ব্যাংক নিয়োগ বিষয়ভিত্তিক প্রস্তুতি

Question: The difference between two positive numbers is 6 and the difference of their squares is 132. The smallest number is- 

Solution: 
Let
the numbers be x and (x + 6)

ATQ,
(x + 6)² - x² = 132
⇒ x² + 12x + 36 - x² = 132
⇒ 12x + 36 = 132
⇒ 12x = 96
∴ x = 8

∴ the smallest number is = 8

Question: The ratio of alcohol to water in a chemical solution is 3 : 2. If 4 liters of alcohol are added to the solution, the new ratio of alcohol to water becomes 7 : 4. Find the final amount of alcohol in the new solution.

Solution: 
Let
the initial amount of alcohol be 3x liters and the amount of water 2x liters.

ATQ,
Ratio of alcohol and water after adding 4 liters of alcohol
(3x + 4)/2x = 7/4
⇒ 12x + 16 = 14x
⇒ 2x = 16
∴ x = 8

∴ Final amount of alcohol in solution = 3x + 4 = (3 × 8) + 4 = 28 liters.

Question: Arif started a business with Tk. 60,000. After some months, Nehal joined with Tk. 40,000. At the end of the year, the profit was divided in the ratio 9 : 5. For how many months was Nehal in the business?

Solution: 
Let,
Nehal joined for x months.
 
ATQ,
(60,000 × 12)/(40,000 × x) = 9/5
⇒ x = (60,000 × 12 × 5)/(40,000 × 9)
∴ x = 10

∴ Nehal joined for 10 months.

Question: A watch is sold for Tk. 840, and the seller makes a profit of 20% on the cost price. What will be the new selling price if he reduces the profit to 5%?

Solution:
At 20% profit,
Selling Price = 120% of Cost Price

∴ Cost Price= (100/120) × Selling Price
= (100/120) × 840
= 700 Tk.

Now, if the seller wants 5% profit,
∴ New Selling Price = 700 + (5% of 700)
= 700 + {(5/100) × 700}
= 700 + 35
= 735 Tk.

Question: Ruma is as much older than Jui as she is younger than Lima. If the sum of the ages of Jui and Lima is 50 years, and Jui is 20 years old, then what is the difference between Ruma and Jui's age?

Solution:
Given,
Jui's age is 20 years
the sum of the ages of Jui and Lima is 50 years
∴ The age of Lima is = 50 - 20 year
= 30 years

let,
the age of Ruma is x years

ATQ,
x - 20 = 30 - x
⇒ 2x = 50
∴ x = 25

So, the difference between Ruma and Jui's age is = (25 - 20) years
= 5 years

Question: The average of four numbers is 72. The sum of the second and third numbers is twice the sum of the first and fourth. What is the value of the sum of the first and fourth numbers?

Solution:
Given,
Average of four numbers is 72
∴ The sum of four numbers is = (4 × 72) = 288

Let
the sum of the first and fourth numbers be x
Then the sum of the second and third numbers = 2x

ATQ,
x + 2x = 288
⇒ 3x = 288
∴ x = 96

∴ the value of the sum of the first and fourth numbers = 96

Question: If 12 men or 18 boys can make 90 chairs in 9 days, then how many chairs will be made by 6 men and 9 boys in 10 days?

Solution:
Here, 
12 men = 18 boys
∴ 6 men = (18 ÷ 2) boys
= 9 boys

∴ 6 men and 9 boys = (9 + 9) = 18 boys

Now,
18 boys can make 90 chairs in 9 days
In 1 day, 18 boys can make (90/9) =10 chairs
∴ In 10 day, 18 boys can make = (10 × 10) chairs
= 100 chairs

Question: A bag contains 5 red balls, 7 green balls, and 9 blue balls. One ball is picked at random. What is the probability that the ball is not green?

Solution:
Total number of balls = 5 + 7 + 9
= 21

Number of non-green balls = 5 (red) + 9 (blue) = 14

∴ the probability of not getting a green ball = 14/21
= 2/3

Question: A train 180 meters long takes 9 seconds to pass a man standing on the platform. How much time will it take to pass a platform 420 meters long?

Solution:
Length of the train = 180 m
Time to pass the man = 9 sec

∴ Speed of the train =180/9 m/sec
= 20 m/sec

Now, total length of the train and platform = (180 + 420) m
= 600 m

∴ Time taken = 600/20 sec
= 30 sec

Question: A large tanker can be filled by two pipes A and B in 30 minutes and 20 minutes, respectively. How many minutes will it take to fill the empty tanker if only B is used in the first-half of the time and A and B are both used in the second-half of the time?

Solution:
Let
the total time to fill the tank = x minutes

Part filled by (A + B) in 1 minute = 1/30 + 1/20
= (2 + 3)/60
= 1/12

(A + B) will take half of total time= (x/2)​ × (1/12)
The rest half will be filled by B only in half of total time = (x/2)​ × (1/20)

ATQ,
x/24 + x/40 = 1
⇒ (5x + 3x)/120 = 1
⇒ 8x/120 = 1
⇒ x/15 = 1
∴ x = 15 

Question: If in a certain language, GARDEN is coded as 714526, and SAND is coded as 9165, then how is DANGERS coded in the same language?

Solution:
GARDEN is coded 714526
G = 7
A = 1
R = 4
D = 5
E = 2
N = 6

SAND is coded as 9165
S = 9
A = 1
N = 6
D = 5 

∴ DANGERS is coded as:
D = 5
A = 1
N = 6
G = 7
E = 2
R = 4
S = 9

∴ DANGERS = 5167249

Question: If a circle has a diameter of 6 units, what is its area?

Solution: 
Given,
diameter = 6 units
∴ radius r = 6/2 = 3 units

We know,
area = πr²
=  π × 3²
= 9π square units

Question: What is the greatest number that can be subtracted from 1000 so that the remainder may be divisible by 32, 36, 48, and 54?

Solution:
L.C.M of 32, 36, 48, and 54 is = 864

Now,
1000 ÷ 864 = 1.15 (approx)

So, required number  is = (1000 - 864) = 136

∴ 136 is the greatest number that can be subtracted from 1000 so that the remainder is divisible by 32, 36, 48, and 54.

Question: If log₇14 + log₇(5x + 1) - 1 = log₇(x + 5), then what is the value of x?

Solution: 
log₇14 + log₇(5x + 1) - 1 = log₇(x + 5)
⇒ log₇14 + log₇(5x + 1) - log₇7 = log₇(x + 5)
⇒ log₇[{14 × (5x + 1)}/7] = log₇(x + 5)
⇒ 2(5x + 1) = x + 5
⇒ 10x + 2 = x + 5
⇒ 10x - x = 5 - 2
⇒ 9x = 3
⇒ x = 3/9
∴ x = 1/3

Question: There are 9 non-collinear points. How many triangles can be drawn by joining these points?

Solution:
We know,
A triangle is formed by joining any three non-collinear points in pairs.

Given.
There are 9 non-collinear points. অর্থাৎ নয়টি বিন্দু সমরেখ নয়।

∴ The number of triangles formed = ⁹C₃
= (9 × 8 × 7 × 6!)/(9 - 3)! × 3!
= (9 × 8 × 7 × 6!)/(6! × 3 × 2)
= 84

Question: If √x = √7 - √5 then the value of x² - 24x + 8 = ? 

Solution: 
Given,
√x = √7 - √5 
⇒ x = 7 + 5 - 2.√7.√5 (Squaring both sides)
⇒ x = 12 - 2√35
⇒ x - 12 = - 2√35  
⇒ x² + 144 - 24x = 140 (Squaring both sides)
⇒ x² + 4 - 24x = 0
⇒ x² + 8 - 24x = 4
∴ x² - 24x + 8 = 4

Question: Which man is the tallest?
When
A is smaller than B
B is smaller than C
C is smaller than D
D is taller than E

Solution:
প্রশ্নানুযায়ী, 
B থেকে A ছোট
C থেকে B ছোট
D থেকে C ছোট
কিন্তু, E থেকে D লম্বা।

∴ A, B, C, D ও E এর মধ্যে D সবচেয়ে লম্বা।

Question: If cos(θ - 15°) = √3/2 then, sin²θ = ?

Solution:
Given,
cos(θ - 15°) = √3/2
⇒ cos(θ - 15°) = cos30°
⇒ θ - 15° = 30°
∴ θ = 45°

Now,
sin²θ = (sin 45°)²
= (1/√2)²
= 1/2

Question: A number is tripled, then 7 is subtracted from it. If the result is then doubled, it becomes 58. What is the number?

Solution:
Let,
the number be x

ATQ,
2(3x - 7) = 58
⇒ 6x - 14 = 58
⇒ 6x = 58 + 14
⇒ 6x = 72
⇒ x = 72/6 
∴ x = 12

So the number is 12.

Question: Find the LCM of the fractions 2/3, 5/9, 1/3.

Solution:
Given,
the fractions are 2/3, 5/9 and 1/3

We know,
LCM of the fraction = LCM of numerator/HCF of denominator

LCM of numerators = LCM of (2, 5, 1) = 2 × 5 × 1 = 10
HCF of denominators = HCF of ( 3, 9, 3) = 3

∴ LCM of fraction = 10/3

Question: Liton’s age after 30 years would be equal to 5 times his age 10 years ago. Find his age 6 years hence.

Solution: 
Let
Liton’s present age be ‘x’ years.

ATQ,
x + 30 = 5(x - 10)
⇒ x + 30 = 5x – 50
⇒ 4x = 80
⇒ x = 20

∴  Liton’s present age = 20 years
Therefore, Liton’s age 6 years hence = 20 + 6 = 26 years

Question: A book costs Tk. 750. It is sold after giving three successive discounts of 20%, 15%, and 10%. Find the final price of the book after all discounts.

Solution:
20% discount on 750 = 750 × 20% = Tk. 150

∴ Price after 20% discount = (750 - 150) = Tk. 600

Again,
15% discount on 600 = 600 × 15% = Tk. 90

∴ Price after 15% discount on Tk. 600 = (600 - 90) = Tk. 510

And,
10% discount on 510 = 510 × 10% = Tk. 51

∴ Price after 10% discount on Tk. 510 = (510 - 51) = Tk. 459

∴ the final price of the book after all discounts = Tk. 459

Question: A, B and C play cricket. The ratio of A’s runs to B’s runs is 5 : 3 and B’s runs to C’s is 3 ∶ 2. They made a total of 350 runs. How many runs did "B" make?

Solution:
Given,
A + B + C = 350
and,
A : B = 5 : 3
B : C = 3 : 2

∴ A : B : C = 5 : 3 : 2

∴ Runs made by "B" = 3/10 × 350 = 105

Question: 2/5 of a tank is filled with water. After 36 litres are added, the tank becomes full. What is the total capacity of the tank?

Solution:
Let,
the total capacity of the tank be x litres.

Already filled = (2/5) × x = (2x/5)

ATQ,
After adding 36 litres, the tank is full
∴ x - 2x/5 = 36
⇒ (5x - 2x)/5 = 36
⇒ 3x/5 = 36
⇒ x = (36 × 5)/3
∴ x = 60

∴ the total capacity of the tank be 60 litres.

Question: If '+' means '-', '-' means '×', '×' means '÷', and '÷' means '+', then what is the value of: 16 - 4 × 2 + 6 ÷ 3?

Solution:
Given,
'+' means '-', '-' means '×', '×' means '÷', and '÷' means '+'

Now,
16 - 4 × 2 + 6 ÷ 3

∴ the transformed expression will be: 16 × 4 ÷ 2 - 6 + 3
= 16 × 2 - 6 + 3
= 32 - 6 + 3
= 35 - 6
= 29

Question: A man can row at 6 kmph in still water. If the velocity of the current is 3 kmph and it takes him 12 hours to row to a place and comes back, how far is the place?

Solution:
Given,
man can row in still water = 6 kmph 
the velocity of the current = 3 kmph

Downstream speed = (6 + 3) = 9 kmph
Upstream speed = (6 - 3) = 3 kmph

Let,
the required distance be = x km

ATQ,
(x/9) + (x/3) = 12
⇒ (x + 3x)/9 = 12
⇒ 4x = 108
∴ x = 27

∴ the required distance be = 27 km

Question: Rahim buys a Tk. 50 share of a company that pays 12% profit. He buys the share at such a price that he earns 20% on his investment. At what price did Rahim buy the share?

Solution:
Here
profit given by the company on 1 share = 12% of  50
= Tk. 6

Let
Rahim buys one share for x Tk

ATQ,
20% of x = 6
20x/100 = 6
⇒ 20x = 600
⇒ x = 600/20
∴ x = 30

Rahim bought the share at Tk. 30

Question: Which of the following fractions is the largest?

Solution:
A. 5/6 = 0.83
B. 11/14 = 0.79
C. 12/15 = 0.8
D. 17/21 = 0.81

So, 5/6 = 0.83 is the largest.

Question: Find the value of the polynomial 6x³ - 2x² - x + 3 when x = - 1

Solution:
Given,
x = - 1

We get,
6x³ - 2x² - x + 3 = 6(- 1)³ - 2(- 1)² - (- 1) + 3
= - 6 - 2 + 1 + 3
= - 4

Question: If a and b are natural numbers such that a^b = 144, the value of (a - 1)^{b - 2} is:

Solution: 
Given,
a^b = 144
⇒  a^b = 12²

∴ a = 12, b = 2

Now,
(a - 1)^{b - 2} = (12 - 1)^{2 - 2}
= (11)⁰
= 1

Question: How many different ways can the letters of the word “BALLOON” be arranged?

Solution: 
The given word contains 7 letters, where L and O is taken 2 times.

∴ Required number of ways = 7!/(2! × 2!)
= 5040/4
= 1260

∴ 1260 distinct arrangements

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

Solution:
7sin²θ + 3cos²θ = 4
⇒ 7sin²θ + 3(1 − sin²θ) = 4
⇒ 7sin²θ + 3 − 3sin²θ = 4
⇒ 4sin²θ = 1
⇒ sin²θ = 1/4
⇒ sinθ = 1/2
⇒ sinθ = sin30°
⇒ θ = 30°

∴ tanθ = tan30° = 1/√3

Question: Find the missing number-

Solution:
In 1st figure:
31 + 18 = 49 ⇒ √(49) = 7

In 2nd figure:
67 + 14 = 81 ⇒ √(81) = 9

In 3rd figure:
99 + 70 = 169 ⇒ √(169) = 13