ব্যাংক নিয়োগ বিষয়ভিত্তিক প্রস্তুতি

Question: A trader mixes 26 kg of rice at Tk. 20 per kg with 30 kg of rice of other variety at Tk. 36 per kg and sells the mixture at Tk. 28 per kg. His profit or loss percent is-

Solution: 
C.P. of 56 kg rice = (26 × 20 + 30 × 36)
= (520 + 1080)
=1600

S.P. of 56 kg rice = (56 × 28)
= 1568

∴ Loss = 1600 - 1568
= 32

 ∴ Loss percentage = (32/1600) × 100% = 2%

Question: 8, 11, 17, 29, 53, 101,....... what number should come next?

Solution:
Given series,
8, 11, 17, 29, 53, 101......

8 
8 + 3 = 11
11 + 6 = 17
17 + 12 = 29
29 + 24 = 53
53 + 48 = 101

∴ next number = 101 + 96 = 197

Question: If dividing Q(x) = 4x³ - 3x² + bx - 5 by (x + 1) results the remainder 8 then find the value of b.

Solution:
Dividing $Q(x)$ by x + 1 we will get the remainder
∴  Q(- 1) = 4(- 1)³ - 3(- 1)² + b(- 1) - 5
= - 4 - 3 - b - 5
= - 12 - b

ATQ,
- 12 - b = 8
⇒ b = - 12 - 8
∴ b = - 20

Question: In what ratio should a mixture of 20% alcohol strength be mixed with a mixture of 60% alcohol strength to get a mixture of 50% alcohol strength?

Solution:
Let the ratio be 1 : x
Then according to the question
⇒ 20 × 1 + 60x = (1 + x) 50
⇒ 20 + 60x = 50 + 50x
⇒ 60x - 50x = 50 - 20
⇒ 10x = 30
⇒ x = 3

∴ the ratio is 1 : 3

$Question:$If x = 2 , what is the value of 5x²√(x⁴ - x²) = ​?

Solution:
Given that,
x = 2

Now,
5x²√(x⁴ - x²)
= 5 × (2)² × √(2⁴ - 2²)
= 5 × 4 × √(16 - 4)
= 20 × √(12)
= 20 × √(4 × 3)
= 20 × 2 × √3
= 40√3

Question: If HOUSE = 81521195 and TREE = 201855, then GARDEN will be equal to-

Solution:

According to the English alphabet,
HOUSE = 8 15 21 19 5
And,
TREE = 20 18 5 5

Now,
GARDEN = 7 1 18 4 5 14
So the code is- 71184514

∴ GARDEN = 71184514

Question: The equation 2x² + kx + 8 = 0 has two equal roots, then the value of k is-

Solution:
Here
a = 2, b = k and c = 8
Since the equation has two equal roots
: b² - 4ac = 0
⇒ (k)² - 4 × 2 × 8 = 0
⇒ k² = 64
⇒ k = ± √(64)
k = ± 8

$Question:$If x = a + 1/a ​ and y= a - 1/a ​, find the value of x² + y² + 2xy = ?

Solution:
Given that,
x = a + 1/a ​ and y= a - 1/a

Now,
x + y = a + 1/a + a - 1/a
∴ x + y = 2a

Now, the given expression is,
x² + y² + 2xy
= x² + 2xy + y²
= (x + y)²
= (2a)²
= 4a²

Question: What is the value of,



Solution:

Question: Solve: x⁴ - 5x² + 4 = 0

Solution:
Given,
x⁴ - 5x² + 4 = 0
(x²)² - 5x² + 4 = 0
Let
x² = a

∴ a² - 5a + 4 = 0
⇒ a² - 4a - a + 4 = 0
⇒ a(a - 4) - 1(a - 4) = 0
⇒ (a - 4)(a - 1) = 0

∴ a - 4 = 0
⇒ a = 4
⇒ x² = 4
∴ x = ± 2

and,
a - 1 = 0
⇒ a = 1
⇒ x² = 1
∴ x = ± 1

∴ x = ± 1, ± 2

Question: What is the ratio of one-fourth of 9/20​ and three-fourths of the same number?

Solution:
one-fourth of 9/20 is {(1/4) × (9/20)} = 9/80
three-fourth of 9/20 is {(3/4) × (9/20)} = 27/80

∴ ratio = 9/80 : 27/80 = 9 : 27 = 1 : 3

Question: Four girls are sitting on a bench to be photographed. Asma is to the left of Rani. Rubi is to the right of Rani. Rita is between Rani and Rubi. Who would be third from the left in the photograph?

Solution:
Asma is to the left of Rani.
Asma ⇔ Rani

Rubi is to the right of Rani.
Rani ⇔ Rubi

Rita is between Rani and Rubi.
Rani ⇔ Rita ⇔ Rubi

∴ Asma ⇔ Rani ⇔ Rita ⇔ Rubi.

Rita would be third from the left in the photograph

Question: How many real roots does the equation have?
x² - 4x + 5 = 0

Solution:
Given,
x² - 4x + 5 = 0
Here,
a = 1, b = - 4 and c = 5

Discriminant of the given equation,
(- 4)² - 4 × 1 × 5
= 16 - 20
= - 4 < 0

∴ There is no real root of the equation.

দ্বিঘাত সমীকরণের মূলের প্রকৃতি:
1. যদি b² - 4ac = 0 হয় তবে দ্বিঘাত সমীকরণের মূলদ্বয় বাস্তব ও সমান হবে।
2. যদি b² - 4ac > 0 হয় তবে দ্বিঘাত সমীকরণের মূলদ্বয় বাস্তব ও অসমান হবে।
3. যদি b² - 4ac < 0 হয় তবে দ্বিঘাত সমীকরণের মূলদ্বয় অবাস্তব ও অসমান হবে।
4. যদি b² - 4ac পূর্ণবর্গ সংখ্যা হয় তবে দ্বিঘাত সমীকরণের মূলদ্বয় মূলদ ও অসমান হবে।

Question: An alloy contains Copper, Zinc and Nickel in the ratio of 5 : 3 : 2. The quantity of Nickel that must be added to 100 kg of this alloy to have the new ratio 5 : 3 : 3 is-

Solution:
Given that,
An alloy contains Copper : Zinc : Nickel = 5 : 3 : 2
Total weight = 100 kg

Now, Sum of the ratio, 5 + 3 + 2 = 10
Copper = (5/10) × 100 = 50 kg
Zinc = (3/10) × 100 = 30 kg
Nickel = (2/10) × 100 = 20 kg

Now, Let the amount of Nickel to be added be x kg
Then, new quantity of Nickel = 20 + x
And Copper : Zinc : Nickel = 5 : 3 : 3

ATQ,
50/5 : 30/3 : (20 + x)/3
Now taking last two ratio, then we get,
⇒ 30/3 = (20 + x)/3
⇒ 30 = 20 + x
⇒ x = 30 - 20
∴ x = 10

∴ Nickel that must be added 10 kg.

Question: If p + q = √7 and p - q = √5 the, 8pq(p² + q²) = ?

Solution:
Given,
p + q = √7
and p - q = √5

Now,
8pq(p² + q²)
= 4pq × 2(p² + q²)
= {(p + q)² - (p - q)²}{(p + q)² + (p - q)²}
= {(√7)² - (√5)²}{(√7)² + (√5)²}
= (7 - 5)(7 + 5)
= 2 × 12
= 24

Question: If A = 2, B = 4, C = 6, D = 8, E = 10 and so on, what is the meaning of following number 40, 18, 14, 10, 36

Solution:
Given,
A = 2, B = 4, C = 6, D = 8, E = 10.......

∴ Each code = Letter position × 2

So
40 ÷ 2 = 20 → T
18 ÷ 2 = 9 → I
14 ÷ 2 = 7 → G
10 ÷ 2 = 5 → E
36 ÷ 2 = 18 → R

the meaning of following number = TIGER

Question: α and β are the roots of 5x² + 3x + 1 = 0 then, the value of (1/α) + (1/β) is-

Solution:
Here,
5x² + 3x + 1 = 0
where, a = 5, b = 6 and c = 1

∴ α + β = - (b/a) = - 3/5
and αβ = c/a = 1/5

∴ (1/α) + (1/β)
= (β + α)/αβ
= {- (3/5)}/(1/5)
= - 3

Question: A solution contains 25% sugar by weight is made sweeter by doubling the amount of sugar. The percent of sugar, by weight, in the new solution is-

Solution: 
let, solution is 100 unit
amount of sugar = 100 × 25%
= 100 × (25/100)
= 25 unit 

by doubling, amount of sweet = 50 unit

∴ solution now = 100 + 25 = 125 unit 

∴ percent of sugar = {(50 × 100)/125}% = 40%

Question: If x + 1/x = 7, find the value of 3x/(x² - 6x + 1)

Solution:
Given,
x + 1/x = 7
⇒ (x² + 1)/x = 7
∴ x² + 1 = 7x

Now,
3x/(x² - 6x + 1)
= 3x/(x² + 1 - 6x)
= 3x/(7x - 6x)
= 3x/x
= 3

Question: If a = b = 4c and abc = 1024, then what is the value of c =?

Solution:
Given that,
a = b = 4c
⇒ a = 4c, b = 4c

Now,

Question: Find the roots of the quadratic equation, p² - 17p + 72 = 0

Solution:
Given,
p² - 17p + 72 = 0
⇒ p² - 9p - 8p + 72 = 0
⇒ p(p - 9) - 8(p - 9) = 0
⇒ (p - 9)(p - 8) = 0
∴ p = 9 or 8