Bank Math Master

Question: secA + tanA = 4/3, then find, secA - tanA = ?

Solution:
Given that,
secA + tanA = 4/3

We know,
sec²A - tan²A = 1
⇒ (secA + tanA)(secA - tanA) = 1
⇒ (4/3)(secA - tanA) = 1
⇒ (secA - tanA) = 1/(4/3)
∴ secA - tanA = 3/4

Question: Today is Thursday. After 44 days, it will be-

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

44 ÷ 7 = 6 (remainder 2)

So, after (7 × 6) = 42 days it will be Thursday.
∴ after 44 days it will be (Thursday + 2 days) = Saturday

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

Solution:
Number of letter in word = 6
Repeated letter O = 2, and rest of the letters are unique.

∴ The number of arrangement = 6!/2! = 720/2 = 360

Question: tan3A = 1/√3, then find the value of A = ?

Solution:
Given that,
tan3A = 1/√3
⇒ tan3A = tan30°
⇒ 3A = 30°
⇒ A = 30°/3
∴ A = 10°

Question: An accurate clock shows 4 : 30 PM. Through how many degrees will the hour hand rotate when the clock shows 9 : 30 PM?

Solution:
From 4 : 30 pm to 10 : 30 pm,
total time = 9 : 30 - 4 : 30 = 5 hours

We know,
Angle traced by hour hand in 12 hours = 360°

∴ Angle traced by hour hand in 5 hours,
= (360 × 5)/12
= 150°

Question: There are 8 true-false questions in an examination, these questions can be answered in-

Solution:
Total number of question = 8
Each question has 2 answer.

These question can be answered in 2⁸ ways = 256 ways

Question: It was Friday on January 1, 2016. What was the day of the week on January 1, 2017?

Solution: 
The year 2016 is a leap year. So, it has 2 odd days.
Given,
1st day of the year 2016 is Friday
So, 1st day of the year 2017 is 2 days beyond Friday.
Friday + 1 day = Saturday
Saturday + 1 day = Sunday
Hence, it will be Sunday.

Question: In how many ways can a committee of 5 people be chosen out of 10 people?

Solution:
Total number of people, n = 10
Number of committee members, r = 5

The numbers of ways of chosen committee = ⁿC_r =
¹⁰C₅ = 10!/5!(10 - 5)!
= (10 × 9 × 8 × 7 × 6 × 5!)/(5 × 4 × 3 × 2 × 1)5!
= 252

Question: cos(θ + 16) = 1/2, then the value of θ?

Solution:
Given that,
⇒ cos(θ + 16) = 1/2
⇒ cos(θ + 16) = cos60°
⇒ θ + 16 = 60°
⇒ θ = (60 - 16)°
∴ θ = 44°

Question: A committee of 4 members is to be formed by selecting out of 6 man and 5 women. In how many different ways the committee can be formed if it should have 2 men and 2 women?

Solution:
2 men can be selected out of 6 men in ⁶C₂ = 6!/2!(6 - 2)! = 15 ways
And
2 women can be selected out of 5 women in ⁵C₂ = 5!/2!(5 - 2)! = 10 ways

∴ Required number of ways = 15 × 10 = 150

∴ 150 different ways to form the committee with 2 men and 2 women.

Question: What is the minimum value of 4sin²θ + 5cos²θ is-

Solution:
Let,
⇒ x = 4sin²θ + 5cos²θ
⇒ x = 4sin²θ + 4cos²θ + cos²θ
⇒ x = 4(sin²θ + cos²θ) + cos²θ
⇒ x = 4 + cos²θ     [sin²θ + cos²θ = 1]
⇒ x = 4 + 0
∴ x = 4

The minimum value of x depends on the minimum value of cos²θ
Since the minimum value of cos²θ is 0, So the minimum value of x is 4.

Question: The angle of elevation of a ladder leaning against a wall is 60° and the foot of the ladder is 8.5 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 = 8.5m

We know,
⇒ cos∠ACB = AC/BC 
⇒ cos60° = AC/BC 
⇒ AC/BC = 1/2
⇒ BC = 2AC
⇒ BC = 2 × 8.5
∴ BC = 17 m

∴ The length of the ladder is 17 m

Question: In your bookshelf, you have five favorite books. If you decide to arrange these five books in every possible combination and moved just one book in every half a minute. How much time it will take you to arrange?

Solution:
5 books can be arranged in = 5! ways
= 5 × 4 × 3 × 2 × 1 = 120 ways

Now,
Time per arrangement 1 move = 1/2 minute

So, total time required is = 120 × (1/2) = 60 minute = 1 hour

Question: If θ be an acute angle and 7sin²θ + 3cos²θ = 4, then the value of tanθ is?

solution:
7sin²θ + 3cos²θ = 4
⇒7sin²θ + 3(1 - sin²θ) = 4    [cos²θ = 1 - sin²θ]
⇒7sin²θ + 3 - 3sin²θ = 4
⇒ 4sin²θ = 1
⇒ sin²θ = 1/4
⇒ sinθ = 1/2
⇒ sinθ = sin30°
∴ θ = 30°

Now,
tanθ
= tan30^∘
= 1/√3

Question: An observer 2.8 m tall is 14√3 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 = AB
Now,
tan∠c = AE/CE
⇒ tan30° = AE/14√3
⇒ 1/√3 = AE/14√3
∴ AE = 14

∴ AB = AE + BE = 14 + 2.8 = 16.8m

Question: If 15 people attend a conference and each person shakes hands with every other person exactly once, find the total number of handshakes.

Solution:
Total handshakes = ¹⁵C₂ = 15!/2!(15 - 2)!
= (15 × 14 × 13!)/(2 × 13!)
= 105

Thus, the total number of handshakes is 105.

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