Bank Math Master

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: How many diagonals can be drawn in a pentagon?

Solution: 
A pentagon has 5 sides. We obtain the diagonals by joining the vertices in pairs.
Total number of sides and diagonals,
= ⁵C₂
= 10
This includes its 5 sides also.

∴ Diagonals = 10 – 5 = 5

Question: If 16th March, 2005 is Wednesday, what was the day of the week on 16th March, 2004?

Solution:
Since 16th March is coming after February leap year will not count in 2004. 
Odd day is 1.

∴ 16th March, 2004 is Tuesday.

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: In how many ways can the letters of the word 'LEADER' be arranged?

Solution:
The word 'LEADER' has total 6 letters among them letter L = 1, E = 2, A = 1, R = 1, D = 1

Ways = 6!/2! = 720/2 = 360

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

Solution:
Total number of letters in the word = 6
Letters have to be taken = 2 

Number of ways = ⁶P₂ = 6!/(6 - 2)! = 6!/4! = 30

Question: From the top of a lighthouse which is 90 m above the sea, the angle of depression of a ship is 60°. How far is the ship from the lighthouse?

Solution:

Let the height of the lighthouse above sea be AC and it is given 90 m.
Ship is at point B so the distance between the base of lighthouse A and ship is AB.
In ΔABC,
tan60° = AC/AB
⇒ √3 = 90/AB
⇒ AB = 90/√3 = 30√3 m

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: In your bookshelf, you have five favorite books. If you decide to arrange these five books in every possible combination. How many combination will be possible?

Solution:
5 books can be arranged in 5! ways 
= 120 ways 

Question How many days are there in n weeks n days?

Solution:
Number of days in n weeks = 7 × n = 7n days

∴ Number of days in n weeks n days = 7n + n days
= 8n days

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: 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:
Time between 8 o'clock to 2 o'clock = 6 hours

Hour hand rotate in 12 hours 360°
∴ Hour hand rotate in 6 hours (360° × 6)/12
= 180°

Question: In how many different ways can the letters of the word BANANA be arranged where all A will be together?

Solution:
If all the A remain together than we count them as 1 letter.
Three A can be arranged in 3!/3! = 1 way
∴ Number of letter in the word where three A's are counted as 1,
So, total letters will be = 4
Repeated letters N = 2

∴ The number of arrangement = 4!/2! = 12

Question: An observer 1.6 m tall is 20√3 m away from a tower. The angle of elevation from his eye to the top of the tower is 30°. The height of the tower is:

Solution:

Here, 
Height of the observer AB = 1.6 m 
Distance  between observer and tower AC = 20√3 m
∴ CE = AB = 1.6 m 
Height of the tower is CD = CE + DE.

From ΔBDE we get,
BE = AC = 20√3 m

Now,
tan30° = DE/BE 
⇒ 1/√3 = DE/(20√3)
⇒ DE = (20√3)/√3 
∴ DE = 20 

∴ The height of the tower is = CE + DE = 1.6 + 20 m = 21.6 m

Question: How many rotations will the short hand of a clock complete in 72 hours?

Solution:

Here,
Short hand = Hour hand

The short hand rotates a full rotation in 12 hours.

Number of rotations
= 72/12
= 6

Question: In how many ways can 5 balls can be chosen from 9 different balls?

Solution: 

Here,
Total number of different balls, n = 9
Chosen balls from different balls, r = 5

The number of ways 5 balls can be chosen is
= ⁿC_r
= n!/r!(n - r)!
= 9!/5!(9 - 5)!
= 9!/(5! × 4!)
= (9 × 8 × 7 × 6 × 5!)/(5! × 4!)
= (9 × 8 × 7 × 6)/4!
= 3024/(4 × 3 × 2 × 1)
= 3024/24
= 126

∴ 5 balls can be chosen from 9 different balls in 126 ways.