Question 1

1. A wire in the form of a square with a side of 22 cm is reshaped into the form of a circle. What is the area of the circle formed? (Take $\pi = \frac{22}{7}$)

A) $484cm^2$B) $616cm^2$C) $154cm^2$D) $308cm^2$

Answer

✓ Correct Answer: Option B

Explanation: Solution: 1. Perimeter of the square = $4 \times side = 4 \times 22 = 88cm$. 2. When reshaped into a circle, the circumference of the circle is equal to the perimeter of the square. $$2\pi r = 88$ $2 \times \frac{22}{7} \times r = 88 ⇒ r = \frac{88 \times 7}{44} = 14cm$ 3. Area of the circle =\pi r^2 = \frac{22}{7} \times 14 \times 14 = 22 \times 28 = 616cm^2$.

Question 2

2. A solid metallic cylinder of radius 6 cm and height 15 cm is melted to form a number of small solid cones, each of radius 3 cm and height 9 cm. How many such cones can be formed?

15

Question 3

3. The length of a rectangular park exceeds its breadth by 8 m. If the area of the park is $240m^2$, find the perimeter of the park.

A) 60 mB) 64 mC) 68 mD) 72 m

Answer

✓ Correct Answer: Option B

Explanation: Solution: Let breadth = B. Then length L = B + 8. Area $= L \times B = (B + 8)B = 240$. $$B^2 + 8B - 240 = 0$ (B + 20)(B - 12) = 0 ⇒ B = 12m(Breadth cannot be negative). LengthL = 12 + 8 = 20m. Perimeter= 2(L + B) = 2(20 + 12) = 2(32) = 64m$.

Question 4

4. A sphere is exactly inscribed inside a cube. What is the ratio of the volume of the cube to the volume of the sphere?

A) $6 : \pi$B) $\pi : 6$C) $3 : 4\pi$D) $4 : 3\pi$

Answer

✓ Correct Answer: Option A

Explanation: Solution: 1. Let the side of the cube be a. 2. Since the sphere is inscribed in the cube, its diameter is equal to the side of the cube, so radius $r = \frac{a}{2}$. 3. Volume of Cube $= a^3$. 4. Volume of Sphere $= \frac{4}{3} \pi r^3 = \frac{4}{3} \pi (\frac{a}{2})^3 = \frac{4}{3} \pi \frac{a^3}{8} = \frac{\pi a^3}{6}$. 5. Ratio $= a^3 : \frac{\pi a^3}{6} = 1 : \frac{\pi}{6} = 6 : \pi$.

Question 5

5. The diagonals of a rhombus are 16 cm and 12 cm. Find the height of the rhombus if its area is considered with one of its sides as the base.

A) 8.4 cmB) 9.2 cmC) 9.6 cmD) 10.4 cm

Answer

✓ Correct Answer: Option C

Explanation: Solution: 1. Area of rhombus $= \frac{1}{2} \times d_1 \times d_2 = \frac{1}{2} \times 16 \times 12 = 96cm^2$. 2. Side of rhombus $s = \sqrt{(\frac{d_1}{2})^2 + (\frac{d_2}{2})^2} = \sqrt{8^2 + 6^2} = \sqrt{100} = 10cm$. 3. Area can also be expressed as Base $\times$ Height: $s \times h = 96$. $$10 \times h = 96 ⇒ h = 9.6cm$$.

Question 6

6. A right prism has a triangular base with sides 3 cm, 4 cm, and 5 cm. If the height of the prism is 10 cm, what is the total surface area?

A) $120cm^2$B) $126cm^2$C) $132cm^2$D) $144cm^2$

Answer

✓ Correct Answer: Option C

Explanation: Solution: 1. Base is a right-angled triangle (since $3^2 + 4^2 = 5^2$). 2. Base Area $= \frac{1}{2} \times 3 \times 4 = 6cm^2$. 3. Perimeter of Base = 3 + 4 + 5 = 12cm. 4. Lateral Surface Area $= Perimeter \times Height = 12 \times 10 = 120cm^2$. 5. Total Surface Area $= LSA + 2 \times Base Area = 120 + 2(6) = 132cm^2$.

Question 7

7. Find the area of an isosceles triangle whose base is 12 cm and each of its equal sides is 10 cm.

A) $48cm^2$B) $60cm^2$C) $36cm^2$D) $54cm^2$

Answer

✓ Correct Answer: Option A

Explanation: Solution: 1. Base b = 12, equal sides a = 10. 2. Height $h = \sqrt{a^2 - (\frac{b}{2})^2} = \sqrt{10^2 - 6^2} = \sqrt{100 - 36} = \sqrt{64} = 8cm$. 3. Area $= \frac{1}{2} \times base \times height = \frac{1}{2} \times 12 \times 8 = 48cm^2$.

Question 8

8. A hollow cylindrical pipe is 7 cm long. The external radius of the pipe is 5 cm and the volume of the metal used in making the pipe is $352cm^3$. Find the thickness of the pipe. (Use $\pi = \frac{22}{7}$)

A) 1 cmB) 1.5 cmC) 2 cmD) 2.5 cm

Answer

✓ Correct Answer: Option C

Explanation: Solution: 1. Volume of metal $= \pi(R^2 - r^2)h$, where R=5, h=7, V=352. $$352 = \frac{22}{7} \times (5^2 - r^2) \times 7$ $352 = 22 \times (25 - r^2) ⇒ 16 = 25 - r^2$ $r^2 = 9 ⇒ r = 3cm$. 2. Thickness= R - r = 5 - 3 = 2cm$.

Question 9

9. The length of the arc of a sector of a circle is 44 cm and the radius of the circle is 14 cm. What is the area of the sector?

A) $154cm^2$B) $308cm^2$C) $616cm^2$D) $440cm^2$

Answer

✓ Correct Answer: Option B

Explanation: Solution: 1. Area of a sector can be calculated using the arc length (L) and radius (r): $$Area = \frac{1}{2} \times L \times r$ $Area = \frac{1}{2} \times 44 \times 14 = 22 \times 14 = 308cm^2$$.

Question 10

10. The radii of the top and bottom of a bucket (frustum of a cone) are 20 cm and 8 cm respectively. If the height of the bucket is 16 cm, find its capacity in terms of $\pi$.

A) $3120\pi cm^3$B) $3224\pi cm^3$C) $3328\pi cm^3$D) $3416\pi cm^3$

Answer

✓ Correct Answer: Option C

Explanation: Solution: 1. Volume of Frustum $= \frac{1}{3} \pi h (R^2 + r^2 + Rr)$. 2. Here R = 20, r = 8, h = 16. $$V = \frac{1}{3} \pi \times 16 \times (20^2 + 8^2 + 20 \times 8)$ $V = \frac{1}{3} \pi \times 16 \times (400 + 64 + 160) = \frac{1}{3} \pi \times 16 \times 624$ $V = \pi \times 16 \times 208 = 3328\pi cm^3$$.

Accepted Answer

✓ Correct Answer: Option C

Explanation: Solution: 1. Volume of Cylinder = $\pi R^2 H = \pi \times 6^2 \times 15 = 540\pi cm^3$. 2. Volume of one Cone = $\frac{1}{3} \pi r^2 h = \frac{1}{3} \pi \times 3^2 \times 9 = 27\pi cm^3$. 3. Number of cones = $\frac{Volume of Cylinder}{Volume of Cone} = \frac{540\pi}{27\pi} = 20$.

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

1. A wire in the form of a square with a side of 22 cm is reshaped into the form of a circle. What is the area of the circle formed? (Take $\pi = \frac{22}{7}$)

2. A solid metallic cylinder of radius 6 cm and height 15 cm is melted to form a number of small solid cones, each of radius 3 cm and height 9 cm. How many such cones can be formed?

3. The length of a rectangular park exceeds its breadth by 8 m. If the area of the park is $240m^2$, find the perimeter of the park.

4. A sphere is exactly inscribed inside a cube. What is the ratio of the volume of the cube to the volume of the sphere?

5. The diagonals of a rhombus are 16 cm and 12 cm. Find the height of the rhombus if its area is considered with one of its sides as the base.

6. A right prism has a triangular base with sides 3 cm, 4 cm, and 5 cm. If the height of the prism is 10 cm, what is the total surface area?

7. Find the area of an isosceles triangle whose base is 12 cm and each of its equal sides is 10 cm.

8. A hollow cylindrical pipe is 7 cm long. The external radius of the pipe is 5 cm and the volume of the metal used in making the pipe is $352cm^3$. Find the thickness of the pipe. (Use $\pi = \frac{22}{7}$)

9. The length of the arc of a sector of a circle is 44 cm and the radius of the circle is 14 cm. What is the area of the sector?

10. The radii of the top and bottom of a bucket (frustum of a cone) are 20 cm and 8 cm respectively. If the height of the bucket is 16 cm, find its capacity in terms of $\pi$.