ব্যাংক ডেইলি কুইজ [লং কোর্সের অংশ]

Question: The greatest value of sin⁴θ + cos⁴θ + 2sin²θcos²θ is?

Solution:
We know,
sin²θ + cos²θ = 1

Squaring both sides,
(sin²θ + cos²θ)² = 1²
⇒ sin⁴θ + cos⁴θ + 2sin²θcos²θ = 1

∴ The greatest value of sin⁴θ + cos⁴θ + 2sin²θcos²θ is 1.

প্রশ্ন: The value of cos1° cos2° cos3° ............... cos88° cos89° cos90° is?

সমাধান:
 cos1° cos2° cos3° ............... cos88° cos89° cos90°
= cos90°
= 0 [0 will make whole series 0]
= 0

Question: The value of sin30° + cos60° = ?

Solution:
We know,
sin30° = 1/2
cos60° = 1/2

So,
sin30° + cos60° = 1/2 + 1/2
= 1

∴ The value of sin30° + cos60° is 1.

Question: A pole of 60 metre long breaks into two parts without complete separation and makes an angle 30° with the ground. Find the length of the broken part of the pole.

Solution:

sin30° = x/(60 - x)
⇒ 1/2 = x/(60 - x)
⇒ 60 - x = 2x
⇒ 3x = 60
⇒ x = 60/3 = 20

∴ The length of the broken part of the pole = 60 - 20 = 40 m

Question: If θ is a positive angle and 9sin²θ - 9 = 0, then the value of tan(θ - 30°) is equal to?

Solution:
Given,
9sin²θ - 9 = 0
⇒ 9sin²θ = 9
⇒ sin²θ = 1
⇒ sinθ = 1
⇒ sinθ = sin90°
∴ θ = 90°

Now,
tan(θ - 30°) = tan(90° - 30°)
= tan60°
= √3

প্রশ্ন: If θ be an acute angle and 5sin²θ + 3cos²θ = 4, then the value of tanθ is?

সমাধান:
5sin²θ + 3cos²θ = 4
⇒ 5sin²θ + 3(1 - sin²θ) = 4
⇒ 5sin²θ + 3 - 3sin²θ = 4
⇒ 2sin²θ = 1
⇒ sin²θ = 1/2
⇒ sinθ = √(1/2)
⇒ sinθ = 1/√2
⇒ sinθ = sin45°
⇒ θ = 45°

∴ tanθ = tan45° = 1

Question: If sec²θ - tan²θ = 1 and tan²θ = 3, then the value of θ when 0° ≤ θ ≤ 90° is?

Solution:
Given,
tan²θ = 3
⇒ tanθ = √3
⇒ tanθ = tan60°

∴ θ = 60°

Question: The angle of elevation of the sun, when the height of a tower is √3 times the length of its shadow, is-

Solution:

Let, ∠ACB = θ
Then, AB/AC = √3
⇒ tan θ = √3 = tan60°

∴ θ = 60°

Question: If asinθ = 2 and acosθ = 2√3, then the value of √3tanθ - 1 is?

Solution:
asinθ = 2
acosθ = 2√3

Now,
asinθ/acosθ = 2/(2√3)
⇒ tanθ = 1/√3
⇒ √3tanθ = 1

∴ √3tanθ - 1 = 0

Question: If sec(3x - 40°) = cosec(50° - x), then the value of x is?

Solution:
sec(3x - 40°) = cosec(50° - x)
⇒ sec(3x - 40°) = cosec{90° - (40° + x)}
⇒ sec(3x - 40°) = sec(40° + x)
⇒ 3x - 40° = 40° + x
⇒ 2x = 80°

∴ x = 40°

Question: A boy of height 1.5 m is walking away from the base of a lamp post at a speed of 0.8 m/sec. Find the height of the lamp post from the ground, if the shadow of the boy is 2.0 m after walking for 4 sec.

Solution:

Given that,
Height of the boy = 1.5 m
Speed of the boy = 0.8 m/s
Distance travelled by boy in 4 sec = 0.8 × 4 = 3.2 m
Total distance of shadow of boy and distance from base of lamp post = 2.0 + 3.2 = 5.2 m

Let the height of lamp post be 'h' m
According to question,
⇒ 1.5/2.0 = h/5.2
⇒ h = (5.2 × 1.5)/2.0
⇒ h = 3.9 m

So, The height of the lamp post is 3.9 meters.

Question: The value of 1 + {(tan 30° - tan 45°)/(cot 45° - cot 60°)} is -

Solution:
1 + (tan 30° - tan 45°)/(cot 45° - cot 60°)
= 1 + (tan 30° - tan 45°)/{cot (90° - 45°) - cot (90° - 60°)}
= 1 + (tan 30° - tan 45°)/(tan 45° - tan 30°)
= 1 + (tan 30° - tan 45°)/(-1)(tan 30° - tan 45°)
= 1 - 1
= 0

Question: A circus artist is climbing a 30 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 30°.

Solution:

By observing the figure, AB is the pole.

In triangle ABC,
⇒ AB/AC = sin30°
⇒ AB/30 = 1/2
⇒ AB = 15

Therefore, the height of the pole is 15 m.

Question: A ladder 26 m long is placed against a wall of height 13 m such that it just touches the top of the wall. Find the angle of elevation made by the ladder with the ground.

Solution:

AC = 26 meter
AB = 13 meter
∠ACB = θ

∴ sin θ = AB/AC = 13/26 = 1/2
⇒ sin θ = sin 30°

∴ θ = 30°

Question: A man looks into a mirror placed on the ground and sees the top of a tower. The mirror is 120 m away from the tower. If the man stands 0.6 m away from the mirror and his height is 1.8 m, find the height of the tower.

Solution:

Given that,
Distance from the mirror to the tower = 120 m
Distance from the man to the mirror = 0.6 m
Height of the man = 1.8 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
⇒ 1.8/0.6 = H/120
⇒ 3 = H/120
⇒ H = 120 × 3 = 360 m