Question 1

1. A rectangular field is 32 m long and 24 m wide. A path of uniform width 3 m is constructed inside the field along its boundary. Find the cost of paving this path with bricks at the rate of ₹25 per square meter.

A) ₹8,250B) ₹7,500C) ₹6,900D) ₹9,100

Answer

✓ Correct Answer: Option B

Explanation: Solution: Area of the outer rectangle = $32 \times 24 = 768 m^2$. Width of path = 3 m. Dimensions of inner rectangle = $(32 - 2 \times 3) \times (24 - 2 \times 3) = 26 \times 18 = 468 m^2$. Area of the path = Outer Area - Inner Area = $768 - 468 = 300 m^2$. Total cost = $300 \times 25 = ₹7,500$.

Question 2

2. A solid metallic cylinder of base radius 12 cm and height 15 cm is melted and recast into small solid cones, each of radius 3 cm and height 5 cm. How many such cones can be formed?

124

Question 3

3. A wire is in the form of a circle of radius 28 cm. If it is bent into the shape of a square, what is the difference between the area of the circle and the area of the square? (Use $\pi = \frac{22}{7}$)

A) $528 cm^2$B) $484 cm^2$C) $612 cm^2$D) $506 cm^2$

Answer

✓ Correct Answer: Option A

Explanation: Solution: Length of wire = Circumference of circle = $2\pi r = 2 \times \frac{22}{7} \times 28 = 176 cm$. Area of circle = $\pi r^2 = \frac{22}{7} \times 28 \times 28 = 2464 cm^2$. Perimeter of square = 176 cm \Rightarrow Side of square (a) = $\frac{176}{4} = 44 cm$. Area of square = $a^2 = 44^2 = 1936 cm^2$. Difference = $2464 - 1936 = 528 cm^2$.

Question 4

4. If the total surface area of a sphere is equal to the total surface area of a hemisphere, find the ratio of their volumes.

A) $1 : \sqrt{2}$B) $3\sqrt{3} : 8$C) $\sqrt{3} : 2$D) $3\sqrt{3} : 4$

Answer

✓ Correct Answer: Option D

Explanation: Solution: Let radius of sphere be R and hemisphere be r. TSA of Sphere = $4\pi R^2$; TSA of Hemisphere = $3\pi r^2$. Given: $4\pi R^2 = 3\pi r^2 \Rightarrow \frac{R^2}{r^2} = \frac{3}{4} \Rightarrow \frac{R}{r} = \frac{\sqrt{3}}{2}$. Ratio of Volumes = $\frac{\frac{4}{3}\pi R^3}{\frac{2}{3}\pi r^3} = \frac{2 R^3}{r^3} = 2 \times \left(\frac{\sqrt{3}}{2}\right)^3 = 2 \times \frac{3\sqrt{3}}{8} = \frac{3\sqrt{3}}{4}$.

Question 5

5. The perimeter of a rhombus is 100 cm and one of its diagonals is 30 cm. Find the area of the rhombus.

A) $500 cm^2$B) $750 cm^2$C) $600 cm^2$D) $450 cm^2$

Answer

✓ Correct Answer: Option C

Explanation: Solution: Perimeter = $4 \times side = 100 \Rightarrow side (s) = 25 cm$. Diagonals of a rhombus bisect each other at $90^\circ$. Let diagonals be $d_1 = 30$ and $d_2$. Half-diagonals are 15 and $\frac{d_2}{2}$. Using Pythagoras: $15^2 + (\frac{d_2}{2})^2 = 25^2 \Rightarrow 225 + (\frac{d_2}{2})^2 = 625$. $(\frac{d_2}{2})^2 = 400 \Rightarrow \frac{d_2}{2} = 20 \Rightarrow d_2 = 40 cm$. Area = $\frac{1}{2} \times d_1 \times d_2 = \frac{1}{2} \times 30 \times 40 = 600 cm^2$.

Question 6

6. A rectangular water tank of base dimensions $8 m \times 6 m$ and height 2.5 m is completely full of water. If 3000 liters of water are drawn out, find the drop in the water level.

A) 5.5 cmB) 6.25 cmC) 7.5 cmD) 4.75 cm

Answer

✓ Correct Answer: Option B

Explanation: Solution: Volume drawn out = 3000 liters. Since 1000 liters = $1 m^3$, Volume = $3 m^3$. Let the drop in water level be h meters. Volume of water removed = Area of base $\times$ Drop in height. $3 = (8 \times 6) \times h \Rightarrow 3 = 48h \Rightarrow h = \frac{3}{48} = \frac{1}{16} m$. In cm: $h = \frac{1}{16} \times 100 = 6.25 cm$.

Question 7

7. The area of a trapezium is $480 cm^2$. The ratio of its parallel sides is 3:5 and the distance between them is 12 cm. Find the length of the shorter parallel side.

A) 30 cmB) 50 cmC) 45 cmD) 24 cm

Answer

✓ Correct Answer: Option A

Explanation: Solution: Let parallel sides be 3x and 5x. Height h = 12. Area = $\frac{1}{2} \times (sum of parallel sides) \times h$. $480 = \frac{1}{2} \times (3x + 5x) \times 12 \Rightarrow 480 = 6 \times 8x$. 480 = 48x \Rightarrow x = 10. Shorter side = $3x = 3 \times 10 = 30 cm$.

Question 8

8. A solid right circular cone is cut into two parts at one-third of its height from the top by a plane parallel to its base. Find the ratio of the volume of the small cone to the volume of the resulting frustum.

A) 1 : 27B) 1 : 8C) 1 : 26D) 1 : 9

Answer

✓ Correct Answer: Option C

Explanation: Solution: Let the height of the original cone be H and the small cone be $h = \frac{H}{3}$. By similarity, the ratio of radii is also 1:3. Ratio of Volumes = $(ratio of heights)^3 = (\frac{1}{3})^3 = \frac{1}{27}$. If Volume of small cone = 1 unit, then Volume of original cone = 27 units. Volume of frustum = Original Volume - Small Cone Volume = 27 - 1 = 26 units. Ratio = 1 : 26.

Question 9

9. Find the length of the arc of a sector of a circle with radius 21 cm and a central angle of $120^\circ$.

A) 22 cmB) 33 cmC) 66 cmD) 44 cm

Answer

✓ Correct Answer: Option D

Explanation: Solution: Length of arc = $\frac{\theta}{360} \times 2\pi r$. Length = $\frac{120}{360} \times 2 \times \frac{22}{7} \times 21$. Length = $\frac{1}{3} \times 2 \times 22 \times 3 = 44 cm$.

Question 10

10. The lateral surface area of a hollow cylinder is $4224 cm^2$. It is cut along its height and formed into a rectangular sheet of width 33 cm. Find the perimeter of the rectangular sheet.

A) 256 cmB) 322 cmC) 294 cmD) 310 cm

Answer

✓ Correct Answer: Option B

Explanation: Solution: LSA of cylinder = $2\pi rh = 4224$. When cut along height, the width of the sheet equals the height of the cylinder (h = 33 cm). The length of the sheet (L) equals the circumference of the base ($2\pi r$). $L \times 33 = 4224 \Rightarrow L = \frac{4224}{33} = 128 cm$. Perimeter of sheet = 2(L + W) = 2(128 + 33) = 2(161) = 322 cm.

Accepted Answer

✓ Correct Answer: Option C

Explanation: Solution: Volume of Cylinder = $\pi R^2 H = \pi \times 12^2 \times 15$. Volume of one Cone = $\frac{1}{3} \pi r^2 h = \frac{1}{3} \pi \times 3^2 \times 5$. Number of cones (n) = $\frac{Volume of Cylinder}{Volume of Cone}$ $n = \frac{\pi \times 144 \times 15}{\frac{1}{3} \pi \times 9 \times 5} = \frac{144 \times 15}{3 \times 5} = 144$.

Competitive Exams : Quantitative Aptitude - Mensuration (2D/3D) (Set 1)

1. A rectangular field is 32 m long and 24 m wide. A path of uniform width 3 m is constructed inside the field along its boundary. Find the cost of paving this path with bricks at the rate of ₹25 per square meter.

2. A solid metallic cylinder of base radius 12 cm and height 15 cm is melted and recast into small solid cones, each of radius 3 cm and height 5 cm. How many such cones can be formed?

3. A wire is in the form of a circle of radius 28 cm. If it is bent into the shape of a square, what is the difference between the area of the circle and the area of the square? (Use $\pi = \frac{22}{7}$)

4. If the total surface area of a sphere is equal to the total surface area of a hemisphere, find the ratio of their volumes.

5. The perimeter of a rhombus is 100 cm and one of its diagonals is 30 cm. Find the area of the rhombus.

6. A rectangular water tank of base dimensions $8 m \times 6 m$ and height 2.5 m is completely full of water. If 3000 liters of water are drawn out, find the drop in the water level.

7. The area of a trapezium is $480 cm^2$. The ratio of its parallel sides is 3:5 and the distance between them is 12 cm. Find the length of the shorter parallel side.

8. A solid right circular cone is cut into two parts at one-third of its height from the top by a plane parallel to its base. Find the ratio of the volume of the small cone to the volume of the resulting frustum.

9. Find the length of the arc of a sector of a circle with radius 21 cm and a central angle of $120^\circ$.

10. The lateral surface area of a hollow cylinder is $4224 cm^2$. It is cut along its height and formed into a rectangular sheet of width 33 cm. Find the perimeter of the rectangular sheet.