Bank Math

Question: If you toss a coin twice, what is the probability that you will get heads the second time?

Solution:
When we toss a coin the sample outcomes are = {HH, HT, TH, TT}
Total number of outcome = 4

Heads in the second time {HH, TH} = 2

∴ Probability that I will get heads the second time = 2/4 = 1/2 = (1 × 100)/2 % = 50%

Question: A man covers half of his journey at 15 km/h and the remaining half at 5 km/h. His average speed is-

Solution: 
Here, x = 15 km/h and y = 5 km/h

We know, 
Average speed = 2xy/(x + y)
= (2 × 15 × 5)/(15 + 5)  
= 150/20
= 7.5 km/h

Choose 1 person for the single room & from the remaining choose 2 for the double room & from the remaining choose 4 people for the four person room,
Then, 7C1 x 6C2 x 4C4
= 7 x 15 x 1
= 105

প্রশ্ন: একটি দৌড় প্রতিযোগিতায় A এবং B এর গতিবেগের অনুপাত 3 : 4। গন্তব্যে পৌঁছাতে A এর চেয়ে B এর সময় 30 মিনিট কম লাগে। তাহলে A গন্তব্যে পৌঁছাতে কত ঘণ্টা সময় নেয়?

সমাধান:
ধরি,
A ও B এর গতিবেগ যথাক্রমে 3x ও 4x কিমি/ঘণ্টা
এবং উভয়ের জন্য দূরত্ব = D কিমি

আমরা জানি, 
সময় = দূরত্ব/গতি

∴ A এর সময় = D/3x
∴ B এর সময় = D/4x

প্রশ্নমতে,
D/3x - D/4x = 30 মিনিট
⇒ D/3x - D/4x = 30/60 ঘণ্টা
⇒ (4D - 3D)/12x = 1/2
⇒ D/12x = 1/2
⇒ D = 6x

তাহলে, A এর সময় = D/3x = 6x/3x = 2 ঘণ্টা

Question: If rsinθ = 5/2 and rcosθ = (5√3)/2, what is the value of r?

Solution:
দেওয়া আছে, 
rsinθ = 5/2 ……………..(i)
rcosθ = (5√3)/2 …………….(ii) 

এখন, (i) এবং (ii) এর বর্গ করে যুক্ত করে পাই,
(rsinθ)² + (rcosθ)² = (5/2)² + ((5√3)/2)² 
⇒ r²(sin²θ + cos²θ) = (25/4) + (75/4) 
⇒ r²(sin²θ + cos²θ) = 100/4
⇒ r² = 25     [∵ sin²θ + cos²θ = 1]
∴ r = ± 5

Let the 3rd number is 100
According to the question,
1st :2nd: 3rd= 20:50:100
Required % = (20/50) × 100 = 40%

Let x and y be the ten's and unit's digits respectively of the numeral denoting the woman's age.

Then, woman's age = (10X + y) years;
So, husband's age = (10y + x) years.
Therefore (10y + x) - (10X + y) = (1/11) (10y + x + 10x + y)
⇒ (9y - 9x) = (1/11)(11y + 11x) = (x + y)
⇒ 10x = 8y
⇒ x = (4/5)y

Clearly, y should be a single-digit multiple of 5, which is 5.
So, x = 4, y = 5.
Hence, woman's age = 10x + y = 45 years.

Question: If 1 - sinθ = n cosθ, then find the value of cotθ.

Solution:
Given,
1 - sinθ = n cosθ
⇒ (1 - sinθ)/cosθ = n
⇒ (1/cosθ) - (sinθ/cosθ) = n
∴ secθ - tanθ = n ...............(i)

We know,
(secθ + tanθ)(secθ - tanθ) = 1
⇒ (secθ + tanθ) × n = 1
⇒ secθ + tanθ = 1/n ................(ii)

Now, (ii) - (i) ⇒
(secθ + tanθ) - (secθ - tanθ) = 1/n - n
⇒ secθ + tanθ - secθ + tanθ = 1/n - n
⇒ 2tanθ = (1 - n²)/n
⇒ tanθ = (1 - n²)/2n
⇒ 1/cotθ = (1 - n²)/2n
∴ cotθ = 2n/(1 - n²)

প্রশ্ন: একটি ঘড়ির মিনিটের কাঁটা ১০ মিনিটে কত ডিগ্রি কোণ অতিক্রম করে?

সমাধান:
আমরা জানি,
মিনিটের কাঁটা ৬০ মিনিটে অতিক্রম করে = ৩৬০°
∴ ১০ মিনিটে অতিক্রান্ত কোণ = (৩৬০ × ১০)/৬০
= ৩৬০০/৬০
= ৬০°

Question: Find the amount of compound interest on Tk. 2000 for 1.5 years at a 20% annual rate, with interest added semi-annually.

Solution: 
Principal P = 2000
Rate, R = 20%
Time, T = 1.5 years

Now,
Compound interest for half-yearly:
A = P{1 + R/(2×100)}^2T
= 2000{1 + (20/200)}^2×1.5
= 2000 × {1 + (1/10)}³
= 2000 × (11/10)³ 
= 2000 × (1331/1000)
= 2662

Compound Interest = A – P
= 2662 – 2000 = 662 Tk

In 12 hours, they are at right angles 22 times.
∴ In 24 hours, they are at right angles 44 times.

Question: If p ⊗ q = p - q + (p × q), what is the value of 5 ⊗ 2?

Solution:
দেওয়া আছে, p ⊗ q = p - q + (p × q)

এখন, p = 5 এবং q = 2 বসিয়ে পাই,
⇒ 5 ⊗ 2 = 5 - 2 + (5 × 2)
⇒ 5 ⊗ 2 = 3 + 10
⇒ 5 ⊗ 2 = 13

∴ সুতরাং, নির্ণেয় মান হলো 13।

Question: log₃(x² + 3x) - log₃(x + 3) = 2, Then what is the value of x?

Solution:
Given that, 
log₃(x² + 3x) - log₃(x + 3) = 2
⇒ log₃[(x² + 3x)/(x + 3)] = 2 ; [log₃A - log₃B = log₃(A/B)] 
⇒ log₃[x(x + 3)/(x + 3)] = 2 
⇒ log₃x = 2
⇒ x = 3² 
∴ x = 9

Question: A boy agrees to work at the rate of one Taka on the first day, two Taka on the second day, and four Taka on third day and so on. How much will the boy get if he started working on the 1^st of February and finishes on the 20^th of February?

Solution:
Given that,
1st term, a = 1
Common ratio, r = 2   ; r > 1

We know, 
Sum Sn = a × {(rⁿ - 1)/(r - 1)}
= 1 × {(2²⁰ - 1)/(2 - 1)}. ; [Putting, a = 1, r = 2 and n = 20]
= 2²⁰ - 1

So the boy will get 2²⁰ - 1 Takas if he works from February 1^st to February 20^th.

Cost Price of 1st book = Tk. 450.
Profit % of 1st book = 20% profit
Selling price of 1st book = Cost Price of 1st book + Profit % of 1st book
= Tk. 450 + 20% of 450
= Tk. 540
Cost price of 2nd book = Tk. 540
Loss % of 2nd book = 5%
Selling price of 2nd book = Cost Price of 1st book + Profit % of 1st book
= Tk. 540 - 5% of 540 = Tk. 513
Overall profit = Tk. 513 - Tk. 450
= Tk. 63.

মনে করি,
হেঁটে যায় = x km
∴ সাইকেলে যায় = (61 - x) km
প্রশ্নমতে, x/4 + (61−x)/9 = 9
⇒ (9x + 244 - 4x)/36 = 9
⇒ 5x + 244 = 324
⇒ 5x = 80
⇒ x = 16 km

Question: Which value of x will satisfy the given inequality,
2(x - 4) ≥ 3x - 5 ?

Solution:
Given,
2(x - 4) ≥ 3x - 5
⇒ 2x - 8 ≥ 3x - 5 
⇒ 2x - 3x ≥ -5 + 8
⇒ - x ≥ 3
⇒ x ≤ - 3 [ Multiplying both sides of an inequality by a negative number reverses the inequality sign ]

Let the number be x and the quotient is q.
Then, x = 899q + 63 = (29 × 31q) + (29 × 2) + 5
= 29(31q + 2) + 5
So, the given number when divided by 29 gives 5 as remainder.

Question: The ratio of the speed of a boat in still water to the speed of the stream is 5 : 3. The boat covers 16 km upstream in 4 hours and x km downstream in 2 hours. Find the value of x.

Solution:
ধরি, স্থির পানিতে নৌকার গতিবেগ = 5k কিমি/ঘণ্টা এবং স্রোতের গতিবেগ = 3k কিমি/ঘণ্টা।
স্রোতের প্রতিকূলে (Upstream) নৌকার গতিবেগ = (5k - 3k) = 2k কিমি/ঘণ্টা।
স্রোতের অনুকূলে (Downstream) নৌকার গতিবেগ = (5k + 3k) = 8k কিমি/ঘণ্টা।

প্রশ্ন অনুযায়ী,
স্রোতের প্রতিকূলে 16 কিমি যেতে সময় লাগে 4 ঘণ্টা।
∴ স্রোতের প্রতিকূলে গতিবেগ = 16 / 4 = 4 কিমি/ঘণ্টা।

শর্তমতে,
2k = 4
⇒ k = 4/2
⇒ k = 2

এখন, স্রোতের অনুকূলে নৌকার গতিবেগ = 8k = 8 × 2 = 16 কিমি/ঘণ্টা।
∴ স্রোতের অনুকূলে 2 ঘণ্টায় অতিক্রান্ত দূরত্ব, x = 16 কিমি/ঘণ্টা × 2 ঘণ্টা
= 32 কিমি

∴ x = 32 কিমি

Question: A fruit seller buys lemons at 2 for Tk. 1 and sells them at 5 for Tk. 3. What is his gain percent?

Solution:
Cost price,
2 lemons = Tk. 1
So, cost price of 1 lemon = 1/2

Selling price, 
5 lemons = Tk. 3
So, selling price of 1 lemon = 3/5

Profit per lemon = SP - CP
= (3/5) - (1/2)
= (6 - 5)/10
= 1/10

Gain percent = (Profit/CP) × 100%
= (1/10)/(1/2) × 100%
= (1/10) × (2/1) × 100%
= (2/10) × 100%
= 100/5%
= 20%

So his gain percent is 20%.

Question: A man can row at 10 kmph in still water. If the speed of the current is 2 kmph and it takes him 1.5 hours to row to a place and come back, how far is the place?

Solution:
Speed downstream = 10 + 2 = 12 kmph
Speed upstream = 10 - 2 = 8 kmph

Let the required distance = x km

ATQ,
(x/12) + (x/8) = 1.5
Or, (2x + 3x)/24 = 1.5
Or, 5x/24 = 1.5
Or, 5x = 36
∴ x = 36/5
∴ x = 7.2 km

∴ The place is 7.2 km away.

Relative speed = (90 + 72) × (5/18) = 162 × (5/18) = 45 m/s

∴ Total length = 45 × 9 = 405 m

∴ Length of Train B is, x = 405 - 225 = 180 m
∴ x = 180 m

ⁿP_r = n!/(n-r)!
⁶P₃ = 6!/(6-3)!
⁶P₃ = 6!/3!
⁶P₃ = 120

Question: If 45% of a certain number is 72, then find the number-

Solution:
Let the number be x.
Then,
⇒ 45% of x = 72
⇒ (45/100) × x = 72
⇒ 9x/20 = 72
⇒ x = (20 × 72)/9
⇒ x = 1440/9
∴ x = 160

Question: The length of a rectangle is 25% more than its breadth. What will be the ratio of the area of the rectangle to that of a square whose side is equal to the breadth of the rectangle?

 
Solution: 
Let,
breadth = X metres
∴ length = 125% of X metres
= 125X/100 metres
= 5X/4 metres

∴ Area of the rectangle = (5x/4 × X) m²
∴ Area of the square = (X × X) m²

∴ The ratio = (5X/4 × X) : (X × X)
= 5 : 4

Question: Which of the following numbers is a prime number?

Solution:
A prime number is a number that can only be divided by itself and 1 without remainders.

ক) 213 is divisible by 3.
খ) 191 has no divisors other than 1 and itself → prime
গ) 275 is divisible by 5.
ঘ) 333 is divisible by 3.

∴ 191 is the required prime number as it is not divisible by any prime number.

Question: Three times a whole number is equal to four less than the square of the number. Find the number.

Solution:
Let the number be x.
Then, 3x = x² - 4
⇒ x² - 3x - 4 = 0
⇒ x² - 4x + x - 4 = 0
⇒ x (x - 4) + 1 (x - 4) = 0
⇒ (x - 4) (x + 1) = 0
⇒ x = 4, - 1 
Solving, x = 4 is the only whole number solution.