NCERT PRACTICE QUESTIONS

Surface areas and volumes

1. 2 cubes each of volume 64 cm3 are joined end to end. Find the surface area of the resulting
cuboid. [ans: 160]
2. Mayank made a bird-bath for his garden in the shape of a cylinder with a
hemispherical depression at one end. The height of the cylinder is 1.45 m and
its radius is 30 cm. Find the total surface area of the bird bath. [ans: 33000]
3. A wooden article was made by scooping out a hemisphere from each end of a
solid cylinder, as shown in Fig. 13.11. If the height of the cylinder is 10 cm, and
its base is of radius 3.5 cm, find the total surface area of the article. [ans: 374]

4. A cubical block of side 7 cm is surmounted by a hemisphere. What is the greatest diameter

the hemisphere can have? Find the surface area of the solid. [ans: 332.5 ]
5. A vessel is in the form of an inverted cone. Its height is 8 cm and the radius of its top, which
is open, 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 in the vessel. [ans: 100]
6. A juice seller was serving his customers using glasses as shown in Fig. 13.13. The inner
diameter of the cylindrical glass was 5 cm, but the bottom of the glass had a hemispherical
raised portion which reduced the capacity of the glass. If the height of a glass was 10 cm,
find the apparent capacity of the glass and its actual capacity. (Use 𝜋𝜋 = 3.14.)
[ans: 163.54]

7. 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. [ans: 1131428.57cm3]
8. A copper rod of diameter 1 cm and length 8 cm is drawn into a wire of length 18 m of
uniform thickness. Find the thickness of the wire. [1/15 cm]
4
9. A hemispherical tank full of water is emptied by a pipe at the rate of 3 litres per second.
7
22
How much time will it take to empty half the tank, if it is 3 m in diameter? (Take 𝜋𝜋 = )
7
[ans: 16.5 mins]
10. A cylindrical bucket, 32 cm high and with radius of base 18 cm, is filled with sand. This
bucket is emptied on the ground and a conical heap of sand is formed. If the height of the
conical heap is 24 cm, find the radius and slant height of the heap. [ans: 36 and 12√13]
11. Water in a canal, 6 m wide and 1.5 m deep, is flowing with a speed of 10 km/h. How much
area will it irrigate in 30 minutes, if 8 cm of standing water is needed? [ans: 562500 𝑚𝑚2 ]
12. A farmer connects a pipe of internal diameter 20 cm from a canal into a cylindrical tank in
her field, which is 10 m in diameter and 2 m deep. If water flows through the pipe at the
rate of 3 km/h, in how much time will the tank be filled? [ans: 100 mins]

Applications of trigonometry

13. An observer 1.5 m tall is 28.5 m away from a chimney. The angle of elevation of the top of
the chimney from her eyes is 45∘ . What is the height of the chimney? [ans: 30 m]
14. From a point P on the ground the angle of elevation of the top of a 10 m tall building is 30∘ .
A flag is hoisted at the top of the building and the angle of elevation of the top of the
flagstaff from 𝑃𝑃 is 45∘ . Find the length of the flagstaff and the distance of the building from
the point 𝑃𝑃. (You may take √3 = 1.732 ) [ans: 7.32 m]
15. The shadow of a tower standing on a level ground is found to be 40 m longer when the
Sun's altitude is 30∘ than when it is 60∘ . Find the height of the tower. [ans: 20√3 𝑚𝑚]
16. The angles of depression of the top and the bottom of an 8 m tall building from the top of a
multi-storeyed building are 30∘ and 45∘ , respectively. Find the height of the multi-storeyed
building and the distance between the two buildings. [ans: 4(3 + √3)m]
17. From a point on a bridge across a river, the angles of depression of the banks on opposite
sides of the river are 30∘ and 45∘ , respectively. If the bridge is at a height of 3 m from the
banks, find the width of the river. [ans: 3(√3 + 1)m.]
18. A 1.5 m tall boy is standing at some distance from a 30 m tall building. The angle of
elevation from his eyes to the top of the building increases from 30∘ to 60∘ as he walks
towards the building. Find the distance he walked towards the building. [ans: 19√3 𝑚𝑚]
19. Two poles of equal heights are standing opposite each other on either side of the road,
which is 80 m wide. From a point between them on the road, the angles of elevation of the
top of the poles are 60∘ and 30∘ , respectively. Find the height of the poles and the distances
of the point from the poles. [ans: 20√3 𝑚𝑚 𝑎𝑎𝑎𝑎𝑎𝑎 20𝑚𝑚 𝑎𝑎𝑎𝑎𝑎𝑎 60 𝑚𝑚]
20. Check whether 301 is a term of the list of numbers 5,11,17,23, … [ans: no]
21. How many two-digit numbers are divisible by 3 ? [ans: 30]
22. Two APs have the same common difference. The difference between their 100 th terms is
100 , what is the difference between their 1000 th terms? [ans: 100]
23. How many three-digit numbers are divisible by 7 ? [ans: 128]
24. How many multiples of 4 lie between 10 and 250 ? [ans: 60]
25. Which term of the AP : 3,15,27,39, … will be 132 more than its 54 th term? [ans: 65th ]
26. Find the 20 th term from the last term of the AP: 3,8,13, … ,253. [ans: 158]
27. Find three numbers in A.P. whose sum is 15 and whose product is 105 ? [ans: 3,5,7 and
7,5,3]
28. Find three numbers in A.P whose sum is 24 and whose product is 440. [ans: 5,8,11 and
11,8,5]
29. Find four numbers in A.P. whose sum is 20 and the sum of whose squares is 120. [ans: 2,
4, 6, 8 or 8, 6, 4, 2]
30. If the sum of the first 14 terms of an AP is 1050 and its first term is 10, find the 20th term.
[ans: 200]
31. How many terms of the AP : 24, 21, 18, . . . must be taken so that their sum is 78? [ans: 4 or
13]
32. Find the sum of the first 40 positive integers divisible by 6 . [ans: 4920 ]
33. Find the sum of the first 15 multiples of 8 . [ans: 960]
34. Find the sum of the odd numbers between 0 and 50 . [ans: 625]
35. If 𝑆𝑆𝑆𝑆, the sum of first n terms of an A. P., is given by 𝑆𝑆𝑆𝑆 = 5𝑛𝑛2 + 3𝑛𝑛, find the AP. [ans:
8,18,28,…]

Construction

36. Draw two concentric circles of radii 3 cm and 5 cm. Construct a tangent to smaller circle
from a point on the larger circle. Also measure its length.
37. Draw a circle of radius 4 cm. Draw two tangents to the circle inclined at an angle of 60° to
each other.
38. Draw a line segment of length 10.4 cm and divide it in the ratio 3: 5.
39. Draw a line segment of length 5.6 cm and divide it in the ratio 4: 3.