ব্যাংক নিয়োগ বিষয়ভিত্তিক প্রস্তুতি

Question: If p and q are odd numbers, which of the following is always odd?

Solution:
Let p = 1 and q = 3 (both are odd numbers)

a) p + q + 2 = 1 + 3 + 2 = 6 ............. Even

b) pq + 2 = (1 × 3) + 2 = 5 ......... Odd

c) 2p + q + 1 = (2 × 1) + 3 + 1 = 2 + 4 = 6 ......... Even

d) p² + q = (1)² + 3 = 1 + 3 = 4 .......... Even

Question: What is the least number which when tripled is exactly divisible by 10, 12, 15, and 18?

Solution:
Let the number be x.
tripled the number is 3x.

10 = 2 × 5
12 = 2 × 2 × 3
15 = 3 × 5
18 = 2 × 3 × 3

∴ LCM = 2 × 2 × 3 × 3 × 5
= 180

∴ x = 180/3 = 60

Question: Average age of 15 students of a class is 15 years. Out of these the average age of 5 students is 14 years and that of the other 9 students is 16 years. The age of the 15th student is-

Solution:
Given that
Average age of 15 students = 15 years
Average age of 5 students = 14 years
Average age of 9 students = 16 years

Now,
Total age of all 15 students = 15 × 15 = 225 years
Total age of 5 students = 5 × 14 = 70 years
Total age of 9 students = 9 × 16 = 144 years 

∴ Age of the 15th student = 225 - (70 + 144) = 225 - 214 = 11 years

Question: Sachin is younger than Rohan by four years. If their ages are in the respective ratio of 7 : 9, how old is Sachin?

Solution:
Given that,
Sachin is 4 years younger than Rohan
Their ages are in the ratio 7 : 9

Let their ages are,
Sachin = 7x
Rohan = 9x

ATQ,
9x - 7x = 4
⇒ 2x = 4
∴ x = 2

Sachin age is = 7x = 7 × 2 = 14​ years

Question: If P and Q together can complete a piece of work in 16 days and P alone in 24 days, in how many days can Q alone complete the work?

Solution: 
P and Q complete a work in = 16 days
One day's work of (P + Q) = 1/16

P complete the work in = 24 days;
One day's work of P = 1/24

Then, Q's one day's work = (1/16) - (1/24)
= (3 - 2)/48
= 1/48

So, Q alone can complete the work in = 48 days.

Question: A motorboat, whose speed in 18 km/h in still water goes 36 km downstream and comes back in a total of 4 hours 30 minutes. The speed of the stream is-

Solution:
Let, the speed of the stream = x km/h
Then,
Speed downstream = (18 + x) km/h
Speed upstream = (18 - x) km/h

ATQ,
36/(18 + x) + 36/(18 - x) = 4.5
⇒ 36{(18 + x) + (18 - x)}/(18 + x)(18 - x) = 4.5
⇒ 1296/(324 - x²) = 9/2
⇒ 9(324 - x²) = 2 × 1296
⇒ 2916 - 2592 = 9x²
⇒ 9x² = 324
⇒ x² = 36 = 6²
∴ x = 6

∴ The speed of the stream is 6 km/h

Question: Two bicycles start at the same time from points P and Q, moving toward each other. If the distance between P and Q is 120 km and their speeds are 12 km/h and 8 km/h respectively then after how much time will they meet each other?

Solution:
Given,
The distance between P and Q = 120 km
Speed of 1^st bicycle = 12 km/h
Speed of 2^nd bicycle = 8 km/h

The bicycles are moving toward each other,
So their relative speed = (Speed of 1^st bicycle + Speed of 2^nd bicycle)
= (12 + 8) km/h
= 20 km/h

Time = (Distance ÷ relative speed)
= (120 ÷ 20)
= 6 h

Question: Two pipes P and Q can fill a cistern in 15 and 20 minutes respectively. Both pipes are opened together, after how many minutes should Q be turned off, so that the cistern be fill in 12 minutes?

Solution:
P can fill the cistern in 15 minutes
So in 1 min P can fill the cistern = 1/15 th part
In 12 min, P can fill the cistern = 12/15
= 4/5 part

Remaining part = 1- (4/5) part
= 1/5 part

As Q can fill full cistern in 20 minutes
So it will fill 1/5 part in = (1/5) × 20 = 4 minutes.

∴ Pipe Q should be turned off after 4 minutes.

Question: Working 5 hours a day, A can complete a work in 9 days and working 9 hours a day, B can complete the same work in 10 days. Working 6 hours a day, they can jointly complete the work in - 

Solution: 
Working 5 hours a day, for 9 days, A can finish the work in = (5 × 9) = 45 hours
Working 9 hours a day, for 10 days, B can finish the work in = (9 × 10) = 90 hours

both together can do in one hour = 1/45 + 1/90
= (2 + 1)/90
= 3/90
= 1/30

so, it will take them 30 hours to do the work.

hence, working 6 hours a day, they need = 30/6 = 5 days

So they will complete the work in 5 days working 6 hours each day.

Question: Which one of the following is a rational number?

Solution:
ক) √7 × √2 = √14 .......... irrational
খ) √5 × √6 = √30 .......... irrational
গ) √5 × √20 = √100 = 10 ......... rational
ঘ) √3 × √8 = √24 ................ irrational

Question: Three numbers are in the ratio 2 : 3 : 5, and their H.C.F. is 15. What is their L.C.M.?

Solution:
দেওয়া আছে,
সংখ্যা তিনটির গ. সা. গু ১৫
সংখ্যা তিনটি হচ্ছে (২ × ১৫) বা ৩০, (৩ × ১৫) বা ৪৫, (৫ × ১৫) বা ৭৫

এখন,
৩০, ৪৫, ৭৫ এর ল. সা. গু = ৪৫০

Question: There are 71 members in group A, 18 members in group B and 53 members in group C. All the members of these groups went to a restaurant. The average amount spent on each member of group A, B and C  is Tk.397, Tk.421 and Tk.137 respectively. The total average amount (in Tk.) spent per member is-

Solution:
Given that,

Members in A = 71, B = 18, C = 53

Average spent per member: A = Tk. 397, B = Tk. 421 and C = Tk. 137

Now,
Total amount by A = 71 × 397 = 28187
Total amount by B = 18 × 421 = 7578
Total amount by C = 53 × 137 = 7261

∴ Total amount = 28187 + 7578 + 7261 = 43026

∴ Total members = 71 + 18 + 53 = 142

∴ Average = 43026/142 = 303

Total average amount spent per member = Tk. 303

Question: After 15 years, Rahul’s age will be five times what it was 5 years before. How old is Rahul now?

Solution:
Given that,
Rahul’s age after 15 years will be five times his age 5 years ago.

Let Rahul’s present age = x years

According to the question:
⇒ x + 15 = 5(x - 5)
⇒ x + 15 = 5x - 25
⇒ 5x - x = 15 + 25
⇒ 4x = 40
∴ x = 10

∴ Rahul is 10 years old now.

Question: A can complete a piece of work in 6 days working 8 hours a day, and B can complete the same work in 12 days working 8 hours a day. How long will it take them to complete the work together if they work 4 hours a day?

Solution:
A can complete the work in 8 × 6 = 48 hours
1 hour's work of A = 1/48 part

B can complete the work in 8 × 12 = 96 hours
1 hour's work of B = 1/96 part

(A + B)'s 1 hour's work = (1/48) + (1/96) part
= (2 + 1)/96 part
= 3/96 part
= 1/32

∴ Time taken by (A + B) working 4 hours daily = 32/(1 × 4)
= 8 days

Question: A man rows downstream at 32 km/h and rows upstream at 22 km/h. At what speed can he row in still water?

Solution:
Given,
Man rows downstream = 32 km/h
Man rows upstream = 22 km/h
We know that,
Speed in still water = (Downstream speed + Upstream speed​) ÷ 2
= (32 + 22) ÷ 2
= 54 ÷ 2
= 27 km/h

Question: Three mechanics A, B, and C can manufacture 120 units in 12, 20, and 30 hours respectively. What is the ratio of the time taken by A alone to complete the work to the time taken by all three working together to complete the same work?

Solution:
(A + B + C) together can do in 1 hour = (1/12) + (1/20) + (1/30)
= (5 + 3 + 2)/60
= 10/60
= 1/6
So, working together they complete the work in 6 hours.

And A alone takes 12 hours.

∴ Ratio of the time taken by A and (A + B + C) =12 : 6 = 2 : 1

Question: A canteen requires 105 kgs of wheat for a week. How many kgs of wheat will it require for 58 days? 

Solution:
For 7 days it is required = 105 kgs
∴ For 1 days it is required = (105/7) kgs
∴ For 58 days it is required = (105 × 58)/7 = 870 kgs

∴ 870 kilograms of wheat will be required for 58 days.

Question: The difference between two numbers is 960. When the larger number is divided by the smaller, the quotient is 5 and the remainder is 12. What is the smaller number?

Solution:
Given that,
The difference of two numbers = 960
Quotient when the larger number is divided by the smaller number = 5
Remainder when the larger number is divided by the smaller number = 12

Now,
Let the smaller number be x.
Larger number = 5x + 12

ATQ,
⇒ 5x + 12 - x = 960
⇒ 4x + 12 = 960
⇒ 4x = 960 - 12
⇒ 4x = 948
⇒ x = 948/4
∴ x = 237

So the smaller number is 237.

Question: The least number by which 108 must be multiplied to make it a perfect square is-

Solution:
একটি সংখ্যা পূর্ণবর্গ সংখ্যা হতে হলে তার মৌলিক গুণনীয়কগুলোকে অবশ্যই জোড় সংখ্যায় (even power) থাকতে হবে।

108 = 2 × 2 × 3 × 3 × 3
= 2² × 3³
জোড়া গঠন করে পাই,(2 × 2) × (3 × 3) × 3
এখানে জোড়া বিহীন সংখ্যা 3

∴ 108 কে 3 দ্বারা গুণ করলে এটি পূর্ণবর্গ সংখ্যা হবে।

Question: The batting average for 27 innings of a cricket player is 47 runs. His highest score in an innings exceeds his lowest score by 157 runs. If these two innings are excluded, the average score of the remaining 25 innings is 42 runs. Find his highest score in an innings.

Solution:
Given that,
The batting average for 27 innings of a cricket player is 47 runs.
His highest score exceeds his lowest score by 157 runs.
If these two innings are excluded, the average of the remaining 25 innings is 42 runs.

Now,
Sum of runs for 27 innings of a cricket player = 47 × 27 = 1269
Sum of runs for 25 innings of a cricket player = 42 × 25 = 1050

∴ Sum of remaining 2 innings = 1269 - 1050 = 219

Let,
The minimum score be x and the maximum score be x + 157

According to the question,
x + x + 157 = 219
⇒ 2x = 219 - 157
⇒ 2x = 62
∴ x = 31

So, highest score = 157 + 31 = 188

Question: Find the H.C.F. of p(x) = 2x³ - 3x² - 2x + 3 and q(x) = 3x² + 8x + 5.

Solution:
Given that,
p(x) = 2x³ - 3x² - 2x + 3 and q(x) = 3x² + 8x + 5

Now,
The factors of p(x) = 2x³ - 3x² - 2x + 3
⇒ x²(2x - 3) - 1(2x - 3)
⇒ (x² - 1)(2x - 3)
⇒ (x + 1)(x - 1)(2x - 3)

And,
The factors of q(x) = 3x² + 8x + 5
⇒ 3x² + 5x + 3x + 5
⇒ x(3x + 5) + 1(3x + 5)
⇒ (3x + 5)(x + 1)

∴ The required H.C.F. is (x + 1).

Question: 3/7 part of the tank is full of water. When 42 liters of water is taken out, the tank becomes empty. The capacity of the tank is -

Solution:
Let us consider,
The tank has 7x liters of total capacity and holds 3x litres of water.
And if 42 liters of water is taken out, then the tank becomes empty.

It means 3x litres of water is taken out.
∴ 3x = 42 liters
⇒ x = 14 liters

∴ Capacity of tank = 7x = 7 × 14 = 98 liters

Question: A 120 meters long train is running at a speed of 108 km per hour. It will cross a railway platform 210 m long in-

Solution:
Here,
Speed of the running train = 108 km/hr
= {108 × (5/18)} m/sec
= 30 m/sec

And length of the train is = 120 metres
Length of platform = 210 m

So, the time will taken by the train = (Length of train + Length of platform)/Speed
= (120 + 210)/30
= 330/30 
= 11 sec

Question: Two cars A and B travel from point P to point Q. Car A starts 1 hour before car B and reaches Q 2 hours after B when travelled at a speed 30 km/hr. If speed of car B is 50 km/hr, then find the distance between point P and point Q.

Solution:
Given that,
Car A starts 1 hour earlier than Car B.
Car A reaches 2 hours later than Car B.
Speed of Car A = 30 km/h
Speed of Car B = 50 km/h

Let time taken by Car B = x hours
Then,
Distance by Car A = 30 × (x + 3)
Distance by Car B = 50x

ATQ,
⇒ 30(x + 3) = 50x
⇒ 50x - 30x = 90
⇒ 20x = 90
∴ x = 90/20 = 4.5 hours

Use Car B's values,
∴ Distance = 50 × 4.5 = 225 km