Bank Math Master

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

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

Therefore x will be the minimum when cosθ = 0. i.e. minimum value of x will 2

Question: In how many ways 4 boys and 3 girls can be seated in a row so that they are alternate.

Solution: 
Let the Arrangement be,
B G B G B G B
4 boys can be seated in 4! Ways
Girl can be seated in 3! Ways

∴ Required number of ways = 4! × 3!
= 24 × 6
= 144

Question: What is the angle between the hour and minute hands of a clock when it is 20 minutes past 8?

Solution:
20 minutes past 8 অর্থাৎ, 8 টা 20 মিনিট।
= 8 + (20/60) ঘন্টা
= 8 + (1/3)
= 25/3 ঘন্টা

আমরা জানি,
ঘণ্টার কাঁটা 12 ঘণ্টায় 360° ঘোরে।
∴ 1 ঘণ্টায় ঘোরে = 360°/12 = 30°
∴ 25/3 ঘন্টায় ঘোরে = (30° × 25)/3 = 250°

আবার,
মিনিটের কাঁটা 60 মিনিটে 360° ঘোরে।
∴ 1 মিনিটে ঘোরে = 360°/60 = 6°
∴ 20 মিনিটে ঘোরে = 20 × 6° = 120°

∴ ঘড়ির কাঁটা দুটির মধ্যবর্তী কোণ = | 250° - 120° | = 130°

Question: The equation cos²θ = (x + y)²/4xy is only possible when ?

Solution: 
Given that,
cos²θ = (x + y)²/4xy
⇒ 1 = (x + y)²/4xy   ; [Max value of cos²θ = 1 ]
⇒ 4xy = (x + y)²
⇒ 4xy = x² + y² + 2xy
⇒ 0 = x² + y² - 2xy
⇒ (x - y)² = 0
⇒ x - y = 0
⇒ x = y

Therefore, the given equation is possible only when x = y.

Question: If ⁷P_r = 210, what is the value of r?

Solution:
We know,
ⁿP_r = n!/(n - r)!

Given that,
⁷P_r = 210
⇒ 7!/(7 - r)! = 210
⇒ 5040/(7 - r)! = 210 (7! = 5040)
⇒ (7 - r)! = 5040/210
⇒ (7 - r)! = 24
⇒ (7 - r)! = 4!   ;(4! = 4 × 3 × 2 × 1 = 24)
⇒ 7 - r = 4
⇒ r = 7 - 4
∴ r = 3

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,
1982 ÷ 4 = 495.5 → বিভাজ্য নয় → Leap year নয়। 
1984 ÷ 4 = 496 → বিভাজ্য → Leap year. 
1998 ÷ 4 = 499.5 → বিভাজ্য নয় → Leap year নয়। 
2002 ÷ 4 = 500.5 → বিভাজ্য নয় → Leap year নয়। 

অতএব, 1984 সালটি অধিবর্ষ।

Question: If 1 + tan²θ = 4 and θ < 90°, than what is the value of θ = ?

Solution:
Given that,
1 + tan²θ = 4 and θ < 90°
⇒ sec²θ = 4    ; [sec²θ = 1 + tan²θ]
⇒ (secθ)² = (2)²
⇒ secθ = 2
⇒ secθ = sec60°
⇒ θ = 60°

Question: At a conference, everyone shakes hands with everybody else. If there were 190 handshakes, how many people were at the conference?

Solution:
Let the number of people at the conference be x.
A handshake occurs between any two people, which can be expressed using combinations.

According to the question,
^xC₂ = 190
⇒ x!/(2!(x - 2)!) = 190
⇒ {x(x - 1)(x - 2)!}/{2 × 1 × (x - 2)!} = 190
⇒ x(x - 1)/2 = 190
⇒ x(x - 1) = 380
⇒ x² - x - 380 = 0
⇒ x² - 20x + 19x - 380 = 0
⇒ x(x - 20) + 19(x - 20) = 0
⇒ (x - 20)(x + 19) = 0

So x - 20 = 0 or x + 19 = 0
∴ x = 20 or x = - 19

Since the number of people cannot be negative, x = 20 

Therefore, there were 20 people at the conference.

Question: A 60-meter cable is attached from the top of a vertical pole down to the ground. If the cable makes an angle of 30 degrees with the ground, find the height of the pole.

 Solution:
 
Let,
Height, AB = h 

Given that,
AC = 60m
∠ACB = 30°

∴ sin30°= AB/AC
⇒ 1/2 = h/60 
⇒ h = 60 × 1/2 
∴ h = 30 m

Question: If θ be a positive acute angle satisfying cos²θ + cos⁴θ = 1, then the value of tan²θ + tan⁴θ is?

Solution:
Given that,
cos²θ + cos⁴θ = 1 ..........(1)
⇒ cos⁴θ = 1 - cos²θ
⇒ cos⁴θ = sin²θ
⇒ cos²θ.cos²θ = sin²θ
⇒ cos²θ = sin²θ/cos²θ
⇒ cos²θ = tan²θ

Now, putting cos²θ = tan²θ in equation (1) than we get,
⇒ tan²θ + tan⁴θ = 1

Question: There are 5 men and 4 women. Two members are to be selected for a committee. In how many ways can two  people be selected?

Solution:
Total candidates = 5 + 4 = 9

Number of ways to select 2 people = ⁹C₂ 
= 9!/2!(9 - 2)!
= (9 × 8 × 7!)/(2  × 7!)
= 36

Question: It was Saturday on January 1, 2015. What was the day of the week on Jan 1, 2016?

Solution:
2015 সালটি 4 দ্বারা বিভাজ্য নয়।  (2016 ÷ 4 = 503.75), তাই এটি একটি অধিবর্ষ নয়।

• অতিরিক্ত দিন নির্ণয়:
365 ÷ 7 = 52 সপ্তাহ এবং 1 দিন অতিরিক্ত হয়।

এখন,
1 লা জানুয়ারি 2015 দিনটি ছিল Saturday।
যেহেতু 2015 সালটি অধিবর্ষ নয়, তাই 1লা জানুয়ারি 2016 দিনটি হবে Saturday এর থেকে 1 দিন বেশি।
⇒ Saturday + 1 দিন = Sunday

∴ 1লা জানুয়ারি 2016 দিনটি হবে Sunday।

Question: If θ = 60° , then what is the value of (1 - sec²θ)/(1 + sec²θ)?

Solution:
Here, θ = 60°

Now,
(1 - sec²θ)/(1 + sec2θ)
= {1 - (sec60°)²}/{1 + (sec60°)²}
= (1 - 2²)/(1 + 2²)
= (1 - 4)/(1 + 4)
= - 3/5

Question: In how many ways can the letters of 'PARALLEL' be arranged if the first letter is always P?

Solution:
The word 'PARALLEL' has a total of 8 letters.

Condition, The first letter is fixed as 'P'.
Now, the remaining 8 - 1 = 7 positions need to be filled with the remaining letters.

Among these 7 letters, there are repeated letters:
A (2 times) and L (3 times)

∴ Number of arrangements for the remaining 7 letters= 7!/(3! × 2!)
= 5040/12
= 420

Question: From a point P on a level ground, the angle of elevation of the top of a tower is 30°. If the tower is 100 m high, the distance of point P from the foot of the tower is-

Solution:

Given that,
Hight of tower, AB = 100 m
Angle of elevation from point P, ∠APB = 30°

We know,
tanθ = opposite/adjacent ​= AB/PA
⇒ tan30° = 100/PA
⇒ 1/√3 = 100/PA
∴ PA = 100√3 m

Thus, the distance from point P to the foot of the tower is 100√3 m.

Question: sin²12° + sin²78° = ?

Solution: 
Given that,
sin²12° + sin²78° 
= sin²12° + sin²(90° - 12°)
= sin²12° + cos²12°
= 1

Question: How many possible two-digit numbers can be formed by using the digits 3, 5 and 7 (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: If tan(θ + 15°) = √3, what is the value of sinθ?

Solution:
Given that,
tan(θ + 15°) = √3
⇒ tan(θ + 15°) = tan 60°
⇒ θ + 15° = 60°
⇒ θ = 60° - 15°
⇒ θ = 45°

Now,
sinθ
= sin45°
= 1/√2

Question: How many times are the hands of a clock at right angle in a day?

Solution:
In 12 hours, they are at right angles 22 times.
∴ In 24 hours, they are at right angles 44 times.

Question: Today is Wednesday. After 86 days, what day of the week will it be?

Solution:
Each day of the week is repeated after 7 days.

So, after (7 × 12) = 84 days, it will be Wednesday.

After 85 days, it will be Thursday
After 86 days, it will be Friday.

∴ After 86 days, it will be Friday.