Every Book Back multiple-choice question from Mensuration (10th Standard Mathematics, Samacheer Kalvi) — each with the correct option highlighted and a clear, worked explanation. Free to read in English and Tamil.
Q1
The curved surface area of a right circular cone of height 15 cm and base diameter 16 cm is
- A. 60\\(\pi\) \\(\text{ cm}^2\)
- B. 68\\(\pi\) \\(\text{ cm}^2\)
- C. 120\\(\pi\) \\(\text{ cm}^2\)
- D. 136\\(\pi\) \\(\text{ cm}^2\)Correct
Explanation. The radius is 8 cm. Slant height is the square root of height squared plus radius squared, which equals 17 cm. The curved surface area is \\(\pi\) times radius times slant height, yielding 136\\(\pi\) \\(\text{ cm}^2\).
Q2
If two solid hemispheres of same base radius r units are joined together along their bases, then curved surface area of this new solid is
- A. 4\\(\pi r^2\) \\(\text{ sq. units}\)Correct
- B. 6\\(\pi r^2\) \\(\text{ sq. units}\)
- C. 3\\(\pi r^2\) \\(\text{ sq. units}\)
- D. 8\\(\pi r^2\) \\(\text{ sq. units}\)
Explanation. Joining two solid hemispheres along their identical bases forms a complete sphere. The curved surface area of a sphere of radius r is 4\\(\pi r^2\).
Q3
The height of a right circular cone whose radius is 5 cm and slant height is 13 cm will be
- A. 12 \\(\text{ cm}\)Correct
- B. 10 \\(\text{ cm}\)
- C. 13 \\(\text{ cm}\)
- D. 5 \\(\text{ cm}\)
Explanation. Using the Pythagorean relationship for a cone, height equals the square root of slant height squared minus radius squared, which is the square root of 144, or 12 cm.
Q4
If the radius of the base of a right circular cylinder is halved keeping the same height, then the ratio of the volume of the cylinder thus obtained to the volume of original cylinder is
- A. 1:2
- B. 1:4Correct
- C. 1:6
- D. 1:8
Explanation. The volume is proportional to the square of the radius. Halving the radius scales the volume by a factor of one-fourth. Thus, the ratio of the new volume to the original is 1:4.
Q5
The total surface area of a cylinder whose radius is \\(\frac{1}{3}\) of its height is
- A. \\(\frac{9}{8}\)\\(\pi h^2\) \\(\text{ sq. units}\)
- B. 24\\(\pi h^2\) \\(\text{ sq. units}\)
- C. \\(\frac{8}{9}\)\\(\pi h^2\) \\(\text{ sq. units}\)Correct
- D. \\(\frac{56}{9}\)\\(\pi h^2\) \\(\text{ sq. units}\)
Explanation. TSA of cylinder is 2\\(\pi r(r + h)\). Substituting r = h/3 gives 2\\(\pi (h/3)(4h/3)\), which simplifies to 8/9 \\(\pi h^2\).
Q6
In a hollow cylinder, the sum of the external and internal radii is 14 cm and the width is 4 cm. If its height is 20 cm, the volume of the material in it is
- A. 5600\\(\pi\) \\(\text{ cm}^3\)
- B. 1120\\(\pi\) \\(\text{ cm}^3\)Correct
- C. 56\\(\pi\) \\(\text{ cm}^3\)
- D. 3600\\(\pi\) \\(\text{ cm}^3\)
Explanation. Volume of a hollow cylinder is \\(\pi (R^2 - r^2)h =\) \\(\pi (R - r)(R + r)h\). Substituting width 4 cm, sum 14 cm, and height 20 cm gives \\(\pi\) \\(\times 4\) \\(\times 14\) \\(\times 20 = 1120\)\\(\pi\) \\(\text{ cm}^3\).
Q7
If the radius of the base of a cone is tripled and the height is doubled then the volume is
- A. made 6 times
- B. made 18 timesCorrect
- C. made 12 times
- D. unchanged
Explanation. Volume is proportional to the square of the radius and the height. Tripling the radius scales volume by nine, and doubling height scales it by two. The overall increase is 18 times.
Q8
The total surface area of a hemi-sphere is how much times the square of its radius.
- A. \\(\pi\)
- B. 4\\(\pi\)
- C. 3\\(\pi\)Correct
- D. 2\\(\pi\)
Explanation. The total surface area of a hemisphere includes the curved surface area and the base area, totaling 3\\(\pi r^2\). Therefore, it is 3\\(\pi\) times the square of its radius.
Q9
A solid sphere of radius x cm is melted and cast into a shape of a solid cone of same radius. The height of the cone is
- A. 3x \\(\text{ cm}\)
- B. x \\(\text{ cm}\)
- C. 4x \\(\text{ cm}\)Correct
- D. 2x \\(\text{ cm}\)
Explanation. Equating the volume of a sphere of radius x, 4/3 \\(\pi x^3\), to the volume of a cone with radius x and height h, 1/3 \\(\pi x^2 h\), gives h = 4x.
Q10
A frustum of a right circular cone is of height 16cm with radii of its ends as 8cm and 20cm. Then, the volume of the frustum is
- A. 3328\\(\pi\) \\(\text{ cm}^3\)Correct
- B. 3228\\(\pi\) \\(\text{ cm}^3\)
- C. 3240\\(\pi\) \\(\text{ cm}^3\)
- D. 3340\\(\pi\) \\(\text{ cm}^3\)
Explanation. Using the formula for frustum volume, V = 1/3 \\(\pi h(R^2 + r^2 +\) Rr). Here h = 16, R = 20, r = 8. Calculating gives 1/3 \\(\pi\) \\(\times 16\) \\(\times 624 = 3328\)\\(\pi\) \\(\text{ cm}^3\).
Q11
A shuttle cock used for playing badminton has the shape of the combination of
- A. a cylinder and a sphere
- B. a hemisphere and a cone
- C. a sphere and a cone
- D. frustum of a cone and a hemisphereCorrect
Explanation. A shuttlecock consists of a flared, cone-like skirt, which is a frustum of a cone, connected to a rounded base, which is a hemisphere.
Q12
A spherical ball of radius \(r_1\) units is melted to make 8 new identical balls each of radius \(r_2\) units. Then \(r_1 : r_2\) is
- A. 2:1Correct
- B. 1:2
- C. 4:1
- D. 1:4
Explanation. Since the sum of volumes of the 8 smaller balls equals the volume of the larger ball\(, r_1^3 = 8 r_2^3\). Taking the cube root, we find \(r_1 = 2 r_2\), or 2:1.
Q13
The volume (in \\(\text{cm}^3\)) of the greatest sphere that can be cut off from a cylindrical log of wood of base radius 1 cm and height 5 cm is
- A. \\(\frac{4}{3}\)\\(\pi\)Correct
- B. \\(\frac{10}{3}\)\\(\pi\)
- C. 5\\(\pi\)
- D. \\(\frac{20}{3}\)\\(\pi\)
Explanation. The largest sphere is limited by the cylinder's diameter, which is 2 cm. This sphere has a maximum radius of 1 cm, so its volume is 4/3 \\(\pi\) \\(\text{ cm}^3\).
Q14
The height and radius of the cone of which the frustum is a part are \(h_1\) units and \(r_1\) units respectively. Height of the frustum is \(h_2\) units and radius of the smaller base is \(r_2\) units. If \(h_2 : h_1 = 1 : 2\) then \(r_2 : r_1\) is
- A. 1:3
- B. 1:2Correct
- C. 2:1
- D. 3:1
Explanation. By similarity of triangles\(, r_2/r_1 = (h_1 - h_2)/h_1 = 1 - h_2/h_1\). Since \(h_2/h_1 = 1/2\), the ratio \(r_2:r_1\) is 1:2.
Q15
The ratio of the volumes of a cylinder, a cone and a sphere, if each has the same diameter and same height is
- A. 1:2:3
- B. 2:1:3
- C. 1:3:2
- D. 3:1:2Correct
Explanation. Let diameter be 2r and height h = 2r. The volumes of the cylinder, cone, and sphere are 2\\(\pi r^3, 2/3\) \\(\pi r^3\), and 4/3 \\(\pi r^3\). The ratio simplifies to 3:1:2.
