Maths Assignment – 2016-2017 - CBSE Today

[Pages:18]Class - XI Maths Assignment ? 2016-2017

Topic : Trigonometry

Q.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 cm

Q.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, 4

Q.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, 8

Q.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?

11 C 90

Q.6 cos24?+cos55?+cos125?+cos204?+cos300?=?

Q.7

Prove that sin2 sin2 sin2 7 sin2 4 2 .

18

9

18

9

Q.8 Prove that :

Q.9

Q.10 Q.11

sec 3 sec 5 tan 5 tan 3 1 2 2 2 2

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)=0 Prove that : tan3A tan2A tanA=tan3A?tan2A?tanA. Prove that : sin2A=cos2(A?B)+cos2B?2cos(A?B)cosA cosB.

Q.12

Prove

that

:

sin2 8

A sin2

2

8

A 2

1 sin A 2

Q.13 Q.14

If

cos 4 ,

5

sin 5

13

and ,

lie between 0 and

4

,

prove

that

tan 2

56 33

.

Prove that : tan70?=tan20?+2tan50?.

Q.15

If

tan

(

cos

)=cot(

sin

),

prove

that

cos 1 4 2 2

.

Q.16 Q.17

If

cos

cos

cos

3 2

,

prove

that

cos

cos cos

sin

sin sin

0.

Prove

that

:

cos11 sin11 cos11 sin11

tan 56 .

Q.18

If

tan

A

+

tan

B=a

and

cot

A

+

cot

B

=

b,

prove

that

cot(A B)

1 a

1 b

.

Q.19 Q.20

Q.21

If

tan x tan x tan x 2 3 , then prove that

3 3

3 tan x tan3 x 1 3 tan2 x

1.

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

24 8sin= 9, find the value of sin(+).

25 If sin + sin = a and cos + cos = b, show that

(i)

sin

2ab a2 b2

ii.

1

cos

b2 b2

a2 a2

Q.22 Q.23 Q.24 Q.25

Q.26 Q.27

Prove that : 2 cos cos 9 cos 3 cos 5 0

13 13

13

13

Prove that : cos 20 cos 40 cos 60 cos 80 1

16

Prove that : sin10 sin 30 sin 50 sin 70 1 .

16

Prove that :

i.

cos cos 2 sin sin2 4 cos2

2

ii.

cos cos cos cos 4 cos cos cos

2

2

2

Prove

that :

cos 4x cos 3x cos 2x sin 4x sin 3x sin 2x

cot 3x .

Prove

that

:

cos A sin A

cos B n sinB

SinA sinB n cos A cosB

2 cotn A B

2

0

, if n is even

.

, if n is odd

Q.28 Q.29 Q.30 Q.31

Prove

that

:

sin sin sin sin 4 sin sin sin

2 2 2

If

sin cos

1 1

m m

,

,

prove

that

4

tan

4

m

.

If cosec A + sec A = cosec B + sec B, prove that : tan A tanB cot A B .

2

If

sin2A sin2B, , prove that

tanA tanA

B B

1

1.

Q.32 Show that : 2 2 21 cos 8 2 cos

Q.33 Q.34

Q.35 Q.36 Q.37 Q.38 Q.39 Q.40

Prove that :

sec sec

8 4

1 1

tan 8 tan 2

.

Prove that (i) cos4 cos4 3 cos4 5 cos4 7 3

8

8

8

82

ii.

sin4 sin4 3 sin4 5 sin4 7 3

8

8

8

82

Prove that : cos2 A cos2 A 120 cos2 A 120 3

2

Prove

that

:

1 cos 1 cos 3 1 cos 5 1 cos 7

8

8

8

8

1 8

Prove

that

:

cos A cos 2A cos 22 A cos 23 A...cos 2n1A

sin 2n A 2n sin A

.

Prove that : cos5A=16cos5A?20cos3A+5cosA.

Prove

that

:

cos A cos60 Acos60 A

1 4

cos

3A

.

Prove that : cos3 A cos3 120 A cos3 240 A 3 cos 3A

4

2

Q.41

Q.42 Q.43 Q.44 Q.45 Q.46 Q.47

Prove that :

i.

cot 7 1 2 3 4 6

2

ii. tan111 4 2 2 2 1 4

iii. tan142 1 2 2 3 6

2

Prove that : 1 cos2 2 2 cos4 sin4 Prove that : cos3 2 3 cos 2 4 cos6 sin6

Prove that : tan tan 60 tan tan 60 tan 60tan 60 3 .

Show that 2 cos ec20 sec 20 4 . Prove that : tan 2 tan 2 4 tan 4 8 cot 8 cot If 2 tan 3 tan , prove that

tan sin 2B

5 cos 2

Q.48

If

tan 2

a a

b b

tan

2

,

prove

that

Q.49

cos

a a

cos b b cos

.

If cos cos .cos , , prove that

tan . tan tan2 .

2

2

2

Q.50

If

cos cos cos 1 cos cos

then prove that one of the values of

tan 2

is

tan

2

cot

2

.

Q.51

If

tan p q

where = 6, being

acute angle,

prove that

1 p cos ec2 q sec 2

2

p2 q2 .

Q.52

Prove

that

sin 14

sin 3 sin 5 14 14

1 8

.

Q.53

Q.54

Q.55 Q.56 Q.57

Prove that 5 cos 3 cos 3 lies in [?4, 10]. 3

Prove that tan620??33tan420?+27tan220?=3 Evaluate : cosec48?+cosec96?+cosec192?+cosec384?. Prove that : sin212?+sin221?+sin239?+sin248?=1+sin29?+sin218?. Prove that :

Q.58

sin x sin 3x sin9x 1 tan 27x tan x

cos 3x cos 9x cos 27x 2 Prove that cos10x + cos8x + 3cos4x + 3cos2x = 8cosx cos33x

3

Q.59 Prove that :

Q.60 Q.61

4 cos 36 cot 7 1 2

1

2

3

4

5

6.

Prove that tan(x?y)+tan(y?z) + tan(z?x)=tan(x?y) tan(y?z) tan(z?x).

Prove that ?

tan 4 tan 4

tan 4 tan 4

cos ec2

xy Q.62 If sinx+siny=a and cosx+cosy=b. Find the value of tan 2 .

Q.63

Prove

that

sin sin 2 sin 3 sin 4

5

5

5

5

516 .

Q.64 Q.65 Q.66

Prove that cos3 cos 3 sin3 sin 3 3.

cos

sin

Show that

tan 3x tan x

never lies between

1 3

&

3

.

Pove that

cos3x cos5x cos7x 1 cosec2x cosec8x

sin2xsin4x sin4xsin6x sin6xsin8x 2sinx

Q.67 Q.68 Q.69

Prove that cos2 33 - cos2 57 1 .

sin21 - cos21

2

Determine the smallest positive value of x? for which tan(x? + 100?)=tan(x+50?)tanx?tan(x?-50?) Sketch the group of the following functions :

i.

y sin x 2

ii. y=4cos2x

(x=30?)

Q.70 Q.71

iii.

y x y 4

iv. y=sinx+cosx

v. y=2?sinx

vi. y=cosx

vii. y=sin2x

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

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

Q.72 Q.73

Solve

cos cos 2 cos 3

1 4

Solve 2sin2x?5sinxcosx?8cos2x=?2.

n , 2n 1

3

8

x=n tan =2

Q.74 Show that tan9??tan27??tan63?+tan81?=4.

x=ntan = ? 3 n 2 4

4

Maths Assignment ? 2016-2017

TRIGONOMETRICAL FUNCTIONS AND IDENTITIES

Q.1 The value of 3 cos ec 20 sec 20 is equal to

a. 2

c.

2. sin 20 sin 40

b. 4

d.

4. sin 20 sin 40

Q.2

The

maximum

value

of

1 sin 4

2 cos

4

for

real

values

of

is

a. 3

b. 5

c. 4

d. none of these

Q.3 The minimum value of cos 2+cos for real values of is

a.

9

8

b. 0

c. -2

d. none of these

Q.4 The value of cos ec 10 3 sec 10 is equal to

1

a.

2

b. 2

c. 0

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

d. 8

a. 4 10 c. 0

b. 4 10 d. none of these

Q.6

If

tan , x 9

and

tan 5 , x 18

are in AP and

tan , y 9

and

tan 7 18

, y are also

in AP

then

a. 2x=y

b. x> y

c. x=y

d. none of these

Q.7 If cos 20? - sin 20? = p then cos 40? is equal to

a.

p 2 p2

b.

p 2 p2

c.

p 2 p2

d. none of these

Q.8

The

value

of

sin .sin 14

3 14

.sin

5 14

.sin

7 14

.sin

9 14

.sin 11 14

.sin 13 14

is equal to

a. 1

1

b.

16

1

c.

64

d. none of these

Q.9

The value of

cos cos 3 cos 5 cos 7 cos 9

11 11

11

11

11

is

a. 0

b. 1

1

c.

2

d.

Q.10

n 1

cos 2

r

r 1

n

is equal to

n

a.

2

n 1

b.

2

c.

n 1 2

d.

Q.11

The

value

of

sin

n

sin

3 n

sin

5 n

... to

n

terms

is

equal

to

none of these none of these

a. 1

b. 0

n c.

2

5

d. none of these

Q.12 If ABCD is a convex quadrilateral such that 4 sec A + 5 = 0 then the quadratic equation whose roots 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 equation whose 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 Q.17

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

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

cos A cos B cosC is equal to

a

b

c

169 a. 144

61 b. 72

Q.18 Q.19 Q.20

61 c. 144

169 d. 72

The straight roads intersect at an angle of 60?. A bus on one road is 2 km. away from the intersection and a car on the other road is 3 km. away from the intersection. then the direct distance between the two vehicles is

a. 1 km

b.

2 km

c. 4 km

d.

7 km

If in a triangle ABC

2 cos A cos B 2 cosC a b

a

b

c bc ca

then the value of the angle A is

a.

3

b.

4

c.

2

d.

6

If in a triangle ABC, b c c a a b then cos A is equal to 11 12 13

1 a. 5

5 b. 7

19 c.

35

d. none of these

6

Q.21 Q.22 Q.23 Q.24

The sides of a triangle are sina, cosa and

1 sin cos

for

some

0

2

.

Then

the

greatest

angle of the triangle is a. 150? c. 120?

b. 90? d. 60?

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

In a ABC, A:B:C=3:5:4. Then a b c 2 . is equal to

a) 2b

b) 2c

c) 3b

d) 3a

2

93

In a ABC, A= 3 , b-c= 3 3 cm and ar ( ABC)= 2 cm2. Then a is

a) 6 3 cm

b) 9cm

c) 18cm

d) none of these

7

Class - XI Maths Assignment ? 2016-2017

Topic : Sequences and Series

Section-A

Q.1 Show that the sequence 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, ..... upto

60 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

s1 s2

...... sm

mn mn 1 .

2

Q.5

Q.6 Q.7 Q.8 Q.9

Q.10

13 The sum of two numbers is . An even no. of A.M.'s are being inserted between them and their

6

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

(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.

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

The sum of three terms of an A.P. is 33 and their product is 792. Find the least term.

(4)

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 terms

is 11, find the no. of terms.

(11)

The numbers t(t2+1), ??t2 and 6 are 3 consecutive terms of an A.P. If t be real, then find the next

two terms of the A.P.

(14, 22)

Q.11

Q.12

Q.13 Q.14 Q.15 Q.16

If

3 5 7 ..... n 5 8 11 ..... 10 terms

7 , find n.

(36)

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)

Find the sum of 11 terms of an A.P. whose middle term is 30.

(330)

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

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

(2401)

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

i.

1

1

........

1

a1 a2

a2 a3

an an1

N ?1 a1 a n

ii.

a1 a 2n a 2 a 2n1 ........ an an1

a1 a2

a2 a3

an an1

(n ?1)(a1 2an ) a1 an

Q.17

P.T.

a

2 1

a

2 2

a

2 3

a

2 4

..............

a

2 2k

k 2k 1

a12

a

2 2k

if 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. iff

1,1, 1 bc ca ab

are in A.P.

8

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