Bank Math Master

Question: In a certain conference, if all the n attendees shake hands 105 times, what is the value of n?

Solution:
a handshake took place between 2 people.
as there are n persons.
ⁿC₂ = 105
or, n(n - 1)/2 = 105
or, n² - n = 210
or, n² - n - 210 = 0
or, n² - 15n + 14n - 210 = 0
or, n(n - 15) + 14(n - 15) = 0
or, (n - 15)(n + 14) = 0

∴ n = 15 or - 14

hence the number of attendance can't be negative. 
n = 15

Question: How many times in a day are the hours and minutes hand of a clock in a straight line and in opposite directions?

Solution: 
ঘড়ির মিনিট এবং ঘণ্টার কাটা প্রতি বারো ঘণ্টায় সোজা এবং পরস্পর বিপরীত দিকে অবস্থান করে এগারোবার।
কারণ, ৫ টা থেকে ৭ টা এর মধ্যে শুধু মাত্র ছয়টা বাজার সময়ই ঘড়ির কাটা সরলরেখায় এবং পরস্পর বিপরীত দিকে অবস্থান করে।
অর্থাৎ, একদিনে মোট ২২ বার ঘড়ির ঘণ্টা এবং মিনিটের কাটা সরলরেখায় থাকে এবং পরস্পর বিপরীত দিকে অবস্থান করে।

Question: If a 30 meter ladder is placed against a 15 metre wall such that it just reaches the top of the wall, the angle of elevation of the wall is:

Solution: 

let, the length of the ladder is AC = 30m
the height of the wall is AB = 15m
elevation angle = θ

we know that,
sinθ = AB/AC
sinθ = 15/30
sinθ = 1/2
sinθ = sin30°
θ = 30°

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

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

the minimum value of x depends on the minimum value of cos²θ.
Since the minimum value of cos²θ is 0, the minimum value of x is 2.

Question: Committee X has 4 members, committee Y has 5 members, and these committees have no members in common. If a task force is to be formed consisting of one member of X and one member of Y, how many different task forces are possible?

Solution:
Committee X has 4 members: Choosing one out of 4 = ⁴C₁ =4
Committee Y has 5 members: Choosing one out of 5 = ⁵C₁ =5

total = 4 × 5 = 20

Question: Through what angle does the minute hand of a clock turn in 7 minutes?

Solution:
Angle traced by the minute hand in 7 minutes is
= (360°/60) × 7
= 42°

Question: If A = π/4, what is the value of sin2A?

Solution: 
given A = π/4

sin2A = 2sinAcosA
= 2sin(π/4)cos(π/4)
= 2sin(45°)cos(45°)
= 2 × 1/√2 × 1/√2
= 2/2
= 1

Question: January 1, 2008, is Monday. What day of the week lies on Jan 1, 2009?

Solution: 
কোন বছরের প্রথম দিন যদি শনিবার হয় তাহলে পরবর্তী বছরে প্রথম দিন একদিন বেশি বা বরিবার হবে।

কিন্তু ২০০৮ সাল লিপ ইয়ার হওয়ার কারনে এখানে অতিরিক্ত ১ দিন যোগ হবে। অর্থাৎ ২ দিন বেশি হবে।
তাই, ২০০৮ সালের প্রথম দিন সোমবার হওয়ায় ২০০৯ সালের প্রথম দিন বুধবার হবে।

Question: If sinA = 4/5 find the value of tanA.secA = ?

Solution:
sinA = লম্ব/অতিভুজ = 4/5
ভূমি = √{(5)² - (4)²}
= 3

tanA.secA = (4/3)(5/3)
= 20/9

Question: A 5-digit identification code is to be created as a sequence that contains each integer in the set {1, 2, 3, 4, 5} exactly once. If all identification codes are possible except those containing even integers next to each other, how many different identification codes are possible?

Solution: 
Total ways to create 5-digit identification code = 5!

Both even number cannot be together.
When both the even digits are taken together as one digit, we have 4 digits, that is 1, 3, 5 and (24) or, 4! to arrange 4 digits
and 2 & 4 can be arranged in 2! ways within themselves. Hence, 4! × 2

Ways we are looking at = 5! - 4 × 2! = 120 - 48 = 72

Question: What is the angle between the hour hand and the minute hand when the clock says 3:15?

Solution: 
আমরা জানি,
মধ্যবর্তী কোণ = | (11M - 60H)/2 |°
=  | {(11 × 15) - (60 × 3)} /2 |°
=  | (165 - 180) /2 |°
= 7.5 °

Question: If Asinθ = 1 and Acosθ = √2, find the value of (3√2/tanθ) + 1 =?

Solution: 
Asinθ = 1
Acosθ = √2

∴ Asinθ/Acosθ = 1/√2
tanθ = 1/√2

(3√2/tanθ) + 1
= 3√2/(1/√2) + 1
= 6 + 1
= 7

Question: Today is Friday, after 147 days, it will be

Solution: 
147/7 = remainder 0

so after 147 days, it will be Friday.

if the remainder is two, the day will be 2 days past Friday or Sunday

Question: If 4 points are indicated on a line and 5 points are indicated on another line that is parallel to the first line, how many triangles can be formed whose vertices are among the 9 points?

Solution: 
There are two types of triangles possible:
With two vertices on the line with 4 points and the third vertex on the line with 5 points = ⁴C₂ × ⁵C₁ = 30
With two vertices on the line with 5 points and the third vertex on the line with 4 points = ⁴C₁ × ⁵C₂ = 40

total waya = 30 + 40 = 70

Question: How many rotations will the hour hand of a clock complete in 96 hours?

Solution: 
Number of rotation = 96/12 = 8

Question: If a 1.5 m tall girl stands at a distance of 3 m from a lamp-post and casts a shadow of length 4.5 m on the ground, then the height of the lamp-post is-

Solution:

let, 
height of the girl DE = 1.5m
distance of the shadow CE = 4.5m
height of the lamp-post AB = h
BC = 3 + 4.5 = 7.5m

here,
tanθ = DE/CE
tanθ = AB/BC

∴ DE/CE = AB/BC
1.5/4.5 = h/7.5
h = 2.5m

Question: sin(180° + θ) = ?

Solution: 

sin(180° + θ) হলে sin তৃতীয় ভাগে পরবে।
তাই sin এর মান ঋণাত্নক হবে।
∴ sin(180° + θ) = - sinθ

Question: A clock is 120 seconds slow in 24 hours. How many seconds will it take to complete one hour?

Solution: 
as the clock is slow, that means it takes more time.

to complete 24 hours or 86400 seconds it takes = (86400 + 120) = 86520 seconds
to complete 1 hour or 3600 seconds it takes = (86520 × 3600)/86400 seconds
= 3605 seconds

Alternate:
in 24 hours it is slow for 120 seconds
in 1 hour it is alow for = (120/24) = 5 seconds

so it will take (3600 + 5) or, 3605 seconds to complete an hour.

Question: A small company employs 3 men and 5 women. If a team of 4 employees is to be randomly selected to organize the company retreat, what is the probability that the team will have exactly 2 women?

Solution: 
the ways of choosing 4 from 8 employs = 8C4 = 70
choosing 2 women from 5 women = 5C2 = 10
choosing 2 men from 3 men = 3C2 = 3

probability = (3 × 10)/70 = 3/7