Four circles of equal radius are drawn with centers, A, B, C and D such that ABCD is a square of side 14 cm and the circles touch externally as in the figure. The area of the shaded region bounded by the 4 circles is: (Take )
1
24 cm
2
42 cm
3
96 cm
4
54 cm
Official Solution
Correct Option: (2)
Step 1: Understanding the Concept:
The shaded area can be found by calculating the area of the square ABCD and subtracting the areas of the four circular sectors that are inside the square. Step 2: Key Formula or Approach:
1. Determine the radius of the circles.
2. Calculate the area of the square.
3. Calculate the area of the four sectors inside the square.
4. Area of Shaded Region = Area of Square - Area of 4 Sectors. Step 3: Detailed Explanation:
The side of the square ABCD is given as 14 cm. Since the circles with centers A and B (or A and D) touch externally, the side of the square is the sum of the radii of two circles.
Side of square = radius + radius = 2 radius.
cm.
The radius of each circle is 7 cm.
Area of the square ABCD = cm .
The four sectors inside the square are at the corners. Since ABCD is a square, the angle of each corner is 90 . So, each sector has a central angle of 90 .
The sum of the angles of the four sectors is , which is the angle of a complete circle.
Therefore, the combined area of the four sectors is equal to the area of one full circle with radius r = 7 cm.
Area of 4 sectors = Area of one circle = .
Area of 4 sectors = cm .
Now, calculate the area of the shaded region:
Area of Shaded Region = Area of Square - Area of 4 Sectors
Area of Shaded Region = cm . Step 4: Final Answer:
The area of the shaded region is 42 cm .
02
PYQ 2025
medium
general-aptitudeID: cuet-ug-
The base diameter of a cylinder is 21 cm and the height is 28 cm, then: (A) Radius of cylinder = 10.5 cm (B) Volume = 12936 cm (C) Curved Surface Area = 1848 cm (D) Total surface area = 2541 cm Which of the following is/ are correct? Choose the correct answer from the options given below:
1
(A), (B) and (D) only
2
(A), (C) and (D) only
3
(B), (C) and (D) only
4
(A), (B) and (C) only
Official Solution
Correct Option: (2)
Step 1: Understanding the Concept:
This question requires calculating the radius, volume, curved surface area (CSA), and total surface area (TSA) of a cylinder and verifying the given statements. Step 2: Key Formula or Approach:
Given: Diameter (d) = 21 cm, Height (h) = 28 cm.
Radius (r) = d/2
Volume (V) =
Curved Surface Area (CSA) =
Total Surface Area (TSA) =
Use . Step 3: Detailed Explanation: (A) Radius of cylinder:
Radius r = cm. Statement (A) is correct. (B) Volume:
V =
V =
Simplify by cancelling terms: .
V = cm .
The statement says Volume = 12936 cm . Thus, statement (B) is incorrect. (C) Curved Surface Area:
CSA =
CSA =
Simplify: .
CSA = cm .
Statement (C) is correct. (D) Total surface area:
TSA = = CSA + 2( )
TSA =
TSA =
TSA = cm .
Statement (D) is correct. The correct statements are (A), (C), and (D). Step 4: Final Answer:
The correct option is (2) which includes statements (A), (C), and (D) only.
