ব্যাংক ডেইলি কুইজ [লং কোর্সের অংশ]

Question: Add 2√2 + 5√3 and √2 - 3√3.

Solution:
(2√2 + 5√3) + (√2 - 3√3)
= 2√2 + √2 + 5√3 - 3√3
= (2 + 1)√2 + (5 - 3)√3
= 3√2 + 2√3

Question: 36 - [18 - {14 - (15 - 4 ÷ 2 × 2)}]. Simplify the expression.

Solution:
36 - [18 - {14 - (15 - 4 ÷ 2 × 2)}]
= 36 - [18 - {14 - (15 - 2 × 2)}]
= 36 - [18 - {14 - (15 - 4)}]
= 36 - [18 - {14 - 11}]
= 36 - [18 - 3]
= 36 - 15
= 21

Question: When an integer m is divided by 6, the remainder is 4. What is the remainder when 7m is divided by 3?

Solution:
m = 6 × quotient + remainder
⇒ m = 6n + 4    [Let, quotient = n]
⇒ 7m = 7 × 6n + 7 × 4
⇒ 7m = 7 × 6n + 28

since remainder can not be greater than or equal to divisor
= 6 ×(7n) + 27 + 1
= 3(14n + 9) + 1

∴ Remainder is 1

Question: Find the number of prime factors in expression (35)⁶ × (38)⁴?

Solution:
35 = 5 × 7
∴ 35⁶ = 5⁶ × 7⁶ 

38 = 2 × 19
∴ 38⁴ = 2⁴ × 19⁴ 

∴ Total number of prime factors in expression:  6 + 6 + 4 + 4 = 20

Question: When x is added to each of 8, 14, 20 and 30 the numbers obtained are in proportion. What is the mean proportional between the numbers (4x - 4) and (8x + 1)?

Solution:
If x is added to each of 8, 14, 20 and 30 then, the number so obtained in this form a proportion
According to the question
(8 + x) : (14 + x) : : (20 + x) : (30 + x)
⇒ (8 + x)/(14 + x) = (20 + x)/(30 + x)
Using Componendo and Dividendo Rule
⇒ (8 + x + 14 + x)/(8 + x - 14 - x) = (20 + x + 30 + x)/(20 + x - 30 - x)
⇒ (22 + 2x)/(- 6) = (50 + 2x)/(- 10)
⇒ (11 + x)/3 = (25 + x)/5
⇒ 5(11 + x) = 3(25 + x)
⇒ 55 + 5x = 75 + 3x
⇒ 5x - 3x = 75 - 55
⇒ 2x = 20
∴ x = 20/2 = 10

As we know,  
Mean proportion of two numbers a and b = √(a × b).

Mean proportion:
√{(40 - 4).(80 + 1)}
= √(36 × 81)
= 6 × 9
= 54

Question: A whole number is added to 100 and the same number is subtracted from 100. The sum of the resulting numbers is-

Solution:
Let the whole number be x.

∴ (x + 100) + (100 - x)
= x + 100 + 100 - x
= 200

Question: 8 + 4 ÷ 2 × 5 =? 

Solution:
8 + 4 ÷ 2 × 5 
= 8 + 2 × 5
= 8 + 10
= 18.

Question: In number 235749, what is the face value of the numeral 5?

Solution:
: In number 235749, the face value of 5 is 5.

Question: The largest 5 digit number exactly divisible by 91 is-

Solution:
Largest 5 digit number = 99999

99999 ÷ 91 (Quotient = 1098; Remainder = 81)

∴ Required number =(99999 - 81)
= 99918

Question: A number in which one-sixth part is increased by 30 is equal to one-eighth part is decreased by 100. Find the number.

Solution:
Let the number be x. 

ATQ,
x/6 + 30 = 100 - x/8
⇒ x/6 + x/8 = 70
⇒ 7x/24 = 70
∴ x = 240

The number is 240. 

Question: Find the sum of 2 + 4 + 6 + …………. + 38?

Solution:
2 + 4 + 6 + …………. + 38
= 2(1 + 2 + 3 + …………. + 19)
Sum of first 19 natural numbers = (19 × 20)/2
= 190

∴ 2 + 4 + 6 + …………. + 38 = 2 × 190 = 380

Question: In 87659_21 what is the least number which can be filled in blank so that the number is divisible by 11.

Solution:
Divisibility rule of 11:
If the sum of digits at odd and even places are equal or differ by a number divisible by 11, then the number is also divisible by 11.
Let the number of blank place be x 
Now,
(8 + 6 + 9 + 2) - (7 + 5 + x + 1) = 0 or 11
⇒ 25 - 13 - x = 0 or 11 
⇒ 12 - x = 0 or 11 

12 - x = 0
∴ x = 12 

Or
12 - x = 11
⇒ x = 12 - 11 
∴ x = 1 

Question: 45 - [38 - {60 ÷ 3 - (6 - 9 ÷ 3) ÷ 3}]. Simplify the expression.

Solution:
45 - [38 - {60 ÷ 3 - (6 - 9 ÷ 3) ÷ 3}]
= 45 - [38 - {60 ÷ 3 - 3 ÷ 3}]
= 45 - [38 - {20 - 1}]
= 45 - [38 - 19]
= 45 - 19
= 26.