Bank Math Master

Question: 30° =?

Solution: 
1° = π/180 radian 
30° = (π/180) × 30° 
= π/6 radian

Question: In how many ways 6 students can be chosen from the class of 10 students?

Solution:
ways 6 students can be chosen from the class of 10 students is = ¹⁰C₆
= 10!/(6! 4!)
= 210 ways

Question: An accurate clock shows 8 o'clock in the morning. Through how many degrees will the hour hand rotate when the clock shows 2 o'clock in the afternoon?

Solution: 
From 8 o'clock to 20'clock total time is 6 hours.

Hours hand makes,
in 12 hours = 360° angle
in 6 hours = (360° × 6)/12
= 180°

Question: Today is Monday. After 61 days, it will be:

Solution:
Each day of the week is repeated after 7 days.
So, after 63 days, it will be Monday.
∴ After 61 days, it will be Saturday.

Question: The angle between the minute hand and the hour hand of a clock when the time is 4:20, is -

Solution:
কোণ = ।11M - 60H।/2
= ।(11 × 20) - (60 × 4)।/2
= ।220 - 240। /2
= 10°

Question: Find the angle of elevation of the sun when the shadow of a pole of 18 m height is 6√3 long.

Solution: 


shadow = base = 6√3 m
height =18 m

let, angle of elevation θ

tanθ = 18/6√3
⇒ tanθ = 3/√3
⇒ tanθ = √3
⇒ tanθ = tan60°
∴ θ = 60°

প্রশ্ন: If (sinθ + cosθ)/(sinθ - cosθ) = 5 , then, tanθ =? 

সমাধান:
5(sinθ - cosθ) = sinθ + cosθ
বা, 5sinθ - 5cosθ = sinθ + cosθ
বা,5sinθ - sinθ = cosθ + 5cosθ
বা, 4sinθ = 6cosθ
বা, sinθ/cosθ = 6/4
∴ tanθ = 3/2

Question: What is the lowest value of sinθ?

Solution: 
θ কোণ নির্দেশ করলে
sinθ এর বৃহত্তম মান = 1
sinθ এর ক্ষুদ্রতম মান = -1

প্রশ্ন: ⁴P₂ - ⁴C₂ =?

সমাধান: 
⁴P₂
= 4!/(4 - 2)!
= 4!/2!
= 12

⁴C₂
= 4!/2!(4 - 2)!
= 4!/2! 2!
= 6

∴ ⁵P₂ - ⁵C₂ = 12 - 6
= 6

Question: Which of the following is not a leap year?

Solution: 
অধিবর্ষ বের করার নিয়ম:

শর্ত-১: সালটি যদি ৪ দিয়ে নিঃশেষে বিভাজ্য হয় এবং ১০০ দিয়ে না হয় তাহলে অধিবর্ষ। যেমন, ২০১৬, ২০২০ এবং ২০২৪ ।

অথবা, শর্ত-২: সালটি যদি ৪, ১০০ এবং ৪০০ সবগুলো দিয়েই নিঃশেষে বিভাজ্য হয় তাহলে অধিবর্ষ। যেমন, ১৬০০, ২০০০ এবং ২৪০০।

Question: A committee of 3 members is to be formed by selecting out of 5 men and 4 women. In how many different ways can the committee be formed if it should have 1 man and 2 women?

Solution:
We can select 1 man from 5 men in
= ⁵C₁ ways
= 5 ways 

Selecting 2 women from 4 women in
= ⁴C₂
= 6

∴ The committee can be formed
= 5 × 6
= 30 ways

Question: If sinA + sin²A = 1 then, cos²A + cos⁴A =? 

Solution: 
sinA + sin²A = 1
বা, sinA = 1 - sin²A
বা, sinA = cos²A

cos²A + cos⁴A
= cos²A + (cos²A)²
= cos²A + sin²A
= 1

প্রশ্ন: If sinA = cos2A,  A =?

সমাধান:
sinA = cos2A
⇒ sinA = 1 - 2 sin²A
⇒ 2 sin²A + sinA - 1 = 0
⇒ 2 sin²A + 2 sinA - sinA -1 = 0
⇒ 2 sinA ( sinA + 1) -1 (sinA + 1) = 0
⇒ (sinA + 1) (2 sinA -1) = 0

হয়, sinA + 1 = 0 
⇒ sinA = -1 
⇒ A = 270° 

অথবা 2 sinA - 1 = 0
⇒ sinA = 1/2
⇒ A = 30°

Question: tanA√(1 - sin²A) = ? 

Solution: 
tanA√(1 - sin²A)
= tanA√(cos²A)
= (sinA/cosA) × cosA
=  sinA

Question: The angle of elevation of a ladder leaning against a wall is 60° and the foot of the ladder is 4.5 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 = 4.5 m.

Here, 
AC/BC = cos60°
⇒ AC/BC = 1/2 
⇒ BC = 2 × AC 
⇒ BC = 2 × 4.5
∴ BC = 9 

∴ The length of the ladder is 9 m.

প্রশ্ন: How many days are there in n weeks n days?

সমাধান:  
In one week there are 7 days.
So in n weeks, there are 7n days

∴ total days = 7n + n = 8n