Trigonometry Test

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Trigonometry Test - 8

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Weekly Quiz Competition

Question 1
1 / -0

Two straight roads intersect at an angle of 60^o. 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. The direct distance between the two vehicles is

A
1 km

B
km

C
4 km

D
km
Question 2
1 / -0

Each side of an equilateral triangle subtends an angle of 60⁰ at the top of a tower h m high located at the centre of the triangle. If `a` is the length of each side of the triangle, then

A
3a² = 2 h²

B
2a² = 3h²

C
a² = 3 h²

D
3a² = h²

Solution

Let O be the centre of the equilateral triangle ABC
and OP be the tower of height h.
Then each of the triangles PAB, PBC and PCA are equilateral.
Thus, PA = PB = PC = a.
Therefore, from right - angled triangle POA,
we have PA² = PO² + OA².
Question 3
1 / -0

If sin + cos = p and sec + cosec= q, then q (p²-1) =

A
2

B
2P

C

D
none of these

Solution

q(p²-1)
= (sec+cosec)[(sin+cos)² -1]
= ( + ) ( 2sincos)
= sin + cos
= 2p
Question 4
1 / -0

The angle of elevation of the top of an unfinished tower at a point 120 m above its base is 45°. How much must the tower's height be increased so that the angle of elevation becomes 60°?

A
120() m

B
() m

C
120() m

D
None of these

Solution

Let the height of unfinished tower be h, distance from the base of the tower to the point A be y and the length to be increased be x.

In ABC

tan 45° = h/y

∴ y = 120 m Now,

In ABD:

tan 60° = (120 + x)/120

∴ x = 120( - 1) m
Question 5
1 / -0

The angles of elevation of the top of the tower observed from each of the three points A, B and C on the ground forms a triangle at the same angle . If R is the circum-radius of the triangle ABC, then the height of the tower is:

A
R sin

B
R cos

C
R cot

D
R tan
Question 6
1 / -0

If tan = 2 sin sin cosec ( + g), then cot , cot , and cot are in

A
A.P.

B
G.P.

C
H.P.

D
None of these
Question 7
1 / -0

If a flagstaff subtends the same angle at the points A, B, C and D on the horizontal plane through its foot, then A, B, C and D form a

A
square

B
cyclic quadrilateral

C
rectangle

D
none of these

Solution

If O is the foot of the flagstaff, then OA = OB = OC = OD,
so that the points A, B, C and D lie on a circle with centre O and
therefore, ABCD is a cyclic quadrilateral. (It may not be a square or
a rectangle.)
Question 8
1 / -0

If x = r sincosy = r sinsinand z = r cos , then x²+Y² =

A
r² - z²

B
r²+ z²

C
r²z²

D
none of these

Solution

x²+y²+z²
r²sin²cos²+ r²sin²sin²+ r² cos²=
= r²sin²+ r² cos²
= r²
x² + y² = r² - z²
Question 9
1 / -0

The angle of elevation of top of a vertical tower from a point P on the ground is 60⁰. At a point Q, 40 m vertically above P, the angle of elevation is 45⁰. Find the height of tower.

A
m

B
m

C
m

D
m

Solution

h =
=
=
Question 10
1 / -0

An observer on the top of a tree finds the angle of depression of a car moving at a constant speed towards the tree to be 30⁰. After 3 min, this angle becomes 60⁰. After how much more time will the car reach the tree?

A
4 min

B
4.5 min

C
1.5 min

D
6 min

Solution

Let x be the length of the pole.
by proportionally.
x/50 = 4/6, 50 = x = 75 metre
Question 11
1 / -0

The value of cos20° cos 40° cos 60° cos 80° is equal to

A

B

C

D
none of these

Solution

cos 20^o cos40^ocos60^o cos80^o
= ½ . ½ (2cos 40^o cos20^o) cos80^o
= ¼ (cos60^o + cos20^o) cos 80^o
= 1/8 (cos80^o + 2 cos 20^o cos80^o)
= 1/8 (cos 80^o + cos 100^o + cos 60^o) =
Question 12
1 / -0

If sin= , then cos =

A

B

C

D
none of these

Solution

cos=
=
= =
Question 13
1 / -0

If tan + sin = m and tan - sin = n, then m²- n² =

A

B

C
4

D
None of these

Solution

m² - n² = (m + n)(m - n) = 2 tan 2 sin = 4 tan sin

Now, going by options:

(1) = = =

Hence, = = m² - n²

So, m² - n² =

(2) = ≠ m² - n²
Question 14
1 / -0

The angle of elevation of a cloud from a point 60 m above a lake is 30^o and from the same point, the angle of depression of its image in the lake is 60^o. The height of the cloud is

A
100 m

B
80 m

C
40 m

D
120 m

Solution

h =

=

=

= 120 m
Question 15
1 / -0

Two vertical poles of equal heights are 120 m apart. A and B are two points on the line joining their bottoms. Angle of elevation of the top of one pole from A is 45⁰ and that of the other pole from B is also 45⁰. If AB = 30 m, then the height of each pole is:

A
40 m

B
45 m

C
50 m

D
42 m

Solution

ADE BFC
So, DE = FC
And EA = BC = x = 45 m
As AB + EA + BC = 120 m = EC
2x = 90
x = 45
= 1
DE = 45 m
So, FC = 45 m
So, option 2 is correct.
Question 16
1 / -0

The value of sin 10° sin 30° sin 50° sin 70° is equal to

A

B

C

D

Solution

sin 10^o sin 30^o sin 50^o sin 70^o
= ½. ½ (2sin 10^o sin50^o) sin 70^o
= ¼ (cos 40^o - cos 60^o) sin 70^o
= (1/8) (2sin 70^o cos 40^o - sin 70^o)
= (1/8) (sin110^o + sin 30^o - sin 70^o)
= (1/8)(sin 70^o + (1/2) - sin 70^o)
= .
Question 17
1 / -0

If tan= , then sec =

A

B
2

C

D
None of these

Solution

sec =
(since sec²= 1 + tan²)
=
=
= 2.
Question 18
1 / -0

If 3sin+ 5cos= 5, then 5sin- 3cos=

A
3

B
- 3

C
Both (1) and (2)

D
None of these

Solution

We have
(3sin+ 5cos)2 + (5sin- 3cos)² = 34
5² + (5sin- 3cos)² = 34
5sin- 3 cos= 3
Question 19
1 / -0

An aeroplane at an altitude of 300 m observes the angles of depression of opposite points on the two banks of a river to be 45⁰ and 60⁰. The width of the river is

A
(300 - 100 ) m

B
(300 + 100 ) m

C
(300 +100) m

D
(300- 100) m

Solution

a =
=
=
=
=
= (300+100)m.
Question 20
1 / -0

A tree 6 m tall, casts a 4 m long shadow. At the same time, a flag pole casts a shadow 50 m long. How long is the flag pole?

A
75 m

B
100 m

C
150 m

D
50 m

Solution