Bank Math Master

Question: If, cosec A + cot A = 5/2, find the value of, cosec A - cot A.

Solution:
Given that,
cosec A + cot A = 5/2

We know,
cosec²A - cot²A = 1
⇒ (cosec A + cot A)(cosec A - cot A) = 1

⇒ (5/2)(cosec A - cot A) = 1
⇒ cosec A - cot A = 1 ÷ (5/2)

∴ cosec A - cot A = 2/5

Question: Today is Monday. After 59 days, it will be-

Solution:
We know that each day of the week is repeated after 7 days.

59 ÷ 7 = 8 (remainder 3)

So, after (7 × 8) = 56 days, it will be Monday.
∴ After 59 days, it will be (Monday + 3 days) = Thursday

Question: In how many different ways can the letters of the word ‘BALLOON’ be arranged?

Solution:
Number of letters in the word = 7

Repeated letters:
L = 2 times
O = 2 times
The remaining letters (B, A, N) are different.

∴ Number of arrangements
= 7!/(2! × 2!)
= 5040/4
= 1260

Question: sin2A = √3/2, then find the value of A = ?

Solution:
Given that,
sin 2A = √3/2

We know,
sin 60° = √3/2
⇒ sin2A = sin60°
⇒ 2A = 60°
⇒ A = 60°⁄2
∴ A = 30°

Question: There are 6 true-false questions in an examination. In how many ways can these questions be answered?

Solution:
Total number of questions = 6
Each question has 2 possible answers (True or False).

∴ Total number of ways = 2⁶ = 64 ways 

Question: If the current month is January, what month will it be after 50 months?

Solution:
We know that there are 12 months in a year, and the months repeat after every 12 months.

প্রথমে 50 কে 12 দিয়ে ভাগ দেই,
50 ÷ 12 = 4 remainder 2,

January মাসের এক মাস পরে হবে February.
January মাসের দুই মাস পরে আসবে March.

Question: In how many ways can a team of 4 people be selected from 8 people?

Solution:
Total number of people, n = 8
Number of team members, r = 4

The number of ways to choose the team = ⁿC_r =
⁸C₄ = 8!/[4! × (8 - 4)!]
= (8 × 7 × 6 × 5 × 4!)/(4 × 3 × 2 × 1 × 4!)
= 70

Question: cos(θ + 25°) = √3/2, then the value of θ?

Solution:
Given that,
cos(θ + 25°) = √3/2
⇒ cos(θ + 25°) = cos30°
⇒ θ + 25° = 30°
⇒ θ = 30° - 25°
∴ θ = 5°

Question: You have 4 favorite novels on your shelf. If you decide to arrange these 4 novels in every possible order and it takes 45 seconds to move one novel, how much time will it take to arrange all possible combinations?

Solution:
4 novels can be arranged in = 4! ways
= 4 × 3 × 2 × 1 = 24 ways

Time required to move one novel = 45 seconds
time to arrange 4 novels = 45 × 4 = 180 seconds

So, total time required = 24 × 180 seconds
= 4320 seconds
= 4320/60
= 72 minutes

∴ It will take 72 minutes to arrange all possible combinations.

Question: How many two-digit numbers can be formed using the digits 3, 5, and 7 if repetition of digits is allowed?

Solution:
Number of possible two-digit numbers which can be formed by using the digits 3, 5 and 7 = 3 × 3.

∴ 9 possible two-digit numbers can be formed.

The 9 possible two-digit numbers are-
33, 35, 37, 53, 55, 57, 73, 75, 77 

Question: From a point Q on level ground, the angle of elevation of the top of a building is 45 degrees. If the building is 50 m high, find the distance of point Q from the foot of the building.

Solution:

Height of building AB = 50 m, angle of elevation ∠AQB = 45°

We know, tanθ = opposite/adjacent = AB/AQ
⇒ tan45° = 50/AQ
⇒ 1 = 50/AQ
⇒ AQ = 50 m

Thus, the distance from point Q to the foot of the building is 50 m.

Question: If θ = 45°, then what is the value of (1 - sec²θ)/(1 + sec²θ)?

Solution:
Here, θ = 45°
(1 - sec²θ)/(1 + sec²θ) = {1 - (sec 45°)²}/{1 + (sec 45°)²}
⇒ sec 45° = 1/cos 45° = 1/(1/√2) = √2
⇒ (1 - (√2)²)/(1 + (√2)²) = (1 - 2)/(1 + 2) = - 1/3

∴ The value of (1 - sec²θ)/(1 + sec²θ) = - 1/3

Question: If θ is a positive acute angle satisfying sin²θ + sin⁴θ = 1, then find the value of cot²θ + cot⁴θ.

Solution:
Given that,
sin²θ + sin⁴θ = 1 ..........(1)
⇒ sin⁴θ = 1 - sin²θ
⇒ sin⁴θ = cos²θ
⇒ sin²θ.sin²θ = cos²θ
⇒ sin²θ = cos²θ/sin²θ
⇒ sin²θ = cot²θ 

Now, putting sin²θ = cot²θ ..........(2)
∴  sin⁴θ = cot⁴θ ..........(3)
From equation (2) & (3) we get
sin²θ + sin⁴θ = cot²θ + cot⁴θ
∴ cot²θ + cot⁴θ = 1

Thus, the value of cot²θ + cot⁴θ = 1.

Question: Which of the following is a leap year?

Solution:
অধিবর্ষ বা লিপ ইয়ার নির্ণয়ের দুটি প্রধান নিয়ম রয়েছে:
১. সাধারণ বছর: বছরটি 4 দ্বারা নিঃশেষে বিভাজ্য হতে হবে।
২. শতাব্দী বছর (100 দ্বারা বিভাজ্য): বছরটি 400 দ্বারা নিঃশেষে বিভাজ্য হতে হবে।

এখন,
1996 ÷ 4 = 499 → নিঃশেষে বিভাজ্য → Leap year।
1982 ÷ 4 = 495.5 → বিভাজ্য নয় → Leap year নয়। 
2003 ÷ 4 = 500.75 → নিঃশেষে বিভাজ্য নয় → Leap year নয়।
2005 ÷ 4 = 501.25 → নিঃশেষে বিভাজ্য নয় → Leap year নয়।

অতএব, 1996 সাল অধিবর্ষ।

Question: sin135° + sin45° = ?

Solution:

Given that,
sin135° + sin45°
= sin(180° - 45°) + sin45°
= sin45° + sin45°
= 2 × sin45°
= 2 × (1/√2)
= √2

∴ sin135° + sin45° = √2

Question: If 12 people attend a meeting and each person shakes hands with every other person exactly once, find the total number of handshakes.

Solution:

Total handshakes = ¹²C₂ = 12!/2!(12 - 2)!
= (12 × 11 × 10!)/(2 × 10!)
= 66

∴ The total number of handshakes is 66.

Question: A committee of 4 members is to be formed by selecting out of 7 men and 6 women. In how many different ways can the committee be formed if it should have 3 men and 1 woman?

Solution:
3 men can be selected out of 7 men in
⁷C₃ = 7!/[3!(7 - 3)!]
= (7 × 6 × 5)/(3 × 2 × 1)
= 35 ways

1 woman can be selected out of 6 women in
⁶C₁ = 6 ways

∴ Required number of ways = 35 × 6 = 210

∴ 210 different ways to form the committee with 3 men and 1 woman.

Question: If tan(θ + 30°) = 1, find the value of cosθ.

Solution:
Given,
tan(θ + 30°) = 1
⇒ tan(θ + 30°) = tan 45°
⇒ θ + 30° = 45°
⇒ θ = 45° - 30°
⇒ θ = 15°

Now,
cosθ = cos 15°
∴ cos 15° = cos(45° - 30°)
= cos45° cos30° + sin45° sin30°
= (1/√2 × √3/2) + (1/√2 × 1/2)
= (√3/2√2) + (1/2√2)
= (√3 + 1)/(2√2) 

Question: What is the angle between the hour and minute hands of a clock when it is 10 minutes past 3?

Solution:
10 minutes past 3 অর্থাৎ, 3 টা 10 মিনিট।
= 3 + (10/60) ঘন্টা
= 3 + 1/6
= 19/6 ঘন্টা

আমরা জানি,
ঘণ্টার কাঁটা 12 ঘণ্টায় 360° ঘোরে।
∴ 1 ঘণ্টায় ঘোরে = 360°/12 = 30°
∴ 19/6 ঘন্টায় ঘোরে = 30 × (19/6) = 570/6 = 95°

মিনিটের কাঁটা 60 মিনিটে 360° ঘোরে।
∴ 1 মিনিটে ঘোরে = 360°/60 = 6°
∴ 10 মিনিটে ঘোরে = 10 × 6 = 60°

∴ ঘড়ির কাঁটা দুটির মধ্যবর্তী কোণ = |95° - 60°| = 35°

Question: In how many ways can 5 boys and 4 girls be seated in a row so that they sit alternately?

Solution:
Let the arrangement be,
B G B G B G B G B

5 boys can be seated in 5! ways
4 girls can be seated in 4! ways

∴ Required number of ways
= 5! × 4!
= 120 × 24
= 2880