Bank Math Master

Question: It requires Taka 20 for fencing 1 meter. How much will it cost to fence a plot having length of 30 meters and breadth of 10 meters? 

Solution: 
প্লটটির দৈর্ঘ্য = ৩০ মিটার 
প্লটটির প্রস্থ= ১০ মিটার 

প্লটটির পরিসীমা = ২(দৈর্ঘ্য + প্রস্থ) একক
= ২ (৩০ + ১০)
= ২ × ৪০
= ৮০ মিটার

নির্ণেয় খরচ = ৮০ × ২০ টাকা
= ১,৬০০ টাকা।

Question: The side of an equilateral triangle is 2m. What is the area of the triangle? 

Solution: 
Area = (√3/4)2²
=  (√3/4)× 4 m²
= √3 m²

প্রশ্ন: i^-49 =?

সমাধান:
আমরা জানি, i = √-1;
i²= -1; 
i³ = i²i = - i; 
i⁴ = i².i² = (-1).(-1) = 1
 
i^{- 49}
= 1/i⁴⁹
= 1/{i⁴⁸.i}
= 1/{(i⁴)¹².i}
= 1/i
= i⁴/i
= i³
= - i

Question: The difference between two positive numbers is 4 and the difference of their squares is 96. The largest number is - 

Solution: 
Let the numbers be X and (X + 4)

then,
(X + 4)² - X² = 96
⇒ X² + 8X + 16 - X² = 96
⇒ 8X + 16 = 96
⇒ 8X = 80
⇒ X = 10

hence, the largest number is = (10 + 4) = 14

Question: If ÷ means ×, × means +, + means - and - means +, find the value of 16 × 3 + 5 - 2 ÷ 4.

Solution: 
16 × 3 + 5 - 2 ÷ 4
=16 + 3 - 5 + 2 × 4
= 27 - 5
= 22

Question: How many different selections of 4 books can be made from 10 different books, if 2 particular books are never selected?

Solution:
Number of different books =10
Number of books to be formed = 4

If two particular books are never selected,
∴ Number of ways = ⁸C₄​
=8!​/4!(8 -4)!
= 8 × 6 × 7 × 5 × 4!/4! × 4 × 3 × 2
= 70 ways

Question: If 3²×9² = 3^x, what is the value of x? 

Solution: 
3² × 9² = 3^x
⇒ 3² × (3²)² = 3^x
⇒ 3² × 3⁴ = 3^x
⇒ 3^{2 + 4} = 3^x 
⇒ 3⁶ = 3^x
∴ x= 6

Question: A man purchased a shirt at Taka 300 after availing a discount of 25%. What is the catalog price of the shirt?

Solution: 
25% ডিসকাউন্টে, 
তালিকা মূল্য 100 টাকা হলে ক্রয়মূল্য = 100 - 25 = 75 টাকা

75 টাকা ক্রয়মূল্য হলে তালিকামূল্য = 100 টাকা
1 টাকা ক্রয়মূল্য হলে তালিকামূল্য = 100/75 টাকা
300 টাকা ক্রয়মূল্য হলে তালিকামূল্য = (100 × 300)/75 টাকা
 = 400 টাকা

প্রশ্ন : 3cotA = 4 হলে, sinA এর মান কত?

সমাধান :
দেওয়া আছে,
3 cotA = 4 
⇒ cot = 4/3
⇒ cot²A = 16/9
⇒ cosec²A  - 1  = 16/9
⇒ cosec²A = (16/9) + 1
⇒ cosec²A = (16 + 9)/9
⇒ cosec²A = 25/9
⇒ cosecA = 5/3
⇒ 1/sinA = 5/3
⇒ sinA = 3/5 

Question: The average of 5 consecutive number integers starting with m as the first integer is n. Then n =?

Solution: 
দেয়া আছে,
প্রথম সংখ্যাটি = m

প্রশ্নমতে 
{m + (m +1) + (m + 2) + (m + 3) + (m + 4)}/5 = n
⇒ m + m + 1 + m + 2 + m + 3 + m + 4 = 5n 
⇒ 5m + 10 = 5n
⇒ 5n = 5(m + 2) 
⇒ n = m + 2

Question: The difference of the two numbers is 20% of the large number, if the smaller number is 20, then the larger number is-

Solution: 
Let the large number be x.
Then,
x - 20 = 20% of x = 20x/100 = x/5
⇒ x - x/5 = 20
⇒ 5x - x = 100
⇒ 4x = 100
∴ x = 25

Question: A mother said to her daughter "I was as old as you are at present at the time of your birth". If the mother's age is 40 years now, the daughter's age five years back was-

Solution: 
ধরি,
মেয়ের বর্তমান বয়স = x বছর।
মাতার বর্তমান বয়স (x + x) = 2x বছর 
                              
শর্তমতে,
2x = 40
x = 20

∴ 5 বছর পূর্বে মেয়ের বয়স ছিল = (20 - 5) বছর 
= 15 বছর

Question: A sum of Tk. 312 was divided among 100 boys and girls in such a way that each boys gets Tk. 3.6 and each girl Tk. 2.4. The number of girls is :

Solution: 
ধরি,
বালিকার সংখ্যা = x জন
বালকের সংখ্যা = (100 - x) জন

1 জন বালক পায় 3.60 টাকা 
(100 - x) জন বালক পায় = 3.60 (100 - x) টাকা 

1 জন বালিকা পায় = 2.40 টাকা
x জন বালিকা পায় = 2.40x টাকা 

প্রশ্নমতে,
 3.60(100 - x) + 2.40x = 312
⇒ 360 - 3.60x + 2.40x = 312
⇒ 360 - 1.2x = 312 
⇒ 360 - 312 = 1.2x
⇒ 1.2x = 48 
⇒ x = 48/1.2 
⇒ x  = 40

Question: The value of (0.125 + 0.027)/(0.5 × 0.5 + 0.09 - 0.15) is- 

Solution: 
let,
0.5 = a
0.3 = b 

(0.125 + 0.027)/(0.5 × 0.5 + 0.09 - 0.15)
= {(0.5)³ + (0.3)³}/(0.5 × 0.5 + 0.3 × 0.3 - 0.5 × 0.3)
= (a³ + b³)/(a² + b² - ab)
= (a + b)(a² + b² - ab)/(a² + b² - ab)
= a + b
= 0.5 + 0.3
= 0.8

Question: The price of a pen is 25% more than the price of a book. The price of a pen holder is 50% more than the price of the book. How much is the price of the pen holder more than the price of the pen?

Solution: 
Let, the price of the book = 100 tk
Price of pen = 100 + 100 × 25%
= 125 tk
Price of pen-holder = 100 + 100 × 50%
= 150 tk

Difference is = 150 - 125 = 25 tk

∴ Percentage = (25 × 100)/125
= 20%

Question: A train 130 m long passes a man, running at 5 km/hr in the same direction in which the train is going, in 10 seconds. The speed of the train is:

Solution:
Speed of the train relative to man = 130/10 m/sec
= 13 m/sec
= (13 × 3600)/1000 km/hr
= 46.8 km/hr 

Let,
The speed of the train be x km/hr.
∴ Relative speed = (x - 5) km/hr.
⇒ x - 5 = 46.8         
∴ x = 51.8 km/hr.

Question: There is 60% increase in an amount in 6 years at simple interest. What will be the compound interest of Tk.1400 after 3 years at the same rate?

Solution: 
Let P = Tk. 100.
Then, S.I. Tk. = 60 and n = 6 years

Rate =(100 × 60)/(100 × 6) = 10%

Now, P = Tk.. 1400.
n = 3 years and R = 10% p.a.

C.I. = 1400(1 + 10/100)³ - 1400
= 1400(1.1)³ - 1400
= 1863.4 - 1400
= 463.4

Question: A motorboat, whose speed is 15 km/hr in still water goes 30 km downstream and comes back in a total of 4 hours 30 minutes. The speed of the stream (in km/hr) is:

Solution: 
Let
The speed of the stream is x km/hr. Then,
Speed downstream = (15 + x) km/hr,
Speed upstream = (15 - x) km/hr.

Now,
{30/(15 + x)} + {30/(15 - x)} = 9/2
⇒ 30(15 - x) + 30(15 + x)/(15 + x)(15 - x) = 9/2
⇒ (450 - 30x + 450 + 30x)/(225 - x²) = 9/2
⇒ 900/(225 - x²) = 9/2
⇒ 100/(225 - x²) = 1/2
⇒ 200 = 225 - x²
⇒ x² = 225 - 200
⇒ x² = 25 
⇒ x² = 5² 
⇒ x = 5

the speed of the stream is 5 km/hr.