Bank Math Master

Question:  (2ⁿ)^{2a + 3} = 4⁵ⁿ, a = ?  

Solution :
(2ⁿ)^{2a + 3} = 4⁵ⁿ
⇒ (2ⁿ)^{2a + 3} = 2^2.5n
⇒ (2ⁿ)^{2a + 3} = (2ⁿ)¹⁰
⇒ 2a + 3 = 10
⇒ 2a = 7
⇒ a = 7/2

Question: 4^{2x + 2} × 8^{x - 2} = 512,  x=?

Solution:
4^{2x + 2} × 8^{x - 2} = 512
⇒ 2^{2(2x + 2)} × 2^{3(x - 2)} = 2⁹
⇒ 2^{4x + 4} × 2^{3x - 6} = 2⁹
⇒ 2^{4x + 4 + 3x - 6} = 2⁹
⇒ 4x + 4 + 3x - 6 = 9
⇒ 7x - 2 = 9
⇒ 7x = 11
∴ x = 11/7

Question : log_4√3 48 = x², x³ = ? 

Solution : 
log_4√3 48 = x²
⇒ log_4√3 (4√3)² = x²    
⇒ 2log_4√3 4√3 = x²       [ log_a a = 1] 
⇒ x² = 2
⇒ x = √2
⇒ x³ = (√2)³ 
∴ x = 2√2

Question: log₂√6 + log₂√(2/3) = ?

Solution:
log₂√6 + log₂√(2/3)
= log₂6^(1/2) + log₂(2/3)^(1/2)
= (1/2) log₂6 + (1/2) log₂(2/3)
= (1/2) {log₂6 + log₂(2/3)}
= (1/2) log₂{6 × (2/3)}
= (1/2) log₂ 4
= (1/2) log₂2²
= (1/2) . 2 log₂2
= log₂2
= 1

Question: log₂3 × log₃2 = ?

Solution:
log₂3 × log₃2
= log₂3 × (1/log₂3)
= log₂3 / log₂3
= 1

Question: Rupa got married 6 years ago. Today her age is 5/4 times her age at the time of marriage. Her son's age is (1/10) times her age. Her son's age is-

Solution:
Let
the age of Rupa before 6 years ago is x.
Then today her age is x + 6 

Now
x + 6 = 5x/4
4x + 24 = 5x
5x - 4x = 24
x = 24 

Now age of  Rupa = 24 + 6 =30 years
Her son's age is = 30/10 ears
= 3 years

Question: If the mean of a, b, c is P and ab + bc + ca = 0, then the mean of a², b², c² is- 

Solution :
Given, 
ab + bc + ca = 0 
and, the mean of a, b, c is P 
⇒ (a + b + c)/3 = P 
⇒ (a + b + c) = 3P ..............(1) 

Now,
(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
⇒ (3P)² = a² + b² + c² + 2 × 0
⇒ 9P² = a² + b² + c²... 
 
∴ Mean = (a² + b² + c²)/3
⇒ (9P²)/3 
⇒ 3P² 

Question: The average age of A, B, C, D and E is 25 years. The average age of A and B is 28 years and the average of C and D is 32 years. Age of E is :

Solution :
The average age of A, B, C, D and E is = 25 years.
The total age of A, B, C, D and E is = (25 × 5) = 125 years.

The average age of A and B is = 28 years
The total age of A and B is = (28 × 2) = 56 years

The average of C and D is = 32 years
The total age of C and D is = (32 × 2) = 64 years

∴ Age of E is = (125 - 56 - 64) = 5 years

Question: The average of the reciprocals of 4 and 5 is: 

Solution:
The reciprocals of 4 and 4 are = 1/4 and 1/5

∴ Required average = {(1/4) + (1/5)}/2
= {(5 + 4)/20}/2
= {9/20}/2
= 9/40

Question: The ratio of the present ages of P and Q is 3 : 4. Five years ago, the ratio of their ages was 5 : 7. Determine how old Q is right now.

Solution:
As the ratio of their present ages is 3 : 4 ,

let,
their present ages be 3X and 4X.
So, 5 years ago, as the ratio of their ages was = 5 : 7,

we can write,
(3x - 5) : (4x - 5) = 5 : 7
⇒ (3x - 5)/(4x - 5) = 5/7
⇒ 21x - 35 = 20x - 25
⇒ x = 10

∴ their present age of Q = 4X = 40

Question: The sum of the first four prime numbers is –

Solution :
the first four prime numbers are = 2, 3, 5, 7 

∴ their sum = (2 + 3 + 5 + 7)
= 17

Question: The product of two co-prime numbers is 820. Then their LCM is =? 

Solution: 
If hcf of two or more numbers is 1, then two or more numbers are co-prime numbers. 
HCF of co-prime number is always 1 

∴ Product of number = LCM × HCF
⇒ LCM × 1 = 820
⇒ LCM = 820

Question: When 4 is added to 1/2 of a number, the result is 14. What is the number?

Solution: 
let the number be = x

ATQ,
(x/2) + 4 = 14
⇒ x/2 = 10
∴ x = 20

Question: If a and b are odd numbers. Which number is even?

Solution:
Let
a = 1
b = 3

∴ ab = (1 × 3)
= 3, which is odd.

(a + 2b + 2) = 1 + 2 × 3 + 2
= 1 + 6 + 2 = 9, which is odd.

(a + b + 1) = 1 + 3 + 1
= 5, which is odd.

(2a + 4b) = 2 × 1 + 4 × 3
= 2 + 12
= 14, which is even.

Question:  What is the least square number which is exactly divisible by 2, 3, 10, 18 and 20? 

Solution: 
The least or smallest number which is exactly divisible by 2, 3, 10, 18, and 20 is the LCM of 2, 3, 10, 18 and 20. 

LCM of (2, 3, 10, 18, 20)  = 180
∴ 180 = 2² × 3² × 5

So, to become a perfect square 180 needs to be multiplied by 5 .

Now, the least square number which is exactly divisible by 2, 3, 10, 18 and 20
⇒ (180 × 5) = 900

Question: Which of the following fractions is the largest? 

Solution: 
3/2 = 1.5
7/4 = 1.75
5/3 = 1.66
6/5 = 1.2 
∴ the largest fraction is = 7/4

Question (.1 × .01 × .002)/(.2 × .02 × .002) = ?

Solution: 
(.1 × .01 × .002)/(.2 × .02 × .002)
= 0.000002/0.000008
= 2/8
= 1/4
= 0.25

Problem: 

Solution :

Question: 2 + x√3 = 1/(2 + √3), x = ? 

Solution: 
2 + x√3 = 1/(2 + √3) 
⇒ 2 + x√3 = (2 - √3)/(2 + √3)(2 - √3) 
⇒ 2 + x√3 = (2 - √3)/{22 - (√3)2}
⇒ 2 + x√3 = (2 - √3)/(4 - 3)
⇒ 2 + x√3 = 2 - √3
⇒  x√3 = - √3
∴ x = - 1

Question: A mother is 20 years older than her daughter. In 10 years, her age will be twice the age of her daughter. What is the present age of the daughter?

Solution:
Let the daughter’s present age be x  years.
Then, the mother’s present age is = x + 20 years.

ATQ
In 10 years, the mother’s age will be twice the daughter’s age.
So, (x + 20) + 10 = 2 (x + 10)
⇒ 2x + 20 = x + 30
⇒ 2x - x = 30 - 20
∴ x = 10

Therefore, the present age of the daughter is 10 years.