CLASS XI SAMPLE OF MATHS

BLUE PRINT

S.No	Topics	VSA(1)	SA(2)	LA-I(3)	LA-II(5)	Total/G.Total
1	a) Sets (b) Relations and Functions (c) Trigonometric Functions	1+1(case study) 3 3	- 2 1	1 - 1	- - -	03(8M) 05(7M) 05(8M)	13(23M)
2	(a) Complex Numbers & Quadratic Equation (b) Linear Inequalities (c) Permutation & Combination (d) Sequence & Series	2 - 2 1	1 1 1 -	1 - 1 1	- 1 - 1	04(7M) 02(7M) 04(7M) 03(9M)	13(30M)
3	(a) Straight Lines (b) Conic Sections (c) Introduction to 3-Dimensional Geometry	- 1 2	1 1 -	1 - -	- - -	02(5M) 02(3M) 02(2M)	06(10M)
4	Limits & Derivatives	-	2	1		03(7M)	03(7M)
5.	a) Statistics (b) Probability	- 1+1(case study)	- -	- -	1 -	01(5M) 02(5M)	03(10M)
	Total	16+2(case study)(24M)	10(20M)	07(21M)	03(15M)	38(80M)

SET-I

COMMON SESSION ENDING EXAMINATION 2021

CHANDIGARH REGION

CLASS: XI

SUBJECT: Mathematics

MAX. MARKS : 80

TIME : 3 Hours

General Instructions :

This question paper contains two parts A and B. Each part is compulsory. Part A carries 24 marks and Part B carries 56 marks.

Part-A has Objective Type Questions and Part –B has Descriptive Type Questions

Both Part A and B have choices.

Part-A :

It consists of two sections I and II.

Section I comprises of 16 very short answer type questions.

Section II contains 2 case studies. Each case study comprises of 5 case-based MCQs. An examinee is to attempt any 4 out of 5 MCQs.

Part-B :

It consists of three sections – III, IV and V.

Section III comprises of 10 questions of 2 marks each.

Section IV comprises of 7 questions of 3 marks each.

Section V comprises of 3 questions of 5 marks each.

Internal choice is provided in 3 questions of Section-III , 2 questions of Section-IV and

3 questions of Section-V. You have to attempt only one of the alternatives in all such questions.

Sr.

No.

Part – A

Marks

Section – I

Q.1

Write the set A={x:x is an integer and -3<x<7} in roster form.

List all subsets of the set {-1,0,1}

Q.2

If A and B are finite sets such that n(A)=m and n(B)=k , find the number of relations from A to B.

Q.3

Determine the domain and range of the relation R defined by R={(x+1,x+3) :xϵ (0,1,2,3,4,5)}.

Q.4

If A={-1,0,1} and B = {3,5} , find A×B .

Q.5

Convert -47°30´ into radian measure.

Q.6

Find the value of -1410° .

Find the value of sin313 .

Q.7

Prove that cosπ+x cos-xsinx -sinx=cot2x

Q.8

Express 1+i6+1-i3in the form of a+ib .

Q.9

Find the multiplicative inverse of 5+3i .

Solve :x2-2x+32=0

Q.10

How many 3-digit even numbers can be formed using the digits 1,2,3,4,5,6 if no digit is repeated ?

How many 4-digit numbers can be formed by using the digits 1 to 9 if repetition of digits is not allowed?

Q.11

In how many ways can a team of 3 boys and 3 girls can be selected from 5 boys and 4 girls?

Q.12

For what values of x ,the numbers -27,x,-72 are in G. P.?

Q.13

Find the equation of the ellipse whose vertices are (±6,0) and foci are (±4,0) .

Find equation of the parabola with focus (0,-3) and directrix y=3.

Q.14

Find the octant in which the points (-3 ,1, 2) and(-3,1,-2) lie.

Q.15

Find the ratio in which the YZ-plane divides the line segment formed by joining the points (-2,4,7) and (3, -5,8).

Q.16

An experiment consists of tossing a coin and then throwing it second time if a head occurs. If a tail occurs on the first toss, then a die is rolled once. Find the sample space.

Section – II

Both the case study-based questions are compulsory. Attempt 4 sub parts from each questions Each question carries 1 mark.

Q.17

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. Based on the above information answer the following:

How many students had taken Chemistry only?

3 (b) 5 (c) 4 (d)1

How many students had taken Mathematics only?

(a) 4 (b) 2 (c) 3 (d) 6

How many students had taken Physics only?

(a) 1 (b) 5 (c) 2 (d) 3

How many students had taken Physics and Chemistry but not Mathematics?

(a) 3 (b) 4 (c) 2 (d)1

How many students had taken Mathematics and Physics but not Chemistry?

(a) 6 (b) 5 (c) 4 (d)2

Q.18

A team of medical students doing their internship have to assist during surgeries at a hospital in Chandigarh. The probabilities of surgeries rated at very complex, complex, routine, simple or very simple are respectively, 0.15,0.20,0.31,0.26,0.08. Based on the above information answer the following:

Find the probability that a particular surgery will be rated complex or very complex.

(a) 0.35 (b) 0.15 (c)0.20 (d)0.26

Find the probability that a particular surgery will be rated routine or simple.

(a) 0.35 (b) 0.31 (c) 0.57 (d)0.56

Find the probability that a particular surgery will be rated routine or complex

(a) 0.57 (b) 0.51 (c) 0.15 (d)0.50

Find the probability that a particular surgery will be rated neither very complex nor very simple.

(a) 0.56 (b) 0.57 (c) 0.75 (d)0.77

Find the probability that a particular surgery will be rated neither complex nor simple.

(a) 0.56 (b) 0.54 (c) 0.75 (d)0.77

Part B

Section III

Q.19

Find the domain of the function fx=9-x2 .

Q.20

Let f=1,1,2,30,-1,-1,-3 be a function from Z→Z defined by fx=ax+b ,for some integers a and b. Determine a and b.

Q.21

Prove that:cos 4x=1-8sin2x cos2 x

Q.22

Find the modulus of 1+i1-i-1-i1+i .

Q.23

Solve:x4<5x-23-7x-35 .

Q.24

How many words, with or without meaning, can be formed using all letters of the word EQUATION at a time so that the vowels and consonants occur together?

In how many ways can 4 books on mathematics and 3 books on physics be placed on a shelf so that books on the same subject always remain together?

Q.25

Find the equation of a line that cuts off equal intercepts on the coordinate axes and passes through the point (2,3).

Find the distance of the point (-1,1) from the line 12x+6=5(y-2) .

Q.26

Find the equation of a circle with Centre (2,2) and passes through the points (4,5) .

Find the Centre and radius of the circle x2+y2 -2x+4y=8 .

Q.27

Evaluate: cos2x-1cosx-1 .

Q.28

Find the derivative of fx=cosx1+sinx .

Section IV

Q.29

Let A and B be sets. If A∩X= B∩X=∅ and A∪X= B∪X for some set X, show that A=B.

Q.30

Prove that: cos2x+cos2 x+3+cos2x-3=32

Prove that:cos4x+cos3x+cos2xsin4x+sin3x+sin2x=cot3x

Q.31

If (x+iy)3=u+iv , then show that ux+vy=4(x2+y2)

Q.32

In how many of the distinct permutations of the letters in MISSISSIPPI do the four I’s not come together ?

Q.33

If an+bnan-1+bn-1 is the A.M. between a and b, then find the value of n.

Insert six numbers between 3 and 24 such that the resulting sequence is an A.P.

Q.34

If the angle between two lines is 4and slope of one of the line is 12 , find the slope of the other line.

Q.35

Find the derivative of x sinx from first principle .

Section V

Q.36

The sum of two numbers is 6 times their geometric mean, show that numbers are in the ratio (3+22 ) :(3-22 ) .

Find the sum of the sequence 7,77,777, 7777 ,…….to n terms.

Q.37

Calculate the mean, variance and standard deviation for the following data:

Classes	30-40	40-50	50-60	60-70	70-80	80-90	90-100
frequencies	3	7	12	15	8	3	2

Find the mean deviation about the mean for the following data:

Income per day	0-100	100-200	200-300	300-400	400-500	500-600	600-700	700-800
Number of persons	4	8	9	10	7	5	4	3

Q.38

Solve the system of inequalities graphically:

3x+2y≤150 ,x+4y≤80 , x≤15 ,x≥0,y≥0 .

How many litres of water will have to be added to 1125 litres of the 45% solution of acid so that the resulting mixture will contain more than 25% but less than 30% acid content?

*************************END OF PAPER*************************

ANSWER KEY

COMMON SESSION ENDING EXAMINATION 2021

CHANDIGARH REGION

CLASS : XI

SUBJECT: Mathematics

MAX. MARKS : 80

TIME : 3 Hours

Marking Scheme

Sr. No.

PART-A

SECTION-I

Q.1

A= {-2,-1,0,1,2,3,4,5,6}

∅,{-1},{0}{1}{-1,0}{0,1}{-1,1}{-1,0,1}

Q.2

n(A×B)=mk

number of relations =2mk

Q.3

Domain= {1,2,3,4,5,6} , Range={3,4,5,6,7,8}

Q.4

A×B = {(-1,3), (-1,5), (0,3), (0,5), (1,3), (1,5)}

Q.5

-47°30'=-4712°=-952180radian=-1972 radian

Q.6

-1410° =cosec-1410°+4×360°=cosec-1410°+1440°=cosec30°=2

sin313=sin10π+3=sin3=32

Q.7

cosπ+x cos-xsinx -sinx=-cosx cosxsinx -sinx=cos2xsin2 x=cot2x

Q.8

1+i6+1-i3

=1+i23 +1-i3-3i+3i2

=(1+i2+2i)3-2-2i=(2i)3-2-2i=-2-10i

Q.9

Z=5+3i , Z=5-3i ,∣Z∣2=5+9=14 ,Z-1=514-314i

x2-2x+32=0

2x2-4x+3=0

D=(-4)2-4×2×3=-8=8i2

x=4±8i22×2=1±2i2 .

Q.10

Even numbers=2,4,6

3-digit even numbers=5×4×3=60

4-digit numbers=9×8×7×6=3024

Q.11

Number of ways=5c34c3

Q.12

-27,x,-72 are in G.P.

∴x2=-72-27 ,x=1

Q.13

V(±6,0) and f(±4,0)

a=6, c=4 ,b2=36-16=20 , x236+y220=1

focus (0,-3) and directrix y=3

a=3 ,x2=-4ay=-4×3y=-12y

Q.14

Point (-3 ,1, 2) lies in II octant and the point (-3,1,-2) lies in VI octant.

Q.15

Let ratio be k:1

On YZ-plane ,the x- coordinate of any point is zero.

∴3k-2k+1=0,k=23

Ratio is 2:3

Q.16

Sample space ={ HH,HT,T1,T2,T3,T4,T5,T6}

SECTION-II

Q.17

(i) (b) 5

(ii)(a) 4

(iii)(c) 2

(iv)(d) 1

(v) (a) 6

Q.18

(i) (a) 0.35

(ii) (c) 0.57

(iii) (b) 0.51

(iv) (d) 0.77

(v) (b) 0.54

PART-B

SECTION-III

Q.19

fx=9-x2

For Df , 9-x2 must be a real number.

⤇9-x2≥0

⤇-x2-9≥0

⤇x2-9≤0

⤇x+3x-3≤0

⤇-3≤x≤3

∴Df=[-3,3]

Q.20

f=1,1,2,30,-1,-1,-3 be a function from Z→Z

fx=ax+b

y=ax+b

Put x=0 ,y=-1

-1=0+b

b=-1

Put x=1 ,y=1

1=a+b

a=1-b=1+1=2

∴a=2 ,b= -1

Q.21

cos 4x=cos22x

=1-2sin22x

=1-2(2sinx cosx)2

=1-8sin2x cos2 x

Q.22

1+i1-i-1-i1+i=1+i1+i-(1-i)(1-i)(1-i)(1+i)

=1+i2+2i-1-i2+2i1-i2

=1-1+2i-1+1+2i2

=4i2=2i=0+2i

Modulus of Z=02+22 =2

Q.23

x4<5x-23-7x-35

x4-5x3+7x5<-23+35

-x60<-115

x>4

Solution is (4,∞ )

Q.24

EQUATION

There are 5 vowels and 3 consonants.

Vowels can be arranged among themselves=5! Ways

Consonants can be arranged among themselves=3! Ways

Two groups of vowels and consonants can be arranged in 2! Ways

Required number of ways =2!×5!×3!=1440

4 books on mathematics can be arranged in 4!ways

3 books on physics can be arranged in 3!ways

Two groups on mathematics and physics can be arranged in 2!ways

Required number of ways =2!×4!×3!=288

Q.25

Equation of line in intercept form is

xa+yb=1

a=b

xa+ya=1

x+y=a ……..(1)

(1) passes through the point (2,3)

∴2+3=a

a=5

Equation of line is x+y=5

Equation of line is

12x+6=5(y-2)

12x-5y+82=0 …….(1)

Distance of the point (-1,1) from given line is

d=12-1+-5+82(12)2+(-5)2 units

=6513 units=5 units

Q.26

Centre of the circle is (2,2).

Equation of circle is

(x-h)2+(y-k)2=r2

(x-2)2+(y-2)2=r2

Circle passes through the point (4,5)

∴(4-2)2+(5-2)2=r2

(2)2+(3)2=r2

r2=13 ,r=13

∴Equation of circle is

(x-2)2+(y-2)2=13

x2+y2-4x-4y-5=0

Equation of circle is

x2+y2 -2x+4y=8

x2-2x+y2 +4y=8

x2-2x+1+y2 +4y+4=13

(x-1)2+(y+2)2=13

Centre is (1, -2) and radius is 13

Q.27

cos2x-1cosx-1 =1-2sin2x-11-2sin2x2-1

=sin2xsin2x2

sin2x. x2x2sin2x2.x24x24

=4(sinxx)2 (sinx2x2)2 =4

Q.28

fx=cosx1+sinx

=1+sinx.-sinx-cosx .cosx(1+sinx)2

=-sinx-sin2x-cos2x(1+sinx)2

=-sinx-(sin2x+cos2x)(1+sinx)2

=-(1+sinx)(1+sinx)2=-11+sinx

SECTION-IV

Q.29

A∩X= B∩X=∅ and A∪X= B∪X

A∪X= B∪X

A∩(A∪X)=A (B∪X)

(A∩A)∪(A∩X)=(A∩B)∪(A∩X)

A∪∅=(A∩B)∪∅

A=(A∩B) ……..(1)

Again A∪X= B∪X

B∩(A∪X)=B (B∪X)

(B∩A)∪(B∩X)=(B∩B)∪(B∩X)

(B∩A)∪∅=B∪∅

B=(A∩B) ……..(2)

FROM (1) AND (2)

A=B.

Q.30

cos2x+cos2 x+3+cos2x-3=32

LHS=cos2x+cos2 x+3+cos2x-3

=1+COS2X2+1+COS(2X+23)2+1+COS(2X-23)2

=12[3+cos2x+2cos2x cos23]

=12[3+cos2x+2cos2x cos(π-3)]

=12[3+cos2x-2cos2x cos3]

=12[3+cos2x-cos2x ]

=32=R.H.S.

cos4x+cos3x+cos2xsin4x+sin3x+sin2x=cot3x

LHS=cos4x+cos3x+cos2xsin4x+sin3x+sin2x

=cos4x+cos2x+cos3xsin4x+sin2x+sin3x

=2cos(4x+2x2)cos(4x-2x2)+cos3x2sin4x+2x2cos4x-2x2+sin3x

=2cos(3x)cos(x)+cos3x2sin3xcosx+sin3x

=cos3x[2cos(x)+1]sin3x[2cosx+1]

=cos3xsin3x

=cot3x

Q.31

(x+iy)3=u+iv

x3+(iy)3+3x2iy+3xiy2= u+iv

x3-iy3+3x2iy-3xy2= u+iv

∴x3-3xy2=u and -y3+3x2y=v

∴ux+vy=x3-3xy2x+3x2y-y3y

=4x2-4y2

=4(x2-y2)

Q.32

MISSISSIPPI

Total number of letters=11

I appears 4times ,S appears 4times , P appears 2 times and M one time.

Total number of permutations=11!4!4!2! =34650

Number of permutations all I’s occur together=8!4!4! =840

Number of permutations all I’s not occur together=34650-840=33810

Q.33

an+bnan-1+bn-1 is the A.M. between a and b

∴an+bnan-1+bn-1 =a+b2

a+ban-1+bn-1=2(an+bn)

abn-1-bn=an-an-1b

bn-1(a-b)=an-1a-b

abn-1=1=ab0

n-1=0

n=1

Let A1,A2,A3,A4,A5,A6 be the six numbers between 3 and 24.

∴3,A1,A2,A3,A4,A5,A6,24 are in A.P.

a=3 ,b=24 ,n=8

24=3+8-1d

d=3

A1=a+d=6

A2=a+2d=9

A3=a+3d=12

A4=a+4d=15

A5=a+5d=18

A6=a+6d=21

∴Six numbers between 3 and 24 are 6,9,12,15,18,21

Q.34

Let θ be the angle between two lines and m1 , m2 be the slopes of two lines.

Let m1=12

∴tanθ= m2-m11+m1m2

tan4= m2-121+m212

1= m2-121+m212

1= m2-121+m212 or-1= m2-121+m212

∴ m2=3 or m2=-13

∴slope of other line is 3 or -13

Q.35

fx=xsinx

f'x=fx+h-f(x)h

x+hsin(x+h)-xsinxh

x+h(sinx cosh+sinhcosx)-xsinxh

xsinxcosh-1+xcosxsinh+h(sinxcosh+sinhcosx)h

xsinx(cosh-1)h+ xcosxsinhh +(sinxcosh+sinhcosx)

=xcosx+sinx

SECTION-V

Q.36

Let two numbers be a and b.

G.M.=ab

a+b=6ab (given)………..(1)

a+b2=36ab

a-b2=a+b2-4ab

= 36ab-4ab=32ab

a-b=42ab ……………(2)

Adding (1) and (2), we get

a=(3+22)ab

Subtracting (1) and (2), we get

b=(3-22)ab

∴ab=(3+22)ab(3-22)ab=(3+22)(3-22)

Sn=7+77+777+7777+……….to n terms

=799+99+999+9999+……….to n terms

=7910-1+(102-1)+(103-1)+(104-1)+….to n terms

=79(10+(102-1)+(103-1)+(104-1)+….to n terms

=7910(10n-1)10-1-n

=7910(10n-1)9-n

Q.37

Classes	fi	xi	fixi	(xi-x)2	fi(xi-x)2
30-40	3	35	105	729	2187
40-50	7	45	315	289	2023
50-60	12	55	660	49	588
60-70	15	65	975	9	135
70-80	8	75	600	169	1352
80-90	3	85	255	529	1587
900-100	2	95	190	1089	2178
	50		3100		10050

Mean=1NI=17fixi=310050=62

Variance=1Ni=17fi(xi-x)2=1005050=201

Standard deviation=201=14.18

Classes	fi	xi	fixi	xi-x	fixi-x
0-100	4	50	200	308	1232
100-200	8	150	1200	208	1664
200-300	9	250	2250	108	972
300-400	10	350	3500	8	80
400-500	7	450	3150	92	644
500-600	5	550	2750	192	960
600-700	4	650	2600	292	1168
700-800	3	750	2250	392	1176
	50		17900		7896

Mean=1NI=18fixi=1790050=358

M.D.=1NI=18fixi-x =789650=157.92

Q.38

3x+2y≤150 ,x+4y≤80 , x≤15 ,x≥0,y≥0

Let x litres of water isto be added.

Total mixture =( x+1125)litres

∴25%of x+1125<45%of1125<30%of(x+1125)

25x+1125<45×1125<30(x+1125)

5x+1125<9×1125<6(x+1125)

5x+5625<10125<6x+6750

5x+5625<10125 and 10125<6x+6750

5x<10125-5625 and 10125-6750<6x

5x<4500 and 3375<6x

x<900 and 562.5<x

∴ number of litres of water that is to be added will have to be more than562.5 but less than 900 litres.

***************************************************************************

SET-II

KENDRIYA VIDYALAYA SANGHATHAN,CHANDIGARH REGION

COMMONSESSION ENDING EXAMINATION 2020-21

CLASS : XI SUBJECT: Mathematics

MAX. MARKS : 80 TIME : 3 Hours

General Instructions :

This question paper contains two parts A and B. Each part is compulsory. Part A carries 24 marks and Part B carries 56 marks.

Part-A has Objective Type Questions and Part –B has Descriptive Type Questions

Both Part A and B have choices.

Part-A :

It consists of two sections I and II.

Section I comprises of 16 very short answer type questions.

Section II contains 2 case studies. Each case study comprises of 5 case-based MCQs. An examinee is to attempt any 4 out of 5 MCQs.

Part-B :

It consists of three sections – III, IV and V.

Section III comprises of 10 questions of 2 marks each.

Section IV comprises of 7 questions of 3 marks each.

Section V comprises of 3 questions of 5 marks each.

Internal choice is provided in 3 questions of Section-III , 2 questions of Section-IV and

3 questions of Section-V. You have to attempt only one of the alternatives in all such questions.

S.No

Part- A

Marks

SECTION 1

In a class,25 students play cricket , 20 students play tennis and 10 students play both the games. How many play at least one of these two games?

If A = {1, 2, 3}, B = {1, 4, 6, 9} and R is a relation from A to B is defined by “x is greater than y”. Find the range of R?

Find the value of: tan 1 tan 2 tan 3 tan 4 tan 89 .

Find the value of (1+i)(1+i2)(1+i3)(1+i4) ?

In how many ways a committee consisting of 3 men and 2 women, can be chosen from 7 men and 5 women?

Given 4 flags of different colours, how many different signals can be generated, if a signal requires the use of 2 flags one below the other?

In an A.P. the pth term is q and the (p + q)th term is 0. Find the qth term?

Find the probability of having 53 Sundays in a leap year?

A card is selected from a pack of 52 cards. Calculate the probability that the card is an ace of spades.

If L is the foot of the perpendicular drawn from a point (6, 7, 8) on x-axis. Find the coordinates of L?

Name the function , f : R R defined as: {1 if x >0 0 if x=0 -1 if x <0

Q10

If sin + =2 then sin2+cosec2 is equal to …

The minute hand of a watch is 1.5 cm long. How far does its tip move in40 minutes? (Use = 3.14).

Q11

-2-3 is equal to …

Q12

What is the locus of a point for which y = 0, z = 0?

Q13

Let A = {1, 2} and B = {3, 4}. Then number of relations from A to B.

If fx=x3-1x3 , then find the value of f(2)

Q14

If tan =3 and lies in third quadrant, then find the value of sin ?

Q15

Find the total number of words formed by 2 vowels and 3 consonants taken from

4 vowels and 5 consonants?

Q16

Find the equations of the lines joining the vertex of the parabola y2 = 6x to the points on it which have abscissa 24.

Find the equation of the circle in the first quadrant touching each coordinate axis at a distance of one unit from the origin.

Section – II

Both the case study based questions are compulsory.

Attempt 4 sub parts from each question. Each question carries 1 marks

Q17

In a group of 50 students, the number of students studying French, English,

Sanskrit were found to be as follows:

French = 17, English = 13, Sanskrit = 15

French and English = 09, English and Sanskrit = 4

French and Sanskrit = 5, English, French and Sanskrit = 3.

Find the number of students who study

1 x 4

(i)

French only

(A)6 (B) 3 (C) 9 (D) 20

(ii)

English only

(A)6 (B) 3 (C) 9 (D) 20

(iii)

Sanskrit only

(A)16 (B) 4 (C) 9 (D) 12

(iv)

at least one of the three languages

(A)16 (B) 30 (C) 24 (D) 12

(v)

none of the three languages

(A)6 (B) 30 (C) 24 (D) 20

Q18

On her vacations Veena visits four cities (A, B, C and D) in a random order. What is the probability that she visits

1 x 4

(i)

A before B?

(A)1 6 (B)13 (C) 1 2 (D) 35

(ii)

A before B and B before C?

(A)16 (B)13 (C)12 (D) 35

(iii)

A first and B last?

(A)1 6 (B) 112 (C) 12 (D) 35

(iv)

A either first or second?

(A)12 (B) 112 (C)13 (D) 35

(v)

A just before B?

(A)12 (B) 35 (C)112 (D) 13

Part B

Section III

Q19

Letf=x ,x21+x2:x R be a function from R into R. Determine the range

of f.

Find the domain of the function f(x) =x2+2x+1x2-8x+12

Q20

Find the principal solutions of the equation : tan x = -13

Q21

Solve 5x2+x+5=0

Find the multiplicative inverse of 2 – 3i.

Q22

Solve the system of inequalities:

3x – 7 < 5 + x

11 – 5x ≤ 1

and represent the solutions on the number line.

Q23

Evaluate 1x-2-2(2x-3)x3-3x2+2x

Evaluate : x4-812x2-5x-3

Q24

P (a, b) is the mid-point of a line segment between axes. Show that equation of the line is xa+yb= 2

Q25

Let A = {1, 2, 3, 4} and B = {5, 7, 9}. Determine

(i) Is A × B = B × A ? (ii) Is n (A × B) = n (B × A) ?

Q26

Find the derivative of function x2cos x

Q27

Find the centre and the radius of the circle x2 + y2 + 8x + 10y – 8 = 0

Q28

In how many ways can this diagram be coloured subject to the following two conditions?

(i) Each of the smaller triangle is to be painted with one of three colours: red, blue or

green.

(ii) No two adjacent regions have the same colour.

Section IV

Q29

Let A,B and C are the sets such that A∪B= A∪C and AB= AC.Using properties of sets, show that B=C.

Q30

Prove that: sin x + sin 3x + sin 5x + sin 7x = 4 cos x cos 2x sin 4x

Q31

If x+iy3=u+iv , then show that ux+vy=4x2-y2

If 1+i1-im=1 ,then find the least positive integral value of m.

Q32

A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways the committee consisting at most 3 girls can be formed?

In an examination, a question paper consists of 12 questions divided into two

parts i.e., Part I and Part II, containing 5 and 7 questions, respectively. A student

is required to attempt 8 questions in all, selecting at least 3 from each part. In

how many ways can a student select the questions ?

Q33

Find the sum to n terms of the sequence, 8, 88, 888, 8888… .

Q34

The owner of a milk store finds that, he can sell 980 litres of milk each week at Rs 14/litre and 1220 litres of milk each week at Rs 16/litre. Assuming a linear relationship between selling price and demand, how many litres could he sell weekly at Rs 17/litre?

Q35

Find the derivative of tan x from first principle.

Section V

Q36

Solve the following system of inequalities graphically:

3x + 2y ≤150, x + 4y ≤ 80, x ≤ 15, y ≥ 0, x ≥ 0

A solution of 8% boric acid is to be diluted by adding a 2% boric acid solution to

it. The resulting mixture is to be more than 4% but less than 6% boric acid. If we have

640 litres of the 8% solution, how many litres of the 2% solution will have to be added?

Q37

150 workers were engaged to finish a job in a certain number of days. 4 workers dropped out on second day, 4 more workers dropped out on third day and so on. It took 8 more days to finish the work. Find the number of days in which the work was completed.

The ratio of the A.M. and G.M. of two positive numbers a and b, is m : n. Show that

a:b=m+m2-n2:m-m2-n2

Q38

Calculate mean, variance and standard deviation for the following distribution.

Classes	30-40	40-50	50-60	60-70	70-80	80-90	90-100
Frequency	3	7	12	15	8	3	2

Calculate mean, variance and standard deviation for the following distribution using short-cut method.

Classes	0-10	10-20	20-30	30-40	40-50	50-60
Frequency	11	29	18	4	5	3

******************************* ENDOF PAPER****************************************

ANSWER KEY

KENDRIYAVIDYALAYA SANGHATHAN,CHANDIGARH REGION MARKING SCHEME SESSION ENDING EXAMINATION 2020-21
CLASS : XI SUBJECT: Mathematics
S.No	Part- A	Marks
Q1	35	1
Q2	{1}	1
Q3	1	1
Q4	0	1
Q5	350 OR 12	1
Q6	p	1
Q7	37 OR 152	1
Q8	(6, 0, 0)	1
Q9	Signum Function	1
Q10	2 OR 6.28 cm	1
Q11	-6	1
Q12	equation of x-axis	1
Q13	16 OR 63/64	1
Q14	-310	1
Q15	7200	1
Q16	2y ± x = 0 OR x2+y2-2x-2y+1=0	1
	Section – II Both the case study based questions are compulsory. Attempt 4 sub parts from each questions Each question carries 1 marks
Q17		1 x 4
(i)	A
(ii)	B
(iii)	C
(iv)	B
(v)	D
Q18		1 x 4
(i)	C
(ii)	A
(iii)	B
(iv)	A
(v)	D
	Part B
	Section III
Q19	OR	1 1 1 1
Q20		1 1
Q21	OR	1 1 1 1
Q22		1 1
Q23	OR	1 1 1 1
Q24		1 1
Q25	(1) No (11) Yes	1mark esch
Q26		1 1
Q27		1 1
Q28	These conditions are satisfied exactly when we do as follows: First paint thecentral triangle in any one of the three colours. Next paint the remaining 3 triangles,with any one of the remaining two colours. By the fundamental principle of counting, this can be done in 3 × 2 × 2 × 2 = 24 ways.	1+1
	Section IV
Q29	To prove (AB)U (BC) = C To prove (AB)U (BC) = B	(1.5M) (1.5M)
Q30		1 1 1
Q31	OR	1 1 1 1 1 1
Q32	OR	1 1 1 1 1 1
Q33		1 1 1
Q34		1 1 1
Q35		1 1 1
	Section V
Q36	OR	1 1 3 1 1 1 1 1
Q37	OR	1 1 1 1 1 1 1 1 1 1
Q38	Let the assumed mean A = 65. Here h = 10 OR	2 1 1 1 2 1 1 1

******************************* ****************************************

SET-III

Sr. No.

Part-A

Marks

Section-I

All questions are compulsory. In case of internal choices attempt any one.

Given that N = {1, 2, 3, ..., 50}, then Write the subset B of N, whose element are represented by x + 2, where x ∈ N.

Find the unionof the following pairs of sets:

A = {x :x is a natural number and 1 <x ≤6 }

B = {x :x is a natural number and 6 <x<10}

If A = {2, 4, 6, 9} and B = {4, 6, 18, 27}, a ∈A, b∈B, find the set of ordered pairs such that 'a' is factor of 'b' and a <b.

Let A = {1, 2} and B = {3, 4, 5}. Find the number of relations from A to B.

Let N be the set of natural numbers and the relation R be defined on N such that R = {(x, y): y = 2x, x, y ∈N}. Is this relation a function?

Find x and y if:(x – y, x + y) = (6, 10)

If the arcs of the same lengths in two circles subtend angles 65°and 110° at the centre, find the ratio of their radii.

The minute hand of a watch is 1.5 cm long. How far does its tip move in40 minutes? (Use π= 3.14).

Prove that: cos7x+cos5xsin7x-sin5x=cotx

Find the values of cosec (– 1410°)

Find the multiplicative inverse of 2 – 3i.

Find the modulus of the complex numbers: 1+i1-i

Find the number of permutations of the letters of the word ALLAHABAD.

How many 4-digit numbers can be formed by using the digits 1 to 9 if repetition of digits is not allowed?

How many chords can be drawn through 21 points on a circle?

What is the 20th term of the sequence defined by

an = (n – 1) (2 – n) (3 + n) ?

Find the equation of the parabola with vertex at (0, 0) and focus at (0, 2).

Find the point on y-axis which is at a distance 10 from the point (1, 2, 3).

What is the locus of a point for which y = 0, z = 0?

A die is thrown repeatedly until a six comes up. What is the sample space for this experiment?

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?

Section-II

Both the Case study-based questions are compulsory. Attempt any 4 sub parts from each question 17(i-v) and 18(i-v). Each question carries 1 mark

In a survey of 200 students of a school, it was found that 120 study Mathematics, 90 study Physics and 70 study Chemistry, 40 study Mathematics and Physics, 30 study Physics and Chemistry, 50 study Chemistry and Mathematics and 20 none of these subjects.

Based on the above information answer the following:

(i) Value of A

(a) 20 (b) 25 (c) 30 (d) 35

(ii)Value of (E+F+D)

(a) 120 (b) 100 (c) 130 (d) 110

(iii) Value of (B+G+C)

(a) 60 (b) 65 (c) 70 (d) 100

(iv) Students who study all the three subjects

(a) B (b) A (c) C (d) G

(v) Students who study Physics & Chemistry Not Math’s

(a) B (b) A (c) C (d) G

Veena visits four cities (A, B, C and D). She reports total visiting routes which can be used for visiting order as follows:

{ABCD, ABDC, ACBD, ACDB, ADBC, ADCB, BACD, BADC, BDAC, BDCA, BCAD, BCDA, CABD, CADB, CBDA, CBAD, CDAB, CDBA, DABC, DACB, DBCA, DBAC, DCAB, DCBA}

Based on the above information answer the following:

(i)Veena’s Total possible order of visiting cities (A, B, C and D). called as:

(a) Events (b) Good Choice (c) Sample Space (d) None of these

(ii) What is the probability that she visits A before B and B before C?

(a) 1/6 (b) 2/5 (c) 1/2 (d) 4/5

(iii) What is the probability that she visits A first and B last?

(a) 1/6 (b) 1/12 (c) 1/3 (d) 2/6

(iv) What is the probability that she visits A either first or second?

(a) 1/2 (b) 1/24 (c) 1/4 (d) 2/3

(v) What is the probability that she visits A just before B?

(a) 1 (b) 1/6 (c) 1/4 (d) 2/5

Part-B

Section-III

Let f = {(1,1), (2,3), (0, –1), (–1, –3)} be a linear function from Z into Z. Find f(x).

Find the domain and range of the following real functions: fx=-x-1

Prove that: 2sin234+2cos24+2sec23=10

If x+iy=a+iba-ib ,prove that x2+y2=1

Solve: 5x2+x+5=0

Find all pairs of consecutive odd natural numbers, both of which are larger than 10, such that their sum is less than 40.

A committee of 3 persons is to be constituted from a group of 2 men and 3 women. In how many ways can this be done? How many of these committees would consist of 1 man and 2 women?

In how many ways can 4 red, 3 yellow and 2 green discs be arranged in a row if the discs of the same colour are indistinguishable?

Find the equation of the line passing through (–3, 5) and perpendicular to the line through the points (2, 5) and (–3, 6).

A rod AB of length 15 cm rests in between two co-ordinate axes in such a way that the end point A lies on x-axis and end point B lies on y-axis. A point P(x, y) is taken on the rod in such a way

that AP = 6 cm. Show that the locus of P is an ellipse.

Find the equation of the circle having centre (1, –2) and passing through the point of intersection of the lines 3x + y = 14 and 2x + 5y = 18.

Suppose fx={ax+b, x<1 4, x=1 b-ax, x>1

and if fx=f(1) , what are possible values of a and b ?

Find the derivative of fx=secx-1secx+1

Section-IV

All questions are compulsory. In case of internal choices attempt any one.

Assume that P ( A ) = P ( B ). Show that A = B

Prove that: cos2x.cosx2-cos3x.cos9x2=sin5x.sin5x2

If x+iy13=a+ib, where a, b, x, y ∈ R. Show that xa-yb=-2a2+b2

A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected if the team has (i) no girl ? (ii) at least one boy and one girl ? (iii) at least 3 girls?

Find the number of different 8-letter arrangements that can be made from the letters of the word DAUGHTER so that (i) all vowels occur together (ii) all vowels do not occur together.

Find the sum to n terms of the sequence, 8, 88, 888, 8888… .

The product of three numbers in A.P. is 224, and the largest number is 7 times the smallest. Find the numbers.

Find the image of the point (3, 8) with respect to the line x +3y = 7 assuming the line to be a plane mirror.

Find the derivative of f (x) using first principle, where f is given by fx=2x+3x-2.

Section-V

All questions are compulsory. In case of internal choices attempt any one.

Solve the following system of inequalities graphically:

4x + 3y ≤ 60,

y ≥ 2x,

x ≥ 3,

x, y ≥ 0

How many litres of water will have to be added to 1125 litres of the 45% solution of acid so that the resulting mixture will contain more than 25% but less than 30% acid content?

Between 1 and 31, m numbers have been inserted in such a way that the resulting sequence is an A. P. and the ratio of 7th and (m – 1)th numbers is 5 : 9. Find the value of m.

If a and b are the roots of x2 – 3x + p = 0 and c, d are roots of x2 – 12x + q = 0, where a, b, c, d form a G.P. Prove that (q + p) : (q – p) = 17:15.

Calculate mean and Standard Deviation for the following distribution.

Classes	30-40	40-50	50-60	60-70	70-80	80-90	90-100
Frequency	3	7	12	15	8	3	2

The mean and standard deviation of 100 observations were calculated as 40 and 5.1, respectively by a student who took by mistake 50 instead of 40 for one observation. What are the correct mean and standard deviation?

___________END OF PAPER____________________________________

ANSWER KEY

Marking-Scheme

Sr. No.

Part-A

Marks

Section-I

B={3,4,5…….,50}

U={0,1,2,3,4,5,6,8}

{(2,4), (2,6), (2,18), (6,18), (9,18), (9,27)}

YES

X=8 & Y=2

22:13

6.28cm

For correct prove

(2+3i)/13

7560

3024

210

-7866

X2=8y

(0,2,0)

x-axis

S={6, (1,6), (2,6), (3,6), (4,6), (5,6), (1,1,6), (1,2,6)…..(2,1,6), (2,2,6)………

Section-II

17 (i)

17(ii)

17(iii)

17(iv)

17 (v)

18(i)

18(ii)

18(iii)

18(iv)

18(v)

Part-B

Section-III

Y=f(x)=ax+b

At (1,1) ⇒ 1=a+b

At (0, -1) ⇒ b=-1

⇒ a=2

F(x)=2x-1

Domain: R

Range: R-

2.122+2.122+2. 22

1+1+8=10

(x+iy) =a+iba-ib(x-iy) =a-iba+ib

(x+iy). (x-iy)=a+iba-ib.a-iba+ib=1

x2+y2=1

X=-1±1-4.525

= -1±19i25

Let x, x+2

x>10

x+2>10

x+x+2<40

x<19

(11,13), (13,15), (15,17), (17,19)

5C3=10

2C1.3C2=6

9!4!.3!

=2520

112

1/2

M1=-1/5

M2=5

5=y-5x+3

5x-y+20=0

y6=sinθ &x9=cosθ

x281+y236=1

Intersection point: (4,2)

R=5

x-12+y+22=25

Rsl=b-a

Lsl=a+b

a=0&b=4

112

1/2

f(x)/=secx+1secx.tanx-secx-1secx.tanxsecx+12

fx/=2secx.tanxsecx+12

Section-IV

Prove for A⊂B

Prove for B⊂A

A=B

122cos2x.cosx2-2cos3x.cos9x2

=12cos5x2-cos15x2

=-sin5x.sin -5x2 =sin5x.sin 5x2

x+iy=a+ib3

X=a3-3ab2

Y=3a2b-b3

xa-yb=-2a2+b2

(i) 7C5 =21

(ii) 7C1.4C4+ 7C2.4C3+7C3.4C2+7C4.4C1

= 7 + 84 + 210 + 140 = 441

(iii) 7C1.4C4+ 7C2.4C3 =7+84=91

(i) 6!.3!= 4320

(ii) 8! - 6!.3! = 50x720 =36000

112

Sn=899+99+999+…n terms

=8910910n-1-n

Let (a-d), a, (a+d)

(a-d).a.(a+d)=224

(a+d)=7(a-d)

a=8 & d=6

2, 8, 14

equation of PQ: y=3x-1

Point R: (1,2)

Image Point Q: (-1, -4)

F(x)/= 2x+h+3x+h-2-2x+3x-2h

=-7x-22

Section-V

For drawing all correct lines

Correct shaded region

Showing solution region

(1125+x).25% < 1125. 45% <(1125+x).30%

(1125+x).25% < 1125. 45%

x<900

1125. 45% <(1125+x).30%

x>561.5

561.5<x<900

1, A1, A2, ……Am,31

A7Am-1=1+7d1+m-1d=59……………(1)

d=30m+1

by (1) m=14

a+b=3 &a.b=p

c+d=12 &c.d=q

a, b, c, d are in GP

ba=cb=dc=r

b=ar, c=ar2, d=ar3

a+b=3 &c+d=12 ⇒ r =±2

qp=161⇒
q+pq-p=16+116-1=1715

Incorrect sum of observations = 4000

Thus, the correct sum of observations = Incorrect sum – 50 + 40

= 4000 – 50 + 40 = 3990

Hence Correct mean =correct sum/100 = 3990/100 = 39.9

= 162601 – 2500 + 1600 = 161701

Therefore, Correct standard deviation =161701100-39.9)2=25=5

SET-IV

COMMON SESSIONENDING EXAMINATION 2021

CHANDIGARH REGION

CLASS: XI

SUBJECT: Mathematics

MAX. MARKS : 80

TIME : 3 Hours

General Instructions :

This question paper contains two parts A and B. Each part is compulsory. Part A carries 24 marks and Part B carries 56 marks.

Part-A has Objective Type Questions and Part –B has Descriptive Type Questions

Both Part A and B have choices.

Part-A :

It consists of two sections I and II.

Section I comprises of 16 very short answer type questions.

Section II contains 2 case studies. Each case study comprises of 5 case-based MCQs. An examinee is to attempt any 4 out of 5 MCQs.

Part-B :

It consists of three sections – III, IV and V.

Section III comprises of 10 questions of 2 marks each.

Section IV comprises of 7 questions of 3 marks each.

Section V comprises of 3 questions of 5 marks each.

Internal choice is provided in 3 questions of Section-III , 2 questions of Section-IV and

3 questions of Section-V.You have to attempt only one of the alternatives in all such questions.

Sr.

No.

Part – A

Marks

Section – I

Q.1

Find the value of sin 765°

Findthevalueof1-tan215°1+tan215°

Q.2

Express the following in the form a+ib : (1-i)-(-1+6i)

Q.3

If .............

Q.4

Find the total number of ways of answering 6 multiple choice questions, each question having 4 choices.

Q.5

If sinθ + cosθ = 1,Find the value of sin2θ

Q.6

Find the Domain of function f given by .

Find x and y,if (x+y,2)=(3,2x+y)

Q.7

If what will be the value of x?

Q.8

Is 1,3,1,5,(2,5), a funtion? Justify

If some elements of A x B are {(a,1),(a,2),(b,3)}, Find set B

Q.9

If n(A)=m and n(B)=n, find the total number of non-empty relations that can be defined from A to B.

Q.10

If the sum of n terms of an AP is nP+ 12n(n-1)Q,where P and Q are constants,find the common difference.OR

For what values of x ,the numbers -27,x,-72 are in G. P.?

Q.11

Find the centre and radius of the circle x2+y2-8x+10y-12=0.

Q.12

In which plane does the point (2,0,5) lies?

In which octant the point (-1,2,-3) lies ?

Q.13

The centroid of a triangle ABC is the point (1,1,1). If the coordinates of A and B are (3,-5,7) and (-1,7,-6) respectively,find the coordinates of the point C.

Q.14

For any two sets A and B,Find A∩A∪B

Q.15

Convert 40°20' into radian measure.

Q.16

Write the sample space for the experiment , A coin is tossed and then a die is rolled only in a case a head is shown on the coin.

Section – II

Both the case study-based questions are compulsory. Attempt 4 sub parts from each questions Each question carries 1 mark.

Q.17

In a group of 100 people,65 like to play Cricket,40 like to play Tennis and 55 like to play Volleyball. All of them like to play atleast one of the three games. If 25 like to play both cricket and Tennis,24 like to play both Tennis and Volleyball and 22 like to play both Cricket and Volleyball,then

(i) How many like to play all the three games?

(a) 17 (b)11 (c)13 (d)10

(ii)How many like to play Cricket only?

(a) 19 (b)11 (c)29 (d)8

(iii) How many like to play Tennis only?

(a) 9 (b)7 (c)2 (d)8

(iv)How many like to play onlyVolleyball game?

(a) 11 (b) 9 (c)15 (d)21

(v) How many like to play both Tennis and Volleyball but not Cricket?

(a) 12 (b)13 (c)19 (d)27

Q.18

Four friends Yash,Daksh,Raju and Sunil playing cards.Daksh shuffling cards and told Raju

to choose any four cards.

(i)What is the probability that Raju getting all face cards.

(a)c412c452 (b) c416c452 (c) c413c452 (d) none of these

(ii) What is the probability that Raju getting two red cards and two black cards.

c2132c452 (b) c426c452 (c) c2262c452 (d) c4262c452

(iii) What is the probability that Raju getting one card from each suit.

c413c452 (b) 134c452 (c) c4132c452 (d) None of these

(iv) What is the probability that Raju getting all king cards.

1c452 (b) 2c452 (c) 4c452 (d) 6c452

(v) What is the probability that Raju getting two king and two jack cards.

(a) c24c452 (b) 36c452 (c) 6c452 (d) None of these

Part B

Section III

Q.19

Letf=x ,x21+x2:x ϵ RR be a function from R into R. Determine the range

of f.

Q.20

Express i19 + i9 in the form of a + i b

Find the multiplicative inverse of complex number: 2- 3i

Q.21

Solve the linear Inequation: 3x+120. Also represent the solution on number line.

Q.22

Find the number of words which can be formed out of the letters of the words ARTICLE, so that vowels occupy the even places.

Q.23

Find the equation of ellipse whose vertices are (±6,0) and foci(±4,0).

Find the equation of hyperbola whose vertices are (±5,0) and foci(±7,0).

Q.24

Evaluate:2+x-2x

Find the derivative of function :x1+tan x

Q.25

Find the distance of the point (3,-5) from the line 3x-4y-26=0.

Q.26

Prove that : ( cosx + cos y ) 2 + ( sin x – siny ) 2 = 4 cos2

Q.27

Evaluate: sin4xsin2x

Q.28

Let A=1,2,3,B=3,4 and C=4,5,6.FindiA×B∩CiiA×B(A×C)

Section IV

Q.29

Prove that :cos6x=32cos6x-48cos4x+18cos2x-1

Find the value of

Q.30

Find modulus and argument of (1+7i) (2-i)²

Q.31

The sum of two numbers is 6 times their geometric means, show that numbers are in the ratio 3+22:3-22

Q.32

If p and q are the length of perpendicular from the origin to the lines xcosθ +ysinθ = kcos2θ and xsecθ + ycosecθ = k respectively,then prove that p²+ 4q2 =k².

Q.33

Let A, B, and C be the sets such that A B = A C and AB = AC . Show that B=C

Q.34

A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of :

(i) exactly 3 girls

(ii) atleast 3 girls

(iii) atmost three girls

From a class of 25 students , 10 are to be chosen for a “SAVE ENVIRONMENT “ campaign . There are 3 students who decide that either all of them will join or none of them will join . In how many many ways can the selection be made and write any two ways of saving environment.

Q.35

Find the derivative of f(x)= tanx using first principle.

Section V

Q.36

Find the graphical solution of the system of inequalities

2x – y > 1, x – 2y < –1, x0, y≥0

A manufacturer has 600 litres of a 12% solution of an acid. How many litres of a 30% acid solution must be added to it so that acid content in the resulting mixture bemorethan 15% but less than 18%?

Q.37

Find the mean, variance and standard deviation of the following frequency distribution

Class	0-10	10-20	20-30	30-40	40-50
frequency	5	8	15	16	6

The mean and standard deviation of 100 observations were calculated as 40 and 5.1 respectivelyby a student who took by mistake 50 instead of 40 for one observation. What are the correct mean and correct standard deviation.

Q.38

Let S be the Sum, P be the product and R, the sum of reciprocals of n terms in a G.P. Prove that P2Rn= Sn.

If p,q,r are in G.P and the equations px2+2qx+r=0 anddx2+2ex+f=0 have a common root then show that dp,eq,fr are in A.P.

ANSWER KEY

CLASS : XI SUBJECT: Mathematics

PART-A

SECTION-I

Award 1 mark for each correct answer.

1. OR Cos 30°= 12

2. 2-7i

3. n=20

4. . (4)6

5. 0

6. R – {-2,3} OR x= -1 and y = 4

7. x=4

8. NO. not a function since 1 has two images. OR B= {1,2,3}

9. 2mn-1

10.Q OR 1

11. Centre (4,-5) and radius = 53

12. XZ-plane OR 6th octant

13.(1,1,2)

14. A

15.121540 radian.

16.S = T,H1,H2,H3,H4,H5,H6

SECTION-II

1 mark each for four correct options.

17.(i) (b)11

(ii) (c)29

(iii) (c)2

(iv) (b) 9

(v) (b)13

18. (i) (a) c412c452

(ii) (c) c2262c452

(iii) (b) 134c452

(iv) (a) 1c452

(v) (b) 36c452

SECTION III

19.Range=[0,1)………………………………………………..(2M)

20. (i2 )9 .i + (i2 )4. I (1M)

= - i + i = 0=0+0i (1M)

Writing Formula (1M)

Multiplicative inverse (1M)

21.Solution is (-,-4]....................................... (IM)

(1M)

- -4 +

22. In a 7 letter word, there are 3 vowels and 4 consonants.

3 vowels can arrange at three even places in 3! ways and consonants in 4! ways at other four places (1M)

Required no of ways = 3!4!=144 (1M)

23. a=6 c=4 c2 =16

a2-b2 = c2

36 –b2= 16

b2=20 (1M)

Equation of ellipse is x236 + y220 =1 (1M)

Finding a2 = 25 and b2 =24 (1M)

Writing equation of hyperbola is x225 - y224 =1 (1M)

24. Put 2+x=y st as x→0, y→2 (1M)

2+x-2x = y12-212y-2 = 122-12 = 122 (1M)

d/dx (x/(1+tanx) = ddxx1+tanx-xd(1+tanx)dx(1+t1nx)2 =11+tanx-x ( 0+sec2x)(1+tanx)2…………(1M)

= 1+tanx – xsec2x /(1+tanx)2 (1M)

25. Distance = 33-4-5-2632+(-4)2 (1M)

= 35 (1M)

26. ( 2 cos )2 + (2cos )2 (1M)

= 4 cos2)

= 4 cos2 (simplifying and getting ) (1M)

27 .sin4xsin2x =sin4x4x .x02xsin2x (1M)

=2.1.1=2 (1M)

28.(i) ABC=1,4,2,4,(3,4) (1M)

(ii) ABAC=1,4,2,4,(3,4) (1M)

SECTION IV

29. cos6x = cos 3(2x) = 4 cos3(2x) – 3 cos2(2x)

Putting cos2x = 2cos2x-1

= 4(2cos2x-1)3 -3(2cos2x-1) (1M)

= 4(8cos6x-12cos4x+6cos2x-1)-6cos2x +3 (1M)

= 32cos6x-48cos4x+18cos2x-1 (1M)

= R.H.S

Writing 2x = (1M)

Put = y and Getting quadratic equation and solving for (1M)

aslies in the first quadrant (1M)

30.1+7i4-1-4i= 1+7i3-4ix3+4i3+4i=-1+i (1M)

rcosθ = -1 rsinθ =1

r =2 (1M)

getting θ = 3Π/4 (1M)

31.

Let numbers be a and b.

a+b=6ab

a+b2ab=3 1 (1M)

Applying componendo and dividend

a+b+2aba+b-2ab=3+13-1

a+ba-b=2 1(1M)

Applying componendo and dividend

a+b+a-ba+b-a-b=2+12-1

ab=2+12-1

On squaring we get, ab=3+223-22 (1M

32.The perpendicular distance p from (0,0) to line x cosθ + ysinθ – kcos2θ =0 is

P = kcos2cos2+sin2 = kcos2θ (1M)

To find q = k sinθ cos (1M)

To prove p2+4q2= k2 (1M)

33 To prove (AB)U (BC) = C (1.5M)

To prove (AB)U (BC) = B (1.5M)

34.Number of ways of selection :

I. 9C4 ×4C3 = 504 (1M)

II No of ways of selection : (1M)

4C3×9C4+4C4×9C3 =588

III. No of ways of selection :

4C0×9C1+4C1× 9C6+4C2× 9C5+4C3×9C4 (1M)

= 36+336+756+504=1632

If 3 students join then we have to choose 7 student out of 22 which can be done in 22C7 ways (1M)

If 3students do not join then we have to choose 10 out 0f 22 which can be done in 22C10 ways (1M)

Total no of ways = 22C7 +22C10 (1M)

35. f'(x)=fx+h-f(x)h =tanx+h-tan(x)h = 1h sin⁡(x+h)cos⁡(x+h)-sinxcosx (1M)

=1h sin x+h cosx-cos x+h sinxcos⁡(x+h)cosx = 1h sin⁡(x+h-x)cos⁡(x+h)cosx (1M)

= sinhh . 1cos x+h .cosx = 1.1cos2x = sec2x (1M)

SECTION V

36.. for correct graph (4M)

For correct shading of required region….(1M)

Let x litres of 30% solution be added so that Total mixture=(x+600) litres. (1M)

ATQ 30%x+12% of 600 > 15% of (x+600) (1M)

30%x+12% of 600< 18% of (x+600) (1M)

solving to get the range as 120<x<300 (2M)

37.

Class	Frequency	Mid point	Yi =xi-2510	fiyi	Fiyi2
0-10	5	5	-2	-10	20
10-20	8	15	-1	-8	8
20-30	15	25	0	0	0
30-40	16	35	2	32	64
40-50	6	45	2	12	24
	N=50		1	26	116

(2M)

x =25.2 (1M)

Ơ2= fiyi2/fi +(fiyi/N)2

= 116/50 +(26/50)2 = 11650 +6762500 =2.5 (1M)

=2.5 (1M)

Finding correct sum = 3990 (1M)

Finding correct mean=39.9 (1M)

Finding correctxi2=161701 (2M)

Finding correct standard deviation= 5 (1M)

38. S = a+ar+ar2+.............+arn-1= a(1+r+......+rn-1) = a 1(rn-1)(r-1) (1M)

P= a.ar.ar2.............arn-1= an r1+2+.......+(n-1) = anrn(n-1)2 (1M)

R = 1a+1ar+1ar2+……+1arn-1 = 1a1(rn-1)rrn(r-1) (1M)

proving the result P2Rn= Sn. (2M)

Finding roots of the equation px2+2qx+r=0 as x= -qp (2M)

x= -qp is also root of the equation dx2+2ex+f=0 . Therefore,

d(-qp)2+2e-qp+f=0 (1M)

using q2=pr to get dp,eq,fr are in A.P. (2M)

__________________________________________________________________

*************************END OF PAPER*************************

class11 sample paper of maths

CLASS XI SAMPLE OF MATHS

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