Bank Math Master

Question: If in an infinite G.P. first term is equal to the twice of the sum of the remaining terms, then its common ratio is

Solution: 
Let the G.P. = a + ar + ar² + ....

ATQ, 
a = 2 ( ar + ar² + ....)
⇒ 1/2 = r + r² + ....
⇒ 1/2 = r/(1 - r)
⇒ 1 - r = 2r 
⇒ r = 1/3

Question: There is 60% increase in an amount in 6 years at simple interest. What will be the compound interest of 12,000 after 3 years at the same rate?

Solution: let simple interest r% 

I = pnr/100
⇒ I/p = nr/100
⇒ 0.6 = 6r/100 
⇒ r = 10% 

The compound interest = 12000 (1.1)³ - 12000 
= 12 × 1331 - 12000 
= 15972 - 12000 
= Tk. 3972

Question: Painting : Frame :: Book :?

Solution:
We know, Cover is used to protect book in same way a frame is use to protect painting.

প্রশ্ন: A wire can be bent in the form of a circle of radius 7cm. If it is bent in the form of a square, then what will be its area?

সমাধান: 
দেওয়া আছে,
বৃত্তের ব্যাসার্ধ r = 7 cm 
বৃত্তের পরিধি = 2πr 
= 2 × (22/7) × 7 
= 2 × 22 × 1
= 44 cm 

বর্গের এক বাহুর দৈর্ঘ্য = 44/4 cm 
= 11 cm 

∴ বর্গের ক্ষেত্রফল = (11)² cm² 
= 121 cm²

Question: What is the length of the diagonal of a square whose area is 4 times of another square with diagonal as 5√2 cm?

Solution:
Area of square = (1/2) × (length of diagonal)²

Area of square2 =(1/2) × (5√2)²  
= 25 cm²

Area of square1 = 4 × 25 = 100 cm²

Length of diagonal of square1 = √(2 × area)
= √(2 × 100)
= 10√2 cm

Question: From a pack of 52 cards, two cards are drawn at random. What is the probability of getting two queens?

Solution:
Total queen 4
Total possibilities = ⁵²C₂
getting two queens = ⁴C₂

∴ probability = ⁴C₂/⁵²C₂
= 6/1326
= 1/221

Question: One of the diagonals of a rhombus is double the other diagonal. Its area is 36 sq. cm. The sum of the diagonal is?

Solution: 
let, one diagonal is x m
other is 2x

0.5 × x × 2x  = 36 
⇒ x² = 36
⇒ x = 6 

sum of diagonals = x + 2x 
= 3x 
= 3 × 6
= 18  

Question: Each side of a rectangular field diminished by 40%. By how much percent is the area of the field diminished?

Solution: 
Let, length is 80 m and breadth is 40 m 
Area = 80 × 40 = 3200 m² 

diminished by 40%, 
New length = 80 - 80 × .4 
= 48 m 
New breadth = 40 - 40 × .4
= 24 m 
New area = 48 × 24
= 1152 m² 

percent is the area of the field diminished = {(3200 - 1152)/3200} × 100%
= 64% 

Question: A pole of 66 metre long breaks into two parts without complete separation and makes an angle 30° with the ground. Find the length of the broken part of the pole.

Solution

sin30 = x/(66 - x)
⇒ 1/2 = x/(66 - x) 
⇒ 66 - x = 2x 
⇒ 3x = 66
⇒ x = 66/3 = 22

the length of the broken part of the pole = 66 - 22 = 44 m

Question: An accurate clock shows the time as 3.00. After the hour hand has moved 135°, the time would be- 

Solution: 
প্রতি ৫ ঘরের মান ৩০° 

১৩৫° তে ঘন্টার কাঁটা যায় ৪.৫ ঘর 
∴ ৪ ঘন্টা ৩০ মিনিট অতিবাহিত হয়েছে। 

নতুন সময় = ৭ : ৩০ মিনিট 

Question: The area of a triangle is 216 cm² and sides are in the ratio 3 : 4 : 5. The perimeter of the triangle is- 

Solution: 
3² + 4² = 5² 

It is a right-angled triangle.
let, the sides 3x, 4x, 5x

(1/2) × 3x × 4x = 216 
⇒ 12x² = 432 
⇒ x² = 36 
⇒ x = 6

perimeter = (3 × 6) + (4 × 6) + (4 × 6)
= 72 cm

Question: Two stations A and B are on a straight line. One train starts from A at 7 a.m. and travels towards B at 20 Kmph. Another train starts from B at 8 a.m. and travel towards A at a speed of 25 kmph. If they meet at 10 a.m., what is the distance between two stations A and B?

Solution:
train started at 7 a.m. traveled for (10 - 7) = 3 hour
so in 3 hour at 20 kmph, the train travelled (20 × 3) = 60 km

train started at 8 a.m. traveled for (10 - 8) = 2 hour
so in 2 hour at 25 kmph, the train travelled (25 × 2) = 50 km

∴the distance between two station A and B is = (60 + 50) km
= 110 km

Question: Find the value of k if (x - 1) is a factor of 4x{x + (k/4x) - (1/4)}.

Solution:
4x{x + (k/4x) - (1/4)}
= 4x² + k - x
= 4x² - x + k

As x - 1 is a factor,
4 × 1² - 1 + k = 0
⇒ 3 + k = 0
⇒ k = -3

Question: Pointing to a photograph Anik said, "He is the son of the only son of my grandfather." How is the man in the photograph related to Anik?

Solution:
- আনিকের দাদার একমাত্র ছেলের অনিকের বাবা।
- অনিকের বাবার ছেলে অনিকের ভাই।

Question: Anik invested 25,200 taka in two accounts, which pay 5 % and 10% interest annually. The amount invested at 10% rate is 110% of the amount invested at 5% rate. After three years, he earns 5760 taka in interest. How much did he invest at the 10% rate?

Solution: 
5% সুদের জন্য আসল x টাকা 
10% সুদের জন্য আসল 110%x = 1.1x টাকা 

5% হারে সুদ = x × 3 × 5/100 = 3x/20 টাকা 
10% হারে সুদ = 1.1x × 3 × 10/100 = 3.3x/10 টাকা 

(3x/20) + (3.3x/10) = 5760 
⇒ (3x + 6.6x)/20 = 5760
⇒ 9.6x = 5760 × 20 
⇒ x = (5760 × 20)/9.6
= 12000 tk.

10% সুদের হারের জন্য আসল = 1.1 × 12000 
= 13200 tk.

Question: If cosB + cos²B = 1, then sin²B + sin⁴B =?

Solution:
cosB + cos²B = 1
⇒ cosB = 1 -  cos²B
⇒ cosB = sin²B

sin²B + sin⁴B
= sin²B + (sin²B)²
= sin²B + cos²B
= 1

Question: A cloth merchant on selling 33 meters of cloth obtains a profit equal to the selling price of 11 meters of cloth, the profit is = ?

Solution:
Profit = selling price of 11 m of cloth = 1/3 selling price of 33 m of cloth

Let selling price of 33 meters of cloth = 3x
∴ profit = 1/3 (3x) = x

So, cost price = selling price - profit = 3x - x = 2x

 % Profit = (x/2x) × 100% = 50%

Question: What is the measure of each interior angle in a regular hexagon?

Solution: 
সুষম বহুভুজের বাহুর সংখ্যা n হলে তার কোণগুলোর সমষ্টি (2n - 4) সমকোণ।
সুতরাং সুষম ষড়ভুজের ছয় কোণের সমষ্টি = (2 × 6 - 4) সমকোণ
= (12 - 4) × 90°
= 8 × 90°
= 720°

সুতরাং সুষম ষড়ভুজের একটি শীর্ষ কোণ = 720°/6
= 120°