Mensuration: Sphere, Cone, Cylinder and Mixed Solids MCQ Chapter Test CBSE, ICSE

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25 Multiple Choice Questions (MCQs) on Mensuration Chapters Covered: Sphere, Cone, Cylinder and Mixed Solids This question set is designed to strengthen conceptual understanding and problem-solving skills in mensuration, focusing on surface area, volume, and real-life applications of three-dimensional solids. Applicable for: ICSE, CBSE, WBBSE and other State Board curricula Recommended for: Secondary level students (Class IX–X)

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Mensuration Chapter Test

Test your understanding of three dimensional solids- Sphere, Cone & Cylinder

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Question 1

A circular metallic sheet is divided into two parts in such a way that each part can be folded in to a cone. If the ratio of their curved surface areas is \(1 : 2\), then the ratio of their volumes is

A\( 1:8\) B\( 1:\sqrt{6}\) C\( 1:\sqrt{10}\) D\( 2:3\)

Question 2

There is a right circular cone of height \(h\) and vertical angle \(60^{\circ} \) . A sphere when placed inside the cone, it touches the curved surface and the base of the cone. The volume of sphere is

A\(\frac{4}{3} \pi h^{3}\) B\(\frac{4}{9} \pi h^{3}\) C\(\frac{4}{27} \pi h^{3}\) D\(\frac{4}{81} \pi h^{3}\)

Question 3

A sealed bottle containing some water is made up of two cylinders A and B of radius 1.5 cm and 3 cm respectively; as shown in the figure. When the bottle is placed right up on a table, the height of water in it is 15 cm, but when placed upside down, the height of water is 24 cm. The height of the bottle is

A\( 25 \quad cm \) B\( 26 \quad cm \) C\( 27 \quad cm \) D\( 28 \quad cm \)

Question 4

A solid sphere is cut into identical pieces by three mutually perpendicular planes passing through its centre. Increase in total surfae area of all the pieces with respect to the total surface area of the original sphere is

A\(250\)% B\(175\)% C\(150\)% D\(125\)%

Question 5

Let the volume of a solid sphere be \(288\pi \quad cm^3 \). A horizontal plane cuts the sphere at a distance of \(3\) cm from the centre so that the ratio of the curved surface areas of the two parts of the sphere is \(3 : 1\). The total surface area of the bigger part of the sphere (in \(cm^2\) ) is:

A\(36\pi\) B\(108\pi\) C\(135\pi\) D\(1144\pi\)

Question 6

A solid metallic cylinder of height \(10\) cm and diameter \(14\) cm is melted to make two cones in the proportion of their volumes as \(3 : 4\), keeping the height \(10\) cm, what would be the percentage increase in the flat surface area?

A\( 9 \) B\( 16 \) C\( 50 \) D\( 200 \)

Question 7

A copper wire \(3\) mm in diameter is rounded about a cylinder whose length is \(1.2\) m and diameter is \(10\) cm, so as to cover the curved surface of the cylinder. The length of the wire is

A\( 125.6 \)m B\( 1256 \)m C\( 12.56 \)m D\( 1.256 \)m

Question 8

A piece of paper is in the form of a sector, making an angle \(\alpha\). The paper is rolled to form a right circular cone of radius \(5\) cm and height \(12\) cm. Then the value of angle \(\alpha\) is

A\(\frac{10\pi}{13}\) B\(\frac{9\pi}{13}\) C\(\frac{5\pi}{13}\) D\(\frac{6\pi}{13}\)

Question 9

Surface area of a cube is double that of surface area of a sphere. The ratio of their volumes is given by

A\(\sqrt{\frac{3}{\pi}}\) B\(2\sqrt{\frac{\pi}{3}}\) C\(4\sqrt{\frac{3\pi}{2}}\) D\(3\sqrt{\frac{4}{\pi}}\)

Question 10

A right circular cone has for its base a circle having the same radius as a given sphere. The volume of the cone is one-half that of the sphere. The ratio of the altitude of the cone to the radius of its base is:

A\(1:1\) B\(1:2\) C\(2:1\) D\(2:3\)

Question 11

A rectangle of length \(a\) and breadth \(b\) is revolved \(360°\) about its length. The volume of the resulting cylinder is :

A\(\pi ab^2\) cu. units B\(\pi a^2b\) cu. units C\(\pi ab\) cu. units D\(2\pi ab\) cu. units

Question 12

The diameter of a solid metallic right circular cylinder is equal to its height. After cutting out the largest possible solid sphere \(S\) from this cylinder, the remaining material is recast to form a solid sphere \(S_1\). What is the ratio of the radius of sphere \(S\) to that of sphere \(S_1\)?

A\(1:2^{1/3}\) B\(2^{1/3} :1 \) C\(2^{1/3} :3^{1/3}\) D\(3^{1/3}:2^{1/3}\)

Question 13

The radius of a cylindrical box is \(8\) cm and the height is \(3\) cm. The number of cm that may be added to either the radius or the height so that in either case the volume of the cylinder increases by same magnitude is:

A\(1\) B\(5\frac{1}{3}\) C\(7\frac{1}{2}\) D\(24\)

Question 14

The diameter of a right circular cylinder is decreased by \(10\)%. The volume of cylinder remains the same then the percentage increase in height is:

A\( 20 \)% B\( 23.45 \)% C\( 5 \)% D\( 20.5 \)%

Question 15

The volume of a sphere having radius \(\sqrt[3]{2}\) cm is equal to the volume of a right circular cone whose lateral surface area is three times of the area of the base. The altitude of the cone is

A 4 cm B6 cm C8 cm D10 cm

Question 16

A right circular cylinder has its height equal to two times its radius. It is inscribed in a right circular cone having its diameter equal to \(10\) cm and height \(12\) cm, and the axes of both the cylinder and the cone coincide. Then, the volume (in \(cm^3\) ) of the cylinder is approximately

A107.5 B118.6 C127.5 D128.7

Question 17

A solid sphere of redius \(x\) cm is melted and cast into the shape of a solid cone of height \(x\) cm, the radius of the base of the cone is:

A\( x \) cm B\( 4x \) cm C\( 3x \) cm D\( 2x \) cm

Question 18

Four pipes each of \(5\) cm in diameter are to be replaced by a single pipe discharging the same quantity of water. If the speed of water remains same in both the cases, then the diameter (in cm) of the single pipe is:

A5 B6 C10 D12

Question 19

A copper rod of diameter \(1\) cm and length \(8\) cm is drawn into a wire of length \(18\) m of uniform thickness. The thickness of the wire is (1) 0.67 mm (2) 1/30 cm (3) 0.5 mm (4) 0.7 cm

A\(0.67\) mm B\(\frac{1}{30} \) cm C\(0.5\) mm D\(0.7\) cm

Question 20

A solid hemispherical shell of outer and inner radius \(R\) cm and \(r\) cm respectively is cut by a horizontal plane along its centre. The ratio of the total surface area of two new solid hemisphercal shell to the original spherical shell is:

A\(\frac{4R^2+r^2}{2(R^2+r^2)}\) B\(\frac{3R^2+r^2}{2(R^2+r^2)}\) C\(\frac{3R^2+2r^2}{2(R^2+r^2)}\) DNone of the above

Question 21

A right circular cylinder of volume \(1386\) cm³ is cut from a right circular cylinder of radius \(4\) cm and height \(49\) cm, such that a hollow cylinder of uniform thickness, with a height of \(49\) cm and an outer radius of \(4\) cm is left behind. The thickness of the hollow cylinder left behind is:

A\( 0.5 \) cm B\(2\) cm C\( 2 \) cm D\( 1 \) cm

Question 22

A conical cup when filled with ice cream forms a hemispherical shape on its open end. Find the volume of ice cream (approximately), if radius of the base of the cone is \(3.5\) cm, the vertical height of cone is \(7\) cm and width of the cone is negligible.

A120 cm³ B150 cm³ C180 cm³ D210 cm³

Question 23

Shown below is a horizontal water tank composed of a cylinder and two hemispheres. The tank is filled up to a height of \(7\) m. Find the surface area of the tank in contact with water. Use \(\pi = \frac{22}{7}\) ​ .

A\( 784 \quad m^2 \) B\( 648 \quad m^2 \) C\( 728 \quad m^2 \) D\(748 \quad m^2 \)

Question 24

A tent is in the shape of a cylinder surmounted by a conical top. If the height and radius of the cylindrical part are \(7\) m each and the total height of the tent is \(14\) m. Find the radius of a sphere whose volume is equal to the quantity of air inside the tent. Use \(\pi = \frac{22}{7}\)

A\( 7 \quad m \) B\( 6 \quad m \) C\( 14 \quad m \) D\( 8 \quad m \)

Question 25

In the diagram given below, a tilted right circular cylindrical vessel with base diameter \(7\) cm contains a liquid. When placed vertically, the height of the liquid in the vessel is the mean of two heights shown in the diagram. Find the area of wet surface, when the cylinder is placed vertically on a horizontal surface. Use \(\pi = \frac{22}{7}\)

A\(125. 5 \quad m^2\) B\(115. 5 \quad m^2\) C\(110. 5 \quad m^2\) D\(135. 5 \quad m^2\)

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Real Numbers : Assessment for classes IX and X

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25 Question MCQ Chapter on Real Numbers for CBSE, ICSE, Madhyamik and other State Boards Class IX & X

👨‍🏫 Author: Singh

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Real Numbers Quiz

Test your understanding of rational, irrational numbers and their properties. The study of Real Numbers in Classes IX and X establishes the foundation for all higher-level mathematics.Real Numbe: The set including all Rational and Irrational numbers. Rational Numbers: Numbers expressible as p/q, q is non-zero . Their decimal expansions are either terminating or non-terminating recurring.Irrational Numbers: Numbers that cannot be written as fractions. Their decimal expansions are non-terminating non-recurring.Number Line: Every point on a number line represents a unique real number.

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Question 1

Which of the following is an irrational number?

A\(\sqrt{25}\) B\(\sqrt{2} + \sqrt{3}\) C\(\frac{3}{7}\) D\(0.\overline{3}\)

Question 2

The decimal expansion of \(\frac{13}{625}\) will terminate after how many decimal places?

A2 B3 C4 D5

Question 3

Which of the following has a terminating decimal expansion?

A\(\frac{77}{210}\) B\(\frac{23}{30}\) C\(\frac{125}{441}\) D\(\frac{129}{2^2 \times 5^7}\)

Question 4

The simplified value of \(\sqrt{12} + \sqrt{27} - \sqrt{75}\) is:

A\(\sqrt{3}\) B\(2\sqrt{3}\) C\(3\sqrt{3}\) D\(0\)

Question 5

If \(p\) and \(q\) are co-prime numbers, then \(p^2\) and \(q^2\) are:

ACo-prime BNot co-prime CEven numbers DOdd numbers

Question 6

A rational number between \(\sqrt{2}\) and \(\sqrt{3}\) is:

A\(\frac{\sqrt{2} + \sqrt{3}}{2}\) B\(\frac{\sqrt{2} \times \sqrt{3}}{2}\) C\(1.5\) D\(1.8\)

Question 7

The value of \(1.\overline{36} + 0.\overline{9}\) is:

A\(2.\overline{27}\) B\(2.\overline{26}\) C\(2.\overline{3}\) D\(2.3\)

Question 8

Which of the following is not an irrational number?

A\(\sqrt{8} \times \sqrt{2}\) B\(\sqrt{18} \div \sqrt{2}\) C\(\sqrt{5} + \sqrt{10}\) D\(\sqrt{7} \times \sqrt{28}\)

Question 9

If \(n\) is a natural number, then \(\sqrt{n}\) is:

AAlways rational BAlways irrational CRational if \(n\) is a perfect square DIrrational if \(n\) is a perfect square

Question 10

The product of a non-zero rational and an irrational number is:

AAlways rational BAlways irrational CCan be rational or irrational DAlways an integer

Question 11

Which of the following rational numbers have terminating decimal representation?

A\(\frac{16}{125}\) B\(\frac{27}{250}\) C\(\frac{343}{2^3 \times 5^3}\) DAll of these

Question 12

The ascending order of \(\sqrt[3]{2}, \sqrt{3}, \sqrt[6]{5}\) is:

A\(\sqrt[3]{2} < \sqrt[6]{5} < \sqrt{3}\) B\(\sqrt[6]{5} < \sqrt[3]{2} < \sqrt{3}\) C\(\sqrt[6]{5} < \sqrt{3} < \sqrt[3]{2}\) D\(\sqrt{3} < \sqrt[3]{2} < \sqrt[6]{5}\)

Question 13

The decimal expansion of the rational number \(\frac{14587}{1250}\) will terminate after:

AOne decimal place BTwo decimal places CThree decimal places DFour decimal places

Question 14

After rationalizing the denominator of \(\frac{7}{3\sqrt{3} - 2\sqrt{2}}\), we get:

A\(3\sqrt{3} + 2\sqrt{2}\) B\(3\sqrt{3} - 2\sqrt{2}\) C\(7(3\sqrt{3} + 2\sqrt{2})\) D\(7(3\sqrt{3} - 2\sqrt{2})\)

Question 15

An irrational number between \(\frac{1}{7}\) and \(\frac{2}{7}\) is:

A\(\frac{1}{2}\left(\frac{1}{7} + \frac{2}{7}\right)\) B\(\sqrt{\frac{1}{7} \times \frac{2}{7}}\) C\(\frac{1}{\sqrt{14}}\) D\(\sqrt{0.14}\)

Question 16

If \(x = 2 + \sqrt{3}\), then \(x + \frac{1}{x}\) equals:

A\(4\) B\(2\sqrt{3}\) C\(-4\) D\(-2\sqrt{3}\)

Question 17

Which of the following is a rational number?

A\(\pi\) B\(\sqrt{2}\) C\(0.\overline{101}\) D\(\sqrt{5} + 1\)

Question 18

The simplest rationalizing factor of \(\sqrt[3]{54}\) is:

A\(\sqrt[3]{2}\) B\(\sqrt[3]{3}\) C\(\sqrt[3]{4}\) D\(\sqrt[3]{9}\)

Question 19

If \(\frac{p}{q}\) is a rational number with terminating decimal expansion, where \(p\) and \(q\) are co-prime, then \(q\) must be of the form:

A\(2^m \times 3^n\) B\(2^m \times 5^n\) C\(3^m \times 5^n\) D\(2^m \times 3^n \times 5^k\)

Question 20

The sum of two irrational numbers is:

AAlways irrational BAlways rational CCan be rational or irrational DAlways an integer

Question 21

The value of \(\frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \cdots + \frac{1}{\sqrt{99} + \sqrt{100}}\) is:

A\(1\) B\(9\) C\(10\) D\(99\)

Question 22

If \(a\) and \(b\) are rational numbers and \(\frac{3 + 2\sqrt{3}}{3 - 2\sqrt{3}} = a + b\sqrt{3}\), then \(a + b =\)

A\(11\) B\(-11\) C\(13\) D\(-13\)

Question 23

Which of the following is not true?

AEvery integer is a rational number BEvery whole number is a natural number CEvery irrational number is a real number DEvery rational number is a real number

Question 24

The product \(\sqrt[3]{2} \times \sqrt[4]{2} \times \sqrt[12]{32}\) equals:

A\(\sqrt{2}\) B\(2\) C\(\sqrt[12]{2}\) D\(\sqrt[12]{32}\)

Question 25

The value of \(0.\overline{6} + 0.\overline{7} + 0.\overline{4}\) is:

A\(2\) B\(\frac{17}{9}\) C\(\frac{53}{22}\) D\(\frac{59}{22}\)

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