ব্যাংক ডেইলি কুইজ [লং কোর্সের অংশ]

Question: In the figure, AOC is the diameter of the circle and arc AXB = (1/2)arc BYC. Find ∠BOC = ?

Solution:
Given that,
arc AXB = (1/2) arc BYC
∴ ∠AOB = (1/2) ∠BOC

We know that,
 ∠AOB + ∠BOC = 180º

Therefore,
(1/2) ∠BOC + ∠BOC = 180º {linear pair since AOC is the diameter}
⇒ (3/2) ∠BOC = 180º
⇒ ∠BOC = (2/3) × 180º = 120º
∴  ∠BOC = 120º

Question: The area of a square park is 1600 sq. meters. If fencing costs Tk. 5 per meter, what is the total cost to fence the park?

Solution:
Given that,
Area of square park = 1600 m²
∴ Side length of the square = √1600 = 40 meters

Perimeter of the square = 4 × side = 4 × 40 = 160 meters
And cost of fencing = Tk. 5 per meter

∴ Total cost = perimeter × cost per meter
= 160 × 5
= Tk. 800

So the total cost to fence the park is Tk. 800.

Question: If the radius of a sphere is 3 cm, what is its volume?

Solution:
Given that,
Radius of sphere = 3 cm

We know,
Volume of a sphere = (4/3) × πr³
= (4/3) × π(3)³
= (4/3) × π × 27
= 36π cm³

Question: A rectangular hall is 12.5 meters long and 6.4 meters wide. The cost of installing wooden flooring is Tk. 120 per square meter. What is the total cost of flooring the hall?

Solution: 
Given that,
Length of the hall = 12.5 meters
Width of the hall = 6.4 meters
Cost of wooden flooring = Tk. 120 per square meter

We know,
Area = Length × Width
= 12.5 m × 6.4 m
= 80 m²

∴ Total cost = Area × Cost per square meter
= 80 × 120 
= Tk. 9600 

Therefore, the total cost of installing wooden flooring in the hall is Tk. 9600.

Question: A bicycle wheel has a diameter of 70 cm and is making 200 revolutions per minute. What is the speed of the bicycle in km/h?

Solution: 
Given that,
Diameter of wheel, D = 70 cm
Radius, r = 70/2 = 35 cm
And revolutions per minute (rpm) = 200

We know, 
Circumference C = 2πr = 2 × (22/7) × 35
= 44 × 5 = 220 cm

∴ Distance per minute = 220 × 200 = 44000 cm/min
= (44000 × 60)/100000 km/h ; [1 km = 100000 cm and 1 hour = 60 min] 
= (44 × 6)/10  km/h
= 26.4  km/h

Question: If a regular polygon has each of its interior angle equal to 175°, then find the number of sides of the polygon.

Solution:
Given that,
Each interior angle = 175°

We know,
Sum of interior angles = (n - 2) × 180
Each interior angle = Sum of interior angles ÷ n

Now,
175 = [(n - 2) × 180] ÷ n
⇒ 175 × n = (n - 2) × 180
⇒ 175n = 180n - 360
⇒ 180n - 175n = 360
⇒ 5n = 360
⇒ n = 360/5
∴ n = 72

So, the polygon has 72 sides.

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:
Let the length of the rectangle = L cm
Let the breadth of the rectangle = B cm

Given that,
Ratio of perimeter to breadth = 5 : 1
Area of the rectangle = 216 cm²

Perimeter of rectangle = 2(L + B)
So, according to the ratio,
⇒ 2(L + B)/B = 5/1
⇒ 2(L + B) = 5B
⇒ 2L + 2B = 5B
⇒ 2L = 5B − 2B
⇒ 2L = 3B
∴ L = (3/2)B .........(1)

Also given that, 
Area = L × B = 216
∴ L × B = 216 ......... (2)

Substitute L from equation (1) into equation (2). Then we get,
⇒ (3/2)B × B = 216
⇒ (3/2)B² = 216
⇒ B² = 216 × (2/3)
⇒ B² = 144 = 12²
∴ B = 12 cm
Now find length, L = (3/2) × 12 = 18 cm

So, the length of the rectangle is 18 cm.

Question: A rectangular prism is 24 inches by 15 inches by 10 inches. Inside it is a cylinder with radius 5 inches and height 7 inches. Rounded to the nearest cubic inch, what is the volume of the prism not taken up by the cylinder?

Solution: 
Volume of the rectangular prism.
Volume = length × width × height
= 24 × 15 × 10
= 3600 cubic inches

And
Volume of the cylinder.
Volume = π × r² × h
= (22/7) × 5² × 7 ; [r = 5 in, h = 7 in] 
= 22 × 25
= 550 cubic inches

∴ Empty space = prism volume - cylinder volume
= (3600 - 550) cubic inches
= 3050 cubic inches

Question: If the difference between the circumference and diameter of a circle is 90 cm, then the diameter of the circle is-

Solution:
Let,
Radius of the circle = r
Diameter of the circle = 2r
Circumference of the circle = 2πr

According to the question,
2πr - 2r = 90
⇒ 2r(π - 1) = 90
⇒ r(π - 1) = 45
⇒ r = 45/(π - 1)
⇒ r = 45/(22/7 - 1) ; [π = 22/7]
⇒ r = 45/{(22 - 7)/7}
⇒ r = 45/(15/7)
⇒ r = 45 × (7/15)
∴ r = 21 cm

∴ Diameter of the circle = 2r = 2 × 21 = 42 cm

So the diameter of the circle is 42 cm.

Question: The diagonal of a rectangle is √41 cm and its area is 20 sq. cm. The perimeter of the rectangle must be:

Solution:
Let the length and breadth of the rectangle be x cm and y cm.
We are given two conditions are,
Diagonal = √41 cm
By Pythagoras theorem:
√(x² + y²) = √41
⇒ x² + y² = 41 ........ (1)
And Area, xy = 20 cm² .......(2)

We know,
(x + y)² = x² + y² + 2xy
= 41 + 2 × 20
= 41 + 40
= 81
⇒ x + y = √81
∴ x + y = 9

∴ Perimeter = 2(x + y) = 2 × 9 = 18 cm

 So the perimeter of the rectangle is 18 cm.

Question: The volume of a cylinder is 100π and its radius is 5. What is the lateral surface area?

Solution:
Given that,
Volume = 100π cubic units
Radius (r) = 5 units

We know,
Volume of a cylinder, V = πr²h
⇒ 100π = π × (5)² × h 
⇒ 100π = π × 25 × h
⇒ h = 100π/(25π)
⇒ h = 100/25
∴ h = 4 units

∴ Lateral surface area of a cylinder = 2πrh
= 2 × π × 5 × 4
= 40π

So the lateral surface area is 40π square units.

Question: The radius of a circular plate is 20 cm. If the radius is decreased by 20%, what is the percentage decrease in its area?

Solution:
Given that, 
Original radius = 20 cm
And decreased by 20%
 ∴ Decrease = 20% of 20 = 4 cm
∴ New radius = 20 - 4 = 16 cm

We know,
Area of circle = πr²
∴ Original area = π × (20)² = 400π cm²
∴ New area = π × (16)² = 256π cm²

∴ Decrease in area = original area - new area = 400π - 256π = 144π cm²

∴ Percentage decrease in area,
= (decrease/original area) × 100%
= (144π/400π) × 100%
= (144/400) × 100%
= 0.36 × 100%
= 36%

So the area decreases by 36%.

অপশনে সঠিক উত্তর না থাকায় প্রশ্নটি বাতিল করা হয়েছে।
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Question: A trapezium with parallel sides 40 m and 60 m has an area of 750 m². What is the height of the trapezium?

Solution:
Given that,
Length of the parallel sides = 40 m and 60 m
Area = 750 m²

We know,
Area = (sum of the parallel sides) × height ÷ 2
⇒ 750 = (40 + 60) × height ÷ 2
⇒ 750 = 100 × height ÷ 2
⇒ 750 = 50 × height
⇒ height = 750/50
∴ height = 15 meters

So, the height of the trapezium is 15 meters.

Question: If the area of a square garden is 50 sq. meters, what is the maximum distance between two points on its boundary?

Solution:
Given that, 
Area of square = 50 m²
∴ Side of length = √Area = √50 =√(25 × 2)
= 5√2 meters

The maximum distance between any two points on the boundary of a square is the length of the diagonal.
∴ Diagonal of the square = side × √2
= 5√2 × √2
= 5 × 2
= 10 meters

So the maximum distance between two points on the boundary is 10 meters.