Bank Math

Question: If 3x + 2y = 12 and xy = 6 , then find the value of 27x³ + 8y³ = ?

Solution:
দেওয়া আছে, 3x + 2y = 12 এবং xy = 6
এখন,
27x³ + 8y³
= (3x)³ + (2y)³
= (3x + 2y)³ - 3 × 3x × 2y(3x + 2y) [a³ + b³ = (a + b)³ - 3ab(a + b)]
= (3x + 2y)³ - 18xy(3x + 2y)
= (12)³ - 18 × 6 × 12
= 1728 - 1296
= 432

সুতরাং, নির্ণেয় মান হলো 432।

Principle = P, Simple interest = P (since both are same), R = 25/4, time = n
Interest = pnr/100
Or, p = pn(25/4)/100
Or, n = p × 100 × 4/25 × p
Or, n = 16

Let the labeled price of TV = Tk. X
∴ SP of the TV = [X x (100 - 20)]/100
= Tk. 4X/5
But 16,800 - 800 = 4X/5
∴ x = (16,000 x 5)/4
= Tk. 20,000

Question: A number, when divided by 312, leaves a remainder of 47. If the same number is divided by 24, what remainder will it leave?

Solution:
Let the number be x and the quotient be q.

Then,
x = 312q + 47
⇒ x = (24 × 13q) + 47
⇒ x = (24 × 13q) + (24 × 1) + 23
⇒ x = 24(13q + 1) + 23

∴ When the number is divided by 24, the remainder is 23.

Let the speed of car be x km/hr
Distance= Speed × Time
Distance = 8x km
According to the question,
⇒ (x+4)×7.5= 8x
⇒ 7.5x+30= 8x
⇒ 8x−7.5x= 30
⇒ 0.5x= 30
⇒x= (30/0.5)= 60 km/hr
Required distance:
= 8 × 60
= 480 km

প্রশ্ন: What is the slope of a line perpendicular to the line whose equation is 3x + 4y = 8?

সমাধান:
প্রদত্ত সরল রেখার সমীকরণ: 3x + 4y = 8

y = mx + c আকারে লিখি, যেখানে m হলো রেখার ঢাল।
4y = - 3x + 8
y = (- 3/4)x + 2

অতএব, মূল রেখার ঢাল (m) = - 3/4

আমরা জানি, কোনো রেখার উপর লম্ব রেখার ঢাল m = - 1/m
= -1/(- 3/4)
= 4/3

∴ লম্ব রেখার ঢাল = 4/3

Question: sin(A + 45°) = √2/2, find the value of A.

Solution:
sin(A + 45°) = √2/2
⇒ sin(A + 45°) = 1/√2
⇒ sin(A + 45°) = sin45°
⇒ A + 45° = 45°
⇒ A = 45° − 45°
∴ A = 0°

Question: What is the slope of a line perpendicular to the line whose equation is 15x - 3y = 9?

Solution:
সরল রেখার সাধারণ সমীকরণ,
y = mx + c ......(1) (এখানে m = ঢাল)

যদি কোনো রেখার ঢাল হয় m, তবে তার লম্ব (perpendicular) রেখার ঢাল হবে,
m' = - (1/m)

এখন, প্রদত্ত সমীকরণটি হলো,
15x - 3y = 9
⇒ 3y = 15x - 9
⇒ y = (15x - 9) / 3
⇒ y = 5x - 3

(1) নং এর সাথে তুলনা করে পাই,
m = 5

∴ লম্ব (perpendicular) রেখার ঢাল হবে, m' = - (1/m)
= - 1/5

Let C's age be x year. 
Then, B's age = 2x year. 
A's age = (2x + 2) year. 
(2x + 2) + 2x + x = 27 
5x = 25 
=> x = 5 
Hence, B's age = 2x = 10 year.

Question: How many words can be formed by using 3 letters from the word 'DELHI'? 

Solution:
Here we will use the Permutations for this question.
We know,
​ ⁿP_r, for this we have,
n = 5, Total 5 Letters
r = 3, Letters word we required

​Now,
ⁿP_r = n!/(n-r)!
= ⁵P₃ 
​= 5!/2! 
​= 120/2 
​= 60

So, Total we can form 60 different permutation of word from Letter Delhi.

Question: If C is the midpoint of the points A(2, 3) and B(8, 11), find the length of AC.

Solution:
দেওয়া আছে, A(2, 3) এবং B(8, 11), এবং C হলো AB-এর মধ্যবিন্দু।

দূরত্বের সূত্র ব্যবহার করে AB-এর দৈর্ঘ্য নির্ণয় করি।
AB = √{(x₂ - x₁)² + (y₂ - y₁)²}
AB = √{(8 - 2)² + (11 - 3)²}
AB = √(6² + 8²)
AB = √(36 + 64)
AB = √100
AB = 10

যেহেতু C হলো AB-এর মধ্যবিন্দু, তাই AC হবে AB-এর অর্ধেক।
∴ AC = AB/2
= 10/2
= 5

Question: If tanA = 3/4, then sinA = ?

Solution:
tanA = 3/4
লম্ব/ভূমি = 3/4
অতিভুজ = √{(4)² + (3)²} = 5

sinA = লম্ব/অতিভুজ
= 3/5

Question: A worker’s regular pay is Tk. 20 per hour up to 40 hours. Overtime is paid at 1.5 times the regular rate. If he was paid Tk. 1,100, how many hours of overtime did he work?

Solution:
ধরি, তিনি x ঘণ্টা overtime কাজ করেছেন।

সাধারণ বেতন (40 ঘণ্টার জন্য) = 20 × 40 = 800 টাকা
Overtime হার (প্রতি ঘণ্টা) = 20 × 1.5 = 30 টাকা
Overtime বেতন = 30 × x = 30x টাকা

মোট বেতন = সাধারণ বেতন + overtime বেতন
শর্তমতে,
800 + 30x = 1,100
⇒ 30x = 1,100 - 800
⇒ 30x = 300
⇒ x = 300 / 30
∴ x = 10

∴ তিনি 10 ঘণ্টা overtime করেছেন।

Let, these two numbers be 3x and 4x then their LCM = 12x
Now, according to question,
12x =  60
Or, x = 5
Thus, the numbers are (3x = 3 × 5) = 15 and (4x = 4 × 5) = 20

Question: A card is drawn from a pack of 52 cards. The probability of getting a queen of spades or a king of diamonds is:

Solution:
Here, n(S) = 52
Let E = event of getting a queen of spades or a king of diamonds.
Then, n(E) = 2

∴ P(E) = n(E)/n(S)
= 2/52
= 1/26

Question: Two pipes, A and B, can fill a tank in 9 minutes and 18 minutes, respectively. If both pipes are opened together, after how many minutes should pipe B be closed to fill the tank in 7 minutes?

Solution:
ধরা যাক, নল B, x মিনিট চলার পর বন্ধ করা হয়।

ট্যাংকটি সম্পূর্ণ পূর্ণ হতে মোট সময় লাগে 7 মিনিট।
সুতরাং, নল A মোট 7 মিনিট চালু থাকে।
এবং নল B মোট x মিনিট চালু থাকে।

নল A দ্বারা 1 মিনিটে পূরণ হয় = 1/9 অংশ
নল B দ্বারা 1 মিনিটে পূরণ হয় = 1/18 অংশ

প্রশ্নানুসারে,
(নল A দ্বারা 7 মিনিটে পূরণ করা অংশ) + (নল B দ্বারা x মিনিটে পূরণ করা অংশ) = 1 (সম্পূর্ণ ট্যাংক)
⇒ 7/9 + x/18 = 1
⇒ (14 + x)/18 = 1
⇒ 14 + x = 18
⇒ x = 18 - 14
∴ x = 4

অতএব, 4 মিনিট পর নল B বন্ধ করলে ট্যাংকটি 7 মিনিটে পূর্ণ হবে।

Suppose the can initially contains 7x and 5x litres of mixtures A and B respectively
Quantity of A in mixture left = {7x - (7/12) × 9} = 7x - 21/4
Quantity of B in mixture left = {5x - (5/12) × 9} = 5x - 15/4

According to the question,
(7x - 21/4)/{(5x - 15/4) + 9} = 7/9
⇒ (28x - 21)/(20x + 21) = 7/9
⇒ 252x - 189 = 140x + 147
⇒ 252x - 140x = 147 + 189
⇒ 112x = 336
⇒ x = 3.

So, they can contain 21 litres of A.

Question: If A's salary is 25% more than B's salary, then B's salary is how much lower than A's salary?

Solution: 
Let B's Salary is Tk. 100.
Then,
A's Salary = (100 + 25% of 100) = Tk. 125
Difference between A's Salary and B's Salary = 125 - 100 = Tk. 25

% Difference (lower) = (25/125) × 100 = 20%

Question: If (x + 5)² = 64, which of the following can be the value of (x + 2)?

Solution:
Given, (x + 5)² = 64
⇒ (x + 5)² = 8²
∴ x + 5 = ± 8

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

Case 2: x + 5 = - 8
x = -8 - 5 = -13
x + 2 = -13 + 2 = - 11

Possible values of (x + 2) are 5 or - 11.

Correct Answer: ক) 5

Question: Every 2 minutes, 5 litres of water are poured into a 1,500 litre tank. After 3 hours, what percent of the tank will be full?

Solution:
In 2 minutes, 5 liters is poured
In 180 minutes = (180 × 5)/2 = 450 liters

So, percentage filled = (450 × 100)/1500
= 30%

Question: If (m + 4)² = 36, which of the following can be the value of (m - 3)?

Solution:
Given that,
(m + 4)² = 36
or, (m + 4)² = 6²
or, m + 4 = ± 6

Case 1 : m + 4 = 6
m = 6 - 4 = 2
m - 3 = 2 - 3 = - 1

Case 2 : m + 4 = - 6
m = - 6 - 4 = - 10
m - 3 = - 10 - 3 = - 13

Possible values of (m - 3) are - 1 or - 13

Question: If log₁₀x + log₁₀y = 3 and log₁₀x - log₁₀y = 1, then x and y are respectively.

Solution: 
Given that, 
log₁₀x + log₁₀y = 3 ......(1)
log₁₀x - log₁₀y = 1 .......(2)

Now, (1) + (2) then we get,
⇒ log₁₀x + log₁₀y + log₁₀x - log₁₀y = 3 + 1
⇒ 2log₁₀x = 4
⇒ log₁₀x = 4/2 = 2
⇒ x = 10²
∴ x = 100

From (1) we get,
⇒ log₁₀x + log₁₀y = 3
⇒ log₁₀100 + log₁₀y = 3
⇒ 2 log₁₀10 + log₁₀y = 3
⇒ log₁₀y = 3 - 2
⇒ log₁₀y = 1
⇒ y = 10¹
∴ y = 10

∴ x and y are respectively 100 and 10.

Question: Find the smallest number divisible by 7, 8, and 9.

Solution:
The smallest number divisible by several numbers is their LCM.
Factorize the numbers:
7 = 7
8 = 2³
9 = 3²
LCM = product of highest powers of all prime factors:
LCM = 2³ × 3² × 7
= 8 × 9 × 7
= 504

Question: Two runners start running together for a certain distance. One runs at 8 km/h and the other at 5 km/h. The faster runner arrives one and half an hour before the slower runner. What is the distance?

Solution:
Given that,
Speed of first runner v₁ = 8 km/h
Speed of second runner v₂ = 5 km/h
First runner arrives 1.5 hours before the second.
Let the distance be d km.

We know,
Time = Distance​/Speed
∴ t₁ = d/8 and t₂ = d/5
And Difference in time, t₂ - t₁ = 1.5 hours

ATQ,
(d/5) - (d/8) = 1.5
⇒ (8d - 5d)/40 = 3/2
⇒ 3d = 60
∴ d = 20

So the distance is 20 km.

Question: How many kilogram of sugar costing Tk. 9 per kg must be mixed with 27 kg of sugar costing Tk. 7 per kg so that there may be a gain of 10% by selling the mixture at Tk. 9.24 per kg?

Solution:
Let,
Quantity of sugar costing Tk. 9 per kg = x
S.P. of 1 kg of mixture = Tk. 9.24,
Gain 10%.
C.P. of 1 kg of mixture = Tk. (100/110) × 9.24 = Tk. 8.40

By the rule of allilation, we have:

⇒ 27 : x = (9 - 8.40) : (8.40 - 7) = 0.60 : 1.40 = 6 : 14 = 3 : 7
⇒ 27/x = 3/7
⇒ x/27 = 7/3
⇒ x = (7 × 27)/3
∴ x = 63

Question: Find the midpoint of the line segment joining the points A₁(2, 5) and A₂(8, - 3).

Solution:

A train passes a platform in 40 sec and a woman standing on the platform in 30 sec. If the speed of the train is 108 km/hr, what is the length of the platform?
Speed of the train = {108×(5/18)} m/sec = 30 m/sec
Length of the train = (30×30) m = 900 m
Let the length of the platform be x meters
Then, (x+900)/30 = 40
⇒ x + 900 = 1200
⇒ x = 300 m

Question: In a tourist group of 100 people, 55 speak French, 40 speak Spanish, and 20 speak none of the languages. How many of them speak just one language?

Solution:

Let,
Number of people who can speak both languages = x persons
∴ Number of people who speak only French = (55 - x) persons
∴ Number of people who speak only Spanish = (40 - x) persons

Given that,
Number of people who speak none of the languages = 20 persons

According to the question,
Only French + Both + Only Spanish = Total students - Those who speak none
⇒ (55 - x) + x + (40 - x) = 100 - 20 
⇒ 95 - x = 80
⇒ x = 95 - 80
∴ x = 15

∴ Only French = (55 - 15) = 40 persons
∴ Only Spanish = (40 - 15) = 25 persons

∴ Number of people who speak only one language (French or Spanish) = (40 + 25) = 65 persons

Question: In 2 kg mixture of copper and aluminum, 30% is copper. How much aluminum powder should be added to the mixture so that the quantity of copper becomes 20%?

Solution:
According to the question,
Mixture of copper and aluminum = 2 Kg = (2 × 1000) = 2000 gm
30% of this mixture is copper,
= (30/100) × 2000 gm
= 600 gm copper
∴ In 2 kg mixture of copper and aluminum, aluminum is = (2000 - 600) = 1400 gm

Let x be the mass of aluminum added. The new total mass is (2000 + x).
 We want the 600 gm of copper to be 20% of this new total.
⇒ 600 = 0.20(2000 + x)
⇒ 600 = 400 + 0.2x
⇒ 0.2x = 200
∴ x = 1000 gm

Question: A person lends Tk. 10,000 at 10% per annum simple interest and Tk. 5000 at 20% per annum simple interest. Find the total interest earned in 2 years.

Solution:
First Loan,
Principle₁ = 10,000 Taka
Rate₁ = 10% Per Annum
Time = 2 Year

Simple Interest:
SI = (P × R × T) / 100
SI₁ = (10,000 × 10 × 2) / 100 = 2,000 Taka

Second Loan,
Principle₂ = 5,000 Taka
Rate₂ = 20% Per Annum
Time = 2 Year

Simple Interest:
SI₂ = (5,000 × 20 × 2) / 100 = 2,000 Taka

Total Interest:
SI₁ + SI₂ = 2,000 + 2,000 = 4,000 Taka

∴ Total earned interest = 4,000 Taka

Question: A train 200 meters long takes 50 seconds to cross a 300-meter-long bridge. How much time will the train take to cross a 150-meter-long platform?

Solution:
Length of train = 200 m
Length of bridge = 300 m
∴ Total distance to cross bridge = 200 + 300 = 500 m

Time taken = 50 seconds

∴ Speed of train = Total distance/Time
= 500/50
= 10 m/s

Length of platform = 150 m
∴ Total distance to cross platform = 200 + 150 = 350 m

∴ Time taken = Total distance/Speed
= 350/10
= 35 seconds

Sourav invests Tk. 4000 for 24 months, Fahim invests Tk. 3000 for 24 months and Arif invests
Tk. 4000 for 18 months.
Then,
Sourav:Fahim:Arif = (4000x24):(3000x24):(4000x18)
= 4x24 : 3x24 : 4x18
= 4 : 3 : 3
Therefore, Fahim's share = Tk. 5000 x 3/10
= Tk. 1500.

Question: A certain sum amounts to Tk. 12,100 in 2 years at 10% per annum compound interest. Find the principal.

Solution:

Principal, P = Tk. P
Amount, C = Tk. 12,100
Rate, r = 10%
Time, n = 2 years

We know,
C = P(1 + r)ⁿ
⇒ 12,100 = P × (1 + 10/100)²
⇒ 12,100 = P × (110/100)²
⇒ 12,100 = P × (11/10)²
⇒ 12,100 = P × (121/100)

∴ P = (12,100 × 100)/121
= 10,000

Hence, Principal = Tk. 10,000