CBSE-X-MATH-BOARD-EXAM-IMPORTANT-QUESTION-CHAPTER-10-12

 CHAPTER 10 to CHAPTER 12 IMPORTANT QUESTIONS FOR CBSE CLASS X 2026

                                                  CHAPTER 10 CIRCLE

Question 1: Prove that the radius is perpendicular to the tangent.

Question 2: A tangent to a circle intersects the circle at ________ point(s).  

Question 3: The common point of a tangent to a circle and the circle is called ____________.

Question 4: The lengths of tangents drawn from an external point to a circle are equal.

Question 5: Two tangents TP and TQ are drawn to a circle with center O from external point T. Prove that ㄥPTQ = 2ㄥOPQ

Question 6: From a point Q, the length of the tangent to a circle is 24 cm, and the distance of Q from the center is 25 cm. The radius of the circle is

(i) 7 cm   (ii) 12 cm  (iii) 15cm (iv) 24.5cm

Question 7: Prove that the tangents drawn at the ends of a diameter of a circle are parallel.

Question 8: Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle that touches the smaller circle.

 Question 9: A quadrilateral ABCD is drawn to circumscribe a circle (see Fig. 10.12). Prove that AB + CD = AD + BC

Question 10: Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact at the center.

Question 11: Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the center of the circle.

Question 12: A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 8 cm and 6 cm respectively (see Fig). Find the sides AB and AC.

                                                   CHAPTER 11

Chapter 11 is about the area related to the circle. So, it becomes important to learn some useful formulas to start solving.

Circumference of the Circle = 2πr 

Circumference of the Semicircle = πr 

 

Areas Related to a Circle

1. Area of a Circle

The area of a circle is given by:

Area = πr²

where r is the radius of the circle.

2. Area of a Semicircle

A semicircle is half of a circle.

Area = (1/2) × πr²

3. Area of a Minor Sector

A sector is a part of a circle formed by two radii and an arc. If the angle at the center is θ (in degrees):

Area of Minor Sector = (θ / 360) × πr²

4. Area of a Major Sector

The major sector is the larger part of the circle.

Area of Major Sector = [(360 − θ) / 360] × πr²

5. Area of a Segment of a Circle (Minor Segment)

A segment is the region between a chord and the corresponding arc.

Area of Minor Segment = Area of Minor Sector − Area of Triangle

Area of Triangle = (1/2) × r² × sin θ

6. Area of a Major Segment

The major segment is the remaining part of the circle after removing the minor segment.

Area of Major Segment = Area of Circle − Area of Minor Segment

Practice Questions: Areas Related to Circles

1. Find the area of the sector of a circle with radius 4 cm and angle 30°. Also, find the area of the corresponding major sector.
   (Use π = 3.14)
2. Find the area of the segment AYB shown in Fig. 11.6, if the radius of the circle is 21 cm and ∠AOB = 120°.
   (Use π = 22/7)
3. In a circle of radius 21 cm, an arc subtends an angle of 60° at the centre. Find the area of the corresponding sector.
   (Use π = 22/7)
4. A chord of a circle of radius 15 cm subtends an angle of 60° at the centre. Find the areas of the corresponding minor segment and major segment of the circle.
   (Use π = 3.14 and √3 = 1.73)
5. A horse is tied to a peg at one corner of a square-shaped grass field of side 15 m by means of a 5 m long rope (see Fig. 11.8).
   (i) Find the area of that part of the field in which the horse can graze.
   (ii) Find the increase in the grazing area if the rope were 10 m long instead of 5 m.
   (Use π = 3.14)
6. To warn ships for underwater rocks, a lighthouse spreads a red coloured light over a sector of angle 80° to a distance of 16.5 km. Find the area of the sea over which the ships are warned.
   (Use π = 3.14)
7. A round table cover has six equal designs as shown in Fig. 11.11. If the radius of the cover is 28 cm, find the cost of making the designs at the rate of ₹ 0.35 per cm².
   (Use √3 = 1.7)

Chapter 12 – Surface Areas and Volumes

Important Formulae

1. Cube (Side = a)

Total Surface Area = 6a²

Lateral Surface Area = 4a²

Volume = a³

2. Cuboid (Length = l, Breadth = b, Height = h)

Total Surface Area = 2(lb + bh + hl)

Lateral Surface Area = 2h(l + b)

Volume = l × b × h

3. Right Circular Cylinder (Radius = r, Height = h)

Curved Surface Area = 2πrh

Total Surface Area = 2πr(r + h)

Volume = πr²h

4. Right Circular Cone (Radius = r, Height = h)

Slant Height (l) = √(r² + h²)

Curved Surface Area = πrl

Total Surface Area = πr(l + r)

Volume = (1/3)πr²h

5. Mixed Solids

For mixed solids (combination of two or more solids), the total volume is the sum of volumes of individual solids.

Surface area is calculated by adding only the exposed surfaces of all solids.

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Practice Questions

1. Find the total surface area and volume of a cube whose edge is 7 cm.
2. Find the lateral surface area and volume of a cuboid of dimensions 12 cm × 8 cm × 5 cm.
3. The radius of a right circular cylinder is 7 cm and its height is 10 cm. Find its curved surface area and volume.
   (Use π = 22/7)
4. Find the curved surface area and volume of a cone whose radius is 3.5 cm and height is 6 cm.
   (Use π = 22/7)
5. A solid is composed of a cylinder surmounted by a hemisphere. The radius of both the cylinder and hemisphere is 3.5 cm and the height of the cylindrical part is 10.5 cm. Find the total surface area of the solid.
6. A toy is in the shape of a cone mounted on a hemisphere. The radius of the base is 7 cm and the height of the cone is 24 cm. Find the total surface area of the toy.
7. How many solid spheres of radius 3 cm can be made by melting a solid cylinder of radius 6 cm and height 9 cm?
8. A vessel is in the form of an inverted cone. Its height is 8 cm and the radius of its top is 5 cm. Find its capacity.
   (Use π = 3.14)

Chapter 12 – Surface Areas and Volumes

Practice Questions

1. The decorative block shown in Fig. 12.7 is made of two solids — a cube and a hemisphere. The base of the block is a cube with edge 5 cm, and the hemisphere fixed on the top has a diameter of 4.2 cm. Find the total surface area of the block.
   (Take π = 22/7)
2. A wooden toy rocket is in the shape of a cone mounted on a cylinder, as shown in Fig. 12.8. The height of the entire rocket is 26 cm, while the height of the conical part is 6 cm. The base of the conical portion has a diameter of 5 cm, while the base diameter of the cylindrical portion is 3 cm. If the conical portion is to be painted orange and the cylindrical portion yellow, find the area of the rocket painted with each of these colours.
   (Take π = 3.14)
3. A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends (see Fig. 12.10). The length of the entire capsule is 14 mm and the diameter of the capsule is 5 mm. Find its surface area.
4. A tent is in the shape of a cylinder surmounted by a conical top. If the height and diameter of the cylindrical part are 2.1 m and 4 m respectively, and the slant height of the top is 2.8 m, find the area of the canvas used for making the tent. Also, find the cost of the canvas at the rate of ₹ 500 per m².
   (Note: The base of the tent is not covered with canvas.)
5. From a solid cylinder whose height is 2.4 cm and diameter 1.4 cm, a conical cavity of the same height and same diameter is hollowed out. Find the total surface area of the remaining solid to the nearest cm².
6. A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in Fig. 12.11. If the height of the cylinder is 10 cm and its base radius is 3.5 cm, find the total surface area of the article.
7. A juice seller serves his customers using glasses as shown in Fig. 12.13. The inner diameter of the cylindrical glass is 5 cm, but the bottom of the glass has a hemispherical raised portion which reduces the capacity of the glass. If the height of the glass is 10 cm, find the apparent capacity and the actual capacity of the glass.
   (Use π = 3.14)
8. A gulab jamun contains sugar syrup up to about 30% of its volume. Find approximately how much syrup would be found in 45 gulab jamuns, each shaped like a cylinder with two hemispherical ends of length 5 cm and diameter 2.8 cm .
9. A vessel is in the form of an inverted cone. Its height is 8 cm and the radius of its open top is 5 cm. It is filled with water up to the brim. When lead shots, each of which is a sphere of radius 0.5 cm, are dropped into the vessel, one-fourth of the water flows out. Find the number of lead shots dropped into the vessel.
10. A solid iron pole consists of a cylinder of height 220 cm and base diameter 24 cm, which is surmounted by another cylinder of height 60 cm and radius 8 cm. Find the mass of the pole, given that 1 cm³ of iron has approximately a mass of 8 g.
    (Use π = 3.14)
11. A solid consisting of a right circular cone of height 120 cm and radius 60 cm standing on a hemisphere of radius 60 cm is placed upright in a right circular cylinder full of water such that it touches the bottom. Find the volume of water left in the cylinder, if the radius of the cylinder is 60 cm and its height is 180 cm.
12. A spherical glass vessel has a cylindrical neck 8 cm long and 2 cm in diameter. The diameter of the spherical part is 8.5 cm. By measuring the amount of water it holds, a child finds its volume to be 345 cm³. Check whether she is correct.
    (Use π = 3.14)