ব্যাংক ডেইলি কুইজ [লং কোর্সের অংশ]

Question: If α and β are positive acute angles, sin(4α - β) = 1 and cos(2α + β) = 1/2, then the value of (α + 2β) is?

Solution:
Given,
sin(4α - β) = 1
⇒ sin(4α - β) = sin 90°
⇒ 4α - β = 90° ..... (1)

again, cos(2α + β) = 1/2
⇒ cos(2α + β) = cos 60°
⇒ 2α + β = 60° .... (2)

(1) + (2) ⇒ 4α - β + 2α + β = 90° + 60°
⇒ 6α = 150°
⇒ α = 25°
From (2), β = 60° - (2 × 25°) = 10°
∴ (α + 2β) = 25° + (2 × 10°) = 45°

Question: sin(A + 14°) = 1/2, find the value of A?

Solution:
sin(A + 14°) = 1/2
⇒ sin(A + 14°) = sin30°
⇒ A + 14° = 30°
⇒ A = 30° - 14°
∴ A = 16°

Question: The angle of elevation of a ladder leaning against a wall is 60° and the foot of the ladder is 3 m away from the wall. The length of the ladder is:

Solution:
Let,
AB be the wall and BC be the ladder.
Then, ∠ACB = 60° and AC = 3 m.

Here,
AC/BC = cos60°
⇒ AC/BC = 1/2
⇒ BC = 2 × AC
⇒ BC = 2 × 3
∴ BC = 6

∴ The length of the ladder is 6 m.

Question: The minimum value of 2sin²θ + 3cos²θ is?

Solution:
Let,
p = 2sin²θ + 3cos²θ
⇒ p = 2sin²θ + 2cos²θ + cos²θ
⇒ p = 2(sin²θ + cos²θ) + cos²θ
⇒ p = 2 + cos²θ [since sin²θ + cos²θ = 1]

Therefore p will be the minimum when cosθ = 0.
So, the minimum value of p will 2.

Question: Find the value of (1 + tanA - secA) (1 + cotA + cosecA).

Solution:

Question: If sec²θ + tan²θ = 7, then the value of θ when 0° ≤ θ ≤ 90° is?

Solution:
sec²θ + tan²θ = 7
⇒ 1 + tan²θ + tan²θ = 7 - 1
⇒ 2tan²θ = 6
⇒ tan²θ = 3
⇒ tanθ = √3
⇒ tanθ = tan60°
∴ θ = 60°

Question: θ is the positive acute angle and sinθ - cosθ = 0, then the value of secθ + cosecθ is?

Solution:

Question: If sin A + cos A = a and sec A+ cosecA = b then, b(a² - 1) = ?

Solution:

Question: If a pole 6 m high casts a shadow 2√3 m long on the ground, then the elevation of the sun is -

Solution:

ধরি,
AB = 6, BC = 2√3

ABC সমকোণী ত্রিভুজ হতে পাই,
tanθ = AB/BC
⇒ tanθ = 6/2√3
⇒ tanθ = 3/√3
⇒ tanθ = √3
⇒ tanθ = tan60°
∴ θ = 60°

Question: The numerical value of 1 + (1/cot²63°) - sec²27° + (1/sin²63°) - cosec²27° is?

Solution:

Question: From a point at some distance from the base of a tree, the angle of elevation to the top of the tree is 30°. If the tree is 12 meters tall, how far is that point from the tree?

Solution:

Let's assume the tree is located a meters away.

Thus,  tan 30° = AB/AC
⇒ 1/√3 = 12/a
∴ a = 12√3 meters

প্রশ্ন: If sec²θ + tan²θ = 5/12 then, sec⁴θ - tan⁴θ = ?

সমাধান:
sec⁴θ - tan⁴θ
= (sec²θ - tan²θ)(sec²θ + tan²θ)
= 1 × (sec²θ + tan²θ) [cause 1 + tan²θ = sec²θ]
= 1 × (7/12)
= 5/12

Question: If xtan60° + cos45° = sec45°, then the value of x² + 1 is?

Solution:

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

Solution:
Let AB be the tree and AC be its shadow.

Let ∠ACB = θ
Then, AC/AB = √3
⇒ cotθ = √3
⇒ cotθ = cot30°
∴ θ = 30°