Important Questions for ICSE, CBSE Board Examination

📘 A comprehensive, interactive revision tool for Class 10 Mathematics. Features 30 curated problems across all major units with step-by-step hints to aid understanding. Ideal for self‑assessment and board preparation. Test yourself before the finals!
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30 Challenging Problems for ICSE, CBSE and other State Boards

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Chapter Test Pair of Linear Equation in Two Variables CBSE ICSE

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25 Question MCQ Prime Maths - Class X, IX CBSE, Madhyamik and ICSE

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Pair of Linear Equations in Two Variables

This quiz offers a thorough evaluation of key concepts and skills related to Linear Equations in Two Variables. 25 multiple-choice questions designed to test both theoretical understanding and practical application.

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Question 1

For the system of equations \(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\) to have a unique solution, the condition is:

A\(\frac{a_1}{a_2} = \frac{b_1}{b_2}\) B\(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\) C\(\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}\) D\(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\)

Question 2

If a pair of linear equations is consistent, then the lines representing them are:

AParallel BAlways coincident CIntersecting or coincident DAlways intersecting

Question 3

For what value of \(k\) do the equations \(3x - y + 8 = 0\) and \(6x - ky = -16\) represent coincident lines?

A\(k = 2\) B\(k = -2\) C\(k = \frac{1}{2}\) D\(k = - \frac{1}{2}\)

Question 4

The pair of equations \(x = a\) and \(y = b\) graphically represents lines which are:

AParallel BIntersecting at \((b, a)\) CCoincident DIntersecting at \((a, b)\)

Question 5

If the system \(2x + 3y = 5\) and \(4x + ky = 10\) has infinitely many solutions, then \(k\) equals:

A\(1\) B\(\frac{3}{2}\) C\(6\) D\(3\)

Question 6

The value of \(k\) for which the system \(x + 2y = 3\) and \(5x + ky + 7 = 0\) is inconsistent is:

A\(k = 10\) B\(k = \frac{2}{5}\) C\(k = -\frac{2}{5}\) D\(k = -10\)

Question 7

Solve: \(\frac{x}{a}+\frac{y}{b}=2\) and \(ax-by=a^2-b^2\)

A\(x=a, \quad y=b^2\) B\(x=a, \quad y=b\) C\(x=a^2, \quad y=b^2\) D\(x=a^2, \quad y=b\)

Question 8

The value of \(k\) for which the pair of linear equations \((3k+1)x+3y=2\) and \((k^2+1)x+(k-2)y-5=0\) has no solution.

A\(k=-2\) B\(k=2\) C\(k=1\) D\(k=-1\)

Question 9

Solve for \(x\) and \(y\): \((a-b)x + (a+b)y = a^2 -2ab-b^2\) and \((a+b)(x+y) = a^2 + b^2\)

A\(x = a+b, y = -\frac{2ab}{a+b}\) B\(x = a+b, y = \frac{2ab}{a+b}\) C\(x = a-b, y = -\frac{2ab}{a+b}\) D\(x = a-b, y = \frac{2ab}{a+b}\)

Question 10

The value of \(x\) satisfying \(\frac{5}{x-1}+\frac{1}{y-2}=2\) and \(\frac{6}{x-1}-\frac{3}{y-2}-1=0, \quad x \neq 1, y \neq 2\) is:

A2 B3 C4 D5

Question 11

If \(37x + 43y = 123\) and \(43x + 37y = 117\), then:

A\(x = 2, y = 3\) B\(x = 1, y = 2\) C\(x = 3, y = 2\) D\(x = -1, y = -2\)

Question 12

The pair of equations \(225x + 881y = 2^{10}\) and \(450x + 1762y = 2048\) has:

ANo solution BUnique solution CTwo solutions DInfinitely many solutions

Question 13

If \(\frac{b}{a}x + \frac{a}{b}y = a^2+b^2\) and \(x + y = 2ab\) then

A\(x=2y\) B\(x=-y\) C\(x=y\) D\(2x=y\)

Question 14

If \(2^x + 3^y = 17\) and \(2^{x+2} - 3^{y+1} = 5\), find \(x\) and \(y\).

A\(x = 9, y = 5\) B\(x = 3, y = 2\) C\(x = 5, y = 9\) D\(x = 8, y = 6\)

Question 15

Solve for \(x\): \(\frac{2}{x} + \frac{3}{y} = 13\) and \(\frac{5}{x} - \frac{4}{y} = -2\).

A\(x = 1/2\) B\(x = 1/3\) C\(x = 2\) D\(x = 3\)

Question 16

For what value of \(p\) does the pair of equations \(4x + py + 8 = 0\) and \(2x + 2y + 2 = 0\) have a unique solution?

A\(p = 4\) B\(p \neq 4\) C\(p = 2\) D\(p \neq 2\)

Question 17

If \(49x + 51y = 499\) and \(51x + 49y = 501\), then \(x\) is:

A3 B5.5 C4 D6.5

Question 18

The difference between two numbers is 26 and one number is three times the other. Find them.

A30, 4 B36, 10 C39, 13 D40, 14

Question 19

Solve: \( \sqrt{2}x + \sqrt{3}y = 0 \) and \( \sqrt{3}x - \sqrt{8}y = 0 \).

A\(x=1, y=1\) B\(x=\sqrt{2}, y=\sqrt{3}\) C\(x=0, y=0\) D\(x=2, y=3\)

Question 20

Find \(k\) if the lines \(x + 2y - 1 = 0\) and \(2x + ky + 2 = 0\) are parallel.

A\(k=2\) B\(k=3\) C\(k=4\) D\(k=-4\)

Question 21

Five years ago, Ramesh was thrice as old as Sonu. Ten years later, Ramesh will be twice as old as Sonu. What is Ramesh's present age?

A40 years B50 years C60 years D45 years

Question 22

A fraction becomes 9/11 if 2 is added to both numerator and denominator. If 3 is added to both, it becomes 5/6. Find the fraction.

A\(7/9\) B\(9/7\) C\(5/9\) D\(3/7\)

Question 23

The sum of the digits of a two-digit number is 9. Also, nine times this number is twice the number obtained by reversing the order of the digits. Find the number.

A18 B27 C81 D72

Question 24

A taxi charge consists of a fixed charge together with the charge for the distance covered. For 10 km, the charge paid is ₹105 and for 15 km, the charge paid is ₹155. What are the fixed charges?

A₹ 10 B₹ 5 C₹ 15 D₹ 8

Question 25

Ritu can row downstream 20 km in 2 hours, and upstream 4 km in 2 hours. Find her speed of rowing in still water.

A6 km/h B4 km/h C8 km/h D10 km/h

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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)

👨‍🏫 Author: Singh

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Mensuration Chapter Test

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

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• This quiz contains 25 multiple choice questions.
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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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