1

Class - XIMaths Assignment – 2016-2017

Topic : TrigonometryQ.1 If the angular diameter of the moon by 30°, how far from the eye a coin of diameter 2.2 cm be kept

to hide the moon? 252 cmQ.2 Find the angle between the minute hand of a clock and the hour hand when the time is 7:20 AM.

100°Q.3 The angle in one regular polygon is to that in another as 3:2 and the number of sides in first is

twice that in the second. Determine the number of side of two polygons. 8, 4Q.4 The number of sides of two regular polygons are as 5:4 and the difference between their angles is

9°. Find the number of sides of the polygons. 10, 8Q.5 A railway train is travelling on a circular curve of 1500 metres radius at the rate of 66 km/hr.

Through what angle has it turned in 10 seconds? C90

11

Q.6 cos24°+cos55°+cos125°+cos204°+cos300°=½

Q.7 Prove that 294

sin187

sin9

sin18

sin 2222

.

Q.8 Prove that :

12

3tan

25

tan2

5sec

23

sec

Q.9 If A, B, C, D be the angles of a cyclic quardilateral, taken in order, prove that

cos(180°–A)+cos(180°+B)+cos(180°+C)–sin(90°+D)=0Q.10 Prove that : tan3A tan2A tanA=tan3A–tan2A–tanA.Q.11 Prove that : sin2A=cos2(A–B)+cos2B–2cos(A–B)cosA cosB.

Q.12 Prove that : Asin2

12A

8sin

2A

8sin 22

Q.13 If 135

sin,54

cos and , lie between 0 and4

, prove that3356

2tan .

Q.14 Prove that : tan70°=tan20°+2tan50°.

Q.15 If tan ( cos )=cot( sin ), prove that22

14

cos

.

Q.16 If 23

coscoscos , prove that 0sinsinsincoscoscos .

Q.17 Prove that :

56tan11sin11cos11sin11cos

.

Q.18 If tan A + tan B=a and cot A + cot B = b, prove thatb1

a1

)BAcot( .

Q.19 If 332

xtan3

xtanxtan

, then prove that 1

xtan31

xtanxtan32

3

.

Q.20 If , are two different values of lying between 0 and 2 which satisfy the equation 6cos+

8sin= 9, find the value of sin(+).25

24

Q.21 If sin + sin = a and cos + cos = b, show that

(i) 22 ba

ab2sin

ii.

22

22

ab

abcos

2

Q.22 Prove that : 0135

cos133

cos139

cos13

cos2

Q.23 Prove that :161

80cos60cos40cos20cos

Q.24 Prove that :161

70sin50sin30sin10sin .

Q.25 Prove that :

i.

2

cos4sinsincoscos 222

ii. 2

cos2

cos2

cos4coscoscoscos

Q.26 Prove that : x3cotx2sinx3sinx4sinx2cosx3cosx4cos

.

Q.27 Prove that :

oddisnif,0

evenisnif,2

BAcot2

BcosAcosBsinSinA

BsinAsinBcosAcos nnn

.

Q.28 Prove that :

2

sin2

sin2

sin4sinsinsinsin

Q.29 If ,

m1m1

cossin

, prove that m4

tan4

.

Q.30 If cosec A + sec A = cosec B + sec B, prove that :2

BAcotBtanAtan

.

Q.31 If ,B2sinA2sin , prove that 1

1BAtanBAtan

.

Q.32 Show that : cos28cos1222

Q.33 Prove that :

2tan8tan

14sec18sec

.

Q.34 Prove that (i)23

87

cos8

5cos

83

cos8

cos 4444

ii.23

87

sin8

5sin

83

sin8

sin 4444

Q.35 Prove that : 23

120Acos120AcosAcos 222

Q.36 Prove that : 81

87

cos18

5cos1

83

cos18

cos1

Q.37 Prove that :Asin2

A2sinA2cos...A2cosA2cosA2cosAcos

n

n1n32 .

Q.38 Prove that : cos5A=16cos5A–20cos3A+5cosA.

Q.39 Prove that : A3cos41

A60cosA60cosAcos .

Q.40 Prove that : A3cos43

A240cosA120cosAcos 333

3

Q.41 Prove that :

i. 643221

7cot

ii. 1222441

11tan

iii. 632221

142tan

Q.42 Prove that : 442 sincos22cos1

Q.43 Prove that : 663 sincos42cos32cos

Q.44 Prove that : 360tan60tan60tantan60tantan .

Q.45 Show that 420sec20eccos2 .

Q.46 Prove that :

cot8cot84tan42tan2tan

Q.47 If tan3tan2 , prove that

2cos5

B2sintan

Q.48 If2

tanbaba

2tan

, prove that

cosba

bcosacos .

Q.49 If ,cos.coscos , prove that

2tan

2tan.

2tan 2

.

Q.50 If

coscos1coscos

cos then prove that one of the values of2

tan

is2

cot2

tan

.

Q.51 If qp

tan where = 6, being acute angle, prove that 22 qp2secq2eccosp21

.

Q.52 Prove that81

145

sin143

sin14

sin

.

Q.53 Prove that 33

cos3cos5

lies in [–4, 10].

Q.54 Prove that

tan620°–33tan420°+27tan220°=3

Q.55 Evaluate : cosec48°+cosec96°+cosec192°+cosec384°.

Q.56 Prove that : sin212°+sin221°+sin239°+sin248°=1+sin29°+sin218°.

Q.57 Prove that :

xtanx27tan21

x27cosx9sin

x9cosx3sin

x3cosxsin

Q.58 Prove that

cos10x + cos8x + 3cos4x + 3cos2x = 8cosx cos33x

4

Q.59 Prove that :

65432121

7cot36cos4 .

Q.60 Prove that tan(x–y)+tan(y–z) + tan(z–x)=tan(x–y) tan(y–z) tan(z–x).

Q.61 Prove that –

2eccos

4tan4tan4tan4tan

Q.62 If sinx+siny=a and cosx+cosy=b. Find the value of tan2

yx .

Q.63 Prove that 165

54sin

53sin

52sin

5sin .

Q.64 Prove that .3sin

3sinsincos

3coscos 33

Q.65 Show thatxtanx3tan

never lies between 3&31 .

Q.66 Pove that

cosec8xcosec2x2sinx

1

sin6xsin8x

cos7x

sin4xsin6x

cos5x

sin2xsin4x

cos3x

Q.67 Prove that2

1

cos21-sin21

2cos-332cos

57.

Q.68 Determine the smallest positive value of x° for which

tan(x° + 100°)=tan(x+50°)tanx°tan(x°-50°) (x=30°)

Q.69 Sketch the group of the following functions :

i.2x

siny ii. y=4cos2x

iii.

4

yxy iv. y=sinx+cosx

v. y=2–sinx vi. y=cosx

vii. y=sin2x

Q.70 If sin sin = cos cos +1=0, prove that 1+ cot tan = 0.

Q.71 If cos(A+B)sin(C–D)=cos(A–B) sin(C+D), show that tanAtanBtanC+tanD=0.

Q.72 Solve41

3cos2coscos 8

1n2,3

n

Q.73 Solve 2sin2x–5sinxcosx–8cos2x=–2. x=n tan =2

Q.74 Show that tan9°–tan27°–tan63°+tan81°=4. x=ntan = 2n4

3–

5

Maths Assignment – 2016-2017

TRIGONOMETRICAL FUNCTIONS AND IDENTITIES

Q.1 The value of 20sec20cos3 ec is equal to

a. 2 b. 4

c.

40sin

20sin.2 d.

40sin

20sin.4

Q.2 The maximum value of

4

cos24

sin1 for real values of is

a. 3 b. 5 c. 4 d. none of theseQ.3 The minimum value of cos 2+cos for real values of is

a.8

9 b. 0 c. -2 d. none of these

Q.4 The value of 10sec310cosec is equal to

a.2

1b. 2 c. 0 d. 8

Q.5 The least value of cos2-6sin.cos+3sin2+2 is

a. 104 b. 104c. 0 d. none of these

Q.6 If x,9

tan

and x,18

5tan

are in AP and y,

9tan

and

18

7tan

, y are also in AP then

a. 2x=y b. x>y c. x=y d. none of theseQ.7 If cos 20° - sin 20° = p then cos 40° is equal to

a. 22 pp b. 22 pp

c. 22 pp d. none of these

Q.8 The value of14

13sin.

14

11sin.

14

9sin.

14

7sin.

14

5sin.

14

3sin.

14sin

is equal to

a. 1 b.16

1

c.64

1d. none of these

Q.9 The value of11

9cos

11

7cos

11

5cos

11

3cos

11cos

is

a. 0 b. 1 c.2

1d. none of these

Q.10

1

1

2cosn

r n

r is equal to

a.2

nb.

2

1nc. 1

2

nd. none of these

Q.11 The value of ...5

sin3

sinsin nnn

to n terms is equal to

a. 1 b. 0 c.2

nd. none of these

6

Q.12 If ABCD is a convex quadrilateral such that 4 sec A + 5 = 0 then the quadratic equation whoseroots are tan A and cosec A is

a. 12x2-29x+15=0 b. 12x2-11x-15=0

c. 12x2+11x-15=0 d. none of these

Q.13 If ABCD is a cyclic quadrilateral such that 12tanA-5=0 and 5cos B+3=0 then the quadratic equationwhose roots are cos C, tan D is

a. 39x2-16x-48=0 b. 39x2+88x+48=0

c. 39x2-88x+48=0 d. none of these

Q.14 The number of real solutions of the equation sin(ex)=2x+2-x is

a. 1 b. 0 c. 2 d. infinite

Q.15 The number of values of x in the interval[0,5] satisfying the equation 3sin2x-7sin x+2=0 is

a. 0 b. 5 c. 6 d. 10

Q.16 In a triangle ABC, a=4, b=3, A=60°. Then c is the root of the equation

a. c2-3c-7=0 b. c2+3c+7=0

c. c2-3c+7 d. c2+3c-7

Q.17 If the sides a,b,c of a triangle ABC are the roots of the equation x3-13x2+54-72=0, then the value of

c

C

b

B

a

A coscoscos is equal to

a.144169

b.7261

c.14461

d.72

169

Q.18 The straight roads intersect at an angle of 60°. A bus on one road is 2 km. away from the intersectionand a car on the other road is 3 km. away from the intersection. then the direct distance between thetwo vehicles is

a. 1 km b. 2 km

c. 4 km d. 7 km

Q.19 If in a triangle ABC

ca

b

bc

a

c

C

b

B

a

A

cos2

coscos2

then the value of the angle A is

a.3

b.4

c.2

d.6

Q.20 If in a triangle ABC,131211

baaccb

then cos A is equal to

a.51

b.75

c.3519

d. none of these

7

Q.21 The sides of a triangle are sina, cosa and cossin1 for some2

0

. Then the greatest

angle of the triangle isa. 150° b. 90°c. 120° d. 60°

Q.22 If the area of a ABC be then a2 sin 2B+b2 sin 2A is equal to

a) 2 b) c) 4 d) none of these

Q.23 In a ABC, A:B:C=3:5:4. Then 2cba . is equal to

a) 2b b) 2c

c) 3b d) 3a

Q.24 In a ABC, A=32

, b-c= 33 cm and ar ( ABC)=2

39cm2. Then a is

a) 36 cm b) 9cm

c) 18cm d) none of these

8

Class - XIMaths Assignment – 2016-2017

Topic : Sequences and SeriesSection-A

Q.1 Show that the sequence <an> defined by an = 2n2+1 is not an A.P.

Q.2 Find the number of identical terms to the two APs 2, 5, 8..... upto 50 terms and 3, 5, 7, 9, ..... upto60 terms. (20)

Q.3 Find the sum of first 24 terms of the A.P. a1, a2....., a24 if it is known that a1+a5+a10+a15+a20+a24=225.(900)

Q.4 If s1, s2, .... sm are the sum of n terms of m A.P.s whose first terms are 1,2,3... on and common

differences are 1, 3, 5,...... (2m–1) respectively. Show that 1mn2

mns......ss m21 .

Q.5 The sum of two numbers is6

13. An even no. of A.M.'s are being inserted between them and their

sum exceeds their number by 1. Find the no. of means inserted. (6)

Q.6 If a, b, c, d, e, f, are in A.P. then prove that e–c=2(d–c) and a–4b+6c–4d+e=0.

Q.7 In an A.P., t7=15, then find the value of common difference d that would make t2t7t12 greatest. (0)

Q.8 The sum of three terms of an A.P. is 33 and their product is 792. Find the least term. (4)

Q.9 The sum of the first four terms of an A.P. is 56. The sum of the last 4 terms is 112. If the first termsis 11, find the no. of terms. (11)

Q.10 The numbers t(t2+1), –½t2 and 6 are 3 consecutive terms of an A.P. If t be real, then find the nexttwo terms of the A.P. (14, 22)

Q.11 If 7terms10.....1185n.....753

, find n. (36)

Q.12 If the sum of first 2n terms of the A.P. 2, 5, 8,...... is equal to the sum of the first n terms of the A.P.57, 59, 61,..... then find n. (11)

Q.13 Find the sum of 11 terms of an A.P. whose middle term is 30. (330)

Q.14 Find the number of numbers lying between 100 & 500 which are divisible by 7 but not by 21. (38)

Q.15 Find the coefficient x49 in (x+1)(x+3)(x+5).....(x+99). (2401)

Q.16 If a1, a2, ...... an are in A.P., where ai>0, i , then evaluate.

i.1nn3221 aa

1........

aa

1

aa

1

n1 aa

1–N

ii.1nn

1nn

32

1n22

21

n21

aa

aa........

aa

aa

aa

aa

n1

n1

aa

)2a1)(a–(n

Q.17 P.T. 2k2

21

2k2

24

23

22

21 aa

1k2k

a..............aaaa

if <an> is an A.P.

Q.18 There are n A.M.'s between 3 & 29 such that 6th mean : (n-1)th mean=3 : 5. Find n. (12)

Q.19 Evaluate a+b+c+d+e+f if a, b, c, d, e f are A.M.'s between 2 & 12. (42)

Q.20 P.T. a, b, c are in A.P. iffab1

,ca1

,bc1

are in A.P.

9

Q.21 Find the coefficients of x99, x98, x97 in (x+1)(x+2)(x+3)(x+4).....(x+99). (1,4950, 1208775)

Q.22 Find the sum to n terms5.3.1

1+

7.5.3

1+

9.7.5

1.... upto n terms. )3n2)(1n2(3

)2n(n

Q.23 If S is the sum of n A.M.'s between a & b and A is the single arithmetic mean between a & b, thenevaluate S/A. (n)

Q.24 If a1, a2, .......an are in A.P., S.T.n1n1n3221 aa1n

aa1

........aa1

aa1

.

Q.25 In a sequence, the first no. is31

. The 2nd no. is first no. divided by 1 more than the first no. The 3rd

no=2nd no. divided by 1 more than the 2nd no. and so on. What is the 100th term of the sequence?

102

1

Q.26 If the fifth term of a G.P. is 2. Find the product of its 9 terms. 92

Q.27 If ap+q=m and ap–q=n, find ap if these are terms of a G.P. mn

Q.28 If a=1+b+b2+b3+........ . Write b in terms of a.a

1–a

Q.29 Find the sum to infinity

21

........x1

x1

x1

x1

)x1(

x1

x1

14

3

3

2

2

Q.30 If 43.46.49.........43x=(0.0625)–54, find x. (8)

Q.31 If

n

1r

55r , find

n

1r

3r and

n

1r

2r . (3025, 385)

Q.32 Find all sequences which are simultaneously A.P. and G.P. (Constant)

Q.33 In an increasing G.P., the sum of the first and last term is 66, the product of the second and the lastbut one term is 128. If the sum of the series is 126, find the number of terms in the series. (6)

Q.34 If Sn is the sum of n terms of a G.P. {an} and Sn’ is the sum of n terms of the sequence

na1

, then

show that Sn=a1ansn'.

Q.35 Let x be the arithmetic mean and y, z be two geometric means between any two positive numbers,prove that y3+z3=2xyz.

Q.36 If x, y, z are distinct positive numbers, prove that (x+y)(y+z)(z+x)>8xyz. Further if x+y+z=1,show that (1–x)(1–y)(1–z)>8xyz.

Q.37 A Snail starts moving towards a point 3cm away at a pace of 1cm per hour. As it gets tired, itcovers only half the distance compared to previous hour in each succeeding hour. In how muchtime will the snail reach its target? (Never reach)

Q.38 In a set of four numbers, the first three are in G.P. and the last three are in A.P. with a commondifference of 6. If the first number is same as the fourth, find the four numbers. (8, –4, 2, 8)

Q.39 If x>0, prove that 2x1

x .

10

Q.40 Let a, b, c, d, e be five real nos. Such that a, b, c, are in A.P. b, c, d are in G.P.e1

,d1

,c1

are in A.P.

If a=2 and e=18, find all possible values of b, c, d. (–2, –6, –18), (4, 6, 9)

Q.41 Find the sum of the series 1.n+2.(n–1)+3.(n–2)+.......+n.1.6

2)1)(nn(n

Q.42 Find the sum to n terms .................5.4.3

14.3.2

13.2.1

1 upto n terms. 2)1)(n2(n

3)n(n

Q.43 Find the sum of product of first n natural nos. taken two by two.24

4)1)(3n1)n(n–(n

Section B (Objective Type Questions)Q.1 The number of numbers lying between 100 and 500 that are divisible by 7 but not by 21 is

a. 57 b. 19 c. 38 d. none of theseQ.2 The largest term common to the sequences 1, 11, 21, 31, ...... to 100 terms and 31, 36, 41, 46, .....

to 100 terms isa. 381 b. 471 c. 281 d. none of these

Q.3 If 51+x+51–x,2a

and 25x+25–x are three consecutive terms of an A.P., then the values of a are given

bya. a 12 b. a>12 c. a<12 d. a 12

Q.4 If the series of natural numbers is divided into groups (1); (2,3,4); (5,6,7,8,9),....... and so on, thenthe sum of the numbers in the nth group isa. n3+(n+1)3 b. n2+(n+1)2 c. (n–1)2+n2 d. (n–1)3+n3

Q.5 If x, 2y, 3z are in A.P. where the distinct numbers x, y, z are in G.P., then the common ratio ofG.P. isa. 3 b. 1/3 c. 2 d. 1/2

Q.6 If the sum of four numbers in G.P. is 60 and the A.M. of the first and the last is 18, then thenumbers area. 8,12,16,20 b. 32,16,8,4 c. 4,8,16,32 d. none of these

Q.7 Sum to infinity of the series ......2411

32

65

32

is

a.94

b.31

c.92

d. none of these

Q.8 .........321

24

.23

2123

.22

122

.21

333333

upon n terms :

a.2

1n b.

1nn

c.2n1n

d.n

1n

Q.9 (666....6)2 + (888....8) is equal to n digits n digits

a. 11094 n b. 110

94 n2 c. 2n 110

94

d. none of these

Q.10 If the sum of the series ......4.3

7

3.2

5

2.1

3222222 upto n terms

a. 21n

2nn

b. 2

2

1n

2n

c. 21n

1n2

d. 2

2

1n

1n

11

Q.11 In a sequence of (4n+1) terms the first (2n+1) terms are in A.P. whose common difference is 2,and the last (2n+1) terms are in G.P. whose common ratio is 0.5 if the middle terms of the AP andGP are equal then the middle term of the sequence is

a.12

2.nn

1n

b.12

2.nn2

1n

c. n2.n d. none of these

Q.12 The coefficient of x19 in the polynomial (x–1)(x–2)(x–22).......(x–219) isa. 220–219 b. 1–220 c. 220 d. none of these

Q.13 If (1-p)(1+3x+9x2+27x3+81x4+243x5)=1-p6, 1p then the value ofxp

is

a.31

b. 3 c.21

d. 2

Q.14 If the sides of a right angled triangle are in A.P. then the sines of acute angles are

a.54

,53

b. 32

,3

1c.

23

,21

d. none of these

Q.15 In a G.P. the first, third and fifth terms may be considered as the first, fourth and sixteenth termsof an A.P. Then the fourth term of the A.P., knowing that its first term is 5 isa. 10 b. 12 c. 16 d. 20

Q.16 Consider the ten numbers ar, ar2, ar3, .....ar10. If their sum is 18 and the sum of their reciprocals is6 then the product of these ten numbers isa. 81 b. 243 c. 343 d. 324

Q.17 Value of

......

3

11

3

11

31

182 is equal to

a. 3 b.56

c.23

d. none of these

Q.18 The value of 234.0 is

a.999419

b.990419

c.1000423

d. none of these

Q.19 Suppose a, b, c are in A.P. and a2, b2, c2 are in G.P. If a>b>c and a+b+c=23

, then the value of a

is

a.2

1b.

3

1c.

3

121 d.

2

121

Q.20 If

21

1ii 693a , where a1, a2, a3, ......a21 are in A.P., then the value of

10

0r1r2a is

a. 361 b. 363 c. 365 d. 398Q.21 The consecutive odd integers whose sum is 452–212 are

a. 43, 45, ..... 75 b. 43, 45, ..... 79 c. 43, 45, ..... 85 d. 43, 45,...... 89

12

Class - XIMaths Assignment – 2016-2017

Topic : Complex Numbers, Quadratic Equations & Linear Inequations

Q.1 If

1x2

ixiba

2

2

, prove that 22

2222

1x2

1xba

.

Q.2 Find the value of 2z4+5z3+7z2–z+41 if 3i2z . 6

Q.3 Evaluate 21

21

20342034 . 6

Q.4 If z, & z2 be any complex numbers, then show that

i. Re(z1z2)=Re(z1)re(z2)–Im(z1)Im(z2)

ii. Im(Z1,z2)=Re(z1)Im(z2)+Im(z1)Re(z2)

Q.5 Find real such that

sini21sini23

is

a. purely real =n, nz

b. purely imaginary2n,n

4

3sin)1(– 1–

Q.6 Ificic

ba

, ab,cR, show that

a2+b2=1 and1c

c2ab

2 .

Q.7 If a & b are different complex numbers with 1 , then find the value of

1.

Q.8 If (1+i)(1+2i)...(1+ni)=x+iy,S.T. 2.5.10......(1+n2)=x2+y2.

Q.9 Let z=x+iy andiz

iz1w

. If |w|=1, then show that z is purely real.

Q.10 For any two complex numbers z1 & z2, Prove that

221

221

221 2zz2zzzz .

Q.11 Find the modules and principal argument of

i.i21i31

4

3,2

ii.

3sini

3cos

1i

12

5,2

Q.12 If z1 & z2 be two non-zero complex numbers such that 2121 zzzz , find arg(z1)–arg(z2).

Q.13 Find the square root of1631

xy

yx

i21

x

y

y

x2

2

2

2

.

4

i–x

y

y

x

Q.14 Find |z| if iz3+z2–z+i=0. 1

Q.15 If |z1|=|z2|=.....=|zn|=1, prove thatn21

n21 z1

.....z1

z1

z.....zz .

13

Q.16 If |z|<4, prove that 94zi3 .

Q.17 If ibasinicos2

3

, prove that a2+b2=4a–3.

Q.18 Express (1+x2)(1+y2)(1+z2) as the sum of two squares. (xy–x–y–z)2(xy+yz+zx–1)2

Q.19 Show that (x2+y2)4=(x4–6x2y2+y4)2+(4x3y–4xy3)2.Q.20 Show that the area of the triangle on the Argand plane formed by the complex numbers z, iz and

z+iz is 2z

21 .

Q.21 Solve the following equations over C.

i. 0i62xi223x2 22,3–ii. x2–(5–i)x+(18+i)=0 3–4i, 2+3iiii. x6–x5+x4–x2+x–1=0 iv. x4+y4=82, x–y=2.

Q.22 Solve

i. 57x1x3

ii. 75x2,54x

iii. 51x71x2

,28x7x

iv. 4

11x25x8

,27x25x4

Q.23 An electrician can be paid under 2 schemes as given below :I : Rs. 500 and Rs. 70 per hour.II : Rs.120 per hour. If the job take x hours, for what values of x does the scheme II give the

electrician better ways :Q.24 A manufacturer has 600 litres of a 12% solution of acid. How many litres of a 30% acid solution

must be added to it so that acid content in the resulting mixture will be more than 15% but lessthan 18%? 120 l < x < 300 l

Q.25 Solve graphically :i. x+y 1 ii. x+y 4

7x+9y 63 3x+y 4x 6, y 5 x+5y 4x, y 0 x 3, y 3

x, y 4

Q.26 i. x6

2x73

1x7,

43

x49

3x4

ii. 32x1x

iii. 21x1 iv. 12x

x3x

(–5, –2) (–1, )

Q.27 How many litres of water will have to be added to 1125 litres of the 45% solution of acid so that theresulting mixture will contain more than 25% but less than 30% acid content!

Q.28 Solve 63x2x1x . ,40,–

Q.29 Solve 4x,44x

2

2

9,

2

7

Q.30 Solve

i. 2x,Rx,02x

1x

,22,––1,1–

ii. 2x,Rx;12x

1

2,11,–2–

14

Class - XIMaths Assignment – 2016-2017

Permutations & Combinations

Q.1 Find n if nCr :nCr+1 :

nCr+2 = 1 : 2 : 3 (n=14)

Q.2 How many arrangements can be made using the letters of the word MATHEMATICS? In How

many of them are the vowels together?

!2

!4

!2!2

!8

!2!2!2

!11

Q.3 A committee of 5 is to be formed from 6 men and 4 women. In how many ways can this be done if

a. at least 2 women are included. 44332 cc2ccc 54bc4b4

b. at most 2 women are included. 5132 c4ccc bbc4b4

Q.4 Find the rank of the word FARIDABAD as written in the dictionary. (20901)

Q.5 Find the number of proper divisors of 2520. (46)

Q.6 The result of 21 football matches (win, loss, draw) are to be predicted. How many different forecastcan contain exactly 18 correct results. 21C18×23

Q.7 For a set of six true or false questions, no student has written all the correct answers and nostudent have given the same sequence of answers. What is the maximum number of students inthe class, for this to be possible. (63)

Q.8 In how many ways can this diagram be coloured subject to the following conditions?

i. Each of the smaller triangle to be painted with one of the 3 colours : red, blue or yellow. (34)

ii. No two adjacent regions receive the same colour. (24)

Q.9 How many natural nos. not exceeding 4321 can be formed with the digits 1,2,3 and 4, if the digitscan repeat? (313)

Q.10 Find the number of zeros at the end of 60! (14)

Q.11 If all the permutations of the letters of the word INDIA are arranged as in a dictionary, what arethe 14th, 49th, 50th and 60th words. (DAINI, NADIIM, NAIDI, NIIDA)

Q.12 In how many ways can 5 gentlemen & 5 ladies be seated at a round table so that no two gentlemenare together? (2880)

Q.13 If n & r are positive integers nr1 , show that

nCr+2.nCr–1+nCr–2=

n+2Cr

Q.14 A student has 3 library tickets and 8 books of his interest in the library. Of these 8, he does notwant to borrow chemistry part II, unless chemistry part I is also borrowed. In haw many ways canbe choose the 3 books to be borrowed? (6C3+

7C2)

Q.15 A man has 8 children to take them to a zoo. He takes three of them at a time to the zoo as often ashe can without taking the same 3 children together more than once. How many times will he haveto go to the zoo? How many times a particular child will go to the zoo? (8C3+

7C2)

15

Q.16 There are 10 points in a plane. Of these 10 pts, 4 pts are in a straight line and with the exceptionof these 4 pts, no three pts are in the same straight line. Find (i) the no. of triangles formed. (ii) Theno. of straight line formed (iii) the no. of quadrilaterals formed, by joining these ten points.

(i) 10C3–4C3 (ii) 10C2–4C2+1 (iii) 10C4–4C4–4C3×6C1

Q.17 Six ‘X’’s have to be placed in the square of the figure given, such that each row contains at leastone “x”, in how many different ways can this be done? (8C6–2)

Q.18 In a group of 15 boys, there are 6 hockey players. In how many ways can 12 boys be selected soas to include at least 4 hockey players? (6C4×

9C8+6C5×

9C7+6C6×

9C6)

Q.19 How many words with or without meaning each of 2 vowels and 3 consonants can be formed fromthe letters of the word DAUGHTER? (3C2×

5C3×5!)

Q.20 How many 3-letter words can be made using the letters of the word ORIENTAL? (8p3)

Q.21 In the figure, two 4-digit numbers are to be formed by filling the places with digits. The number ofdifferent ways in which the places can be filled by digits so that the sum of the numbers formed isalso a 4-digit number and in no place the addition is with carrying, is (d)

a. 554 b. 220

c. 454 d. none of these

Q.22 The number of proper divisors of 2p.6q.15r is (b)

a. (p+q+1)(q+r+1)(r+1) b. (p+q+1)(q+r+1)(r+1)-2

c. (p+q)(q+r)r-2 d. none of these

Q.23 The number of proper divisors of 1800 which are also divisible by 10, is (d)

a. 18 b. 34

c. 27 d. none of these

Subjective questions :

Q.24 In how many ways can 5 people be seated in a car with two people in the front seat and three in the

rear, if two particular persons out of the five cannot drive? (4p3×2p2)

Q.25 How many non-zero numbers can be formed using the digits 0,1,2,3,4 and 5 if repetition of digits is not

allowed? (1630)

Q.26 What is the largest integer n such that 50 is divisible by 2n? (47)

Q.27 Prove that (i) nPr=r.n–1Pr–1+n–1Pr (ii) 1+1.1P1+2.2P2+3.3P3+.........+n.nPn=

n+1Pn+1.

Q.28 How many even numbers are there with three digits such that if 5 is one of the digits in a number then

7 is the next digit in that number. (365)

Q.29 How many number of 4-digits can be formed with the digit 1,2,3 and 4. Find the sum of those numbers.

Q.30 How many different words can be formed with the letters of the word UNIVERSITY so that (i) all the

vowels are together (ii) all vowels are not together (iii) no two vowels are together (iv) all vowels are

together and the consonants are together. (i)!2

!4!7(ii)

!2

!7!–2!10 (iii)!2

!p!6 47

(iv) 4!6!

Th H T U

16

Q.31 In how many ways can the letters of the word ARRANGE be arranged so that

i. the two R’s are never together? (900)

ii. the two A's are together but not the two R's? (240)

iii. neither the two A's nor the two R's are together? (660)

Q.32 If n–1Cr :nCr :

n+1Cr= 6 : 9 : 13, find n and r.

Q.33 Find the value of the expression

5

1i3

i52r

47 CC . (52C4)

Q.34 Find the number of divisor of 21600.

Q.35 Find the total number of ways of selecting 5-letters from the letters of the word INDEPENDENT.

(70, 3300)Q.36 The letters of the word OUGHT are written in all possible orders and these words are written out as in

a dictionary. Find the rank of the word TOUGH in the dictionary.

Q.37 Find the rank of the word ROBIN when all letters are arranged to form a dictionary.

Q.38 Find the number of (i) permutations (ii) selection of 4-letter words formed with the letters of the wordPROPORTION.

Q.39 Find the rank of the following words when the letters are arranged as in a dictionary (i) INDIA (ii)RUSSIA (iii) MOTHER (iv) SHOBIT.

Q.40 Find the number of (i) permutations (ii) selections of 4- letter words formed with the letters of the word(a) MATHEMATICS (ii) PROPORTION (iii) MISSISSIPPI.

Q.41 There are 7 boys and 5 girls. In how many ways can they be seated in a row so that (i) no restriction(ii) all girls sit together (iii) all girls do not sit together (iv) no two girls sit together.

Q.42 Find the number of sides of a polygon having 90 diagonals.

Q.43 We wish to select 6 persons from 8 persons, but if the person A is chosen, then B must be chosen. Inhow many ways can the selection be made?

Q.44 There are 10 points in a plane of which 4 are collinear. No three of the remaining 6 points are collinear.How many different (i) straight lines (ii) triangles formed by joining these points.

17

Class - XIMaths Assignment – 2016-2017

Binomial Theorem

Q.1 Find the coefficient of x6 in the expansion of9

2

x31x3

. (278)

Q.2 a. Find the middle term in the expansion of (x2+2y2)6.

b. Find the term independent of x in the expansion of

10

2x23

3x

.

5

510

2

3C

Q.3 Show that 881212 is an even positive integer and hence find the integral value of

12 8. (197)

Q.4 The coefficients of 3 consecutive terms in the expansion of (1+x)n are in the ratio 1 : 3 : 5. Find n.(7)

Q.5 If P be the sum of the odd terms and Q that of the even terms in the expansion of (x+a)n prove that

i. (x2–a2)n=P2–Q2

ii. (x+a)2n–(x–a)2n=4PQ

iii. (x+a)2n+(x–a)2n=2(P2+Q2)

Q.6 Prove that

i. (1+x)n-nx-1 is divisible by x2. nN.

ii. 24n-15n-1 is divisible by 225.

Q.7 In the expansion of

n

3

3

3

12 , the ratio of 7th term from the beginning to the 7th term from the

end is 1: 6, find n. (15)Q.8 If the coefficients of xr-1, xr and xr+1 in the binomial expansion of (1+x)n are in A.P., prove that

n2-n(4r+1)+4r2-2=0. (129)Q.9 If a1, a2, a3, a4 be the coefficients of four consecutive terms in the expansion of (1+x)n, then prove

that

32

2

43

3

21

1

aaa2

aaa

aaa

Q.10 Find the number of integral terms in the expansion of (51/2+71/8)1024.

Q.11 If a, b, c and d in any binomial expansion be the 6th, 7th, 8th and 9th terms respectively, then prove

thatc3a4

bdc

acb2

2

.

Q.12 Find the coefficient ofx1

in the expansion of n

n

x

11x1

. (2nCn–1)

Q.13 Find the coefficient of x5 in the expansion of (1+x)21+(1+x)22+........+(1+x)30. (31C6–21C6)

Q.14 Write the number of terms in the expansion of

i. 1010x32x32 (6)

ii. (1–3x+3x2–x3)8 (25)iii. (x+y+z)50 52C2

Q.15 If a & b denote the sum of the coefficients in the expansions of (1–3x+10x2)n and (1+x2)n respectively,then write the relation between a and b. (a=b3)

18

Q.16 Find a, x and n in the expansion of (a+b)n if the 6th, 7th and 8th terms in the expansion of (x+a)n are

112, 7 and41

respectively. (n=8,x=4,a=½)

Q.17 Prove that the coefficient xn in (1+x)2n is twice the coefficient of xn in (1+x)2n–1.

Q.18 Find the coefficient of the term independent of x in the expansion of

10

21

31

32

xx

1x

1xx

1x

. (210)

Q.19 Find the coefficient of x7 in11

2

bx1

ax

and x–7 in

11

2bx

1ax

and find the relation between a &

b so that these coefficients are equal. (ab=1)Q.20 Using binomial theorem. Prove that (101)50>10050+9950.

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

Q.22 Find the sum of rational terms in the binomial expansion of10

51

32

. (41)

Q.23 The sum of the coefficients of the first three terms of the expansion m,0x,x

3x

m

2

being a

natural number, is 559. Find the coefficient of x3. (–12C9(3)3)Q.24 For what values of m the coefficient of the (2m+1)th and (4m+5)th terms in the expansion of (1+x)10

are equal. (m=1)

Q.25 Show that the middle term in the expansion of (1+x)2n is nn x.2!n

)1n2...(5.3.1 .