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 3-dimentional 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

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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. Symbol Meaning Example1. ∈ Belongs to 3 ϵ {1,2 ,3 , 4 ,5 ,6 }2. ∉ Does not belongs to 3∉ {2 , 4 ,6 ,8 }3. ∃ There exists4. ∀ For all5. ∪ Union A∪B={x : x∈ A∨x∈ B}6. ∩ Intersection A ∩ B={x : x∈ A∧x∈B }7. −¿ Difference A−B={x : x∈ A∧x∉B }8. ∆ Symmetric difference A△B=( A−B)∪(B−A)9. ' Complement A'={x : x∉ A }10. ϕ Empty Set11. ⊂∨⊆ Subset12. N Set of natural numbers {1, 2, 3, …}13. W Set of whole numbers {0, 1, 2, 3, …}14. Z Set of integers {0 , ± 1, ± 2, ± 3 ,…}15. Z+¿ ¿ Set of all positive integers16. Z−¿¿ Set of all negative integers17. Q Set of rational numbers { p

q: p , q ϵ Z , q ≠ 0}

18. T∨Q¿∨Q' Irrational Numbers Numbers which cannot be written as

pq

: p , qϵ Z ,q ≠ 0,

e.g √5, 3√−3 , 3√6419. R Set of real numbers All rational and irrational numbers20. R+¿ ¿ Set of all positive real

numbers21. R−¿ ¿ Set of all negative real

numbers22. {} Brackets used to enclose

the elements of a set23. n(A) Cardinal Number Number of elements in a set.

A={a , b , c }⟹n ( A )=3

Representation of Set

Roster / Tabular Form Set Builder FormIn this representation, elements are listed in curly braces separated by commas.For Example, A={1,2,3,4,5 }

In this representation common property of elements is defined.For example,A = Collection of natural numbers less than 6.

3

or A={x : x∈N , x<6 }

4

Types of Set:S. No.

Type of Set Meaning Example

1 Empty Set, Null Set, Void Setϕ∨¿{}

A set which has no element. A={ x∨x∈N , 3<x<4 }

2 Singleton Set A set containing only one element. A= set of vowels in the word ‘SET’ = {E}3 Subset ⊂∨⊆ Set A is said to be subset of set B if all the

elements of A are in B.A⊂B∨A⊆B

Set of vowels is subset of set of Alphabets.

3 Super Set ⊃∨⊇ Set B is said to be subset of set A if B contains all the elements of A

B⊃ A∨B⊇ A

Set of alphabets is super set of set of vowels.

4 Finite Set Set which contains countable number of elements.

A={a , e ,i , o , u }

5 Infinite Set Set which contains uncountable number of elements.

Set of Natural Numbers.

6 Equal Sets Two or more sets which have same elements.

A ={a , e , i , o , u },B ={x∨x is vowel∈the wod EDUCATION }

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.

9 Power Set ,P Set of all possible subsets of a given set. A ={1,2} then P(A)={ ∅,{1},{1, 2},{2} }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 = 2pand number of proper subsets is 2p-1.8. If XϵP(A) ,then X⊂ A9. n[P(A)]=2 n(A)

Properties of Sets:S. No.

Property Meaning

1 Commutative Law A∪B=B∪AA ∩ 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=UU ∩ A=A

5 Idempotent Law A∪A=AA ∩ 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 ( A∪B )'=A ' ∩ B'

5

( A ∩ B )'=A'∪B '9 Law of Double Complementation ( A' )'=A10 Law of Empty Set and Universal Set ∅ '=U

U '=∅

6

Questions for Practice:

1. A={a ,b , c , {a ,b } , d }. Fill in the blanks by putting appropriate symbols.

i) a¿ ii) {a , b }¿ iii) {{a ,b }}¿ iv) d¿ 2. Write in roster form

i)A={x∨x∈Z , x2−5 x+6=0} ii) The set of letters of the word “PANIC”3. Write the followings in set builder form.

i) A={12

, 23

, 34

, …} ii) B= {0,1,2,3,4 }

4. Represent the following in Roster form

(i ) {2 ,4 ,6 ,8 ,10 } (ii) {1 , 4 ,8 , 9 , 16 , 25 ………} (iii) {23

, 34

, 45

, 56

,…………….}(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 56. 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) 1ϵ A (v) {3,4}ϵ AInterval’sQ1. write the below interval’s in set builder form.(i) [ a,b] [ii] [a,b) (iii)[a,b] (iv) (a,b] (v) (-1,3)(vi) (-1,3) (vii) [3,10] (viii) (-4,0) (ix) [2,9) (x) (-1,3]Q2. write the below sets in interval form.(i) {x: 0< x ≪ 10} (ii) ) {x: - 4 < x < 2} (iii){ x: 2≪x≪7}Q.3.Find the intersection of given two intervals.(i) [-1,3] and [0,5] (ii) (2,6) and(-1,2] (iii) [1,4) and (2,5)

Level –I

1. Find the cardinal number of the set

2. Let A= and B= . Find and .3. 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?

4. Let A, B and C be three sets such that and Show that B=C5. If U={1,2,3,4,5,6,7,8,9 }, A={2,4,6,8 }∧B={2,3,5,7 } then verify that

i) ( A∪B )'=A ' ∩ B ' ii) ( A ∩ B )'=A'∪B '6. 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) (iii) (iv) 2. Draw appropriate Venn diagram for each of the following:

(i) (ii) Level III

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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 subjectsiii) None of the subjects

8

Module-2 : Relations and FunctionsS. No. Concept Meaning Example

1 Cartesian Product A×B ¿{( x , y ): xϵA , yϵB } A={a,b,c} and B={e,f},then A×B ={(a,e),(a,f),(b,e),(b,f),(c,e),(c,f)}

2 Relation A subset of cartesian product A×B

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 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)}.

3 Domain Set of all the first elements of the ordered pairs of R

Domain of R = {a : (a ,b ) ϵ R }In the above example,domain of R={2,3,4,5,6}

4 Range of a Relation

Set of all the second elements of the ordered pairs of R

Range of R = {b : (a ,b ) ϵ R }In the above example, range of R={2,3,5}

5 Function 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

6 Range of a function

Set of all the elements of the codomain of the function f which have pre-image in domain under f

7 Real Valued Function

A function whose range is the subset of R

8 Real Function A function whose domain and range are the subsets of R

9 Identity Function f (x)=x, for each xϵR10 Constant Function f (x)=c, for each xϵR, where c

is a constant11 Modulus Function f (x)=|x|, for each xϵR, i.e.

f ( x )={−x ,∧x<0x ,∧x ≥ 0

|−5|=5 |7|=7

|−32 |=3

2|0|=0

12 Signum Functionf ( x )={−1 , x<0

0 , x=01 , x>0

13 Greatest Integer Function

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

Note: (i) If any of the sets A and B is empty then A×B is empty.(ii) If any of the sets A and B is infinite then A×B is infinite.(iii) If both the sets A and B are finite then A×B is finite.(iv) Set A is called domain of f and the set B is called codomain of f.(v) If f : A → B and f (x)= y, then xϵA∧ yϵB.(vi) If n ( A )=m and n ( B )=n, then n( A × B) =mn(vii) If n ( A )=m and n ( B )=n, then the number of subsets of A×B =2mn

(viii) If n( A)=m and n(B)=n, then the number of relations from A to B =2mn

9

10

Exercises for Practice

Level-I1. 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.

3. If (x+1 , y2 )=¿), 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)}

7. If f ( x )=2x−7, then find f (0 ) , f (3 ) , f (−2 ) , f ( 32 ) and f (−3

2).

8. If f ( x )=7 x−8 and If g ( x )=x3, then find If f +g , f −g , f . g∧ fg .

Level-II1. If A ¿ {−2,2 } ,then find A×A ×A.2. If (x ,1) , ( y , 2 ) , ( z , 2 ) are in A×B, then find A and B, where n( A)=3 and n(B)=2.3. 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).

4. Write the relation R = {( x , x3) : x is a prime number less than10} in roster form.

5. If f ( x )=x2, then find f (1.1 )−f (1)

1.1−1.

6. If f ( x )=3 x3−5 x2+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 )=x2+3 , x ϵ R

8. Find the domain of the functions

(i) f ( x )=1x

(ii) f ( x )=√2 x+4Level-III

1. Find domain of the functionf ( x )= x2+2 x+1x2−8 x+12

.

2. Let f ={(x , x2

1+x2 ) : x∈R} be a function from R into R. Determine the range of f .

3. Find domain and range of the real valued functionf ( x )= 4−xx−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|.

6. Find domain and range of the real valued functionf ( x )=| 2−x3 x−6|.

7. Find domain and range of the real functionf ( x )=√16−x2.

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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 andb.

9. If f ( x )=ex and g ( x )=loge 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|

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Module 3 : Principles of Mathematical InductionIntroduction: 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 PracticeLevel-I

1. Prove the following by using mathematical induction, for all n∈N(i) 2n>n(ii) 1+3+5+………..+(2 n−1 )=n2

(iii) 13+23+33+…………+n3=[n(n+1)

2]2

(iv) 1+4+7+………..+(3n−2 )=n(3n−1)2

(v) 1+3+32 + 33+…… ..+3n−1= 3n−12

Level-II1. Prove the following by using mathematical induction, for all n∈N

(i)1

2.5+ 1

5.8+ 1

8.11+………+ 1

(3n−1 ) (3n+2 )= n

6 n+4

(ii)1

1.2.3+ 1

2.3.4+ 1

3.4 .5+… ……+ 1

n (n+1 ) (n+2 )=

n(n+3)4¿¿

(iii) 1+2+3+……… ..+n< 18(2n+1)2

(iv) 12+32+52+…………+(2 n−1 )2=n(2 n−1)(2 n+1)3

(v) 2n+7<(n+3)2

Level-III1. Prove the following by using mathematical induction, for all n∈N

(i) 102 n−1+1 is divisible by 11.(ii) 32n+2−8 n−9is divisible by 8.(iii) (ab)n=an bn

(iv) x2 n− y2n is divisible by x+ y.(v) 7n−3n is divisible by 4 .(vi) (1+x)n ≥ (1+nx ) , where x>−1.

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Module -4 (Complex Numbers)1. A complex number z = x + iy, where x, y are real numbers and i2=−1.2. |x|=|x+iy|=√x2+ y2

3. If z=x−iy is called conjugate of z

4. Arg of z = x + iy , θ=tan−1 yx

.

5. Polar form of z=x+iyis z=r cosθ+ isin θ

where r=√x2+ y2 and θ=tan−1 yx

.

LEVEL 11. Express √−25 in combination of real and imaginary number.2. Express √−36+√25 in a+ ib form, where ' a ' and ' b ' are real numbers.3. Find the value of i−63.4. Compute: i643+i644+i645+i646.

5. Find the conjugate of z=7+2i4−3 i .

6. Find the multiplicative inverse of z=2+3 i5+12i .

7. If arg z = π3 and |z|=2, then find z in standard form.

LEVEL 21. Find the square root of the followings

i) 12−5 i ii) i iii) 8−15 i24+7 i iv) 7−30√2 i

2. Convert the following into polar form.

i) 3√3+3 i ii) 1−i1+i iii) −i

3. If a+ ib= x−ix+i , where x is a real number, prove that a

2+b2=1 , ba= 2 x

x2−14. Solve the following quadratic equations:

i) x2+3x+9=0 ii) √3 x2+√2 x+3√3=0 iii) x2+ x√2

+1=0

iv)x2−x+2=0

LEVEL 3

1. If ( x+iy )3=u+ iv ,thenshow that ux+ v

y=4 ( x2− y2 )

2. If α and β are the different complex numbers. With | β |=1, then find the value of |β−α

1−α β |.

3. Find the number of non zero integral solutions of the equation ¿1−i∨¿x=2x¿

4. If ( 1+i1−i )

n

=1,find the least positive integral value of n.

14

5. Convert the complex number z= i−1

cos π3+isin π

3 in the polar form.

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Module 5 (Linear Inequalities)LINEAR INEQUALITIES

Rules for solving Linear InequalitiesThe 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 Solution1. Draw lines corresponding to each equation treating it as equality.2. Find the feasible region: intersection of all the inequalities.

LEVEL – I1. 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

3−7 x−3

3 > x4

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%?

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Module 6: Binomial TheoremBasic concepts: Factorial : 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

Binomial expansions: (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 1 1 n=1 ` 1 2 1 n=2 1 3 3 1 n=3 1 4 6 4 1 n=4 1 5 10 10 5 1 n=5

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 QuestionsLEVEL 1

1. Expand (a) (1−2x)5 (b )( 2x− x

2 )5

2. Compute 985 using binomial theorem.3. Find the coefficient of x5 in the expansion of(x+3)8.4. Write the general term in the expansion of (2 x−3 y2)6

LEVEL 21. Using binomial theorem, indicate which number is larger:(1.1)10000 or 1000.

2. Find the middle term of the expansion of (3− x3

6)

7

3. Find the term independent of x in the expansion of( 33

x2− 13 x

)6

.

4. Find a positive value of m for which the coefficient x2of in the expansion (1+x)m is 6.LEVEL 3

1. Show that the middle term in expansion of (1+x)2 n is 1.3.5 .7… ..(2n−1)

n! 2n xn, where n is a positive

integer.

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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 x5 in the product (1+2 x)6(1−x )7 using binomial theorem.4. The coefficient of the (r−1)th ,r th∧(r+1)th terms in the expansion of (x+1)n are in the ratio 1:3:5, find

n and r.

18

Module 7: Conic SectionsConic 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 :̊(x−h)2 + ( y−k )2 =r2

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 x2

a2 + y2

b2 = 1 ¿ >b2¿

Vertical ellipse x2

b2 + y2

a2 = 1 ¿ < a2¿

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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 = -4axiii) x2 = 4ayiv) x2 = -4ay

The plane can intersect two nappes of the cone resulting in a hyperbola.

Standard equations of hyperbola:

Horizontal hyperbola x2

a2 - y2

b2 = 1 ¿ > b2¿

Vertical hyperbola y2

a2 - x2

b2 = 1 ¿ > a2¿

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E X E R C I S E for P R A C T I C ELEVEL 1

1. Find the equation of the circle with centre (0, 2) and radius 22. 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

y2 = 8x 4. Find the equation of the ellipse that satisfy the given condition vertex (± 5 ,0¿ and (± 4 ,0 ¿.

LEVEL 21. 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 9x2+4y2 =36.4. Find the equation of hyperbola with foci (0,± 3) and vertices ¿11/2).

LEVEL 31. 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 y2 = 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.

21

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=m x2+n x1

m+n , y=

m y2+n y1

m+n , z=

m z2+n z1

m+n

(ii) If C is external point on AB then

x=m x2−n x1

m−n , y=

m y2−n y1

m−n , z=

m z2−n z1

m−n

Level -1

1. A point is on the z-axis, what are the x & y- coordinate.2. A point is in XY- plane, what is its Z-coordinate?3. Show that point P(-2,3,5),Q(1,2,3) and R(7,0,-1) are collinear.4. Verify that (-1,2,1) ,(1,-2,5),(4,-7,8) and (-2,3,4) are the vertices of a parallelogram.5. 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 -311. 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).

22

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 bylimh→ 0

f (a+h) , here h is positive

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 bylimh→ 0

f (a−h) , here h is positive

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 limh→ 0

f (a+h)=limh→ 0

f (a−h)

Note: If both limits are not equal then the value of the limit of the function cannot be found at that point.

DERIVATIVES1. 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

limh→ 0

f (a+h )−f (a)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

limh→ 0

f (x+h )−f (x)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.

Thus, f’(x) = limh→ 0

f (x+h )−f (x)h

Properties:

1. limx⟶α

[ f ( x ) ± g ( x ) ]=limx→ α

f ( x ) ± limx→ α

g(x )

2. limx⟶α

[ f ( x ) . g ( x ) ]= limx→ α

f ( x ) . limx →α

g(x )

3. limx⟶α [ f ( x )

g ( x ) ]¿limx→ α

f (x)

limx →α

g(x )

4. limx→ a

xn−an

x−a=nan−1

5. limx →0

sin xx

=1

6. limx →0

1−cos xx

=0

7. The derivative of a function f at a is defined by

f ' (a )=limh → 0

f ( a+h )−f (a)h

8. Derivative of a function f at any point x is defined by

f ' ( x )=d f (x )

dx =limh → 0

f ( x+h )−f (x )h

9. For functions u and v, the following holds:a) (u ± v )'=u' ± v '

b) (uv )'=u' v+uv '

c) ( uv )

'

=u' v−uv 'v2

23

10.Standard Derivatives:

a)ddx

(xn)=n xn−1

b)ddx

(sin x )=cos x

c)ddx

(cos x)=−sin x

24

EXERCISE FROM LIMITSLevel 1

1. Evaluate :

(i) limx →3

2x+3 (ii) limz → 1

3 z−4 (iii) limx→ o

ax+bcx+d

LEVEL 21. Evaluate

(i) limx →1

x15−1x10−1

(ii) limz → 1

z13−1

z 16−1

(iii) limx→ o

sin axbx

(iv) ) limx→ o

(1+ x)5−1x

LEVEL 31. Evaluate

(i) limx→ o

sin (π−x)π (π−x )

(ii) limx→ o

sin ax+bxax+sinbx

,a+b≠ 0 (iii) limx→ π

2

tan 2x

x− π2

(iv) )limx→ o

cos2 x−1Cosx−1

2. Suppose f(x)={a+bx , if x<14 ,if x=1

b−ax ,if x>1and lim

x →1f ( x )=f (1 ), then what are the possible values of a and b.

5. Evaluate limx →0

f ( x ), if f(x)={|x|x,if x≠ 0

0 , if x=0EXERCISE FROM DERIVATIVES

LEVEL 11. Find the derivatives of

(i) x3 -27 (ii) 2x -34

(iii) 5x3−¿2x +4 (iv) (7x-3) (4x2-2)

LEVEL 2

1. Find the derivatives of (i) (x-a)(x-b) (ii)x−ax−b

(iii) x+1x

(iv) Sin(x+1)

(v) px2+qx+rax+b

(vi) (px+q)( rx+s) (vii)

1+ 1x

1−1x

(viii) (ax+b)n(cx+d)m

LEVEL 3

1 Find the derivatives of (i) Sinx+CosxSinx−Cosx

(ii) sin(x+a)

cosx(iii) Sin2x

2. Find the derivatives (i) (x + sec x)(x-tanx) (ii) secx−1sec+1

(iii) x2 cos π

4sin x

3. Find the derivatives using first principal (i) tanx (ii) sin (x+ 1)

25

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.

STATEMENTSA 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 STATEMENTSThe denial of a statement is called negation of statements. Negation of p is denoted by ∼ pFor example p : Delhi is the capital of India. ~p : Delhi is not the capital of India.COMPOUND STATEMENTSA 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 thenCONVERSE STATEMENT q⇒ pINVERSE STATEMENT ∼ p⇒∼qCONTRAPOSITIVE 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

26

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.

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 n2 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.

27

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 XIThe important statistics formulas are listed in the chart below:

Mean

In series Data:

X=1n∑i=1

n

x i

In frequency data:

X= 1N ∑

i=1

n

f i x i , where N=∑i=1

n

f i

x = value of observation

n = Total number of observations

Median If n is odd, then

Me=( n+12 ) th termIf n is even, then

Me=[( n

2 )th term+( n2+1)th term]

2In Grouped Data:

n = Total number of itemsNote: To find median, first we arrange the given data in increasing or decreasing order.l=lower limit of median classc=cumulative frequency of the class preceding to the median classf=frequency of median classh= class width of median class

28

Me=l+( n2−c

f )h

Mode

The value which occurs most frequentlyIn Grouped Data:

Mo=l+( f 1−f 0

2 f 1−f 0−f 2)h

l = lower limit of modal class (highest frequency class)f1 = frequency of modal classf0 = frequency of the class preceding modal classf2 = frequency of the class next to modal class

Mean Deviation about mean

In Series Data:

M . D . ( x )=1n∑i=1

n

¿ x i−x∨¿¿

In Frequency Data:

M .D . ( x )= 1N ∑

i=1

n

f i∨xi−x∨¿¿

n = total number of observations in series data

where N=∑i=1

n

f i

Mean Deviation about median

In Series Data:

M . D . ( Me )=1n∑i=1

n

¿ x i−Me∨¿¿

In Frequency Data:

M . D . ( Me )= 1N ∑

i=1

n

f i∨xi−Me∨¿¿

n = total number of observations in series dataMe = median

where N=∑i=1

n

f i

Standard Deviation, σ

For Ungrouped Data:

σ=√ 1n∑i=1

n

(x i− x)2

For Discrete Frequency Data:

σ=√ 1N ∑

i=1

n

f i(x i−x )2

For Continuous Frequency Data,

σ=√ 1N ∑

i=1

n

f i(x i)2−(∑i=1

n

f i x i)2

Shortcut Method

σ= hN √N ∑ f i y i

2−(∑ f i yi)2

x = Items givenx¯ = Meann = Total number of items

where N=∑i=1

n

f i

y=x i−A

hFor series with equal means, the series with lesser standard deviation is more consistent or less scattered.

Variance

For Ungrouped Data,

σ 2 .=1n∑i=1

n

( x i−x )2

For Frequency Data

σ 2.= 1N ∑

i=1

n

f i ( x i−x )2

x = Items givenx¯ = Meann = Total number of items

where N=∑i=1

n

f i

Coefficient of variation C .V .=σ

x× 100 ; x≠ 0

Note : Among two given data , the data having

Where σ is standard deviation and x is arithmetic mean

29

larger value of C.V. is more variable than other.

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, 257. Find coefficient of variation of data: 12, 8, 10, 12, 16, 18, 20, 22, 10, 12, 13, 15, 16, 188. 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 69. 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 ChemistryMean 42 32 40.9Standard Deviation 12 15 20Which 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 1013. Find the mean, variance and standard deviation using short-cut method:

Height in cms 70-75 75-80 80-85 85-90 90-95 95-100 100-105 105-110 110-115No. of Children 3 4 7 7 15 9 6 6 3

30

Module 12: ProbabilityBasic 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∨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∧ω :ωϵ B}

(iv) The event A but not B :A but not B = A∩ B c = A – B = {ω : ωϵ A∧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.

31

(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?

32

KENDRIYAVIDYALAYA SANGATHAN, RAIPUR REGION

Blue-Print Mathematics

Class XI(Half Yearly Examination)

TEMPORARILY ADJUSTIBLE

Time: 3 hours Max Marks: 100

S. No.

Topics Very Short Answer

(1 marks)

Short Answer

(2 marks)

Long Answer-1 (6marks)

Long Answer-2(6 marks)

Total Marks

Unit I 1 Sets 1(1) 2(1) 4(1) 6(1) 13(4)2 Relations and

Functions_ 4(2) 8(2) _ 12(4)

3 Trigonometry 1(1) 2(1) 8(2) 6(1) 17(5)Unit II 4 Principle of

Mathematical Induction

_ _ _ 6(1) 6(1)

5 Complex Numbers and Quadratic Equations

_ 2(1) 8(2) _ 10(3)

6 Linear Inequalities _ _ 4(1) 6(1) 10(2)7 Permutations and

Combinations_ 4(2) 4(1) _ 8(3)

8 Binomial Theorem 1(1) _ 4(1) 6(1) 11(3)9 Sequences and Series 1(1) 2(1) 4(1) 6(1) 13(4)

Total 4(4) 16(8) 44(11) 36(6) 100(29)

33

KENDRIYA VIDYALAYA SANGATHAN, RAIPUR REGION

BLUE PRINT

Mathematics

CLASS-XI

Session Ending Examination

Unit S. No.

Topics Very Short Answer

(1 marks)

Short Answer

(2 marks)

Long Answer-1 (6marks)

Long Answer-2(6 marks)

Total Marks

I 1 Sets 4(1) 6(1) 10(2)2 Relations and Functions 1(1) 2(1) 4(1) 7(3)3 Trigonometry 2(1) 4(1) 6(1) 12(3)

II 4 Principle of Mathematical Induction

6(1) 6(1)

5 Complex Numbers and Quadratic Equations

1(1) 4(1) 5(2)

6 Linear Inequalities 6(1) 6(1)7 Permutations and

Combinations2(1) 4(1) 6(2)

8 Binomial Theorem 4(1) 4(1)9 Sequences and Series 4(1) 6(1) 10(2)

III 10 Straight Lines 4(1) 4(1)11 Conic Sections 2(1) 4(1) 6(2)12 Introduction to Three-

Dimensional Geometry1(1) 2(1) 3(2)

IV 13 Limits and Derivatives 2(1) 4(1) 6(2)V 14 Mathematical Reasoning 1(1) 2(1) 3(2)VI 15 Statistics 6(1) 6(1)

16 Probability 2(1) 4(1) 6(2)Total 4(4) 16(8) 44(11) 36(6) 100(29)

34

THINGS TO REMEMBERTrigonometric Functions

Radian Measure = π

180 x Degree Measure

Degree Measure = 180π x Radian Measure

cos2 x+sin2 x=1 1+ tan2 x=sec2 x 1+cot2 x=cosec2 x cos (2nπ+x )=cos x sin (2 nπ+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

cos (π2 – x) = sin x cos (

π2 + x) = - sin x

sin (π2 – x) = cos x sin (

π2 + x) = cos x

cos (π –x) = - cos x sin (π – x) = sin x cos (2π –x) = cos x sin (2 π – x) = -sin x

If x, y, and (x ± y) are not an odd multiple of π2 , i.e x,y, (x ± y) ≠ π

2, 3π

2, 5 π

2, … , then

tan ( x+ y )= tan x+ tan y1−tan x tan y

tan ( x− y )= tan x−tan y1+ tan x tan y

If x, y, and (x ± y) are not a multiple of π, i.e x,y, (x ± y) ≠ π , 2 π , 3 π … , then

cot ( x+ y )= cot xcot y−1cot y+cot x

cot ( x− y )=cot x cot y+1cot y−cot x

sin 2 x=2sin x cos x = 2 tan x

1+ tan2 x

cos2 x=cos2 x−sin 2 x=2 cos2 x−1=1−2sin2 x=1−tan2 x1+ tan2 x

tan2 x= 2 tan x1−tan2 x

sin 3 x=3 sin x−4 sin3 x cos3 x=4 cos3 x−3 cos x

tan3 x=3 tan x−tan3 x1−3 tan2 x

cos x+cos y=2cos x+ y2

cos x− y2

35

cos x−cos y=−2 sin x+ y2

sin x− y2

sin x+sin y=2 sin x+ y2

cos x− y2

sin x−sin y=2cos x+ y2

sin x− y2

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)

36

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

nPr = n!

(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

of second type, r objects are of third type and so on is given by n !

p ! q!r ! … Number of Combinations of n different things taken r at a time is

nCr = n !

r ! (n−r )!,0≤r ≤ n

nPr = nCr. r!, 0 ≤ r≤ n

nCr + nCr+1 = n+1Cr

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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 termd = common differencel = last termn = number of termsSn = sum of first n termsan = nth term

an = a + (n-1)d

Sn = n2{2a + (n-1)d}=

n2 (a+l)

Arithmetic Mean, AM = a+b

2 Geometric Progression:

A sequence a1, a2, a3, … an is a GP ifak +1

ak=r, k∈N and r is constant.

a1 = first termr = common ratiol = last termn = number of termsSn = sum of first n termsan = nth term

an = a.rn-1

Sn = a(1−rn)1−r

or Sn = a(r n−1)r−1

Geometric Mean, GM = √ab

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Straight Lines General Equation of line : Ax + By + C = 0 Slope of a line through (x1, y1) and (x2, y2)

m=y2− y1

x2−x1=

y1− y2

x1−x2, x1 ≠ x2

Slope of a line making angle α , with the x-axism=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

tanθ=| m2−m1

1+m1 m2|, 1+m1+m2 ≠0

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.

Forms Equation of Line Forms of Ax+By+C=0 General Notations

1 Horizontal Line

y = ay = -a

2 Vertical Line

x = ax= -a

3 Point Slope Form

y- y0 = m (x – x0) m = slope of line(x0, y0) is a point on line

4 Two-point Form

y− y1

x−x1=

y2− y1

x2−x1

(x1, y1) and (x2, y2) lie on line

5 Slope Intercept Form

y = mx + cy = m(x – d) y=−A

Bx−C

Bm = slope of line =

−AB

c = y-intercept of line = −CB

d = x-intercept of line6 Intercept

Formxa+ y

b=1 x

−CA

+ y−CB

=1a = x-interceptb = y-intercept

7 Normal Form

x cosω+ y sinω=p Acosω

= Bsin ω

=−Cp

p = distance of line from originω = angle between normal and positive x-axis

cos ω=± A√ A2+B2

sin ω=± B√ A2+B2

p¿± C√ A2+B2

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