strictly based on latest syllabus issued by cbse

Strictly Based on Latest Syllabus Issued by CBSEfor 2016 Examination

SAMPLEQUESTION

PAPERSMATHEMATICS

OSWAAL CBSE

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0109

CONTENTS

Syllabus v - viii

On Tips Notes 9 - 24

Sample Question Papers (Solved)

Sample Question Paper - 1 25 - 27





Sample Question Papers for self assessment






Solutions






* SOLUTIONS for Sample Question Paper 6 to 10 can be downloaded from

www.OswaalBooks.com

( iii )

Believe in YourselfThis is the foremost barrier to be crossed for scoring high marks in exams. One needs to believe in his / her

ability to learn, memorize and reproduce what has been learnt. Exams are nothing but the test of our faith, confidence and knowledge.

Follow a Time – TableA well set time table allotting specific durations for studying, sleeping, playing/surfing the net and eating

can help every student a lot. Every above quoted thing has to be done every day. A proper schedule can help a student beat examination stress.

Set Every Day GoalsPreparations should be done every day to excel at the day of exams without depending upon any miracle

capsule to bail you out at the last moment. By setting everyday targets and goals, one can achieve incredible results in terms of efficiency and performance.

Take care of your Health Health is wealth. This adage never fails. Only a sound body and sound mind can work effectively towards achieving any objective. Thus to sum up, healthy body is a mandate for rigorous mental exercise that comes up during examinations.

Practice Daily We eat daily, we sleep daily, so why not study daily?Regular practice in every subject will keep students close to a subject. If one avoids any subject for more than three days in a go, he/she is bound to lose interest in it.

1.TIPS N

How to Score

2.

3.

OSWAAL Your Elixir of Positivity & Confidence Positivity and confidence can do wonders to your grades, far more than you can think. By studying from Oswaal Sample Question papers, you develop confidence in yourself which makes you positive and hence gives you the winners advantage!! A bunch of important questions along with their systematic presentation helps you tremendously in studying effectively.

OSWAAL - Your Planner For Examination Preparations You must make a schedule for your studies followed by strict implementation of that schedule. Oswaal SQP's give you questions on the important topics or topics which need more practice or time. Oswaal SQP's include last year exam questions as well as sample papers for the proper schedule of your study. You may also study with your friends and make the entire learning process fun!!

OSWAAL - Your Confidence Booster Just before the Exam One should never try to read, study or cram anything new just before the beginning of your exam. You can just open your Oswaal SQP's and read through the answers highlighted by you a night before for the last time and then put away all your books. This gives you a new wave of confidence just before the commencement of your exam!

4.

5.

High in your Examination?

How to get 6.

7.

8.

9.

10.

1.

2.

3.

Play Games Playing games – both the indoors and outdoors, help in inculcating a practical approach towards dealing with a problem along with beating the examination stress.

Presentation Till now we have discussed the pre-exam tips. This is a crucial tip while writing the examinations. A systematic and neat display of answers can boost your chances of scoring high.

Time Management You need to manage time not only during exam preparations, but also at the time of solving the Question paper. Carrying a wrist watch during exams is an excellent way to manage time well. You should also try to save some time at the end of the paper to recheck the answers.

Sleep Well A sound sleep a night before exam helps us relax and rejuvenates our mind. The tired brain needs and we need to understand the needs of our brain, only then we will be able to make the most out of it especially during examinations.

Relax Yourself This is last but not the least. Relaxing is like meditating. When we are relaxed we are the most efficient in reproducing what we have learnt.

The 'OSWAAL' Advantage?

Highlights of Curriculum Document 2015-16 for March 2016 Exam

Curriculum 2014-15(Printed in 2014)

Curriculum 2015-16 final for the examination to be held in March 2016

Class XI

Unit-I : Sets and Functions

Sub unit-2 : Relations & Functions :

Class XI

Topic(s) Added :

Domain and range of exponential, logarithmic function (Page 95)

Unit-VI : Statistics and Probability

Sub unit-1 : Statistics

Question Paper Design (XI-XII)

Topic Added :

Range (Page 97)

Question Paper Design (XI-XII)

(Page 98,103)

Understanding Based - Number of LA-II questions- 1, Total Marks-16,

% Weightage- 16%

Application Based - Number of LA-I questions-3 , Total Marks-25,

% Weightage- 25%

HOTS - Number of LA-II questions-2, Total Marks-21, %weightage-21%

Evaluation and Multi- disciplinary Based -Number of LA-I questions- 3, Total Marks-18, %Weightage- 18%

Understanding Based - Number of LA-II questions- 2, Total Marks-22,

% Weightage- 22%

Application Based - Number of LA-I questions-4, Total Marks-29,

% Weightage- 29%

HOTS - Number of LA-II questions-1, Total Marks-15, % weightage-15%

Evaluation Based - Number of LA-I questions- 2, Total Marks-14, % Weightage- 14%

SYLLABUSMATHEMATICS Class–XI (Code No. 041)

(2015-16)Total Hour - Periods of 35 Minutes each

One Paper 3 Hours 100 Marks

Unit No. Unit No. of Periods(35 Minutes each)

Marks

I Sets and Functions 60 29

II Algebra 70 37

III Coordinate Geometry 40 13

IV Calculus 30 06

V Mathematical Reasoning 10 03

VI Statistics and Probability 30 12

Total 240 100

*No chapter/unit wise weightage. Care to be taken to cover all the chapters.

Circular No. Acad.-18/2015

6 | OSWAAL CBSE Sample Question Paper, Mathematics, Class – XI

UNIT I : SETS AND FUNCTIONS1. Sets (20) Periods Setsandtheirrepresentations.Emptyset.FiniteandInfinitesets.Equalsets.Subsets.Subsetsofaset

of real numbers especially intervals (with notations). Power set. Universal set. Venn diagrams. Union and Intersection of sets. Difference of sets. Complement of a set. Properties of Complement Sets.

2. Relations & Functions (20) Periods Orderedpairs,Cartesianproductofsets.Numberofelementsinthecartesianproductoftwofinite

sets.Cartesianproductofthesetofrealswithitself(uptoRxRxR).Definitionofrelation,pictorialdiagrams, domain, co-domain and range of a relation. Function as a special type of relation. Pictorial representation of a function, domain, co-domain and range of a function. Real valued functions, domain and range of these functions, constant, identity, polynomial, rational, modulus, signum, exponential, logarithmic and greatest integer functions, with their graphs. Sum, difference, product and quotient of functions.

3. Trigonometric Functions (20) PeriodsPositive and negative angles. Measuring angles in radians and in degrees and conversion from one measuretoanother.Definitionoftrigonometricfunctionswiththehelpofunitcircle.Truthoftheidentity sin2x+cos2x = 1, for all x. Signs of trigonometric functions. Domain and range of trignometric functions and their graphs. Expressing sin (x±y) and cos (x±y) in terms of sin x, sin y, cos x & cos y and their simple applications. Deducing the identities like the following :

tan (x ± y) =

±+

tan tan1 tan tan

x yx y

, cot (x ± y) = ±

cot cot 1cot cot

x yy x

sin a ± sin b = 2sin12

(a ± b) cos12

(a b) =

cos a + cos b = 2cos12

(a + b) cos12

(a – b)

cos a – cos b = – 2sin12

(a + b) sin12

(a – b)

Identities related to sin 2x, cos 2x, tan 2x, sin 3x, cos 3x and tan 3x. General solution of trigonometric equations of the type sin y = sin a, cos y = cos a and tan y = tan a.

UNIT II : ALGEBRA1. Principle of Mathematical Induction (10) Periods Process of the proof by induction, motivating the application of the method by looking at natural

numbers as the least inductive subset of real numbers. The principle of mathematical induction and simple applications.

2. Complex Numbers and Quadratic Equations (15) Periods Needforcomplexnumbers,especially√−1,tobemotivatedbyinabilitytosolvesomeofthequardratic

equations. Algebraic properties of complex numbers. Argand plane and polar representation of complex numbers. Statement of Fundamental Theorem of Algebra, solution of quadratic equations (withrealcoefficients)inthecomplexnumbersystem.Squarerootofacomplexnumber.

3. Linear Inequalities (15) Periods Linear inequalities. Algebraic solutions of linear inequalities in one variable and their representation

on the number line. Graphical representation of linear inequalities in two variables. Graphical method offindingasolutionofsystemoflinearinequalitiesintwovariables.

4. Permutations and Combinations (10) Periods Fundamental principle of counting. Factorial n. (n!) Permutations and combinations, derivation of

formulae for npr and ncr and their connections, simple applications.

Syllabus | 7

5. Binomial Theorem (10) PeriodsHistory, statement and proof of the binomial theorem for positive integral indices. Pascal's triangle, General and middle term in binomial expansion, simple applications.

6. Sequence and Series (10) PeriodsSequence and Series. Arithmetic Progression (A.P.). Arithmetic Mean (A.M.). Geometric Progression (G.P.),generaltermofaG.P.,sumoffirstntermsofaG.P.,infiniteG.P.anditssum,geometricmean(G.M.), relation between A.M. and G.M. Formulae for the following special sums

= = =∑ ∑ ∑2 3

1 1 1

, , ,n n n

k k k

k k and k

UNIT III : CO-ORDINATE GEOMETRY1. Straight Lines (10) Periods

Brief recall of two dimensional geometry from earlier classes. Shifting of origin. Slope of a line and angle between two lines. Various forms of equations of a line : parallel to axis, point-slope form, slope intercept form, two-point form, intercept form and normal form. General equation of a line.Equation of family of lines passing through the point of intersection of two lines. Distance of a point from a line.

2. Conic Sections (20) PeriodsSections of a cone : circle, ellipse, parabola, hyperbola, a point, a straight line and a pair of intersecting lines as a degenerated case of a conic section. Standard equations and simple properties of parabola, ellipse and hyperbola. Standard equation of a circle.

3. Introduction to Three-dimensional Geometry (10) PeriodsCo-ordinate axes and co-ordinate planes in three dimensions. Co-ordinates of a point. Distance between two points and section formula.

UNIT IV : CALCULUS1. Limits and Derivatives (30) Periods

Derivative introduced as rate of change both as that of distance function and geometrically. Intutive idea of limit. Limits of polynomials and rational functions trigonometric, exponential and logarithmic functions. Definition of derivative relate it to scope of tangent of the curve, Derivative of sum,difference, product and quotient of functions. Derivatives of polynomial and trigonometric functions.

UNIT V : MATHEMATICAL REASONING1. Mathematical Reasoning (10) Periods

Mathematically acceptable statements. Connecting words/ phrases - consolidating the understanding of"ifandonlyif(necessaryandsufficient)condition","implies","and/or","impliedby","and","or","there exists" and their use through variety of examples related to real life and Mathematics. Validating the statements involving the connecting words, Difference between contradiction, converse and contrapositive.

UNIT VI : STATISTICS AND PROBABILITY1. Statistics (15) Periods

Measures of dispersion : Range, mean deviation, variance and standard deviation of ungrouped/grouped data. Analysis of frequency distributions with equal means but different variances.

2. Probability (15) PeriodsRandom experiments; outcomes, sample spaces (set representation). Events; occurrence of events, 'not', 'and' and 'or' events, exhaustive events, mutually exclusive events, Axiomatic (set theoretic) probability, connections with other theories studied in earlier classes. Probability of an event, probability of 'not', 'and' and 'or' events.

8 | OSWAAL CBSE Sample Question Paper, Mathematics, Class – XI

MATHEMATICS (CODE NO. 041)QUESTION PAPER DESIGN

Class-XI (2015-16)Time : 3 Hours Max. Marks : 100

S.No.

Typology of Questions

Very Short

Answer(1 Mark)

LongAnswer

I(4 Marks)

LongAnswer

II(6 Marks)

TotalMarks

% (Wei-

ghtage)

1.

Remembering-(Knowldge based Simple recall questions, to know specific facts, terms, concepts,principles,ortheories,Identify,define-or recite, information)

2 3 1 20 20%

2.

Understanding (Comprehension–to be familiar with meaning and to understand conceptually, interpret, compare, contrast, explain, paraphrase information)

2 2 2 22 22%

3.

Application (Use abstract information in concrete situation, to apply knowledge to new situations; Use given content to interpret a situation, provide an example, or solve a problem)

1 4 2 29 29%

4.

High Order Thinking Skills. (Analysis & Synthesis - Classify, compare, contrast, or differentiate between different pieces of information; Organize and/or integrate unique pieces of information from a variety of sources)

1 2 1 15 15%

5.

Evaluation - (Appraise, judge, and/or justify the value or worth of a decision or outcome, or to predict outcomes based on values)

1 + 1(Value Based)

1 14 14%

Total 6 × 1 = 6 13 × 4 = 52 7 × 6 = 42 100 100%

ON TIPSNOTES

Note Making is a skill that we use in many walks of life : at school, university and in the world of work. However, accurate note making requires a thorough understanding of concepts. We, at Oswaal, have tried to encapsulate all the chapters from the given syllabus into the following ON TIPS NOTES. These notes will not only facilitate better understanding of concepts, but will also ensure that each and every concept is taken up and every chapter is covered in total. So, go ahead and use it to your advantage.... go get the OSWAAL ADVANTAGE!!

CHAPTER 1: Sets

¾ Set of Number1. N : The set of Natural numbers 2. W : The set of whole numbers3. Z : The set of integers4. Q : The set of rational numbers5. R : The set of real numbers

6. Z+ : The set of positive Integers7. Q+ : The set of positive rational numbers8. R+ : The set of positive real numbers9. C : The set of all complex number

¾ Examples of Roster and Set Builder Form Roster Form Set Builder Form(i) {P, R, I, N, C, A, L} (a) {x : x is a letter of the word PRINCIPAL}(ii) {1, 2, 3, 6, 9, 18} (b) {x : x is a positive integer and is a divisor of 18}(iii) {3, – 3} (c) {x : x is an integer and x2 – 9 = 0}

¾ Types of Sets

Type Definition ExampleEmpty Set (Null or Void)

A set which does not contain any element

A = {x : 1 < x < 2, x is a natural number}

Finite Set A set which is empty or consists of a definitenumberofelements

A = {1, 2, 3, 4, 5}

InfiniteSet set which consists of an indefinitenumber of elements {x : x Î N and x is prime}

Equal Set Two sets A and B are said to be equal if they have exactly the same elements

A = {1, 2, 3, 4}B = {3, 1, 4, 2} Then A = B

Subset A set A is said to be a subset of a set B if every element of A is also an element of B.

A Ì B if a Î A Þ a Î BA = {1, 3}, B = {1, 5, 9},C = {1, 3, 5, 7, 9}ThenA Ì C B Ì C

Power Set The collection of all subsetsof a set A

if A = {1, 2}, thenP(A) = {f, {1}, {2}, {1, 2}}

Universal Set If there are some sets under consideration, then there happens to be a set which is a superset of each one of the given set. Such a set is called the universal set.

the set of all integers, all the people in the world

10| OSWAAL CBSE Sample Questions Paper, Mathematics Class –11 ON TIPS NOTES | 11

¾ Intervals as subsets of R(a) An open interval denoted by (a, b) is the set

of real numbers {x : a < x < b}(b) A closed interval denoted by [a, b] is the set

of real numbers {x : a £ x £ b}

(c) Intervals closed at one ened and open at the other are given by

[a, b) = {x : a £ x < b} (a, b] = {x : a < x £ b}

a b

( )a, b

a b a b a b

[ ]a, b [ )a, b ( ]a, b

¾ Operation on SetsOperation Venn Diagram Example

Union

A È B = {x : x Î A or x Î B}

If A = {1, 2, 3, 4} and B = {2, 4, 6, 8} A È B = {1, 2, 3, 4} È {2, 4, 6, 8} = {1, 2, 3, 4, 6, 8}.

Intersection UA B

A Ç B = {x : x Î A and x Î B}

If A = {1, 2, 3, 4} and B = {2, 4, 6, 8} A Ç B = {1, 2, 3, 4} Ç {2, 4, 6, 8} = {2, 4}.

Complement

A

A U

A’ {x : x Î U and x Ï A}

U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}A = {1, 3, 5, 7, 9}A’ = {2, 4, 6, 8, 10}

Difference UA B

A – B = {x : x Î A and x Ï B}

If A = {1, 2, 3, 4, 6, 12} B = {1, 2, 4, 8, 16} A – B = {3, 6, 12} B – A = {8, 16}.

Symmetric Differnce

A D B = (A – B) È (B – A) = {x : x Ï A Ç B} = (A È B) – (A Ç B)

If A = {1, 2, 3, 4, 5, 6, 7, 8} B = {1, 3, 5, 6, 7, 8, 9} A – B = {2, 4}, B – A = {9} A D B = {2, 4, 9}.

¾ Properties of Complement(i) Complement Laws :

(a) A È A’ = U(b) A ∩ A’ = f

(ii) De Morgan’s Laws :(a) (A È B)’ = A’ ∩ B’(b) (A Ç B)’ = A’ È B’

(iii) Law of Double Complementation(a) (A’)’ = A

(iv) Complement of f and U :(a) f’ = U(b) U’ = f

¾ Disjoint SetsTwo sets A and B are said to be disjoint, if A Ç B = f. If A Ç B ¹ f, then A and B are called overlapping or interesting sets.Example : If A = {1, 2, 3, 4, 5, 6}, B = {7, 8, 9, 10, 11} and C = {6, 8, 10, 12 14} then A and B are disjoint sets while A and C are intersecting sets.

¾ Cardinal Properties of Sets(a) n(A È B) = n(A) + n(B) – n(A Ç B) If A and B are disjoint, then n(A È B) = n(A) + n(B)(b) n[A È B È C] = n(A) + n(B) + n(C) – n(A ÇB

- n(B Ç C)) – n(C Ç A) + n(A Ç B Ç C)(c) n(A – B] = n(A) – n(A Ç B)(d) n(A D B] = n(A) + n(B) – 2n(A Ç B)(e) n(A’ È B’] = n(A Ç B)’ = n(U) – n(A Ç B)(f) n(A’ Ç B’] = n(A È B)’ = n(U) – n(A È B).


CHAPTER 2 : Relations and Functions ¾ Cartesian Product of Sets

efinition Given two one-empty sets A and B, the set of all ordered pairs (x, y), where x Î A and y Î B is called Cartesian product of A and B; symbolically, we write

A × B = {(x, y) | x Î A and y Î B }If A = {1, 2, 3} and B = {4, 5}, then

A × B = {(1, 4), (2, 4), (3, 4), (1, 5), (2, 5), (3, 5)]and B × A = {(4, 1), (4, 2), (4, 3), (5, 1), (5, 2), (5, 3)]

¾ RelationsA Relation R from a non-empty set A to a non-empty set B is a subset of the Cartesian product set A × B. The subset is derived by describing a relationshipbetween thefirst element and thesecond element of the ordered pairs in A × B.

The set of all first elements in a relation R, iscalled the Domain of the relation R, and the set of all second elements called images, is called the range of R.

¾ Types of Relations(i) Empty Relation : f is a relation of A, called

the empty or void relation, since f Í A × A.(ii) Universal Relation : A × A is a universal

relation, since A × A Í A × A.(iii) Identity Relation : Every element of A is

related to itself Refelxive Relation = {(a, a) : a Î A}

i e exi e elation A relation R on a set A is said to be a symmetric relation, if and only if (iff) (a, b) Î R Þ (b, a) Î R for all a, bÎ A

(v) Antisymmetric Relation :(a) (a, b) Î R but (b, a) Ï R, for all a ¹ b(b) (a, b) Î R but (b, a) Ï R, a = b for all a, b Î A

(vii) Transitive Relation : A relation R on a set A is said to be transitive relation iff (if and only if) (a, b) Î R and (b, c) Î R Þ (a, c) Î R for all a, b, c Î A

¾ FunctionA relation from a set A to a set B is said to be function if every element of set A has one and only one image in set B.

¾ Types of FunctionsFunction Function Definition and Graph Domain Range

Identityy = f(x) = x

–6 –4 –2–2

–4

–6

6

4

2

2 4 6

F( ) =x x

R R

Constanty = f(x) = c

C

Graph if > 0c

Graph if = 0c

Graph if = 0c

R {c}

Polynomial y = f(x) = a0 + a1x + a2x2 + .... + anxn R Depends on

type ofpolynomial

Rational ( )( )

f xg x

where g(x) ¹ 0R Depends on

type of Nr. and Dr.

ModulsY = f(x) = |x|

Y = – xY

y = x

Ox

R R+


Signum

f(x) = >

=− <

1, if 0

0, if 0

1, if 0

x

x

x

1

Y

Y = 1

O– 1Y = – 1

Y'

f x( ) = | |xx

X' X

R {– 1, 0 1}

Greatest Integerf(x) = [x]

–3 –2 –1–1

–2

–3

3

2

1

1 2 3

Y'

4 5OX'

R Z

CHAPTER 3 : Trigonometric Functions

¾ Degree Measure of an angle1° = 60’ [One degree = 60 minute]1 = 60’’ [One minute = 60 seconds]

¾ Relation between Degree and Radian Measure

1. q = lr

where q = angle subtended by arc of length l at the centre of the circle r = readius of the circle.

2. p radians = 180°

3. 1 radians = °

π180

O

I

r

θ

= 57°17’45” (Approx.)

4. 1° = π

180 radian = 0·01746 (app.)

¾ Trigonometric Ratios and Their Q uadrants

I II III IV

sin x + + – –

cos x + – – +

tan x + – + –

cosec x + + – –

sec x + – – +

cot x + – + –

Note : (i) For angle – x, p – x, p + x, 2p – x, 2p + x, the t-ratio remains to be the same.sin (– x) = – sin x, tan (p + x) = tan x, cos (2p – x) = cos x

(ii) For angles the

π π π π− + − +3 3, , ,

2 2 2 2x x x x

the t-ratio changes from cos to sin, tan to cot and cosec to sec and vice versa.

e.g., π π + = + = − sin cos , tan cot2 2

x x x x

¾ T- Ratios of π − 2

x

1. π − = sin cos2

x x

2. π − = cos sin2

x x

3. π − = tan cot2

x x

4. π − = cosec sec2

x x

5. π − = sec cosec2

x x

6. π − = cot tan2

x x

¾ T- Ratios of π + 2

x

1. π + = sin cos2

x x

2. π + = − cos sin2

x x


3. π + = −

tan cot2

x x

4. π + = cosec sec2

x x

5. π + = − sec cosec2

x x

6. π + = − cot tan 2

x x

¾ T- Ratios of ( p – x) :1. sin (p – x) = sin x2. cos (p – x) = – cos x3. tan (p – x) = – tan x4. cosec (p – x) = cosec x5. sec (p – x) = – sec x6. cot (p – x) = – cot x

¾ T- Ratios of ( p + x) :1. sin (p + x) = – sin x2. cos (p + x) = – cos x3. tan (p + x) = tan x4. cosec (p + x) = – cosec x5. sec (p + x) = – sec x6. cot (p + x) = cot x

¾ T- Ratios of ( 2 p – x) :1. sin (2p – x) = – sin x2. cos (2p – x) = cos x

3. tan (2p – x) = – tan x4. cosec (2p – x) = – cosec x5. sec (2p – x) = sec x6. cot (2p – x) = – cot x

¾ T- Ratios of ( 2 p + x) :1. sin (2p + x) = sin x2. cos (2p + x) = cos x3. tan (2p + x) = tan x4. cosec (2p + x) = cosec x5. sec (2p + x) = sec x6. cot (2p + x) = cot x

¾ Domain and Range of Trigonometric Functions

Functions Domain Range

sine R [– 1, 1]

cosine R [– 1, 1]

tanR – {2n + 1}

π2

: n Î Z}R

cotR – {pn : n Î Z}

R

secR – {2n + 1}

π2

:

n Î Z}

R – (– 1, 1)

cosecR – {np : n Î Z}

R – (– 1, 1)

¾ Important Angles and T- Raios0° 15° 18° 30° 36° 45° 60° 90°

sine 0 −6 24

−5 14

12

−10 2 54

12

32

1

cosine 1 +6 24

+10 2 54

32

+5 14

12

12

0

tan 0 −2 3 −25 10 55

13 −5 2 5 1 3

not defined

¾ Function of Negative Anglessin (– q) = – sin q, cos (– q) = cos qtan (– q) = – tan q, cot (– q) = – cot qsec (– q) = sec q, cosec (– q) = – cosec q

¾ Compound Angles(i) sin (A + B) = sin A cos B + cos A sin B(ii) sin (A – B) = sin A cos B – cos A sin B(iii) cos (A + B) = cos A cos B – sin A sin B(iv) cos (A – B) = cos A cos B + sin A sin B

(v) tan (A + B) = +

−tanA tan B

1 tanA tan B

(vi) tan (A – B) = −

+tanA tan B

1 tanA tan B

(vii) cot (A + B) = ++

cot Acot B 1cot A cot B

(viii) cot (A – B) = ++

cot Acot B 1cot A cot B


(ix) sin 2A = 2 sin A cos A = + 22 tanA

1 tan A

(x) cos 2A = cos2A – sin2A = 1 – 2 sin2A

= 2 cos2

A – 1 =

−+

2

21 tan A1 tan A

(xi) tan 2A = − 22 tanA

1 tan A

(xii) sin 3A = 3 sin A – 4 sin3 A(xiii) cos 3A = 4 cos3A – 3 cos A

(xiv) tan 3A = −

−

3

2

3 tanA tan A

1 3tan A

(xv) cos A + cos B = + −A B A B

2 cos cos2 2

(xvi) cos A – cos B = + −A B B A

2 sin sin2 2

(xvii) sin A + sin B = + −A B A B

2 sin cos2 2

(xviii) sin A – sin B = + −A B A B

2 cos sin2 2

(xix) 2 sin A cos B = sin (A + B) + sin (A – B)(xx) 2 cos A sin B = sin (A + B) – sin (A – B)(xxi) 2 cos A cos B = cos (A + B) + cos (A – B)(xxii) 2 sin A sin B = cos (A – B) – cos (A + B)

(xxiii) −

= ±A 1 cosA

sin2 2

A+ if lies in quadrants I or II

2A

- if lies in III or IV quadrants2

(xxiv) += ±A 1 cosA

cos2 2

−

A+ if lies in I or IV quadrants

2A

if lies in II or III quadrants2

(xxv) −= ±+

A 1 cosAtan

2 1 cosA

−

A+ if lies in I or III quadrants

2A

if lies in II or IV quadrants2

¾ General Solution of TrigonometricEquations :

sin x = 0 gives x = np, where n Î Z.

cos x = 0 gives x = (2n + 1)π2

, where n Î Z.

sin x = sin y implies x = np, + (– 1)n y, where n Î Z.

cos x = cos y implies x = 2np ± y, where n Î Z.

tan x = tan y implies x = np + y, where n Î Z.

¾ Sine and Cosine FormulaSine Formula :

= =sinA sin B sinCa b c

¾ Cosine Formulaa2 = b2 + c2 – 2bc cos Ab2 = c2 + a2 – 2ca cos Bc2 = a2 + b2 – 2ab cos C

A

B C

a

bc

CHAPTER 4 : Mathematical Induction

¾ How to prove that statement P( n) holds with the help of principle of mathmatical

induction ?

Following steps are applied :Step 1 : Verify that P(n) holds for n = 1 i.e., P(1)Step 2 : Suppose that P(n) holds for every k Î NStep 3 : Prove that P(n) holds for n = k + 1.

CHAPTER 5 : Complex Number & Quadratic Equations

¾ Integral Powers of ii = −1

i2 = – 1i3 = i2i = – ii4 = (i2)2 = (– 1)2 = 1

¾ Algebra on Complex NumbersLet z1 = a + ib and z2 = c + id• z1 + z2 = (a + c) + i(b + d)

• z1 – z2 = z1 + (– z2)

• z1z2 = (ac – bd) + i(ad + bc)


• 1

2

zz

=

− =

11 2 1

2

1· ·z z z

z

¾ Conjugate of Complex NumberLet z = a + ib be a complex number z = a – ib

¾ Modulus of Complex Number

Let z = a + ib be a complex number |z|= +2 2a b

¾ Multiplicative InversLet z = a + ib be a complex number

z–1 = −= +

+ + +2 2 2 21 a b

ia ib a b a b

= − =+2 2 2| |

a ib za b z

¾ Polar form of a Complex Number

Let z = a + ib be a complex number Step 1 = Find modulus r.

r = +2 2a b

Step 2 : Find a = tan–1 |b/a|.Step 3 : Find Argument q from the table :q = a If I quadrant i.e., z = a + ibq = p – a If II quadrant i.e., z = – a + ibq = a – p If III quadrant i.e., z = – a – ibq = – a If IV quadrant i.e., z = a – ib

Step 4 : The polar form is z = r (cos q + i sin q) :

¾ Solving Q uadratic Equations For quadratic equation ax2 + bx + c = 0;

x = − ± −2 42

b b aca

CHAPTER 6 : Linear Inequalities

¾ Intervals(a) An open interval denoted by (a, b) is the set

of real numbers {x : a < x < b}(b) A closed interval denoted by [a, b] is the set

of real numbers {x : a £ x £ b}

(c) Interval closed at one end open at the other are given by

[a, b) = {x : a £ x £ b}[a, b) = {x : a < x £ b}

a b

( )a, b

a b a b a b

[ ]a, b [ )a, b ( ]a, b

¾ Important Results(a) If a, b Î R and b ¹ 0, then

(i) ab > 0 or ab

> 0 Þ a and b are of the same

sign.

(ii) ab < 0 or ab

< 0 Þ a and b are opposite sign.

(b) If a is any positive real number, i.e., a > 0, then(i) |x| < a Û – a < x < a

|x| £ a Û – a £ x £ a(ii) |x| > a Û x < – a or x > a

|x| ³ a Û x £ – a or x ³ a

CHAPTER 7 : Permutations and Combinations

¾ Intervals1. Whenever a situation “AND” arises

“Multiplication” will be done2. Whenever a situation “OR” arises

“ADDITION” will be done

¾ FactorialThecontinuedproductoffirstn natural numbers is called the ‘n factorial’ and is dnoted by n! or

.n

0 ! = 1

1 ! = 12 ! = 2 × 1 = 23 ! = 3 × 2 × 1 = 64 ! = 4 × 2 × 2 × 1 = 24 etc...

¾ Permutation and Combination

PERMUTATIONS COMBINATIONS

In one word : Arrangement

In one word : Selection

=−

!P

( )!n

rn

n r=

−!

!( )!n

rn

Cr n r


Example : Consider

three letters a, b & c,

then permutation of

2 out of three will be :

ab , ba

cb , bc

ac , ca

P = 6 permutations

Example : Consider

three letters a, b & c,

then combination of

2 out of three will be :

ab or ba

cb or bc

ac or ca

C = 3 combinations

Always remember:P(n, 0) = 1P(n, 1) = nP(n, n) = n!

Always remember:C(n, 0) = 1C(n, 1) = nC(n, n) = 1

Note : The no. of permutations of n things of which p are alike of one kind; q are alike of second kind

and remaining all are distinct is

!! !n

p q.

¾ Important Combination Results(i) nCr = nCn–r(ii) nCr +

nCr – 1 = n + 1Cr(iii) nn – 1Cr – 1 = (n – r + 1)

nCr – 1

CHAPTER 8 : Binaomial Thereom

¾ Binomial Theorem(a + b)n = nC0a

n + nC1an–1b + nC2a

n–2b2 + ... + nCn–1

a·bn–1 + nCnbn (a + b)n = −

=Σ

0

n n n k kk

kC a b

¾ Some Important Cases• (x – y)n = nC0x

n – nC1xn–1y + nC2x

n–2y2 + .... + (– 1)n nCnyn

• (1 + x)n = nC0 + nC1x + nC2x2 + nC3x

3 + .... + nCnxn

• (1 – x)n = nC0 – nC1x + nC2x2 – .... + (– 1)n nCnxn

• 2n = nC0 + nC1 + nC2 + .... + nCn.

¾ The ( r + 1) th termThe general term or (r + 1)th term in the expansion is given by Tr+1 = nCr a

n–rbr

¾ The pth term from the endThe pth term from the end in the expansion of (a + b)n is (n – p + 2)th term from the beginning.

¾ Middle Terms(i) If n is even then the number of terms in the

expansion will be n + 1. Since n is even so n + 1 is odd. Therefore, the middle term is

+ +

th1 12

n , i.e., +

th12

n

term

(ii) If n is odd, then n + 1 is even, so there will be two middle terms in the expansion, namely,

+

th12

nterm and

+ +

th11

2n

term.

CHAPTER 9 : Sequence and Series

¾ Arithmetic Progression• Standard From of AP : a, a + d, a + 2d, .....• nth term : an = a + (n – 1)d.

• Sum of n terms : Sn = [ ]+ −2 ( 1)2n

a n d

Sn = [ ]+2n

a l

• Arithmetic Mean of a & b : A = +2

a b

• Selection of terms in AP :

No. of terms

TermsCommondifference

3 a – d, a, a + d d4 a – 3d, a – d, a + d, a + 3d 2d5 a – 2d, a – d, a, a + d, a + 2d d6 a – 5d, a – 3d, a –d , a + d ,

a + 3d, a + 5d2d

¾ Geometric Progression• Standard Form of GP a, ar, ar2, ...., arn–1,...• nth term : an = arn – 1

• Sum of n terms : Sn = −−

(1 ),

1

na rr

if |r| < 1

Sn = −

−( 1)

,1

na rr

if |r| > 1

• Geometric Mean of a & b : G = ab

• Selection of terms in GPNo. of terms Terms Common difference

3 , ,a

a arr

r

4 33 , , ,a a

ar arrr

r2

5 22 , , , ,a a

a ar arrr

r


¾ Properties of AP1. If a constant is added to each term of an A.P.,

the resulting sequence is also an A.P. with the common difference.

2. If a constant is subtracted from each term of an A.P., the resulting sequence is also an A.P. with the same common difference.

3. If each term of an A.P. is multiplied by a constant. then the resulting series is also and A.P.

4. If each term of an A.P. is divided by a non-zero constant, then the resulting series is also an A.P.

¾ Sum of Special Sequencesum of fir t n natural num er

Sn = 1 + 2 + 3 + ... + n = +( 1)

2n n

um of the e uence of the fir t n natural num er

Sn2 = 12 + 22 + 32 + ... + n2

= + +( 1)(2 1)

6n n n

um of the cu e of the fir t n natural num er

Sn3 = 13 + 23 + 33 + ... + n3

= +

2( 1)2

n n

¾ u o nfinite G

S = −1a

r

C 10 trai ht ine

¾ Slope of a Line [ Passing through ( x1 , y1 ) and ( x 2 , y2 ) ]

m = tan q = −−

2 1

2 1

y yx x

¾ Angle between two Lines

tan q = −±

+1 2

1 2

( )1m m

m m

If we take the acute angle between two lines,

then tan q = −

+1 2

1 21m m

m m

If the lines are parallel, then m1 = m2.

If the lines are perpendicular, then m1m2 = – 1.

¾ Various Forms of Equation of Line1. If a line is at a distance a and parallel to

x-axis, then the equation of the line is y = ± a.

2. If a line is parallel to y-axis at a distance b from y-axis then its equation is x = ± b

3. Point-slope form : The equation of a line having slope m and passing through the point (x0, y0) is given by y – y0 = m (x – x0)

4. Two-point-form : The equation of a line passing through two points (x1, y1) and (x2, y2) is given by

y – y1 = −

−−

2 11

2 1( )

y yx x

x x

5. Slope intercept form : The equation of the line making an intercept c on y-axis and having slope m is given by

y = mx + cNote that the value of c will be positive or negative as the intercept is made on the positive or negative side of the y-axis, respectively.

6. Intercept form : The equation of the line making intercepts a and b on x- and y-axis

respectively is given by + = 1.yx

a b

7. Normal form : Suppose a non-vertical line is known to us with following data :(a) Length of the perpendicular (normal) p

from origin to the line.(b) Angle w which normal makes with the

positive direction of x-axis.Then the equation of such a line is given by x cos w + y sin w = p

¾ Genera uation o a ineAny equation of the form Ax + By + C = 0, where A and B are simultaneously not zero, is called the general equation of a line.

ifferent form of x + By + C = 0The general from of a line can be reduced to various forms as given below :i Slope intercept form : If B ¹ 0, then Ax + By +

C = 0 can be written as

y = − −+A CB B

x


or y = mx + c,

where m = −AB

and c = −CB

If B = 0, then x = −CA

which is a verical line

whoseslopeisnotdefinedandx-interceptis−C

.A

(ii) Intercpt form : If C ¹ 0. then Ax + By + C = 0

can be written as +− −C CA B

x x

= 1 or + = 1

yxa b

,

where a = −CA

and b = −CB

.

If C = 0, then Ax + By + C = 0 can be written as Ax + By = 0 which is a line passing through the origin and therefore has zero intercepts on the axes.

(iii) Normal Form : The normal form of the equaiton Ax + By + c = 0 is x cos w + y sin w = p where.

cos w = ±+2 2

A

A B,

sin w = ±+2 2

B

A B

and p = ±+2 2

C

A B

Note : Proper choice of signs is to be made so that p should be always positive

¾ DistanceDistance of a point from a line The perpendicular distance (or simply distance) d of a point P (x1, y1) from the line Ax + By + C = 0 is given by

d = + +

+1 1

2 2

|A B C|

A B

x y

Distance between two parallel lines

The distance d between two parallel lines y = mx

+ c1 and y = mx + c2 is given by d = 1 2

21

c c

m

−

+Equation of family of lines : Let the two Intersecting lines l1 and l2 be given by

A1x + B1y + C1 = 0 (1)

and A2x + B2y + C2 = 0 (2)

From the equations (i) and (ii), we can form and equation

A1x + B1y + C1 + k(A2x + B2y + C2) = 0 (3)

CHAPTER 11 : Conic Section

¾ Types of Conic Section

αβ

Plane

Cone

l

(a) When b = 90°, the section is a circle.(b) When a < b < 90°, the section is an ellipse.(c) When b = a; the section is a parabola.

(In each of the above three situations, the plane cuts entirely acorss one nappe of the cone).

(d) When 0 £ b < a; the plane cuts through both the nappes and the curves of intersection is a hyperbola.

¾ CircleEquation of Circle

(x – h)2 + (y – k)2 = r2

( )h, kP( )x, y

C

O

Y

X

The cetnre of this circle is (h, k)The radius of the circle is rGeneral Equations of Circle x2 + y2 + 2gx + 2fy + c = 0The centre of this circle is (– g, – f)

The radius of the circle is + −2 2g f c

¾ Parabola

F( 0)a,

y ax2 = 4

xa

= –

Y

Y'

X' X


Main facts about the parabola

Forms of Parabolas y2 = 4ax y2 = – 4ax x2 = 4ax x2 = – 4ay

Axis y = 0 y = 0 x = 0 x = 0

Directix x = – a x = a y = – a y = a

Vertex (0, 0) (0, 0) (0, 0) (0, – 0)

Focus (a, 0) (– a, 0) (0, a) (0, – a)

Length of latus rectum 4a 4a 4a 4aEquations of latus rectum x = a x = – a y = a y = – a

¾ Ellipse

Main facts about the Elipse

Forms of the ellipse + =22

2 2 1yx

a b+ =

22

2 2 1yx

b a

a > b a > b

Equation of major axis y = 0 x = 0

Length of major axis 2a 2aEquation of Minor axis x = 0 y = 0

Length of Minor axis 2b 2b

Directrices x = ± ae

y = ± ae

Equation of latus rectum x = ± ae y = ± ae

Length of latus rectum22b

a

22ba

Centre (0, 0) (0, 0)

In both cases a > b and b2 = a2 (1 – e2), e < 1.

Oswaal CBSE Sample Question PapersFor Class 11 Mathematics

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