Bank Math Master

Question: The sum of the squares of three numbers is 138, while the sum of their products taken two at a time is 131. Their sum is-

Solution:
Let the numbers are a, b and c.
Then,
a² + b² + c² = 138
and ab + bc + ca = 131.

Now,
(a + b + c)² = a² + b² + c² + 2(ab + bc + ca) = 138 + (2 × 131) = 138 + 262 = 400
⇒ (a + b + c)² = 400
∴ a + b + c = 20.

Question: A factory manufactures products in batches of 16, 24, and 32 units. What is the minimum number of units the factory needs to produce so that each batch can be formed exactly?

Solution:
To find the minimum number of units the factory needs to produce so that each batch size (16, 24, and 32) can be formed exactly, we need to find the least common multiple (LCM) of these batch sizes.

The prime factorization of each batch size is -
16 = 2 × 2 × 2 × 2 = 2⁴
24 = 2 × 2 × 2 × 3 = 2³ × 3
32 = 2 × 2 × 2 × 2 × 2 = 2⁵

Now,
So the highest power of 2 is 2⁵
The highest power of 3 is 3¹

So, the LCM of 16, 24, and 32 is = 2⁵ × 31 = 32 × 3 = 96.

So the minimum number of units the factory needs to produce is 96.

Question: A bag costs 20% more than a purse. A wallet costs 30% less than the bag. If the price of the purse is 200 Tk, then by what percentage is the wallet cheaper than the purse?

Solution:
Given,
the price of the purse = 200 tk

∴ Price of the bag = 200 + (200 × 20%)
= 240 tk

Price of the wallet = 240 − (240 × 30%)
= 240 − 72
= 168 Tk

Difference is = (200 - 168) = 32 tk

∴ Percentage = (32 × 100)/200 = 16%

Question: Two trains are running in opposite directions. They cross a man standing on a platform in 28 seconds and 10 seconds respectively. They cross each other in 24 seconds. What is the ratio of their speeds?

Solution:
Given that,
Train one crosses a man in 28 seconds
Train two crosses the man in 10 seconds
They both cross each other in 24 seconds

We know,
Time = Distance/speed
As the trains travel in opposite directions, the speed of the trains added

Now,
Let the speed of the first train & second train be x m/s and y m/s respectively.
Length of the first train is 28x metres
Length of the second train is 10y meters

According to the question,
⇒ 24 = (28x + 10y)/(x + y)
⇒ 24x + 24y = 28x + 10y
⇒ 14y = 4x
⇒ x/y = 7/2

∴ The ratio of the speed of the train is 7 : 2

Question: The side of an equilateral triangle is 6m. What is the height of the triangle?

Solution:
Given,
The side of an equilateral triangle = 6m

We know,
Area of an equilateral triangle = (√3/4) × 6²
= (√3/4) × 36
= 9√3

Let,
the height of the triangle = h

We also know,
(1/2) × base × height = area
⇒ (1/2) × 6 × h = 9√3
⇒ h = 9√3/3
∴ h = 3√3

So, the height of the triangle = 3√3 m

Question: A fruit shop has 12 types of fruits. You don’t like Mango and Papaya. How many ways can you select 5 different fruits from the ones you like?

Solution:
Given that, 
Total fruits = 12
Fruits you don’t like = 2
∴ Fruits you can choose = 12 - 2 = 10
Number of fruits to choose = 5

∴ Number of ways = ¹⁰C₅ = 10!/5!(10 - 5)!
= (10 × 9 × 8 × 7 × 6 × 5!)/(5! × 5!)
= (10 × 9 × 8 × 7 × 6)/(5 × 4 × 3 × 2)
= 252

So, there are 252 ways to select 5 different fruits from the ones you like.

Question: The difference between the ages of a father and his son is 40% of the father’s age. If the son is 18 years old, find the father’s age.

Solution:
Let
Let father’s age = x

Then,
x - 18 = 40% of x
⇒ x - 18 = 40x/100
⇒ x - 18 = 2x/5
⇒ x - 2x/5 = 18
⇒ (5x - 2x)/5 = 18
⇒ 3x = 90
∴ x = 30

So the father’s age is 30 years.

Question: If cosA = 8/17 than, what is the value of tanA = ?

Solution:
Given that, 
cosA = 8/17

We know, 
sin²A = 1 - cos²A = 1 - (8/17)²
= 1 - (64/289)
= (289 - 64)/289
= 225/289
∴ sinA = √(225/289) = 15/17

Now, 
tanA = sinA/cosA = (15/17)/(8/17) = 15/8
∴ tanA = 15/8

Question: If x and y are positive integers satisfying x + y = 7, what is the probability that x < y?

Solution: 
Since both x and y must be positive integers,
total possible ways = (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1) = 6 pairs.
Next, we identify the pairs where x < y,
(1, 6), (2, 5), (3, 4) ; There are 3 pairs satisfying x < y.

∴ Probability = Number of pairs where x < y/Total number of pairs = 3/6 = 1/2

So the probability that x < y is 1/2

Question: A clock seen through a mirror, shows quarter past five. What is the correct time shown by the clock?

Solution:
Quarter past five = 5 : 15

আমরা জানি,
প্রকৃত সময় = 11 : 60 - আয়নার দেখা সময়
= 11 : 60 - 5 : 15
= 6 : 45

Question: Two pipes P and Q can fill a tank in 12 hours and 24 hours, respectively. If both pipes are opened together, how long will it take to fill the tank?

Solution:
Part filled by P in 1 hour = 1/12
Part filled by Q in 1 hour = 1/24

Part filled by (P + Q) in 1 hour
= (1/12) + (1/24)
= (2 + 1)/24
= 3/24
= 1/8

∴ Time to fill the tank = 1/(1/8) = 8 hours

∴ Both pipes can fill the tank in 8 hours

Question: A sum increases by 80% in 8 years at simple interest. If Tk 2500 is invested at the same rate for 2 years under compound interest, what is the final amount?
 
Solution:
let
the principal = Tk. 100
Simple Interest = Tk 80 (since 80% increase in 8 years)

We know,
Simple interest = Pnr/100
⇒ 80 = (100 × r × 8)/100
⇒ 80 = 8r
∴ r = 10%

The compound interest = 2500 {1 + (10/100)}² - 2500
= 2500 × {1 + (1/10)}² - 2500
= 2500 × (11/10)² - 2500
= (2500 × 121)/100 - 2500
= 3025 - 2500
= Tk. 525

So the compound interest on Tk 2500 after 2 years is Tk 525.

Question: 24 workers can complete a construction job in 15 days, working 6 hours a day. How many additional workers are needed to complete the same job in 10 days, working 8 hours a day?

Solution:
দৈনিক 6 ঘণ্টা করে কাজ করে 15 দিনে শেষ করতে লোক লাগে 24 জন
∴ দৈনিক 1 ঘণ্টা করে কাজ করে 1 দিনে শেষ করতে লোক লাগে (24 × 15 × 6) জন
∴ দৈনিক 8 ঘণ্টা করে কাজ করে 10 দিনে শেষ করতে লোক লাগে (24 × 15 × 6)/(10 × 8) জন
= 27 জন

∴ অতিরিক্ত লোক লাগবে = (27 - 24) = 3 জন

Question: 


Solution:

Question: The curved surface area of a cylindrical pillar is 264 m² and its volume is 924 m³. Find the height of a cylindrical pillar.

Solution:
Let the radius of the cylinder be r meters and the height be h meters.
Curved surface area = 2πrh
∴ 2πrh = 264 .......(1)

And Volume = πr2h
∴ πr²h = 924 ......(2)

Now, (2) ÷ (1),
πr²h/2πrh = 924/264
⇒ r/2 = 924/264
⇒ r = (924/264) × 2
∴ r = 7

From (1) we get,
h = 264/2πr = 264 × (7/22 × 2 × 7) = 6m
∴ h = 6m

So the height of the cylindrical pillar is 6 meters

Question: If 2^{n - 1} + 2^{n + 1} = 320, then the value of n is = ?

Solution:
Given that, 
2^{n - 1} + 2^{n + 1} = 320
⇒ 2^{n - 1} + 2^{n - 1} . 2² = 320
⇒ 2^{n - 1}(1 + 2²) = 320
⇒ 2^{n - 1} . 5 = 320
⇒ 2^{n - 1} = 320/5 = 64
⇒ 2^{n - 1} = 26
⇒ n - 1 = 6
⇒ n = 6 + 1
∴ n = 7

So the value of n is 7.

Question: The price of 10 chairs is equal to that of 4 tables. The price of 15 chairs and 2 tables together is Tk. 4000. The total price of 12 chairs and 3 tables is-

Solution:
Let the cost of a chair and a table are x and y respectively.
Then,
10x = 4y
⇒ y = (10/4)x = 5x/2
∴ y = 5x/2 .......(1)
And,
15x + 2y = 4000
⇒ 15x + 2(5x/2) = 4000
⇒ 20x = 4000
⇒ x = 4000/20
∴ x = 200

From (1), 
y = 5x/2 = (5 × 200)/2 = 500
∴ y = 500

Hence, the cost of 12chairs and 3tables is,
= 12x + 3y
= (2400 + 1500)
= 3900

So the total price of 12 chairs and 3 tables is Tk. 3900.

Question: Which of the following statements is not correct?

Solution:
Option ক)
Since log_aa = 1
So, log⁡₁₀10 = 1 
This is correct.

Option খ)
log⁡(2 + 3) = log⁡(2 × 3)
Compute the left side- 2 + 3 = 5, so log⁡(2 + 3) = log⁡5.
Compute the right side- 2 × 3 = 6, so log⁡(2 × 3) = log⁡6.
Logarithm property: log⁡(a⋅b) = log⁡a + log⁡b, not log⁡(a + b).
This is incorrect.

Option গ)
Since log_a1 = 0, so log₁₀1 = 0.
This is correct.

Option ঘ)
log (1 + 2 + 3) = log 1 + log 2 + log 3
Compute the left side- 1 + 2 + 3 = 6, so log⁡(1 + 2 + 3) = log⁡6.
Right side- log⁡1 + log⁡2 + log⁡3 = log(1 × 2 × 3) = log6
Both sides are equal: log⁡6 = log⁡6
This is correct.

Option খ) is the only statement that is not correct.

Question: The mean weight of 100 students in a class is 46 kg. The mean weight of boys is 50 and of girls is 40 kg. Therefore, the number of boys is-

Solution:
Given that, 
Total students = 100
Mean weight of all students = 46 kg
∴ Total weight of all students = 100 × 46 = 4600 kg.

Let,
The number of boys = x. Then, the number of girls = 100 - x 
Mean weight of boys = 50 kg,
∴ total weight of boys = 50x
And, 
Mean weight of girls = 40 kg,
∴ total weight of girls = 40(100 - x)

ATQ,
50x + 40 × (100 - x) = 4600
⇒ 50x + 4000 - 40x = 4600
⇒ 10x = 4600 - 4000
⇒ x = 600/10
∴ x = 60

So the number of boys is 60.

Question: The area of a trapezium is 96 square cm. The length of one of the parallel sides is 12 cm, and the distance between the parallel sides is 8 cm. Find the length of the other parallel side.

Solution:
Given,
Area of the trapezium = 96 cm²
One parallel side a = 12 cm
Distance between the parallel sides h = 8 cm

Let
the other parallel side = b cm

We know,
The area of a trapezium = (1/2) × (a + b) × h
⇒ 96 = (1/2) × (12 + b) × 8
⇒ 96 = (12 + b) × 4
⇒ (12 + b) = 96/4
⇒ 12 + b = 24
⇒ b = 24 - 12
∴ b = 12

∴ The other parallel side is 12 cm.

Question: Find the first term of an arithmetic progression(A.P.) whose 8^th and 12^th terms are respectively 39 and 59.

Solution:
We know,
Arithmetic progression, n^th term is  = a + (n - 1)d,
where a is the first term and d is the common difference.
Given that, 
8^th term = 39 and 12^th term = 59

Now,
8^th term = a + 7d = 39 ........... (i)
12^th term = a + 11d = 59 ........... (ii)
(i) - (ii) ⇒ a + 7d - a - 11d = 39 - 59
⇒ 4d = 20
 ∴ d = 5
Hence, a + 7 × 5 = 39
Thus, a = 39 - 35 = 4

 so the first term of the A.P. is 4.

Question: If α, β are the roots of the equation 2x² + 10x - 12 = 0, then α + β equals to:

Solution:
Given that,
2x² + 10x - 12 = 0
Where, a = 2, b = 10, c = - 12

Now, For a quadratic equation ax2 + bx + c = 0, the sum of roots α + β = - b/a
sum of roots α + β = - b/a = - 10/2 = - 5

Question: The ratio between the perimeter and the breadth of a rectangle is 5 : 1. If the area of the rectangle is 216 sq. cm, what is the length of the rectangle?

Solution: 
et the breadth of the rectangle be B cm.
Perimeter of a rectangle = 2(Length + Breadth) = 2(L + B)
Given the Ratio of perimeter to breadth = 5 : 1
Now,
2(L + B)/B = 5/1
⇒ 2L + 2B = 5B
⇒ 3B = 2L
∴ B = (2/3)L

Area of the rectangle = L . B = 216 sq. cm.
⇒ L . (2/3)l = 216
⇒ L² = (216 × 3)/2
⇒ L² = 324 = 18²
∴ l = 18 cm

So the length of the rectangle is 18 cm.