ব্যাংক ডেইলি কুইজ [লং কোর্সের অংশ]

সিলেবাস

Exam -81

Daily Quiz:
Math: 
 Topic: Trigonometry (Basic Trigonometry, Heights and Distances)

প্রশ্ন দেখুন অনুশীলন

ব্যাংক ডেইলি কুইজ [লং কোর্সের অংশ]

ব্যাংক ডেইলি কুইজ [লং কোর্সের অংশ] · তারিখ অনির্ধারিত · ১০ প্রশ্ন

১.

A mirror is placed on the ground facing upwards. A man sees the top of a tower in the mirror which is at a distance of 100 m from the mirror. The man is 0.5 m away from the mirror, and his height is 1.5 m.

ক
300 m
খ
200 m
গ
50.5 m
ঘ
315 m

ব্যাখ্যা

Question: A mirror is placed on the ground facing upwards. A man sees the top of a tower in the mirror which is at a distance of 100 m from the mirror. The man is 0.5 m away from the mirror, and his height is 1.5 m.

Solution:

Given that,
Distance from the mirror to the tower = 100 m
Distance from the man to the mirror = 0.5 m
Height of the man = 1.5 m
Height of the tower, H = ?

Now,
⇒ Height of the man/Distance from man to mirror = Height of the tower/Distance from tower to mirror
⇒ 1.5/0.5 = H/100
⇒ 3 = H/100
⇒ H = 100 × 3 = 300 m

২.

If tan 2A = cot(A - 30°) and 2A is an acute angle, then find 'A' is-

ক
40°
খ
60°
গ
120°
ঘ
45°

ব্যাখ্যা

Question: If tan 2A = cot(A - 30°) and 2A is an acute angle, then find 'A' is-

Solution:
Given that,
tan 2A = cot (A - 30°)
⇒ tan 2A = tan [90° - (A - 30°)]
⇒ 2A = 90° - A + 30°
⇒ 3A = 120°
⇒ A = 120°/3
∴ A = 40°

৩.

∠B is the right angle of a right angles triangle ABC. If tanA = 3/4, then 5sinACosA =?

ক
5/3
খ
12/3
গ
1
ঘ
12/5

ব্যাখ্যা

Question: ∠B is the right angle of a right angles triangle ABC. If tanA = 3/4, then 5sinACosA = ?

Solution:

দেওয়া আছে,
tanA = 3/4

∴ অতিভুজ = √(3² + 4²) = √(9 + 16) = 5

∴ sinA = AB/AC = 3/5
cosA = BC/AC = 4/5

∴ 5sinACosA = 5 × (3/5) × (4/5)
= 12/5

৪.

A circus artist is climbing a 28 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole, if the angle made by the rope with the ground level is 60°.

ক
32√2 m
খ
24√3 m
গ
11√5 m
ঘ
14√3 m

ব্যাখ্যা

Question: A circus artist is climbing a 28m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole, if the angle made by the rope with the ground level is 60°.

Solution:

By observing the figure, AB is the pole.
In triangle ABC,
⇒ AB/AC = sin60°
⇒ AB/28 = √3/2
⇒ AB = 14√3

Therefore, the height of the pole is 14√3 m

৫.

A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle 30° with it. The distance between the foot of the tree to the point where to top touches the ground is 12 m then the height of the tree is-

ক
14√2 m
খ
12√3 m
গ
24√2 m
ঘ
18√3 m

ব্যাখ্যা

Question: A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle 30° with it. The distance between the foot of the tree to the point where to top touches the ground is 12 m then the height of the tree is-

Solution:

Let the height of the tree be h. Let the part that is still standing on the ground be x, and that part which has fallen be y.
Hence, we have
sin θ = x/y
⇒ sin θ = 1/2 ;[ θ = 30°]
⇒ x/y = 1/2
⇒ y = 2x ...............(1)

Also, it is given that the distance of the tip of the fallen tree to that of the base of the tree is 12 m
⇒ cos θ = 12/y
⇒ cos 30° = 12/y
⇒ √3/2 = 12/y
⇒ y = 24/√3 = (8 × √3 × √3)/√3
⇒ y = 8√3

From (1), x = 8√3/2 = 4√3

∴ Height of tree, h = x + y = 4√3 + 8√3 = 12√3 m

৬.

Two ships are sailing in the sea on the two sides of the lighthouse. The angle of elevation of the top of the lighthouse is observed from the ships are 45° and 45° respectively. If the lighthouse is 120 m high, the distance between the two ships is?

ক
250 m
খ
150 m
গ
300 m
ঘ
240 m

ব্যাখ্যা

Question: Two ships are sailing in the sea on the two sides of the lighthouse. The angle of elevation of the top of the lighthouse is observed from the ships are 45° and 45° respectively. If the lighthouse is 120 m high, the distance between the two ships is?

Solution:

Given that,
Height of the lighthouse = 120m
Now,
In triangle ADC,
AD/DC = tan 45°
⇒ AD/DC = 1 [tan 45° = 1]
⇒ AD = DC = 120m
Again,
In triangle ABD,
AD/BD = tan 45°
⇒ 120/BD = 1 [tan 45° = 1]
⇒ BD = 120 m

Now,
BC = BD + DC
= 120 + 120 = 240 m
∴ Total distance = 240 m

৭.

The angle of elevation of the top of a tower of height x metre from a point on the ground is found to be 60°. By going y metre away from that point, it becomes 30°. Which one of the following relations is correct?

ক
x = y
খ
2x = √3y
গ
2x = 3y
ঘ
None of the above

ব্যাখ্যা

Question: The angle of elevation of the top of a tower of height x metre from a point on the ground is found to be 60°. By going y metre away from that point, it becomes 30°. Which one of the following relations is correct?

Solution:
Given that,
The angle of elevation of the top of a tower of height x meter from a point on the ground is found to be 60°.
By going y metre away from that point, it becomes 30°.

According to the question,
tan60° = AB/BC
⇒ √3 = x/BC
⇒ BC = x/√3 .........(1)

Again,
tan30° = AB/BD
⇒ 1/√3 = AB/BD
⇒ 1/√3 = x/BD
⇒ BD = √3x .........(2)

Now,
BD = BC + CD
⇒ √3x = (x/√3) + y [From equation (1 and 2)]
⇒ √3x = (x + √3y)/√3
⇒ 3x = x + √3y
⇒ 3x - x = √3y
⇒ 2x = √3y

৮.

A boy of height 1.3 m is walking away from the base of a lamp post at a speed of 0.9 m/sec. Find the height of the lamp post from the ground, if the shadow of the boy is 2.4 m after walking for 5 sec.

ক
7.9 m
খ
5.5 m
গ
3.6 m
ঘ
3.74 m

ব্যাখ্যা

Question: A boy of height 1.3 m is walking away from the base of a lamp post at a speed of 0.9 m/sec. Find the height of the lamp post from the ground, if the shadow of the boy is 2.4 m after walking for 5 sec.

Solution:
Given that,
Height of the boy = 1.3m
Speed of the boy = 0.9 m/s

∴ Distance travelled by boy in 5 sec = 0.9 × 5 = 4.5m

∴ Total distance of shadow of boy and distance from base of lamp post = 2.4 + 4.5 = 6.9 m
Let the height of lamp post be 'h' m

According to question,
⇒ 1.3/2.4 = h/6.9
⇒ h = (6.9 × 1.3)/2.4
⇒ h = 3.7375 = 3.74m

So, The height of the lamp post is 3.74 meters.

৯.

Which of these values cannot be the value of sinθ?

ক
√3/2
খ
1/2
গ
3/2
ঘ
-√2/2

ব্যাখ্যা

Question: Which of these values cannot be the value of sinθ?

Solution:
আমরা জানি,
sin⁡θ এর মান সবসময় -1 থেকে 1 এর মধ্যে হয়।
অর্থাৎ, - 1 ≤ sin⁡θ ≤ 1

এখন,
√3/2 = 0.866 → সত।
1/2 = 0.5 → সত্য।
3/2 = 1.5 → এই মান 1-এর চেয়েও বড় সত্য নয়।
-√2/2 = - 0.707 → সত্য।

১০.

A zip wire runs between two posts, 30m apart. The zip wire is at an angle of 60° to the horizontal. Calculate the length of the zip wire.

ক
80 m
খ
60 m
গ
45 m
ঘ
90 m

ব্যাখ্যা

Question: A zip wire runs between two posts, 30m apart. The zip wire is at an angle of 60° to the horizontal. Calculate the length of the zip wire.

Solution:

Given,
Horizontal distance between posts is 30 meters
Angle of elevation, θ = 60°
Then, we find the Length of the zip wire (hypotenuse) = L
Since we know the adjacent side and the angle, and we need to find the hypotenuse, we use the cosine function.

⇒ cos(θ) = Adjacent/Hypotenuse
⇒ cos(60°) = 30/L
⇒ L = 30/(1/2)
= 30 × 2
= 60 m

So, Length of zip wire is = 60 m