Bank Math Master

Question: An accurate clock shows 8 o'clock in the morning. Through how may 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: A committee of 3 members is to be formed by selecting out of 5 men and 4 women. In how many different ways the committee can be formed if it should have 1 man and 2 women?

Solution:
We can select 1 men from 5 men in = ⁵C₁ ways = 5
2 women from 4 women in = ⁴C₂ = 6
Committee can be formed = 5 × 6 = 30

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 = height of tree
BC= Shadow of tree
angle of elevation = C
∴  BC = √3 AB

We know,
tanC = AB/BC
⇒ tanC = AB/√3AB
⇒ tanC = 1/√3
⇒ tanC = tan30°
∴ C = 30°

Question: Express the angle in degree included between hands of hour and minute of the clock at 4 : 30?

Solution:
আমরা জানি,
মধ্যবর্তী কোণ, θ = Ι(11M - 60H)/2Ι°
= Ι(11 × 30) - (60 × 4)/2Ι°
= Ι(330 - 240)/2Ι°
= 45°

Question: How many days are there x weeks 3x days?

Solution:
x weeks = (7 × x) days = 7x days.

∴ x weeks 3x days = (7x + 3x) days
= 10x days.

Question: In how many ways can 3 boys and 3 girls be selected from 12 boys and 9 girls?

Solution:
12 জন বালক হতে প্রতিবার 3 জন বালক বেছে নেয়া যায় = ¹²C₃ = 220 উপায়ে
9 জন বালিকা হতে প্রতিবার 3 জন বালিকা বেছে নেয়া যায় = ⁹C₃ = 84 উপায়ে

∴ মোট বেছে নেয়া যায় = 220 × 84 = 18480 উপায়ে

Question: A ladder rests against a wall that is perpendicular to the ground. If the bottom of the ladder is 4 meter away from the bottom of the wall, while the top of the ladder is at a height of 3 meter. What is the length of the ladder?

Solution:
 
মইয়ের দৈর্ঘ্য, AC = x m
ভূমি হতে মইয়ের শীর্ষবিন্দুর উচ্চতা, AB = 3 m
মই হতে দেয়ালের দূরত্ব, BC = 4 m

পীথাগোরাসের উপপাদ্য হতে পাই,
AC² = AB² + BC²
⇒ x² = 3² + 4²
⇒ x² = 25
⇒ x = √25
∴ x = 5

Question: A pole of 48m long breaks such that parts are not completely separated and the upper part makes on angle 30° with the ground. At what height did the pole break?

Solution:

খুঁটিটি মাটি হতে x মিটার উঁচুতে ভেঙ্গেছিল।

আমরা জানি,
sin30° = AB/AC
⇒ sin30° = x/(48 - x)
⇒ 1/2 = x/(48 - x)
⇒ 48 - x = 2x
⇒ 3x = 48
∴ x = 16

Question: How many ways can the word `MOTHER' be arranged with 2 letters each time?

Solution:
Total number of words, n = 6 and r = 2

∴ Arrangement = ⁿP_r
= ⁶P₂
= 30

Question: If 1st October is Monday, then 1st November will be -

Solution:
Given that 1st October is Monday.
Number of days in October = 31
31 days = 3 odd days. [As we can reduce multiples of 7 from odd days which will not change anything]

∴ 1st November = (Monday + 3 odd days) = Thursday.

Question: A committee is to consist of three members. If there are seven men and five women available to serve on the committee, how many different committees can be formed?

Solution:
Total, n = 7 + 5 = 12
and r = 3

Number of committee = ¹²C₃
= 220

Question: A committee of 5 members is to be formed by selecting out of 7 men and 6 women. In how many different ways the committee can be formed if it should have at least 3 men? 

Solution:
      Men (7)       Women (6)
1)    3                     2
2)    4                     1
3)    5                     0

From (1) Number of ways = ⁷C₃ × ⁶C₂ = 35 × 15 = 525
From (2) Number of ways = ⁷C₄ × ⁶C₁ = 35 × 6 = 210
From (3) Number of ways = ⁷C₅ × ⁶C₀ = 21 × 1 = 21

Total number of ways = 525 + 210 + 21 = 756

Question: Robin wanted to arrange 3 of four plants in a row on a shelf. If each of the plants is in a different colour container, how many different arrangements can he make?

Solution:
চারটি চারাগাছ হতে তিনটি নিয়ে সাজানো যায় = ⁴P₃ = 24

Question: The angle of elevation of a ladder learning against a wall is 60° and the foot of the ladder is 4.6 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 = 4.6 m

We know,
cos∠ACB = AC/BC
⇒ cos60° = AC/BC
⇒ AC/BC= 1/2
⇒ BC = 2 × AC
⇒ BC = 2 × 4.6
∴ BC = 9.2 m

Question: At what angle the hands of a clock are inclined at 15 minutes past 5?

Solution:
Hours hand moves in 15 past.
5 from 12 p.m = (5 + 15/60) hours = 21/4 hours
Angle of hours hand = (360/12) × (21/4)
= 157.5°

Minutes hands makes angle of = (360/60) × 15
= 90°

Angle between hours and minutes hands = (157.5° - 90°)
= 67.5°

Question: How many different six digit numbers can be formed using all of the following digits 3, 3, 4, 4, 4, 5?

Solution:
প্রদত্ত অঙ্ক মোট 6টি যার মধ্যে 2টি 3 এবং 3টি 4 আছে।

∴ নির্ণেয় ছয় অঙ্কবিশিষ্ট মোট গঠিত সংখ্যা = 6!/(2! × 3!) টি
= 60 টি 

Question: The angle of depression of a point situated at a distance of 70 m from the base of a tower is 60°. The height of the tower is -

Solution:

Let,
Height of the tower AB = h meter.
Now, ∠DAC = ∠ACB = 60° and BC = 70 meter.

In ΔABC 
tan60° = AB/BC
⇒ √3 = h/70
∴ h = 70√3

Height of the tower 70√3 meter.

Question: In how many different ways can the letters of the word 'BANANA' be arranged?

Solution:
The word BANANA
Here total letters = 6
No of A = 3
No of N = 2

∴ Arrangement = 6!/(3! × 2!)
= 60

Question: At the end of a banquet 20 people shake hands with each other. How many handshakes will there be in total?

Solution:
Two people can make a one handshake.
Number of handshakes = ²⁰C₂ = 190