Bank Math

Question: A 50-meter cable is attached from the top of a vertical pole down to the ground. If the cable makes an angle of 30 degrees with the ground, find the height of the pole.

Solution: 

ধরি,
উচ্চতা(Height), AB = h

দেয়া আছে,
AC = 50m                    
∠ACB = 30°

∴ sin30°= AB/AC
⇒ 1/2 = h/50
⇒ h = 50 × 1/2
∴ h = 25 m

Question: A ladder is leaning against a wall. It makes a 60° angle with the wall. If the distance between foot of ladder and wall is 7.5 meters, find the length of the ladder.

Solution:

Let BC be the wall and AC be the ladder.
∠BAC = 60° and AB = 7.5 meter
In ΔABC,
cos60° = AB/AC
⇒ 1/2 = 7.5/AC
⇒ AC = 7.5 × 2
∴ AC = 15

Question: If today is Wednesday, which day of the week will it be after 47 days?

Solution: 
Each day of the week is repeated after 7 days.

So, after 49 days, it will be Wednesday.
∴ After 47 days, it will be Monday.

The cost price of 12 toys = Tk. 375
The selling price of 12 toys = 33 × 12
= Tk. 396
profit = 396 - 375
= Tk. 21
Profit percentage = (21 × 100)/375
= (7 × 100)/125
= (7 × 4)/5
= 5.6%

Total volume of water displaced = (4 x 50) m³ = 200 m³.
Rise in water level =200/(40 x 20)m 0.25 m = 25 cm.

দেয়া আছে,
k = 462/n যেখানে k একটি পূর্ণসংখ্যা।
∴ n এর মান এমন হবে যা দ্বারা 462 কে নিঃশেষে ভাগ করা যাবে। অপশন অনুযায়ী একমাত্র 22 দ্বারা 462 কে ভাগ করা যায়।

সুতরাং n এর মান 22.

Age decreased = (5 × 3) years
= 15 years
So the required difference = 15 years

Working hours per day= 24 – 9 = 15 hrs.
So, Total working hours for 40 days = 15 × 40 = 600 hrs.
On doubling the distance, the time required becomes twice but on walking twice as fast, the time required gets halved. Therefore, the two negates each other with respect to time required.

By increasing rest to twice reduces walking hours per day to 24 – (2 × 9) = 6 hrs.
∴ Number of days required to cover twice the distance, at twice speed with twice the rest = 600/6 = 100 days

Principle,
= (P.W. of Tk. 882 due 1 year hence) + (P.W of Tk. 882 due 2 years hence)
= [{882/(1 + 5/100)} + {882/(1 + 5/100)²}]
= [{(882 × 20)/21} + {(882 × 400)/441}]
= Tk. 1640.

Question: If tanθ = 5/12, then find the value of sinθ.

Solution:
Given, tanθ = Perpendicular/Base = 5/12
Here, base = 12 and perpendicular = 5
Let hypotenuse = x

Now, according to Pythagorean theorem,
x² = 5² + 12²
⇒ x² = 25 + 144
⇒ x = √169
⇒ x = 13

∴ sinθ = Perpendicular/Hypotenuse = 5/13

Let man's rate upstream be x km/hr
Then, his rate downstream = 2x km/hr
∴ (speed in still water) : (Speed of stream)
(2x + x)/2 : (2x - x)/2
3x/2 : x/2
3 : 1

Question: A clock loses (falls behind) 12 minutes each day. How many days will it take to reach a point where the clock will indicate the correct time?

Solution:
একটি ঘড়িকে পুনরায় সঠিক সময় দেখাতে হলে তাকে 12 ঘণ্টা সময় হারাতে হবে। কারণ একটি 12 ঘণ্টার ঘড়িতে যখন সঠিক সময় থেকে 12 ঘণ্টা সময় পিছিয়ে পড়বে, তখন এটি আবার সঠিক সময় নির্দেশ করবে।
12 ঘণ্টা = 12 × 60 মিনিট = 720 মিনিট

এখন, যেহেতু ঘড়িটি প্রতিদিন 12 মিনিট করে সময় হারায়,

12 মিনিট সময় হারায় 1 দিনে
∴720 মিনিট সময় হারাবে = (1/12) × 720 দিনে
= 60 দিনে
সুতরাং, 60 দিন পর ঘড়িটি আবার সঠিক সময় দেখাবে।

Let,
The number is x,
therefore, (1/2) x + 4 = 14,
so, x = 20

Question: If (x - 2) is a factor of the polynomial x³ + 4x² - px + 10, what is the value of p?

Solution:
Let f(x) = x³ + 4x² - px + 10
Since (x - 2) is a factor of f(x),
by the Factor Theorem,
When x - 2 = 0 ⇒ x = 2, then f(x) = 0

Now,
f(2) = (2)³ + 4(2)² - p(2) + 10
= 8 + 4(4) - 2p + 10
= 8 + 16 - 2p + 10
= 34 - 2p

According to the condition,
f(2) = 0
⇒ 34 - 2p = 0
⇒ 2p = 34
⇒ p = 34/2
⇒ p = 17

So the value of p is 17.

Question: The list price of a commodity is the price after a 20% discount on the retail price. The festival discount price on the commodity is the price after a 30% discount on the list price. Customers purchase commodities from stores at a festival discount price. What is the effective discount offered by the stores on the commodity on its retail price?

Solution: 
Let r be the retail price.
The list price is the price after a 20% discount on the retail price.
Hence, the list price = r[1 - (20/100)] 
= r[1 - (1/5)]
= 4r/5

The festival discount price is the price after a 30% discount on the list price.
Hence, the festival discount price = (4r/5) [1 - (30/100)]
= (4r/5) × [1 - (3/10)] 
= (4r/5) × (7/10)
= 14r/25

Hence, the total discount offered is [(Original Price - Price after discount) × 100]/Original Price
= [r - (14r/25) × 100]/r 
= (11r × 100)/25r
= 44%

Question: What is the difference between the biggest and the smallest fraction among 3/7, 4/9, 5/11 and 6/13?

Solution:
Converting each of the given fractions into decimal form, we get,
3/7 = 0.4286
4/9 = 0.4444
5/11 = 0.4545
6/13 = 0.4615

Since, 0.4615 > 0.4545 > 0.4444 > 0.4286 So, 6/13 > 5/11 > 4/9 > 3/7

∴ Required difference = 6/13 - 3/7
= (42 - 39)/91
= 3/91

Question: A clock gains (moves ahead) 20 minutes each day. How many days will it take to reach a point where the clock will indicate the correct time?

Solution:
একটি ঘড়িকে পুনরায় সঠিক সময় দেখাতে হলে তাকে 12 ঘণ্টা সময় এগিয়ে যেতে হবে।
কারণ একটি 12 ঘণ্টার ঘড়িতে যখন সঠিক সময় থেকে 12 ঘণ্টা সময় এগিয়ে যাবে, তখন এটি আবার সঠিক সময় নির্দেশ করবে।

12 ঘণ্টা = 12 × 60 মিনিট = 720 মিনিট

এখন, যেহেতু ঘড়িটি প্রতিদিন 20 মিনিট করে সময় এগিয়ে যায়,
20 মিনিট সময় এগিয়ে যায় 1 দিনে
∴ 720 মিনিট সময় এগিয়ে যাবে = (1/20) × 720 দিনে
= 36 দিনে

∴ 36 দিন পর ঘড়িটি আবার সঠিক সময় দেখাবে।

3x + 2y = 7 .... (i)
3x - 2y = 5 .... (ii)

(i) + (ii), 6x = 12
Or, x = 2

From, (i), y = 1/2

So, xy = 2.1/2 = 1

Question: The sum of the ages of a mother and her daughter is 50 years. The mother’s age is four times the daughter’s age. After how many years will the mother’s age be three times the daughter’s age?

Solution:
Let the daughter's current age be x.
Then, the mother's current age is 4x.

ATC,
x + 4x = 50
⇒ 5x = 50
∴ x = 10

∴ Daughter's current age x = 10 years
and Mother's current age is 4x = 4 × 10 = 40 years

Let, the number of years after which mother’s age will be three times the daughter’s age be t.

So after t years,
The daughter’s age will be 10 + t
The mother’s age will be 40 + t

ATC,
40 + t = 3(10 + t)
⇒ 40 + t = 30 + 3t
⇒ 40 - 30 = 3t - t
⇒ 10 = 2t
⇒ t = 5

∴ After 5 years, the mother’s age will be three times the daughter’s age.

Question: A man can row 12 kmph in still water. It takes him thrice as long to row up as to row down the river. Find the rate of the stream-

Solution:
Given that,
Speed of man in still water = 12 km/h
Time upstream = 3 × Time downstream

Let,
Rate of the stream = x km/h

Now,
Downstream speed = 12 + x
Upstream speed = 12 - x

And, let the distance travelled is = D km

According to the Question,
Time taken upstream = 3 × (Time taken downstream)
D/(12 - x) = 3 × {D/(12 + x)}
⇒ 1/(12 - x) = 3/(12 + x)
⇒ 12 + x = 36 - 3x
⇒ 4x = 24
∴ x = 6

So, rate of the stream = 6 km/h

Question: A shopkeeper incurs a loss by selling an article for Tk 800. If he had sold it for Tk 1200, he would have made a profit which is four times the initial loss. At what price should he sell the article to make 25% profit?

Solution:
ধরি, পণ্যের ক্রয়মূল্য = x টাকা
800 টাকায় বিক্রি করলে ক্ষতি = x - 800 টাকা
1200 টাকায় বিক্রি করলে লাভ = 1200 - x টাকা

প্রশ্নমতে,
1200 - x = 4(x - 800)
⇒ 1200 - x = 4x - 3200
⇒ 1200 + 3200 = 4x + x
⇒ 4400 = 5x
∴ x = 880 টাকা

এখন, 25% লাভে,
ক্রয়মূল্য 100 টাকা হলে বিক্রয়মূল্য 125 টাকা
ক্রয়মূল্য 1 টাকা হলে বিক্রয়মূল্য 125/100 টাকা
∴ ক্রয়মূল্য 880 টাকা হলে বিক্রয়মূল্য = (125 × 880)/100 টাকা
= 1100 টাকা

∴ বিক্রয়মূল্য: Tk. 1100

Question: The sum of three number is 330. If the ratio of the first to the second is 3 : 2 and then that of the second to third is 5 : 4, what is the second number?

Solution: 
Let the three numbers be a, b, c.

Here,
a : b = 3 : 2
⇒ a/b = 3/2
⇒ a = 3b/2

And,
b : c = 5 : 4
⇒ b/c = 5/4
⇒ c = 4b/5

Given,
Sum of the three numbers, a + b + c  = 330
⇒ (3b/2) + b + (4b/5) = 330
⇒ (15b + 10b + 8b)/10 = 330
⇒ 33b = 3300
⇒ b = 3300/33
⇒ b = 100

∴ The second number is 100.

Angle traced by the minute hand in 5 minutes. =(360/60×5)^∘ = 30^∘

Average of 11 numbers = 30

Step 1: Calculate total of 11 numbers by multiplying it by average value 30 = 11 x 30 = 330
Step 2: Calculate the total of the first six members by multiplying it by average value 17.5 = 17.5 x 6 = 105
Step 3: Calculate the total of the last six members by multiplying it by average value 42.5 = 42.5 x 6 = 255

Therefore,
we can find the sixth number by adding the value of the first six and last six numbers and subtracting it from the total value of 11 numbers.
Sixth number = (105 + 255) - 330
= 30.

Question: The slope of the line 4x - 2y = 8 is not the same as the slope of which one of the following lines?
​
​Solution:
​প্রথমে, প্রদত্ত রেখাটির ঢাল নির্ণয় করতে হবে। 
​রেখাটির সমীকরণকে y = mx + c তে রূপান্তর করতে হবে। এখানে 'm' হলো ঢাল (Slope)।

প্রদত্ত রেখার সমীকরণ: 4x - 2y = 8
⇒ - 2y = - 4x + 8
⇒ y = (- 4/- 2)x + (8/- 2)
⇒ y = 2x - 4
∴ এই রেখাটির ঢাল (m) হলো 2.

এবার, প্রদত্ত অপশনগুলোর প্রত্যেকটির ঢাল নির্ণয় করি:
ক) 4x - 2y = 12
⇒ -2y = -4x + 12
⇒ y = 2x - 6
∴ ঢাল 2

খ) 2x - y = -5
⇒ -y = -2x - 5
⇒ y = 2x + 5
∴ ঢাল 2

গ) y = 2x - 1
∴ ঢাল 2

ঘ) x + 2y = 6
⇒ 2y = -x + 6
⇒ y = (-1/2)x + 3
∴ ঢাল - 1/2

সুতরাং, দেখা যাচ্ছে যে শুধুমাত্র অপশন (ঘ) এর রেখার ঢাল মূল রেখার ঢাল থেকে ভিন্ন।

Total results = 2x + 3x + 4x = 9x,
Favourable result is (red ball) = 2x
∴ Probability = 2x/9x
= 2/9
Answer: 2/9

Question: Ruma can paint 70 walls in 14 minutes. Salma can paint 45 walls in 9 minutes. Working together, how many walls can they paint in 35 minutes?

Solution:
Ruma can paint in 1 minute = 70/14 = 5 walls
Salma can paint in 1 minute = 45/9 = 5 walls

∴ Working together they can paint in 1 minute = (5 + 5) = 10 walls

∴ they can paint in 35 minutes = (35 × 10) = 350 walls

Question: If y = sin(sinx) then what is the value of dy/dx?

Solution:

Question: If a right-angled isosceles triangle has base 6 cm, then height is:
(Officer Cash 2022 অনুযায়ী)

Solution:
(Right-angled isosceles triangle) সমকোণী সমদ্বিবাহু ত্রিভুজ এর ভূমি = 6 cm.
সমকোণী সমদ্বিবাহু ত্রিভুজ এর ভূমি ও উচ্চতা সমান।
ভূমি = উচ্চতা = 6 cm.
∴ উচ্চতা = 6 cm

Question: What will replace the question mark?


Solution:
এখানে,
উপরের সংখ্যাটি নিচের দুটি সংখ্যার বর্গের সমষ্টির বর্গমূলের সমান।

১ম চিত্রে,
√(5² + 12²) = √169 = 13
২য় চিত্রে,
√(15² + 8²) = √289 = 17
৩য় চিত্রে,
√(6² + 8²) = √100 = 10

প্রশ্ন: If θ be an acute angle and 5sin²θ + 3cos²θ = 4, then the value of tanθ is?

সমাধান:
5sin²θ + 3cos²θ = 4
⇒ 5sin²θ + 3(1 - sin²θ) = 4
⇒ 5sin²θ + 3 - 3sin²θ = 4
⇒ 2sin²θ = 1
⇒ sin²θ = 1/2
⇒ sinθ = √(1/2)
⇒ sinθ = 1/√2
⇒ sinθ = sin45°
⇒ θ = 45°

∴ tanθ = tan45° = 1