ব্যাংক ডেইলি কুইজ [লং কোর্সের অংশ]

Question: The length of the side of a square whose area is four times the area of a square with a side 25m is-

Solution:
Area of given square = 25² = 625 m²
Area of new square = 625 × 4 = 2500 m²
Side of new square = √2500 = 50 m

Question: If the volume of a cube is 2744 cm³, then the surface area of the cube will be -

Solution: 
দেওয়া আছে,
আয়তন, a³ = 2744
⇒ a³ = 14³
⇒ a = 14

পৃষ্ঠের ক্ষেত্রফল = 6a²
= 6 × 14²
= 6 × 196
= 1176 cm²

Question: The area of a square and rectangle are equal. The length of the rectangle is greater than the length of any side of the square by 5 cm and the breadth is less 3 cm. Find the perimeter of the rectangle.

Solution:
Let, the length of each side of the square be x cm.
Then, the length of rectangle = (x + 5) cm
and its breadth = (x - 3) cm

ATQ,
(x + 5)(x - 3) = x²
⇒ x² + 5x - 3x - 15 = x²
⇒ 2x = 15
∴ x = 15/2

Length = (15/2) + 5 = 25/2 cm
Breadth = (15/2) - 3 = 9/2 cm

∴ Perimeter = 2(length + breadth) = 2 {(25/2) + (9/2)}
= 2 (34/2) = 34 cm

Question: Three cubes with sides in the ratio 3 : 4 : 5 are melted fo form a single cube whose diagonal is 12√3 cm. The sides of the cubes are-

Solution:
Let, the sides of the three cubes be 3x, 4x and 5x.
Then, Volume of the new cube = (3x)³ + (4x)³ + (5x)³
= 216x³

Edge of the new cube, ‍a³ = 216x³
⇒ a = (216x³)^1/3
∴ a = 6x

The diagonal of the new cube = 6√3x

ATQ,
6√3x = 12√3
∴ x = 2

So, the side of cubes are (3 × 2) cm, (4 × 2) cm, (5 × 2) cm Or, 6 cm, 8 cm and 10 cm

Question: A rectangular carpet has an area of 120 sq. meters and a perimeter of 46 meters. The length of its diagonal is-

Solution:
Let, the length of carpet be x m and breadth the y m.

ATQ,
2(x + y) = 46
x + y = 23
and xy = 120

Diagonal = √(x² + y²)
= √{(x + y)² - 2xy}
= √(23² - 2 . 120)
= √(529 - 240)
= √289
= 17 m

Question: The area of a square inscribed in a circle is 140 cm². What is the area of the semi-circle?

Solution:
The area of a square inscribed in a circle is 140 cm²
side of square = √140 cm
= 2√35 cm

diagonal of the square = √2 × 2√35
= 2√70 cm

diameter of circle = 2√70 cm
radius of the circle = √70 cm
∴ area of the circle = π (√70)2 cm²
= (22/7) × 70 cm²
= 220 cm²

area of semi-circle = 220/2 
= 110 cm²

Question: The area of a circle is increased by 22 square cm if its radius is increased by 1 cm. The original radius of the circle is -

Solution:
Let the original radius of the circle be r cm.

ATQ,
π(r + 1)² - πr² = 22
⇒ π{(r + 1)² - r²} = 22
⇒ π(r² + 2r + 1 - r²) = 22
⇒ 2r + 1 = 22/π
⇒ 2r + 1 = (22 × 7)/22
⇒ 2r + 1 = 7
⇒ 2r = 6
∴ r = 3 cm

Question: If the radius of the base of a right circular cylinder is halved, keeping the height same, what is the ratio of the volume of the reduced cylinder to that of the original one?

Solution:
Let original radius = R
Then, new radius = R/2

Volume of reduced cylinder/Volume of original cylinder = π(R/2)²h/πR²h
= 1/4
= 1 : 4

Question: A copper sphere of diameter 18 cm is drawn into a wire of diameter 4 mm. Find the length of the wire.

Solution:
ব্যাসার্ধ, r = 18/2 = 9 সেমি
গোলকটির আয়তন = (4/3) × π × r³
= (4/3) × π × 9³
= 972π

তারটি ব্যাসার্ধ = 4/2 = 2 মিমি = 0.2 সেমি
তারটির আয়তন = πr²l
= π × (0.2)² × l
= 0.04πl

শর্তমতে,
0.04πl = 972π
⇒ l = 972/0.04
⇒ l = 24300 cm
⇒ l = 243 m