ব্যাংক নিয়োগ প্রস্তুতি ⎯ লং কোর্স

Question: A table fan is quoted for Tk. 1500. Saadman pays Tk. 1173 for it. If he gets a series of two discounts and the rate of the first discount is 15%, then the rate of the second discount is?

Solution: 
After first discount = 1500 - 1500 × 15% 
= 1500 - 1500 × 15/100 
= 1500 -225
= 1275 taka

let second discount is x%

1275 - 1275 × x/100 = 1173 
⇒ 1275x/100 = 1275 - 1173 = 102
⇒ x = 102 × 100/1275
= 8

Question: A father said to his son, 'I was as old as you are at the present at the time of your birth'. If the father's age is 48 years now, the son's age three years back was-

Solution:
let, at present son is x years old
so, at time of his birth the father was x years old

∴ at present the age of father is = x + x year
= 2x year

2x = 48
⇒ x = 48/2
= 24 year

∴ the son's age five years back was = 24 - 3
= 21 years

Question: A motorboat in still water travels at speed of 26 kmph. It goes 42 km upstream in 1 hour 45 minutes. The time taken by it to cover the same distance down the stream will be:

Solution:
মোটরযানের বেগ ২৬ কিমি/ঘণ্টা 

স্রোতের বিপরীতে ১ঘণ্টা ৪৫ মিনিট বা ৭/৪ ঘণ্টায় যায় ৪২ কিমি 
১ ঘণ্টায় যায়  ৪২/৭/৪ কিমি
= ২৪ কিমি 
স্রোতের বেগ = ২৬ - ২৪ কিমি/ঘণ্টা 
= ২ কিমি/ঘণ্টা 

স্রোতের অনুকূলে বেগ = ২৬ +২
= ২৮ কিমি/ঘণ্টা 

৪২ কিমি যেতে সময় লাগে ৪২/২৮
= ৩/২
= ১.৫ ঘণ্টা
= ১ ঘণ্টা ৩০ মিনিট 

Question: If p is an even integer and q is an odd integer, which of the following must be an odd integer?

Solution: 
2 × even integer = even integer
So, 2p is an even integer

even integer + odd integer = odd integer

So,
2p + q must be an odd integer.

Question:
Solution: 

Question: The difference between the numerator and the denominator of a fraction is 5. If 5 is added to the denominator the fraction is decreased by 5/4 then the value of the fraction will be equal to:

Solution:
ধরি,
ভগ্নাংশটির হর = x
ভগ্নাংশটির লব = x + 5

∴ ভগ্নাংশটি = (x + 5)/x

প্রশ্নমতে,
{(x + 5)/x} - {(x + 5)/(x + 5)} = 5/4
⇒ {(x + 5)/x} - 1 = 5/4
⇒ {(x + 5)/x} = (5/4) + 1
∴ {(x + 5)/x} = 9/4

∴ ভগ্নাংশটি = 9/4

Question: There are three dice each of them having faces with a number from 1 to 6. These dices are rolled. Find the number of possible outcomes such that at least one of the dice shows the number 3.

Solution: 
মোট উপায় = ৬ × ৬ × ৬ 
= ২১৬ 

কোন ছক্কায় ৩ না আসার উপায় = ৫ × ৫ × ৫
= ১২৫ 

কমপক্ষে একটি ৩ আসার উপায় = ২১৬ - ১২৫ 
= ৯১

Question: All possible three-digit numbers are formed by 1, 3, 5. If one number is chosen randomly, the probability that it would be divisible by 5 is – 

Solution: 
1, 3, 5 এই তিনটি অংক দ্বারা 3! = 6 উপায়ে সংখ্যা গঠন করা যায়
5 দ্বারা বিভাজ্য হতে হলে শেষের অংক 5 রেখে সংখ্যার প্রথম দুটি স্থান বাকি দুইটি অংক দিয়ে 2! = 2 উপায়ে গঠন করা যায়।
∴ সংখ্যাটি 5 দ্বারা বিভাজ্য হবার সম্ভাবনা = 2/6 = 1/3

Question: If 10^a = 1/2, then 10^{- 6a}?

Solution: 
10^a = 1/2
⇒ (10^a)^-6 = (1/2)^{- 6}
⇒ 10^{- 6a} = 2⁶
∴ 10^{- 6a} = 64

Question: The ratio of two numbers is 2 : 3 and their sum is 250. The smaller one of the numbers is- 

Solution:
let, the numbers be  2x, 3x

2x + 3x = 250
⇒ 5x = 250 
⇒ x = 250/5
∴ x = 50 

numbers are 100, 150 

Question: The average of the first five multiples of 11 is-

Solution: 
The  first five multiples of 11: (11 × 1), (11 × 2), (11 × 3), (11 × 4), (11 × 5)
their sum = (11 × 1) + (11 × 2) + (11 × 3) + (11 × 4) + (11 × 5)
= 11 (1 + 2 + 3 + 4 + 5)
= 11 × 15

∴ average = (11 × 15)/5
= 33

Question: If 2x - 1 ≥ - 3, then- 

Solution:  
2x - 1 ≥ - 3
⇒ 2x ≥ - 3 + 1
⇒ 2x ≥ - 2 
⇒ x ≥ - 1

Question: If PALE is coded at 2134, EARTH is coded as 41590, then how can PEARL be coded in that language?

Solution: 
P → 2  
A → 1
L → 3
E → 4

এবং
E → 4
A → 1
R → 5
T → 9
H → 0 

 PEARL  → 24153

Question: If three unbiased coins are tossed simultaneously, then the probability of exactly two heads is

Solution: 
n(S) = 2³ = 8 
Let E = Event of getting exactly two heads,
= {(H, H, T), (H, T, H), (T, H, H)}
= n(E)
= 3
Required probability = 3/8

Question: M is a girl and has the same number of brothers as sisters. N is a boy and has twice as many sisters as brothers. M and N are the children of Q. How many children does Q have?

Solution: 
Let d and s represent the number of daughters and sons respectively.
Then, we have :
d - 1 = s
and
2(s - 1) = d
2s - 2 = s + 1
2s - s = 1 + 2
s = 3

d = 3 + 1 = 4 

Total children = 4 + 3 = 7 

প্রশ্ন: If A = {1, 2, 3, 4, 5}, then the number of proper subsets of A is-

সমাধান:
এখানে A সেটের উপাদান সংখ্যা ৫টি
A সেটের উপসেট সংখ্যা হবে ২৫ টি = ৩২টি 

∴ A সেটের প্রকৃত উপসেট এর  সংখ্যা (৩২ - ১) টি = ৩১টি

Question: The wheel of an engine of 300 cm in circumference makes 10 revolutions in 6 seconds. What is the speed of the wheel (in km/h)?

Solution:
total distance = 10 × 300 cm 
= 3000 cm 
= 30 m

speed = 30/6 ms^-1 
= 5 ms^-1
= (5 × 3600)/(1000) kmh^-1
= 18 kmh^-1 

Question: If m² + (1/m²) = 47, what is the value of m + (1/m)? 

Solution: 
m² + (1/m²) = 47
⇒ (m + 1/m)² - 2 . m. (1/m) = 47 
⇒ (m + 1/m)² - 2 = 47 
⇒ (m + 1/m)² = 47 + 2 = 49 
∴ (m + 1/m) = √49 = 7