ব্যাংক নিয়োগ বিষয়ভিত্তিক প্রস্তুতি

Question: 5, 8, 14, 26, 50, 98, ..... What number should come next?

Solution: Given series: 5, 8, 14, 26, 50, 98...

First number: 5
Now, 
5 + 3 = 8
8 + 6 = 14
14 + 12 = 26
26 + 24 = 50
50 + 48 = 98

∴ Next number = 98 + 96 = 194

Question: In a 90-liter mixture of milk and water, the ratio of milk to water is 2 : 1. How many liters of water must be added to make the ratio become 1 : 2?

Solution:
Total mixture = 90 litres
Given ratio (milk : water) = 2 : 1

Milk = 90 × (2/3) = 60 litres
Water = 90 − 60 = 30 litres

To make the ratio 1 : 2, x liters of water need to be added.
milk : water = 60 : (30 + x)

So,
60 / (30 + x) = 1 / 2

Cross-multiplying,
2 × 60 = 30 + x
120 = 30 + x
x = 120 − 30
∴ x = 90

Quantity of water to be added = 90 litres.

Question: If a³ - b³ = 117 and a - b = 3 What is the value of ab?

Solution:
Given,
a³ - b³ = 117
a - b = 3

We know,
⇒ (a - b)³ + 3ab(a - b) = a³ - b³
⇒ 27 + 3ab(3) = 117
⇒ 27 + 9ab = 117
⇒ 9ab = 117 - 27
⇒ 9ab = 90
⇒ ab = 90/9
∴ ab = 10

Question: The clock showed 10 : 30 in the mirror. What is the actual time?

Solution:
আমরা জানি,
প্রকৃতপক্ষে সময়
= 11 : 60 - আয়নায় দেখা সময়
= 11 : 60 - 10 : 30
= 1 : 30

Question: Silver is 12 times as heavy as water, and lead is 8 times as heavy as water. In what ratio should these be mixed to get an alloy 9 times as heavy as water?

Solution:
Let silver be 12x times as heavy and lead 8y times as heavy as water.

12x + 8y = 9(x + y)
⇒ 12x + 8y = 9x + 9y
⇒ 12x - 9x = 9y - 8y
⇒ 3x = y
∴ x/y = 1/3 = 1 : 3

Question: If one root of x² - (q + 2)x + 15 = 0 is 5, then the value of q is:

Solution:
Given equation: x² - (q + 2)x + 15 = 0

One root is x = 3. Substitute:
(5)² - (q + 2)(5) + 15 = 0
⇒ 25 - 5(q + 2) + 15 = 0
⇒ 25 - 5q - 10 + 15 = 0
⇒ 30 - 5q = 0
⇒ 5q = 30
∴ q = 6 

Question: If x² + yz + zx + xy is divided by x + z, the result is-

Solution:
x² + yz + zx + xy
= x² + xy + zx + yz
= x(x + y) + z(x + y)
= (x + y)(x + z)

∴ divided by (x + z) the result = (x + y)(x + z)/(x +z) = (x + y)

Question: Today is Friday. What day will it be after 352 days?

Solution:
এখানে,
৩৫২ কে ৭ দিয়ে ভাগ করলে ভাগশেষ থাকে ২।
৩৫২ দিন থেকে ২ দিন বাদ দিলে হয় ৩৫০ দিন।

আমরা জানি,
যে কোনো তারিখ হতে ৭ দিন পর পর (৮ম দিনে) একই বার পাওয়া যায়।
অর্থ্যাৎ, শুক্রবারের ৭ দিন পর বা ৮ম দিনে গিয়ে আবার শুক্রবার পাওয়া যাবে।

অর্থাৎ,
৩৫১ তম দিন শুক্রবার।
৩৫২ তম দিন শনিবার 
৩৫৩ তম দিন রবিবার 

Question: If the ratio of syrup and water in a mixture is 7 : 3, then the percentage of water in mixture is-

Solution: 
The ratio of syrup and water in a mixture is 7 : 3
Total = 10
∴ percentage of water = (3 × 100)/10 %
= 30%

Question: If (x + 3)² = 64, which of the following can be the value of (x + 2)?

Solution:
Given, (x + 3)² = 64
⇒ (x + 3)² = 8²
∴ x + 3 = ± 8

Case 1: x + 3 = 8
x = 8 - 3 = 5
x + 2 = 5 + 2 = 7

Case 2: x + 3 = - 8
x = - 8 - 3 = - 11
x + 2 = - 11 + 2 = - 9

Possible values of (x + 2) are 7 or - 9.

Question: What should be the value of "P" so that the expression (9 − 24x + Px²) becomes a perfect square?

Solution:
(9 − 24x + Px²)
= (3)² − 2 × 3 × 4x + (4x)² + Px² - (4x)²
= (3 - 4x)² + Px² - 16x²

∴ the expression becomes a perfect square if,
Px² - 16x² = 0
⇒ Px² = 16x²
∴ P = 16

Question: Find the product of two consecutive numbers if three times the first number is 8 more than twice the second number.

Solution:
Let the numbers be a and a + 1.

According to the question:
3 × (first number) = 2 × (second number) + 8
⇒ 3a = 2(a + 1) + 8
⇒ 3a = 2a + 2 + 8
⇒ 3a = 2a + 10
⇒ 3a - 2a = 10
⇒ a = 10

∴ The numbers are 10 and 11.
Product = 10 × 11 = 110

Question: If a + 1/a = √3, then what is the value of a³⁶ + a³⁰ + a⁶ + 2?
 
Solution:
Given, a + 1/a = √3
Now,
a³ + 1/a³ = (a + 1/a)³ - 3a × (1/a)(a + 1/a)
⇒ a³ + 1/a³ = (√3)³ - 3(√3) [∵ a + 1/a = √3]
⇒ a³ + 1/a³ = 3(√3) - 3(√3)
⇒ a³ + 1/a³ = 0 
⇒ a⁶ + 1 = 0 [Multiplying both sides by a³]

Then,
a³⁶ + a³⁰ + a⁶ + 2
= a³⁶ (a⁶ + 1) + (a⁶ + 1) + 1
= (a³⁶ × 0) + 0 + 1
= 0 + 1
= 1

Question: If dividing P(x) = 4x³ - 7x² + bx - 5 by (x - 2) results in the remainder 13, then find the value of b.

Solution:
According to the Remainder Theorem, if a polynomial P(x) is divided by (x - c), then the remainder = P(c).
Here divisor is (x - 2).
So remainder = P(2).

Now,
P(2) = 4(2)³ - 7(2)² + b(2) - 5

= 4 × 8 - 7 × 4 + 2b - 5

= 32 - 28 + 2b - 5

= 32 + 2b - 33
= 2b - 1

According to the question, the remainder is 10.
So, 2b - 1 = 13

⇒ 2b = 13 + 1
⇒ 2b = 14
⇒ b = 7

Question: If x² - 5x + 1 = 0, and x > 1, then what is the value of x - (1/x)?

Solution:
We are given:
x² - 5x + 1 = 0
⇒ x - 5 + 1/x = 0
∴ x + 1/x = 5

Now,
(a - b)² = (a + b)² - 4ab
⇒ (x - 1/x)² = (x + 1/x)² - 4 × x × (1/x) [Here, a = x, b = 1/x]

Substitute the values:
(x - 1/x)² = 25 - 4
⇒ (x - 1/x)² = 21
⇒ (x - 1/x)² = 21
∴ x - 1/x = √21

Question: In what ratio must rice at Tk 8 per kg be mixed with rice at Tk 14 per kg so that the mixture be worth Tk 10 per kg?

Solution:
Let x kg of rice at Tk 8 and y kg of rice at Tk 14 be mixed.

⇒ 8x + 14y = 10(x + y)
⇒ 8x + 14y = 10x + 10y
⇒ 10x - 8x = 14y - 10y
⇒ 2x = 4y
⇒ x/y = 4 : 2 = 2 : 1

The question has been canceled because there is no correct answer in the option.

Question: Five times a whole number is equal to three less than twice the square of the number. Find the number?

Solution: Let the required whole number be x.

According to the question,
5x = 2x² – 3
⇒ 2x² – 5x – 3 = 0
⇒ 2x² - 6x + x - 3 = 0
⇒ 2x(x - 3) + 1(x - 3)
⇒ (2x + 1)(x – 3) = 0

So,
2x + 1 = 0 or x – 3 = 0
⇒ x = –1/2 or x = 3

Since x must be a whole number,
∴ the required number is 3.

Question: If x = 5 + √3 and y = 5 - √3, find the value of (x² + y²)²

Solution:
We are given:
x = 5 + √3, y = 5 - √3

Now,
x + y
= (5 + √3) + (5 - √3)
= 10

And,
⇒ xy
= (5 + √3)(5 - √3)
= 5² - (√3)²
= 25 - 3
= 22

We know,
x² + y² = (x + y)² - 2xy
⇒ x² + y² = (10)² - 2(22) [Substitute the values]
⇒ x² + y² = 100 - 44
⇒ x² + y² = 56

∴ (x² + y²)² = 56² = 3136

Question: If p and q are the roots of the equation 2x² − 9x + 7 = 0, then what is the value of (1/p) + (1/q)?

 
Solution:
Given equation:
2x² − 9x + 7 = 0
⇒ 2x² − 7x − 2x + 7 = 0
⇒ x(2x − 7) − 1(2x − 7) = 0
⇒ (x − 1)(2x − 7) = 0

So the roots are:
x = 1 = p
x = 7/2 = q

Now,
1/p + 1/q
= 1/1 + 1/(7/2)
= 1 + 2/7
= 9/7

Question: If x + (2/x) = 4, what is the value of x³ + (8/x³)?

Solution: 
Here, x + (2/x) = 4

Now, 
x³ + (8/x³)
= (x)³ + (2/x)³
= {(x + (2/x)}³ - 3 . x . 2/x {x + (2/x)}
= 4³ - 3 . 2 . 4
= 64 - 24
= 40

Question: In a class, 25 students play football, 15 students play cricket, and 5 students play both. 10 students play neither football nor cricket. What is the total number of students in the class?

Solution:
Let the number of students who play football = 25
Number of students who play cricket = 15
Number of students who play both football and cricket = 5
Number of students who play neither = 10

Number of students who play football or cricket:
n(F ∪ C) = n(F) + n(C) − n(F ∩ C)
n(F ∪ C) = 25 + 15 − 5 = 35

Add the students who play neither to get the total number of students:
Total students = n(F ∪ C) + neither = 35 + 10 = 45

Question: The sum of the squares of three numbers is 120, and the sum of their products taken two at a time is 140. What is the sum of the three numbers?

Solution:
Let the numbers be a, b, c.
Given:
a2+ b2 + c2 = 120, 
ab + bc + ca = 140,
⇒ (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca) = 120 + 2 × 140 = 120 + 280 = 400

∴ a + b + c = √400 = 20

Question: If 1 < p < 4 and 2 < q < 6, which of the following best describes p - q?

Solution: দেয়া আছে:
1 < p < 4 -------------(1)
2 < q < 6 -------------(2)

এখন, আমরা p - q এর সীমা বের করতে চাই।

(2)⇒
2 < q < 6
⇒ - 2 > -q > - 6 (যদি -1 দ্বারা গুণ করি)
⇒ - 6 < -q < - 2 ----------(3)

(1) এবং (3) যোগ করি,
⇒ (1 + (- 6 )) < p - q < (4 + (- 2))
⇒ - 5 < p - q < 2

Question: Find the missing number:

Solution: 
We can see that,
The elements of the 3rd row are 8, 6, and a missing element.
8 is equal to the summation of 2 and 6.
6 is equal to the summation of 1 and 5.

∴ So the rest of the missing element is the summation of 3 and 4, which is 7.

Question: A chemist has two solutions, one containing 40% acid and the other containing 80% acid. How many liters of each solution should be mixed to get 12 liters of a solution containing 60% acid?

Solution:
Let x liters of 40% solution be used. Then (12 - x) liters of 80% solution will be used.

According to the problem:

40% × x + 80% × (12 - x) = 60% × 12
⇒ 40x + 80(12 - x) = 720
⇒ 40x + 960 - 80x = 720
⇒ -40x + 960 = 720
⇒ -40x = -240
⇒ x = 6

∴ 6 liters of 40% solution and 6 liters of 80% solution are needed.