12
 The plane can intersect one nappe of the cone at an angle to the axis resulting in a parabola.
Standard Equation of a Parabola:
i) y2 = 4ax
ii) y2 = -4ax
iii) x2 = 4ay
iv) x2 = -4ay
 The plane can intersect two nappes of the cone resulting in a hyperbola.
Standard equations of hyperbola:
Horizontal hyperbola
Vertical hyperbola
𝑥2
𝑎2
𝑦2
𝑎2
-
𝑦2
𝑏2
𝑥2
𝑏2
= 1
(𝑎2 > 𝑏 2 )
= 1 (𝑏 2 > 𝑎2 )
13
E X E R C I S E for P R A C T I C E
LEVEL 1
1. Find the equation of the circle with centre (0, 2) and radius 2
2. Find the equation of the circle passing through the point (4,1) and (6,5) and whose centre is on line 4x+ y
= 16.
3. Find the coordinates of the focus, axes of the equation of the directrix and lattus rectum of the parabola
𝑦 2 = 8x
4. Find the equation of the ellipse that satisfy the given condition vertex (±5, 0) and (±4, 0).
LEVEL 2
1. Find the equation of the circle which passes through the points (2, -2) and (3, 4) and whose centre lies on
the line x+y = 2.
2. Find the equation of parabola which is symmetric about the y axis and passes through the point (2,-3).
3. Find the coordinates of the foci, the vertices, the length of major and minor axes and the eccentricity of
the ellipse 9𝑥 2 +4𝑦 2 =36.
4. Find the equation of hyperbola with foci (0, ±3) and vertices (0, ±11/2).
LEVEL 3
1. If a parabolic reflector is 20cm in diameter and 5cm in deep, find the focus.
2. An arch is in the form of semi ellipse. It is 8m wide and 2m high at the centre find the height of the arch
at a point 1.5m from one end.
3. An equilateral triangle is inscribed in the parabola 𝑦 2 = 4ax, where one vertex is at the vertex of the
parabola. Find the length of the side of the triangle.
4. A rod of length 12cm moves with its ends always touching the coordinate axes. Determine the equation
of the locus of a point P on the rod which is 3cm from the end in contact with the axes.
14
Module 8: Three Dimensional Geometry
DISTANCE FORMULA : Let 𝐴(𝑥1 , 𝑦1 , 𝑧1 ) and 𝐵(𝑥2 , 𝑦2 , 𝑧2 ) are two points in space then
AB = |√(𝒙𝟐 − 𝒙𝟏 )𝟐 + (𝒚𝟐 − 𝒚𝟏 )𝟐 + (𝒛𝟐 − 𝒛𝟏 )𝟐 | units
SECTION FORMULA : Let C (𝑥 , 𝑦 , 𝑧) divides the join of 𝐴(𝑥1 , 𝑦1 , 𝑧1 ) and 𝐵(𝑥2 , 𝑦2 , 𝑧2 ) in the ratio of m :
n,A
(i)
If C is internal point on AB then
𝑥=
𝑚𝑥2 +𝑛𝑥1
,
𝑚+𝑛
𝑦=
𝑚𝑦2 +𝑛𝑦1
𝑚+𝑛
, 𝑧=
𝑚𝑧2 +𝑛𝑧1
𝑚+𝑛
(ii) If C is external point on AB then
𝑥=
𝑚𝑥2 − 𝑛𝑥1
𝑚−𝑛
,
𝑦=
𝑚𝑦2 − 𝑛𝑦1
𝑚− 𝑛
, 𝑧=
𝑚𝑧2 − 𝑛𝑧1
𝑚−𝑛
Level -1
1.
2.
3.
4.
5.
A point is on the z-axis, what are the x & y- coordinate.
A point is in XY- plane, what is its Z-coordinate?
Show that point P(-2,3,5),Q(1,2,3) and R(7,0,-1) are collinear.
Verify that (-1,2,1) ,(1,-2,5),(4,-7,8) and (-2,3,4) are the vertices of a parallelogram.
Find the coordinate of the points which divides the line segment joining the points (-2, 4, 7) and (3,-5, 8) in ratio 2:3
internally.
Level -2
6. Find the equation of set of points which are equidistant from the points (1, 2, 3) and (3, 2,-1)
7. Using section formula, prove that the points (-4, 6, 10), (2, 4, 6) and (14, 0,-2) are collinear.
8. Find the ratio in which the line segment joining the points (4, 8, 10) and (6, 10,-8) is divided by the YZ-plane.
9. Using section formula prove that the points (-4, 6, 10), (2, 4, 6) and (14, 0,-2) are collinear.
10. Find the coordinates of the points which trisect the line segment joining the points P (4, 2,-6) and Q (10,-16, 6)
Level -3
11. Find the equation of the set of the point P such that its distance from the points A (3, 4,-5) and B (-2, 1, 4) are equal,
12. Show that the points A(1,2,3), B(-1,-2,-1), C(2,3,2) and D(4,7,6) are the vertices of a parallelogram ABCD, but it is not a
rectangle.
13. Find the coordinates of a point on Y-axis which are at a distance of 5√2 from the point P(-3,2,5).
15
Module 9: (Limit And Derivatives)
LIMITS
Limit of a function at a point is the value of the function at the points just before and after the given point.
Right Hand Limit:The value of a function at a point just immediate after the given point is called the right hand limit of the
function at that given point
RHL of f(x) at x = a is given by lim 𝑓(𝑎 + ℎ) , here h is positive
ℎ→0
Left Hand Limit:The value of a function at a point just immediate before the given point is called the Left hand limit of the
function at that given point
LHL of f(x) at x = a is given by lim 𝑓(𝑎 − ℎ) , here h is positive
ℎ→0
If LHL and RHL of a function at a point are equal then this value of LHL or RHL is called Limit of the function at that point.
Limit of the function f(x) at x = a is given by lim 𝑓(𝑎 + ℎ)=lim 𝑓(𝑎 − ℎ)
ℎ→0
Note:
ℎ→0
If both limits are not equal then the value of the limit of the function cannot be found at that point.
DERIVATIVES
1. Definition: Suppose f is a real valued function and a is a point in its domain of definition. The derivative of f at a is
defined by
𝑓(𝑎 + ℎ) − 𝑓(𝑎)
lim
ℎ→0
ℎ
provided this limit exists. Derivative of f(x) at a is denoted by f ‘(a).
2. Definition: Suppose f is a real valued function ,the function defined by
𝑓(𝑥 + ℎ) − 𝑓(𝑥)
lim
ℎ→0
ℎ
wherever the limit exists is defined to be the derivative of f at x and is denoted by f’(x) . This definition of derivative is also
called the first principle of derivative.
𝑓(𝑥+ℎ)−𝑓(𝑥)
Thus, f’(x) = lim
ℎ→0
ℎ
Properties:
1. lim [𝑓(𝑥) ± 𝑔(𝑥)] = lim 𝑓(𝑥) ± lim 𝑔(𝑥)
𝑥⟶𝛼
𝑥→𝛼
𝑥→𝛼
2. lim [𝑓(𝑥). 𝑔(𝑥)] = lim 𝑓(𝑥). lim 𝑔(𝑥)
𝑥⟶𝛼
3. lim [
𝑓(𝑥)
]=
𝑥⟶𝛼 𝑔(𝑥)
𝑥 𝑛 − 𝑎𝑛
4. lim
𝑥→𝑎
5. lim
𝑥→0
6. lim
𝑥→0
𝑥−𝑎
sin 𝑥
𝑥→𝛼
lim 𝑓(𝑥)
𝑥→𝛼
𝑥→𝛼
lim 𝑔(𝑥)
𝑥→𝛼
= 𝑛𝑎𝑛−1
=1
𝑥
1−cos 𝑥
𝑥
=0
7. The derivative of a function f at a is defined by
𝑓(𝑎 + ℎ) − 𝑓(𝑎)
𝑓 ′ (𝑎) = lim
ℎ→0
ℎ
8. Derivative of a function f at any point x is defined by
𝑑 𝑓(𝑥)
𝑓(𝑥 + ℎ) − 𝑓(𝑥)
𝑓 ′ (𝑥) =
= lim
ℎ→0
𝑑𝑥
ℎ
9. For functions u and v, the following holds:
a) (𝑢 ± 𝑣)′ = 𝑢′ ± 𝑣 ′
b) (𝑢𝑣)′ = 𝑢′ 𝑣 + 𝑢𝑣′
𝑢 ′
𝑢′𝑣−𝑢𝑣′
c) ( ) =
𝑣
𝑣2
10. Standard Derivatives:
𝑑
a)
(𝑥 𝑛 ) = 𝑛 𝑥 𝑛−1
𝑑𝑥
b)
c)
𝑑
𝑑𝑥
𝑑
𝑑𝑥
(sin 𝑥) = cos 𝑥
(cos 𝑥) = −sin 𝑥
16
EXERCISE FROM LIMITS
Level 1
1. Evaluate :
(i) lim 2𝑥 + 3
(ii) lim 3𝑧 − 4
𝑥→3
𝑧→1
(iii) lim
𝑎𝑥+𝑏
𝑥→𝑜 𝑐𝑥+𝑑
LEVEL 2
1. Evaluate
(i) lim
𝑥→1
1
𝑥 15 −1
(ii) lim
𝑧 3 −1
(iii) lim
1
𝑧→1 𝑧6−1
𝑥 10 −1
𝑆𝑖𝑛 𝑎𝑥
(iv) ) lim
𝑏𝑥
𝑥→𝑜
(1+𝑥)5 −1
𝑥→𝑜
𝑥
LEVEL 3
1. Evaluate
𝑆𝑖𝑛(𝜋−𝑥)
(i) lim
𝑥→𝑜 𝜋(𝜋−𝑥)
(ii) lim
𝑆𝑖𝑛 𝑎𝑥+𝑏𝑥
𝑥→𝑜 𝑎𝑥+𝑠𝑖𝑛𝑏𝑥
,𝑎 +𝑏 ≠ 0
(iii) lim𝜋
𝑥→
2
tan 2𝑥
(iv) )lim
𝜋
2
𝐶𝑜𝑠2𝑥 −1
𝑥→𝑜 𝐶𝑜𝑠𝑥−1
𝑥−
𝑎 + 𝑏𝑥, 𝑖𝑓 𝑥 < 1
2. Suppose f(x)={ 4, 𝑖𝑓 𝑥 = 1 and lim𝑓(𝑥) = 𝑓(1) , then what are the possible values of a and b.
𝑥→1
𝑏 − 𝑎𝑥, 𝑖𝑓 𝑥 > 1
|𝑥|
, 𝑖𝑓 𝑥 ≠ 0
5. Evaluate lim𝑓(𝑥), if f(x)={ 𝑥
𝑥→0
0, 𝑖𝑓 𝑥 = 0
EXERCISE FROM DERIVATIVES
LEVEL 1
1. Find the derivatives of
3
(i) 𝑥 3 -27 (ii) 2x - (iii) 5𝑥 3 −2x +4 (iv) (7x-3) (4𝑥 2 -2)
4
LEVEL 2
𝑥−𝑎
1
1. Find the derivatives of (i) (x-a)(x-b)
(ii)
(iii) x+
(v)
𝑝𝑥 2 +𝑞𝑥+𝑟
𝑎𝑥+𝑏
LEVEL 3
1 Find the derivatives of
𝑟
(vi) (px+q)( + 𝑠) (vii)
𝑥
(i)
𝑆𝑖𝑛𝑥+𝐶𝑜𝑠𝑥
𝑆𝑖𝑛𝑥−𝐶𝑜𝑠𝑥
(ii)
1
1+
𝑥
1
1−
𝑥
𝑥−𝑏
𝑥
(viii) (ax+b)n(cx+d)m
𝑆𝑖𝑛(𝑥+𝑎)
(iii) Sin2x
𝑐𝑜𝑠𝑥
2. Find the derivatives (i) (x + sec x)(x-tanx)
3. Find the derivatives using first principal (i) tanx
(iv) Sin(x+1)
𝑠𝑒𝑐𝑥−1
(ii)
sec + 1
(ii) sin (x+ 1)
17
𝜋
(iii)
𝑥 2 𝑐𝑜𝑠 4
sin 𝑥
Module 10: Mathematical Reasoning
INTRODUCTION
Mathematical reasoning is the critical skill that enables a student to make use of all other mathematical skills. With the
development of mathematical reasoning, students recognize that mathematics makes sense and can be understood. There
two types of mathematical reasoning namely inductive and deductive reasoning.
STATEMENTS
A sentence is called a mathematically acceptable statement if it is either true or false but not both. For example Sum of
two even numbers is always even. It is always true statements.
NEGATION OF STATEMENTS
The denial of a statement is called negation of statements. Negation of p is denoted by ∼ 𝑝
For example p : Delhi is the capital of India.
~p : Delhi is not the capital of India.
COMPOUND STATEMENTS
A compound statement is combination of two or more simple statements using a connective. Each statement is called
component statements. Some of the connecting words like ‘OR’ and ‘AND ‘etc. are called connectives.
Compound Statements
(i) Conjunction statement : When two statements are combined by using connective AND then resulting statement is
called as conjunction. Conjunction of p, q is denoted by p ∧ 𝑞
For example p: A square is a quadrilateral.
q: A square has all its sides equal.
Conjunction: A square is a quadrilateral and its four sides are equal.
(ii) Disjoint statement : When two statements are combined by using connective OR then resulting statement is called as
conjunction. Disjunction of p, q is denoted by p ∨ 𝑞
For example p: A square is a quadrilateral.
q: A square has all its sides equal.
Disjunction: A square is a quadrilateral or its four sides are equal.
(iii) Conditional statement : When two statements are combined by using connective IF AND THEN, then resulting
statement is called as conditional statement. Conditional statement of p, q is denoted by p ⇒ 𝑞 and read as If p then q
For example p: A square is a quadrilateral.
q: A square has all its sides equal.
Conditional statement: If a square is a quadrilateral then its four sides are equal.
(iv) Biconditional statement : When two statements are combined by using connective IF AND ONLY IF, then resulting
statement is called as biconditional statement. Biconditional statement of p, q is denoted by p ⇔ 𝑞 and read as p if and
only if q
For example p: A square is a quadrilateral.
q: A square has all its sides equal.
Biconditional statement: A square is a quadrilateral if and only if its four sides are equal.
CONTRAPOSITIVE, CONVERSE AND INVERSE OF A STATEMENT : If p ⇒ 𝑞 is given then
CONVERSE STATEMENT
q⇒𝑝
INVERSE STATEMENT
∼p⇒∼𝑞
CONTRAPOSITIVE STATEMENT
∼q ⇒∼ 𝑝
VALIDATING A CONDITIONAL STATEMENT
BY DIRECT METHOD: By assuming p is true, prove that q must be true.
BY CONTRAPOSITIVE METHOD: By assuming q is false, prove that p must be false
BY CONTRADICTION : Here to check whether a statement p is true, we assume that p is not true. Then we arrive at
some result which contradicts our assumption. Therefore, we conclude that p is true.
18
LEVEL -1
(1) (a) Define statement with example. How it is different from sentence.
(b) What do you mean by negation of statements? Give an example.
(c) Define compound statements with examples.
(d) What are connective and quantifiers? Give a example for each one.
(2) Check whether the following are statements
(a) 8 is less than 6.
(b) Mathematics is fun.
(c) The square of a number is always is an even number.
(3) Write the negation of following statements
(a) New Delhi is a city.
(b) √3 is an irrational number.
(4) Write the conjunction of following statements
(a)There is something wrong with the bulbs.
(b)There is something wrong with the wiring.
LEVEL-2
(1) Find the component statements of following compound statements.
(a) The sky is blue and grass is green.
(b) It is raining or it is cold.
(2) Write the converse of the statement “If a number n is even, then n 2 is even.”
(3) For the given statements identify the necessary and sufficient conditions
If you drive over 80 m per hour, then you will get a fine.
(4)Write the contrapositive and converse of the following statements
“If x is a prime number, then x is odd”
LEVEL-3
(1) Prove √2 is an irrational number by contradiction.
(2) For each of the following statements, determine whether an inclusive ‘Or’ or exclusive ‘Or’
is used. Give reason for your answer.
a.
To enter a country, you need a passport or a voter registration card.
b.
Two lines intersect at a point or are parallel.
19
Module 11: STATISTICS
Statistics deals with the analysis of data; statistical methods are developed to analyze large volumes of data and their
properties.
Frequency: In statistics the frequency (or absolute frequency) of an event is the number of times the event occurred in an
experiment or study. These frequencies are often graphically represented in histograms
Mean or Average: Mean or average, in theory, is the sum of all the elements of a set divided by the number of elements
in the set. Mean could be treated as a collaborative property of the whole set of values. You can get a fairly good idea
about the whole set of data by calculating its mean.
Median: Median is the middle value of a set arranged in increasing or decreasing order. So, if a set consists of odd
number of elements, then the middle value is the median of the set, and if the set consists of an even number of sets, then
the median is the average of the two middle values.
Mode: The mode in a dataset is the value that is most frequent in a dataset. Like mean and median, mode is also used to
summarize a set with a single piece of information.
For example, the mode of the dataset S = {1,2,3,3,3,3,3,4,4,4,5,5,6,7} is 3 since it occurs the maximum number of times
in the set S.
Variance: You may want to measure the deviation of a set of data from the mean value. For example, a huge variance of
the household income data of a country may be interpreted as an economy with high inequality. Many useful
interpretations can be carried out by analyzing the variance in data. The variance is obtained by:



Finding out the difference between the mean value and all the values in the set.
Squaring those differences.
Average of the squared differences obtained above is the variance of the given data.
Thus, we can observe that the variance of the particular dataset is always positive.
Standard Deviation: The standard deviation is calculated by square rooting the variance of the data. The standard
deviation gives a more accurate account of the dispersion of values in a dataset.
Statistics Formula Sheet used in class XI
The important statistics formulas are listed in the chart below:
In series Data:
𝑛
𝑋̅ =
1
∑ 𝑥𝑖
𝑛
x = value of observation
𝑖=1
Mean
In frequency data:
𝑋̅ =
n = Total number of observations
𝑛
𝑛
𝑖=1
𝑖=1
1
∑ 𝑓𝑖 𝑥𝑖 , 𝑤ℎ𝑒𝑟𝑒 𝑁 = ∑ 𝑓𝑖
𝑁
If n is odd, then
Median
𝑛+1
𝑀𝑒 = (
)𝑡ℎ 𝑡𝑒𝑟𝑚
2
If n is even, then
𝑛
𝑛
[( ) 𝑡ℎ 𝑡𝑒𝑟𝑚 + ( + 1) 𝑡ℎ 𝑡𝑒𝑟𝑚]
2
2
𝑀𝑒 =
2
In Grouped Data:
𝑛
−𝑐
𝑀𝑒 = 𝑙 + (2
)ℎ
𝑓
n = Total number of items
Note: To find median, first we
arrange the given data in increasing
or decreasing order.
l=lower limit of median class
c=cumulative frequency of the class
preceding to the median class
f=frequency of median class
h= class width of median class
Mode
The value which occurs most frequently
l = lower limit of modal class
20
In Grouped Data:
(highest frequency class)
f1 = frequency of modal class
f0 = frequency of the class preceding
modal class
f2 = frequency of the class next to
modal class
𝑓1 − 𝑓0
𝑀𝑜 = 𝑙 + (
)ℎ
2𝑓1 − 𝑓0 − 𝑓2
In Series Data:
𝑛
1
∑ |𝑥𝑖 − 𝑥̅ |
𝑛
𝑀. 𝐷. (𝑥̅ ) =
Mean Deviation
about mean
n = total number of observations in
series data
𝑖=1
𝑛
In Frequency Data:
𝑤ℎ𝑒𝑟𝑒 𝑁 = ∑ 𝑓𝑖
𝑛
1
𝑀. 𝐷. (𝑥̅ ) = ∑ 𝑓𝑖 |𝑥𝑖 − 𝑥̅ |
𝑁
𝑖=1
𝑖=1
In Series Data:
𝑛
1
𝑀. 𝐷. (𝑀𝑒) = ∑ |𝑥𝑖 − 𝑀𝑒|
𝑛
n = total number of observations in
series data
Me = median
𝑖=1
Mean Deviation
about median
In Frequency Data:
𝑛
𝑛
1
𝑀. 𝐷. (𝑀𝑒) = ∑ 𝑓𝑖 |𝑥𝑖 − 𝑀𝑒|
𝑁
𝑤ℎ𝑒𝑟𝑒 𝑁 = ∑ 𝑓𝑖
𝑖=1
𝑖=1
For Ungrouped Data:
𝑛
1
𝜎 = √ ∑(𝑥𝑖 − 𝑥̅ )2
𝑛
𝑖=1
For Discrete Frequency Data:
x = Items given
x¯ = Mean
n = Total number of items
𝑛
Standard
Deviation, 𝜎
1
𝜎 = √ ∑ 𝑓𝑖 (𝑥𝑖 − 𝑥̅ )2
𝑁
𝑛
𝑤ℎ𝑒𝑟𝑒 𝑁 = ∑ 𝑓𝑖
𝑖=1
𝑖=1
For Continuous Frequency Data,
𝑛
𝑛
𝑖=1
𝑖=1
1
𝜎 = √ ∑ 𝑓𝑖 (𝑥𝑖 )2 − (∑ 𝑓𝑖 𝑥𝑖 )
𝑁
2
𝑥𝑖 − 𝐴
𝑦=
ℎ
For series with equal means, the
series with lesser standard deviation
is more consistent or less scattered.
Shortcut Method
𝜎=
2
ℎ
√𝑁 ∑ 𝑓𝑖 𝑦𝑖2 − (∑ 𝑓𝑖 𝑦𝑖 )
𝑁
For Ungrouped Data,
𝑛
1
𝜎 . = ∑(𝑥𝑖 − 𝑥̅ )2
𝑛
2
Variance
𝑖=1
𝑛
For Frequency Data
𝑛
𝜎 2. =
x = Items given
x¯ = Mean
n = Total number of items
1
∑ 𝑓𝑖 (𝑥𝑖 − 𝑥̅ )2
𝑁
𝑤ℎ𝑒𝑟𝑒 𝑁 = ∑ 𝑓𝑖
𝑖=1
𝑖=1
Coefficient of
variation
𝜎
× 100 ; 𝑥̅ ≠ 0
𝑥̅
Note : Among two given data , the data having
larger value of C.V. is more variable than other.
C. V. =
21
Where 𝜎 is standard deviation and 𝑥̅
is arithmetic mean
Level – I
1. Find the Arithmetic Mean of the following series data: 7, 8, 9, 12, 14, 18, 20.
2. Find median of the following data: 24, 12, 7, 10, 19, 20, 26, 28.
3. Find the mode of following data: 22, 22, 21, 18, 19, 21, 18, 19, 10, 10, 20, 15, 15, 16, 17, 18, 20, 18, 17, 11,
14, 18, 19, 21, 22.
4. Find mean deviation about mean for the following data: 6, 9, 7, 9, 10, 12, 13, 8 , 12, 15, 2
Level - II
5. Find the mean deviation about median for the following data: 12, 3, 18, 17, 9, 17, 19, 20, 17, 11, 3, 23, 14, 16,
10, 12, 8, 7, 5,6
6. Find standard deviation of the following data: 12, 14, 10, 8, 6, 4, 2, 12, 14, 18, 12, 25, 25
7. Find coefficient of variation of data: 12, 8, 10, 12, 16, 18, 20, 22, 10, 12, 13, 15, 16, 18
8. Find the mean deviation about mean of following data:
X
0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60
f
12
18
27
20
17
6
9. Find the mean, median and mode of the following data:
x
0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60
f
4
12
5
11
8
7
Level – III
10. The mean of 5 observations is 4.4 and their variance is 8.24. If three of the observations are 1, 2 and 6. Find
the other two observations.
11. The means and standard deviation of marks obtained by 50 students of a class in three subjects, Mathematics,
Physics and chemistry are given below:
Subject
Mathematics
Physics
Chemistry
Mean
42
32
40.9
Standard Deviation
12
15
20
Which of the three subjects shows the highest variability in marks.
12. Find the standard deviation and variance of the following data:
X
0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60
f
8
5
6
4
9
10
13. Find the mean, variance and standard deviation using short-cut method:
Height in cms
70-75 75-80 80-85 85-90 90-95 95-100
100-105
105-110
110-115
No. of Children
3
4
7
7
15
9
6
6
3
22
Module 12: Probability
Basic concepts:
Random Experiment: An experiment is called random experiment if it satisfies the following two
conditions:
(i) It has more than one possible outcome.
(ii) It is not possible to predict the exact outcome in advance.
Sample Space, S: The set of all possible outcomes of a random experiment is called sample space of the
experiment.
Examples:
1. Sample space of tossing two coins
S = {HH, HT, TH, TT}
2. Sample space of throwing a die
S = {1, 2, 3, 4, 5, 6}
Types of events:(i) Impossible events, ∅:
An event which never occurs is called an impossible event.
For Example, getting even numbers on tossing a coin
(ii) Sure event:
An event which is certain to happen is called sure event
The whole sample space is a sure event.
(iii) Simple event:
If an event E has only one sample point of a sample space is called simple event.
Example: E = {a}
(iv) Compound event
If an event has more than one sample point, it is called compound event.
Example S = {1, 2, 3, 4, 5, 6}
E1 = {2, 4, 6}, E2= {1, 3}
Algebra of events:(i)
Complementary events :
For every event A there corresponds another event Ac is called complementary event to A.
It is also called the event not A.
For Example, for tossing a coin, S = {1, 2, 3, 4, 5, 6},
For the event of getting even numbers A = {2, 4, 6},
Then Ac = {1, 3, 5}
(ii)
The event A or B :
When the sets A and B are two events associated with a sample space,
then A or B = A U B = {𝜔: 𝜔𝜖 𝐴 𝑜𝑟 𝜔: 𝜔𝜖 𝐵 }
(iii)
The event A and B :
When the sets A and B are two events associated with a sample space,
then A and B = A ∩ B= {𝜔: 𝜔𝜖 𝐴 𝑎𝑛𝑑 𝜔: 𝜔𝜖 𝐵 }
(iv)
The event A but not B :
A but not B = A∩ B c = A – B = {𝜔: 𝜔𝜖 𝐴 𝑎𝑛𝑑 𝜔: 𝜔 ∉ 𝐵 }
Practice Questions:
LEVEL 1
(1) A coin is tossed if it shows head, we draw a ball from a bag consisting of three blue and four white
balls if it shows tail we throw a die .Describe the sample space of this experiment.
(2) A die is rolled. Let E be the event “die shows 4 “and F be the event “die shows even number”. Are E
and F mutually exclusive?
(3) A fair coin with 1 marked on one face and 6 marked on the other and a fair die are both tossed. Find
the probability that the sum of numbers that turn up is (i) 3 (ii) 12
(4) A coin is tossed twice .What is the probability that at least one tail occurs.
LEVEL 2
(1) A box contains one red and three identical white balls. Two balls are drawn at random in succession
without replacement. Write the Sample space for this experiment.
23
(2) Three coins are tossed once. Let A denote the event three heads shows denote the event two heads
and one tail shows ,C denote the event three tails shows and D denote the event head shows on the
first coin. Which events are (a) mutually exclusive (b) Simple (c) Compound?
(3) In a lottery a person choses six different natural numbers at random from 1 to 20, and these six
numbers match with the six numbers already fixed by the lottery committee he wins the prize. What
is the probability of winning the prize in the game?
(4) Find the probability that when a hand of 7 cards is a drawn from a well shuffled deck of 52 cards, it
contains (i) all kings (ii) three kings (iii) at least three kings?
LEVEL 3
(1) A die is thrown repeatedly until a six comes up. What is the sample space for this experiment?
(2) A committee of two persons is selected from two men and two women .What is the probability that
the committee will have (i) no man (ii) one man (iii) two men?
(3) In a class XI of a school 40% of the students study Mathematics and 30% study Biology .10% of the
class study both Mathematics and Biology. If a student is selected at random from the class find the
probability that he will be studying Mathematics or Biology?
(4) A box contains 10 red marbles, 20 blue marbles and 30 green marbles. If 5 marbles are drawn from
the box, what is the probability that (i) All will be blue (ii) At least one will be green?
24
KENDRIYAVIDYALAYA SANGATHAN, RAIPUR REGION
Blue-Print
Mathematics
Class XI
(Half Yearly Examination)
TEMPORARILY ADJUSTIBLE
Time: 3 hours
Unit I
Unit II
S.
N
o.
1
2
3
4
5
6
7
8
9
Max Marks: 100
Topics
Sets
Relations and
Functions
Trigonometry
Principle of
Mathematical
Induction
Complex Numbers and
Quadratic Equations
Linear Inequalities
Permutations and
Combinations
Binomial Theorem
Sequences and Series
Total
Very Short
Answer
(1 marks)
1(1)
_
Short
Answer
(2 marks)
2(1)
4(2)
1(1)
_
4(1)
8(2)
Long
Answer-2
(6 marks)
6(1)
_
13(4)
12(4)
2(1)
_
8(2)
_
6(1)
6(1)
17(5)
6(1)
_
2(1)
8(2)
_
10(3)
_
_
_
4(2)
4(1)
4(1)
6(1)
_
10(2)
8(3)
1(1)
1(1)
4(4)
_
2(1)
16(8)
4(1)
4(1)
44(11)
6(1)
6(1)
36(6)
11(3)
13(4)
100(29)
25
Long Answer1 (6marks)
Total
Marks
KENDRIYA VIDYALAYA SANGATHAN, RAIPUR REGION
BLUE PRINT
Mathematics
CLASS-XI
Session Ending Examination
Unit
I
II
S.
No.
1
2
3
4
5
6
7
III
IV
V
VI
8
9
10
11
12
13
14
15
16
Topics
Sets
Relations and Functions
Trigonometry
Principle of Mathematical
Induction
Complex Numbers and
Quadratic Equations
Linear Inequalities
Permutations and
Combinations
Binomial Theorem
Sequences and Series
Straight Lines
Conic Sections
Introduction to ThreeDimensional Geometry
Limits and Derivatives
Mathematical Reasoning
Statistics
Probability
Total
Very Short
Answer
(1 marks)
Short
Answer
(2 marks)
1(1)
2(1)
2(1)
1(1)
Long Answer-1
(6marks)
4(1)
4(1)
4(1)
Long
Answer-2
(6 marks)
6(1)
6(1)
6(1)
4(1)
1(1)
2(1)
2(1)
1(1)
2(1)
2(1)
4(1)
4(1)
4(1)
4(1)
4(1)
6(1)
4(1)
6(1)
2(1)
16(8)
4(4)
26
4(1)
44(11)
10(2)
7(3)
12(3)
6(1)
5(2)
6(1)
2(1)
Total
Marks
36(6)
6(1)
6(2)
4(1)
10(2)
4(1)
6(2)
3(2)
6(2)
3(2)
6(1)
6(2)
100(29)
THINGS TO REMEMBER
Trigonometric Functions
𝜋
 Radian Measure = 180 x Degree Measure
 Degree Measure =












2
180
𝜋
x Radian Measure
2
𝑐𝑜𝑠 𝑥 + 𝑠𝑖𝑛 𝑥 = 1
1 + 𝑡𝑎𝑛2 𝑥 = 𝑠𝑒𝑐 2 𝑥
1 + 𝑐𝑜𝑡 2 𝑥 = 𝑐𝑜𝑠𝑒𝑐 2 𝑥
cos(2𝑛𝜋 + 𝑥) = cos 𝑥
sin(2𝑛𝜋 + 𝑥) = sin 𝑥
sin (-x) = -sin x
cos (-x) = cos x
cos (x + y) = cos x cos y - sin x sin y
cos (x - y) = cos x cos y + sin x sin y
sin (x + y) = sin x cos y + cos x sin y
sin (x – y) = sin x cos y – cos x sin y
𝜋
𝜋
cos (2 – x) = sin x
cos (2 + x) = - sin x
𝜋
𝜋
 sin (2 – x) = cos x
sin (2 + x) = cos x
 cos (𝜋 –x) = - cos x
 cos (2𝜋 –x) = cos x
sin (𝜋 – x) = sin x
sin (2𝜋 – x) = -sin x
𝜋
 If x, y, and (x ± y) are not an odd multiple of 2 , i.e x,y, (x ± y) ≠
tan 𝑥 + tan 𝑦
tan(𝑥 + 𝑦) =
1 − tan 𝑥 tan 𝑦
tan 𝑥 − tan 𝑦
tan(𝑥 − 𝑦) =
1 + tan 𝑥 tan 𝑦
𝜋 3𝜋 5𝜋
, , ,…
2 2 2
, then
 If x, y, and (x ± y) are not a multiple of 𝜋 , i.e x,y, (x ± y) ≠ 𝜋, 2𝜋, 3𝜋 … , then
cot 𝑥 cot 𝑦 − 1
cot(𝑥 + 𝑦 ) =
cot 𝑦 + cot 𝑥
cot 𝑥 cot 𝑦 + 1
cot(𝑥 − 𝑦 ) =
cot 𝑦 − cot 𝑥
2 tan 𝑥
 𝑠𝑖𝑛 2𝑥 = 2𝑠𝑖𝑛 𝑥 𝑐𝑜𝑠 𝑥 = 1+𝑡𝑎𝑛2 𝑥
 cos 2𝑥 = 𝑐𝑜𝑠 2 𝑥 − 𝑠𝑖𝑛2 𝑥 = 2𝑐𝑜𝑠 2 𝑥 − 1 = 1 − 2𝑠𝑖𝑛2 𝑥 =
 tan 2𝑥 =
2 tan 𝑥
1−𝑡𝑎𝑛2 𝑥
 sin 3𝑥 = 3 sin 𝑥 − 4 𝑠𝑖𝑛3 𝑥
 cos 3 𝑥 = 4 𝑐𝑜𝑠 3 𝑥 − 3 cos 𝑥
3 tan 𝑥− 𝑡𝑎𝑛3 𝑥
1−3𝑡𝑎𝑛2 𝑥
𝑥+𝑦
cos 𝑦 = 2𝑐𝑜𝑠 2
 tan 3𝑥 =
 cos 𝑥 +
𝑥−𝑦
cos 2
𝑥+𝑦
𝑥−𝑦
cos 𝑥 − cos 𝑦 = −2𝑠𝑖𝑛
sin
2
2
𝑥+𝑦
𝑥−𝑦
sin 𝑥 + sin 𝑦 = 2𝑠𝑖𝑛
𝑐𝑜𝑠
2
2
𝑥+𝑦
𝑥−𝑦
sin 𝑥 − sin 𝑦 = 2𝑐𝑜𝑠
𝑠𝑖𝑛
2
2
 2cos x cos y = cos (x+y) + cos(x-y)
-2sin x sin y = cos (x+y) – (cos(x-y)
2sin x cos y = sin (x+y) + sin (x-y)
27
1−𝑡𝑎𝑛2 𝑥
1+𝑡𝑎𝑛2 𝑥
Permutations and Combination
 Fundamental Principle of Counting: If an event can occur in m different ways, following which
another event can occur in n different ways, then total number of occurrence of the events in the given
order is m x n.
 A permutation is an arrangement in a definite order of a number of objects taken some or all at a time.
For example, formation of a code or lock pattern.
 A combination is an arrangement of a number of objects in no definite order. For example, choosing a
group of players.
 𝑛! = 1 × 2 × 3 × … × 𝑛
 𝑛! = 𝑛 × (𝑛 − 1)!
 0! = 1
 1! = 1
 Number of permutations of n different things taken r at a time, when repetition is not allowed is
𝑛!
n
Pr = (𝑛−𝑟)!, 0 ≤ 𝑟 ≤ 𝑛
 Number of permutations of n different things taken r at a time, when repetition is allowed is nr.
 Number of permutations of n objects taken all at a time, where p objects are of one type, q objects are
𝑛!
of second type, r objects are of third type and so on is given by 𝑝!𝑞!𝑟!…
 Number of Combinations of n different things taken r at a time is
𝑛!
 nCr = 𝑟!(𝑛−𝑟)! , 0 ≤ 𝑟 ≤ 𝑛

n

n
Pr = nCr. r!, 0 ≤ 𝑟 ≤ 𝑛
Cr + nCr+1 = n+1Cr
28
Sequnces and Series
 Sequence is a collection of numbers in a proper order.
For Example 2, 4, 6, 8, …
2, 4, 8, 16, 32, …
1, 4, 9, 16, 25,…
 Series: Let a1, a2, a3, … an be a given sequence,
then a1 + a2 + a3 + … +an is the series associated with the given sequence.
 Arithmetic Progression:
A sequence a1, a2, a3, … an is an AP if an+1 - an = d, n∈N and d is constant.
a1 = first term
d = common difference
l = last term
n = number of terms
Sn = sum of first n terms
an = nth term
an = a + (n-1)d
𝑛
𝑛
Sn = 2{2a + (n-1)d}= 2(a+l)
𝑎+𝑏
 Arithmetic Mean, AM = 2
 Geometric Progression:
𝑎
A sequence a1, a2, a3, … an is a GP if 𝑎𝑘+1 = 𝑟, k∈N and r is constant.
𝑘
a1 = first term
r = common ratio
l = last term
n = number of terms
Sn = sum of first n terms
an = nth term
an = a.rn-1
Sn =
𝑎(1−𝑟 𝑛 )
1−𝑟
or Sn =
𝑎(𝑟 𝑛 −1)
𝑟−1
 Geometric Mean, GM = √𝑎𝑏
29
Straight Lines
 General Equation of line : Ax + By + C = 0
 Slope of a line through (x1, y1) and (x2, y2)
𝑦 −𝑦
𝑦 −𝑦
𝑚 = 𝑥2 − 𝑥1 = 𝑥1− 𝑥2 , 𝑥1 ≠ 𝑥2
2
1
1
2
 Slope of a line making angle 𝛼, with the x-axis
𝑚 = tan 𝛼, 𝛼 ≠ 90°
 Slope of horizontal line is zero and slope of vertical line is not defined.
 Acute angle, 𝜃, between line l1 and l2
𝑚2 − 𝑚1
tan 𝜃 = |
| , 1 + 𝑚1 + 𝑚2 ≠ 0
1 + 𝑚1 𝑚2
where m1 and m2 are slopes of lines l1 and l2.
 Two lines are parallel ⇔ their slopes are equal
 Two lines are parallel ⇔ product of their slopes is -1.
 Forms of Equations of lines:
S.
No.
1
2
3
4
5
Forms
Horizontal
Line
Vertical
Line
Point Slope
Form
Two-point
Form
Slope
Intercept
Form
Equation of Line
Forms of Ax+By+C=0
y=a
y = -a
x=a
x= -a
y- y0 = m (x – x0)
m = slope of line
(x0, y0) is a point on line
(x1, y1) and (x2, y2) lie on line
𝑦 − 𝑦1 𝑦2 − 𝑦1
=
𝑥 − 𝑥1 𝑥2 − 𝑥1
y = mx + c
y = m(x – d)
6
Intercept
Form
𝑥 𝑦
+ =1
𝑎 𝑏
7
Normal
Form
𝑥 𝑐𝑜𝑠𝜔 + 𝑦 𝑠𝑖𝑛𝜔 = 𝑝
General Notations
𝑦=−
𝐴
𝐶
𝑥−
𝐵
𝐵
𝑥
𝑦
+
=1
𝐶
𝐶
−
−
𝐴
𝐵
𝐴
𝐵
𝐶
=
=−
𝑐𝑜𝑠 𝜔 sin 𝜔
𝑝
30
m = slope of line =
−𝐴
𝐵
c = y-intercept of line =
d = x-intercept of line
a = x-intercept
b = y-intercept
−𝐶
𝐵
p = distance of line from
origin
𝜔 = angle between normal
and positive x-axis
𝐴
cos 𝜔 = ±
2
√𝐴 + 𝐵2
𝐵
sin 𝜔 = ±
2
√𝐴 + 𝐵2
𝐶
p= ±
2
√𝐴 + 𝐵2