FOR MARCH 2018

EXAM

MATHEMATICS

thStrictly Based on the Latest CBSE Syllabus Dated 4 April 2017 for Academic Year 2017-18

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CBSE CLASS 11

Includes Solved Papers 2017(Kendriya Vidyalaya Sangathan)

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07

04

1. Sets 1 – 12

2. Relations and Functions 13 – 20

3. Trigonometric Functions 21 – 37

4. Principle of Mathematical Induction 38 – 46

5. Complex Numbers and Quadratic Equation 47 – 57

6. Linear Inequalities 58 – 69

7. Permutations and Combinations 70 – 79

8. Binomial Theorem 80 – 91

9. Sequences and Series 92 – 107

10. Coordinate Geometry 108 – 126

11. Conic Sections 127 – 145

12. Introduction to Three Dimensional Geometry 146 – 150

13. Limits and Derivatives 151 – 163

14. Mathematical Reasoning 164 – 171

15. Statistics 172 – 184

16. Probability 185 – 199

qqNote : BSM - Board Supplementary Material, NCT - National Capital Territory, KVS - Kendriya Vidyalaya

Sangathan

• Syllabus 5 – 8

• Solved Paper (KVS), 2017 – Agra Region 9 – 17

• Solved Paper (KVS), 2017 – Guwahati Region 18 – 26

• Chapterwise / Topicwise 2 marks Questions for March Exam 2018 27 – 40

• Solved Paper (KVS), 2016 – Agra Region ix – xviii

• Solved Paper (KVS), 2016 – Mumbai Region xix – xxx

• Solved Paper (KVS), 2016 – Guwahati Region xxxi – xl

• Solved Paper (KVS), 2015 9 – 16

CONTENT

( 3 )

PREFACE

CBSE always believes in Global Trends of Educational Transformation. The CBSE

curriculum gets its lead from National Curriculum Framework – 2005 and Right to Free and

Compulsory Education Act – 2009. The aim of CBSE Curriculum is not just to let learners

obtain basic knowledge but to make them life-long learners. CBSE always updates and

reviews the syllabus to make it more relevant with educational transformation and in last few

years the chapters and topics which CBSE has added are very interesting and increase

practical knowledge.

Oswaal Question Banks are designed to nurture individuality and thus enhance one's

innate potentials which helps in increasing the self-study mode for students. This book

strengthens knowledge and attitude related to subject. It is designed in such a way that students

can set their own goals and can improve their problem solving and thinking skills.

The journey of this book is never ending as this book is reviewed every year and

new questions, previous year's examination questions, new HOTS or any change in

syllabus is updated time to time. Also regular review and readers’ feedback increases the

efficiency of this book gradually.

Moreover every Question Bank strictly follows the latest syllabus and pattern, and

contains more than sufficient questions and brief description of chapters, which help students in

practicing and completing the syllabus. Questions incorporated in this Question Bank

encompass all the ‘Typologies’ mentioned by CBSE namely Remembering, Understanding,

Application, High Order Thinking Skills and Evaluation. Solutions for these have been checked

twice and efforts have been made to align them closely to the Marking Scheme. Practically, this

book provides students everything they need to learn and excel.

At last we would like to thank our authors, editors, reviewers and specially students who

regularly send us suggestions which helps in continuous improvement of this book and makes

this book stand in the category of “One of the Best”. Wish you all Happy Learning.

–Publisher

( 4 )

Latest Syllabus for Academic Year 2017-18

Course Structure

One Paper Total Periods 240 of 35 Minutes each

Three Hours Max. Marks. 100

� No. Units � No. of Periods� 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.

Unit-I : Sets and Functions

1. � Sets � 20 Periods

� Sets and their representations. Empty set. Finite and Infinite sets. Equal sets. Subsets. Subsets of a set 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

� Ordered pairs, Cartesian product of sets. Number of elements in the cartesian product of two finite sets. Cartesian product of the sets of reals with itself (upto R x R x R). Definition of relation, pictorial diagrams, 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 Periods

� Positive and negative angles. Measuring angles in radians and in degrees and conversion from one measure to another. Definition of trigonometric functions with the help of unit circle. Truth of the

2 2identity sin x + cos x=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 and cos y and their simple applications. Deducing the identities like the following :

tan (x ± y) = , cot (x ± y) =

sina ± sinb = 2sin (a ± b ) cos (a����b�)

cosa + cosb = 2cos (a+b ) cos (a - b ),

cosa – cosb = – 2sin (a+b ) sin (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.

±

tan x ± tan y

1 tan x and tan y±

cot x cot y 1

cot y ± cot x

±

12

12

12

12

12

12

( 5 )

Unit-II : Algebra

1. � 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

� Need for complex numbers, especially – , to be motivated by inability to solve some of the quardratic

equations. Algebraic properties of complex numbers. Argand plane and polar representation of

complex numbers. Statement of Fundamental Theorem of Algebra, solution of quadratic equations

(with real coefficients) in the complex number system. Square root of a complex number.

3. � Linear Inequalities� 15 Periods

� Linear inequalities. Algebraic solutions of linear inequalities in one variable and their representation

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

finding a solution of system of linear inequalities in two variables.

4. � Permutations and Combinations � 10 Periods

� Fundamental principle of counting. Factorial n. (n!) Permutations and combinations, derivation of

formulae for and n and n their connections, simple applications.p crr

5. � Binomial Theorem � 10 Periods

� History, 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 Periods

� Sequence and Series. Arithmetic Progression (A.P.). Arithmetic Mean (A.M.) Geometric Progression

(G.P.), general term of a G.P., sum of first n terms of a G.P., infinite G.P. and its sum, geometric mean

(G.M.), relation between A.M. and G.M. Formulae for the following special sum.

�

Unit-III : Coordinate Geometry

1. � 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 Periods

� Sections of a cone: circles, 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 Periods

� Coordinate axes and coordinate planes in three dimensions. Coordinates of a point. Distance between

two points and section formula.

1

Sn

k=1

k,Sn

k=1

2k andSn

k=1

3k

( 6 )

Unit-IV : Calculus

1. � 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, trignometric, exponential and

logarithmic functions. Definition of derivative, relate it to slope of tangent of a curve, Derivative of

sum, difference, product and quotient of functions. Derivatives of polynomial and trignometric

functions.

Unit-V : Mathematical Reasoning

1. � Mathematical Reasoning 10 Periods

� Mathematically acceptable statements. Connecting words/phrases - consolidating the understanding

of “if and only if (necessary and sufficient) condition”, “implies”, “and/or”, “implied by”, “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 Probability

1. � 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 Periods

� Random 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 theories studied in earlier classes. Probability of an event, probability of

‘not’, ‘and’ and ‘or’ events.

ll

( 7 )

Question Paper Design (2017-18)

Mathematics (Code No. 041)

Class XI Time : 3 Hours� Max. Marks. 100

S.� Typology of Questions� Very� Short Long� Long� Marks � %

No.� � Short� Answer Answer I� Answer-II� � Weightage� � Answer� (2 marks) (4 marks)� (6 marks) (1 marks)

1. � Remembering-(Knowledge based Simple�� recall questions, to know specific facts,�� terms, concepts, principles, or theories,� 2� 2� 2� 1 20� 20%� Identify, define, or recite, information)� �

2.� Understanding-(Comprehension to be�� familiar with meaning and to understand� 1� 3� 4� 2 35� 35%

� conceptually, interpret, compare, contrast,

� explain, paraphrase information)

3. � Application (Use abstract information in�� concrete situation, to apply knowledge to � 1� –� 3� 2 25� 25%� new situations, Use given content to � interpret a situation, provide an example, � or solve a problem)

4.� High Order Thinking Skills (Analysis�� & Synthesis- Classify, compare, contrast, � or differentiate between different pieces of� –� 3� 1� – 10� 10%� information, Organize and/or integrate � unique pieces of information from a variety � of sources)

5.� Evaluation -(Appraise, judge, and/or � � � �� justify the value or worth of a decision� –� –� 1� 1 10� 10%� or outcome, or to predict outcomes based � � �� on values)� �

� TOTAL � 1×4=4 � 2×8=16 � 4×11=44 � 6×6=36 100 � 100%

QUESTION WISE BREAK UP

� � Type of Question � Mark per Question � Total No. of Questions � Total Marks

� � VSA � 1 � 4 � 04

SA 2 8 16

� � LA-I � 4 � 11 � 44

� � LA-II � 6� 6 � 36

� � Total � � 29 � 100

1. No chapter wise weightage. Care to be taken to cover all the chapters.

2. Suitable internal variations may be made for generating various templates keeping the overall weightage to different form of questions and typology of questions same.

Choice(s) :

There will be no overall choice in the question paper. However, 30% internal choices will be given in 4 marks and 6 marks questions.

VBQ : One of the LA-1 type question should be to assess the values inherent in the texts.

( 8 )

KENDRIYA VIDYALAYA SANGATHAN[AGRA REGION]

SESSION ENDING EXAMINATION 2017SUBJECT : MATHEMATICS

CLASS–XI(SOLVED PAPER)

Time : 3 Hrs. M.M. : 100

Instructions : 1. All questions are compulsory. 2. The question paper consists of 29 questions divided into 4 sections A, B, C and D. Section A consists of 4 questions of

1 mark each, Section B consists of 8 questions of 2 marks each. Section C consists of 11 questions of 4 marks each, Section D consists of 6 questions of 6 marks each.

3. All questions in Section A are to be answered in one word, one sentence or as per the exact requirement of the questions. 4. Question number 21 is value based question. 5. Calculator is not allowed.

SECTION A

1. If f x x( ) ,= +95

32 then find f(– 10).

1

2. Write the general term in the expansion of (x2 – y)6. 1 3. Find the equation of circle whose centre is (– 1, 2) and radius is 4. 1 4. For the following compound statement, identify the connecting word and then break it into component

statements : 1 "The sand heats up quickly in the sun and does not cool down fast at night."

SECTION B 5. Let A = {All prime numbers less than 10} and B = {all odd natural numbers less than 10}. Find A – (A Ç B). 2 6. Find the value of tan 15 °. 2

7. Represent the complex number z i= +1 3 in the polar form. 2

8. In a G.P., the 3rd term is 24 and 6th term is 192. Find the tenth term. 2 9. In how many ways can select a cricket team of eleven from 17 players in which only 5 players can bowl if each

cricket team of 11 must include exactly 4 bowlers ? 2

10. Evaluate : limsin

sinx

ax bxax bx→

++0

2

11. Show that the statement : "If x is a real number such that x3 + 4x = 0, then x is 0." Is true by method of contra positive. 2

12. Coefficient of variation of two distributions are 60 and 70, and their standard deviations are 21 and 16 respectively. What are their arithmetic means ? 2

SECTION C 13. Out of 25 members in a family, 12 like to take tea, 15 like to take coffee and 7 like to take coffee and tea both. How

many like (i) at least one of the two drinks. (ii) only tea but not coffee. (iii) only coffee but not tea. (iv) neither tea nor coffee. 4

14. Define a relation R on the set N of natural numbers by R = {(x, y) : y = 2x – 1; x, y Î N, x £ 5}. Depict this relationship using roster form. Write down the domain and range. 4

15. Prove that : cos cos( ) cot cot( ) .32

232

2 1π

ππ

π+

+ −

+ +

=x x x x

4

OR Find the general solution for the equation : 3 2cos sin .θ θ+ =

10 ] OSWAAL CBSE Question Bank Chapterwise Solution, Mathematics-XI

16. Find real q such that 3 21 2

+−

iisinsin

θθ

is purely real.

4

17. In how many of distinct permutations of the letters in MISSISSIPPI do the four I's not come together ? 4OR

If nCr : nCr+1 = 1 : 2 and nCr+1 :

nCr+2 = 2 : 3, determine the values of n and r. 18. Find the sum of n terms of the sequence 7, 77, 777, 7777, ......... . 4 19. Assuming that straight lines work as the plane mirror for a point, find the image of the point (3, 8) in the line

x + 3y = 7. 4 20. As arch is in the form of a parabola with its axis vertical. The arch is 10 m high and 5 m wide at the base. How

wide is it 2 m from the vertex of the parabola. 4OR

An arch is in the form of a semi ellipse. It is 8 m wide and 2 m high at the centre. Find the height of the arch at a point 1·5 m from one end.

21. Using section formula, show that the points : A(2, – 3, 4), B(– 1, 2, 1) and C 013

2, ,

are collinear.

4

22. Find from first principle the derivative of sin2x. 4 23. Calculate the mean and standard deviation of the data given below : 4

Class Interval 1 – 5 6 – 10 11 – 15 16 – 20 21 – 25

Frequency 3 8 13 18 23

SECTION D 24. In a survey it was found that 21 persons liked product A, 26 liked product B and 29 liked product C. If 14 people

liked products A and B, 12 people liked products C and A, 14 people liked products B and C and 8 liked all the three products. Find –

(a) the number of people who liked at least one product. (b) the number of people who liked product C only. 6

25. Prove that : cos cos cos2 2 2

3 332

x x x+ +

+ −

=

π π

6

OR

In a triangle ABC, if sin( )sin( )

,A BA B

a ba b

−+

=−+

2 2

2 2 prove that the triangle is either isosceles or right angled.

26. Prove the following by using the principle of mathematical induction for all n Î N : 6

11 2 3

12 3 4

13 4 5

11 2

34 1 2· · · · · ·

.......( )( )

( )( )(

+ + + ++ +

=+

+ +n n nn n

n n ))

27. Solve graphically the following system of inequalities : 6 3x + 2y £ 150, x + 4y £ 80, x £ 15, x ³ 0, y ³ 0. 28. The coefficients of the (r – 1)th, rth and (r + 1)th terms in the expansion of (x + 1)n are in the ratio 1 : 3 : 5. Find n

and r. 6OR

Find the value of a a a a2 24

2 24

1 1+ −( ) + − −( ) .

29. In a class of 60 students, 30 opted for NCC, 32 opted for NSS and 24 opted for both NCC and NSS. If one of these students is selected at random, find the probability that

(a) the student opted for NCC or NSS. (b) the student has opted neither NCC nor NSS. (c) the student has opted NSS but not NCC. 6

OR In a single throw of two dice, determine the probability of obtaining a total of 7 or 9.

Solved Paper - 2017 [ 11

SOLUTIONSSECTION A

1. f(x) = 95

32x +

\ f(– 10) =

95

10 32( )− +

= – 18 + 32 = 14 1 2. General term in the expansion of (x2 – y)6 is Tr+1 = 6Cr(x

2)6–r(– y)r

= (– 1)r·6Crx12–2r·yr 1

3. Equation of circle having centre (– 1, 2) and radius 4 is given by

(x + 1)2 + (y – 2)2 = (4)2

Þ x2 + 2x + 1 + y2 – 4y + 4 = 16 Þ x2 + y2 + 2x – 4y – 11 = 0 1 4. Connecting word = and Statement I : The sand heats up quickly in the sun. Statement II : Sand does not cool down fast at night. 1

SECTION B 5. A = {All prime numbers less than 10} Þ A = {2, 3, 5, 7} Þ B = {all odd numbers less than 10} Þ B = {1, 3, 5, 7, 9} Þ A Ç B = {3, 5, 7} \ A – (A Ç B) = {2, 3, 5, 7} – {3, 5, 7} = {2} 2 6. We know that

tan(A – B) = tan tantan ·tanA BA B−

+1 Putting A = 60° and B = 45°

\ tan (60° – 45°) = tan tantan ·tan60 45

1 60 45° − °

+ ° °

= 3 1

1 3−

+ 2

7. Given Z = 1 3+ i

Let Z = r(cos q + i sin q) Þ r cos q = 1

and r sin q = 3

squaring and adding, we get r2cos2q + r2 sin2q = 1 + 3 r2(sin2q + cos2q) = 4 Þ r2 = 4 or r = 2

Also, rrsincos

θθ

= 31

(Dividing)

Þ tan q = 3

Þ q = π3

\ Complex number in polar form

z = 23 3

cos sinπ π

+

i

2

8. Let the first term of G.P. be a and the common ratio be r.

Given, T3 = 24 Þ ar2 = 24 ...(i) and T6 = 192 Þ ar5 = 192 ...(ii) Dividing (ii) ÷ (i)

Þ

arar

5

2 =

19224

Þ r3 = 8 Þ r = 2 putting in eqn. (i) a(2)2 = 24 Þ a = 6 \ T10 = ar9

= 6(2)9

= 3072 2 9. Total players = 17 Bowlers = 5 Non-Bowlers = 12 Team of 11 players is to be selected having exactly 4

bowlers No. of ways = 5C4 × 12C7 = 5 × 792 = 3960. 2

10. limsin

sinx

ax bxax bx→

++0

Divide both numerator and denominator by x

Þ lim

sin

sinx

axx

b

abxx

→

+

+0

= lim

sin

sinx

a axax

b

ab bxbx

→

+

+0

= a ba b

++

= 1 2 11. To prove statement p to be true by contrapositive

method, we assume that r is false and prove that q must be false.

Here, r is false implies that it is required to consider the negation of statement r. This obtains the following statement.

~ r : x is not 0. It can be seen that x2 + 4 will always be positive.

x ¹ 0 implies that the product of any positive real number with x is not zero.

Let us consider the product x with (x2 + 4). \ x(x2 + 4) ¹ 0 \ x3 + 4x ¹ 0

12 ] OSWAAL CBSE Question Bank Chapterwise Solution, Mathematics-XI

This shows that the statement q is not true. Thus it has been proved that ~ r ® ~ q \ The given statement is true. 2 12. C·V1 = 60, C·V2 = 70, s1 = 21, s2 = 16

We know that C.V. = σx

× 100

Þ 60 = 21

1001x

×

Þ x1 =

21 10060×

Þ x1 = 35

Similarly 70 = 16

1002

x×

Þ

x2 =

16 10070×

Þ x2 = 22·85 2

SECTION C 13. Given that n(T) = 12 n(C) = 15 n(T Ç C) = 7 (i) \ n(T È C) = n(T) + n(C) – n(T Ç C) = 12 + 15 – 7 n(T È C) = 20 20 members like at least one of the two drinks. (ii) Only tea but not coffee = n(T) – n(T Ç C) = 12 – 7 = 5 (iii) Only coffee but not tea = n(C) – n(T Ç C) = 15 – 7 = 8 (iv) Neither tea nor coffee = n(U) – n(T È C) = 25 – 20 = 5 4

14. Given y = 2x – 1 and x £ 5 for x = 1, y = 2 × 1 – 1 = 1 x = 2, y = 2 × 2 – 1 = 3 x = 3, y = 2 × 3 – 1 = 5 x = 4, y = 2 × 4 – 1 = 7 x = 5, y = 2 × 5 – 1 = 9 Relation R = {(1, 1), (2, 3), (3, 5), (4, 7), (5, 9)} \ Domain = {1, 2, 3, 4, 5} Range = {1, 3, 5, 7, 9} 4

15. To prove :

cos ·cos( ) cot cot( )

32

232

2π

ππ

π+

+ −

+ +

x x x x

= 1 L.H.S. =

cos ·cos( ) cot cot( )

32

232

2π

ππ

π+

+ −

+ +

x x x x

= sin ·cos [tan cot ]x x x x+

= sin ·cossincos

cossin

x xxx

xx

+

= sin ·cossin cossin ·cos

x xx xx x

2 2+

= sin ·cos ·sin ·cos

x xx x

1

= 1 = R.H.S. 4

16. z = 3 21 2

+−

iisinsin

θθ

z = ( sin )( sin )

( sin )( sin )

3 21 2

1 21 2

+−

×++

ii

ii

θθ

θθ

z = 3 6 2 41 4

2

2+ + −

+i isin sin sin

sinθ θ θ

θ

z = 3 4 81 4

2

2− +

+sin sin

sinθ θ

θi

z = 3 41 4

81 4

2

2 2−+

++

sinsin

sinsin

θθ

θθ

i

for z to be purely real

81 4 2

sinsin

θθ+

= 0

Þ sin q = 0 Þ q = 0, p, 2p ..... Hence q = np 4 17. In the given word MISSISSIPPI, there are 4I, 4S, 2P

and 1M. Total number of permutations with no restriction

= 11

4 2 2!

! ! !

If we take 4I as one letter then total letters become = 11 – 4 + 1 = 8 If P is the permutation when 4I's are not together,

then

P = 11

4 2 2!

! ! ! −

84 2!! !

= 34650 – 840 = 33,810. 4

OR nCr :

nCr+1 = 1 : 2 and nCr+1 :

nCr+2 = 2 : 3

nr

nr

C

C+1

=

12

Solved Paper - 2017 [ 13

Þ

nr n r

nr n r

!!( )!

!( )!( )!

−

+ − −1 1

=

12

Þ ( )!( )!

!( )!r n r

r n r+ − −

−1 1

=

12

Þ ( )rn r

+−1

=

12

Þ 2r + 2 = n – r

Þ n = 3r + 2 ...(i)

Similarly,

nr

nr

C

C+

+

1

2 =

23

Þ

nr n r

nr n r

!( )!( )!

!( )!( )!

+ − −

+ − −

1 1

2 2

=

23

Þ ( )!( )!( )!( )!r n rr n r

+ − −+ − −2 21 1

=

23

Þ rn r

+− −

21

=

23

Þ 3r + 6 = 2n – 2r – 2

Þ 2n = 5r + 8 ...(ii)

Solving (i) and (ii),

6r + 4 = 5r + 8

Þ r = 4

and n = 14 4

18. S = 7 + 77 + 777 + .... 77 .... 7

(n terms)

S = 7[1 + 11 + 111 + .... 11 ... 1]

(n terms)

S = 799 99 999 99 9[ ... .... ]+ + +

S = 7910 1 10 1 10 1 10 12 3[ ... ]− + − + − + −n

S = 7910 10 101 2+ + − .... n n

= 7910 10 110 1( )n

n−

−−

= 79109

10 1( )n n− −

4

19. Let AB be the line x + 3y = 7 and the image of P(3, 8) be Q(x1, y1)

Mid-point of PQ is Mx y3

2

8

21 1

+ +

, and M lies on

AB.

P(3, 8)

( , )x y1 1

AM

Q

B

x + y =3 7

\

x y1 1

3

23

8

2

++

+

= 7

Þ x1 + 3 + 3y1 + 24 = 14 Þ x1 + 3y1 = – 13 ...(i)

Slope of AB = −13

Slope of PQ = y

x1

1

8

3

−

−

and AB is Perpendicular to PQ

Þ −

−

−

13

8

31

1

y

x = – 1

Þ y1 – 8 = 3(x1 – 3) Þ y1 = 3x1 – 1 ...(ii) Putting value of y1 in eqn. (i) x1 + 3(3x1 – 1) = – 13 Þ 10x1 = – 10 Þ x1 = – 1 and y1 = – 4 4 \ The point Q is (– 1, – 4). 20.

x-axis

y-axis

0

As the axis of parabola is vertical, its equation is x2 = 4ay

Point 52

10, −

lies on it.

\ 254

= 4a(– 10)

Þ 4a =

−58

Þ x2

=

−58

y

Let a point P(x1, – 2) lies on parabola

Þ (x1)2 = − × −

58

2( )

14 ] OSWAAL CBSE Question Bank Chapterwise Solution, Mathematics-XI

Þ x12 = +

54

Þ x1 = ±52

\ Width of parabola 2 m from vertex is

252

× = 5 m

= 2·23 m 4OR

The arch is 8 m wide and 2 m high at centre in the form of a semi-ellipse.

\ Arch is part of ellipse whose semi-major and semi-minor axis are 4 m and 2 m respectively.

i.e., a = 4, b = 2

\ Eqaution of ellipse is x y2 2

16 41+ =

or x2 + 4y2 = 16. A point 1·5 m from one end of arch is (4 – 1·5) m = 2·5 m

from origin. Height of the arch at this point is given by value of y obtained by putting x = 2·5.

4 m 2·5 m

2 m

y

x-axis

y-axis

1·5 m

\ (2·5)2 + 4y2 = 16 \ 4y2 = 16 – 6·25 = 9·75

y = 9 752

3 122

· ·=

= 1·56 m (approx) 4

21. A B C

(2, – 3, 4) (– 1, 2, 1)0, , 21

3

k : 1

Let the ratio in which B divides AC be k : 1 i.e. AB : BC = k : 1

Applying section formula

for x-coordinate – 1 = k

k( ) ( )0 1 2

1++

Þ – k – 1 = 2 Þ k = – 3 ...(i) for y-coordinate

2 = k

k

13

1 3

1

+ −

+

( )

Þ 2k + 2 = k3

3−

Þ 23

kk

− = – 5

Þ 53k

= – 5

Þ k = – 3 ...(ii)

for z-coordinate 1 = k

k( ) ( )2 1 4

1++

Þ k + 1 = 2k + 4 Þ k = – 3 ...(iii) from (i), (ii) and (iii), k has the same value Þ A, B, C are collinear. 22. Let f(x) = sin2x f(x + h) = sin2(x + h)

\ f '(x) = lim( ) ( )

h

f x h f xh→

+ −0

= lim

sin ( ) sinh

x h xh→

+ −0

2 2

= lim

sin( ) sin sin( ) sinh

x h x x h xh→

+ +[ ] + −[ ]0

= lim

sin cos sin cos

h

xh h h

xh

h→

+

+

0

22 2

22 2

= lim sin cos ·cos ·

sin

hx

hx

h hh

h→+

+

04

2 2 22

22

= 4 112

·sin cos · ·x x

= 2sin x cos x. 4 23.

C.I.C.I.

(Continuous)xi fi fi xi fixi

2

1 – 5 0·5 – 5·5 3 3 9 27

6 – 10 5·5 – 10·5 8 8 64 512

11 – 15 10·5 – 15·5 13 13 169 2197

16 – 20 15·5 – 20·5 18 18 324 5832

21 – 25 20·5 – 25·5 23 23 529 12167

Sfi = 65 1095 20735

Oswaal CBSE Chapterwise/TopicwiseQuestion Bank For Class 11

Mathematics (Mar. 2018 Exam)

Publisher : Oswaal Books ISBN : 9789386339690 Author : Panel Of Experts

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