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CBSE 10th Maths Some Applications of Trigonometry Sample Question Papers Study Material - QB365

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Questions:

Fill up / 1 Marks

35 x 1 = 35

A circle artist is climbing a 10m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground, then the height of pole, if the angle made by the rope with the ground level is 60°, is..........

The angle of elevation of a ladder leaning against a wall is 60° and the foot of the ladder is 19 m away from the wall. The length of the ladder is...........

The angle of elevation of the top of a tower from a distance of 100m from its foot is 30°. The height of tower is ..........

The angle of elevation of the Sun, when the length of a shadow of a vertical pole is double to its height, is .............

If the horizontal distance between the two trees 20m and 28m high is 15m, then distance between their tops is........

A 6m long pole casts a shadow of 4m long.At the same time a tree casts a shadow of 28m long, then length of tree is.......

If height of a tower and distance of the point of observation from its foot, both are increased by 50%, then angle of elevation of its top..............

The ................ is the line drawn from the eye of an observer to the point in the object viewed by the observer.

If the ratio of the length of a pole and its shadow is \(1:\sqrt{3}\), then sun's elevation is ..............

10.

If sun's elevation is 45⁰, then ratio of the length of a pole and its shadow is .............

11.

The length of the shadow of a tree 10 high, when the sun's elevation is 30⁰ , is ..........

12.

A tower stands vertically on the ground.From a point on the ground, which is 100m away from the foot of the tower, the angle of elevation of the top of the tower is found to be 60⁰, then the height of the tower is .........

13.

If the length of the shadow of a tower is increasing, then the angle of elevation of the sun is .......

14.

If the ratio of the length of a pole and its shadow is \(\sqrt{3}:1\), then the sun's elevation is ........

15.

If the angle of elevation of the top of a tower from two points distance and t from its foot are complementary, the height of the tower is ........

16.

The angle of ....... of an object viewed, is the angle formed by the line of sight with horizontal when it is below the horizontal level.

17.

The angle of ....... of an object viewed, is the angle formed by the line of sight with the horizontal when it is above the horizontal level.

18.

If the ratio of the length of a pole and its shadow is 1 : 1, then angle of elevation of the sun is.......... .

19.

A 6m tall tree casts a shadow of length 4m.If at the same time a flagpole casts a shadow 50m in length, then the length of the flagpole is...........

20.

The height of a tower is 10m.The height of its shadow when sun's altitude is 45⁰ , is ...........

21.

If the height of a tower and the distance of the point of observation from its foot, both are increased by 10% then the angle of elevation of its top remains.........

22.

The length of the shadow of a tree 8m high, when the sun's elevation is 45⁰ , is .......

23.

Three points A, B and C are collinear, if any one of the following takes place:
ar \((\Delta ABC)\) = .............

24.

Three points A, B and C are collinear, if any one of the following takes place:
........... + AB = CB

25.

Three points A, B and C are collinear, if any one of the following takes place:
AC + ............ = AB

26.

Three points A, B and C are collinear, if any one of the following takes place:
AB + ............. = AC

27.

The point of intersection of the medians of a triangle is called the .................. of the triangle.

28.

The bisector of angles of triangle are ...........................

29.

The distance between the point (2,5) and (7,5) is ...........

30.

In the fourth quadrant for a point, the abscissa is ....... and the ordinate is ........

31.

The abscissa and ordinate of a point taken together is known as ............... of a point.

32.

Area of triangle formed by the vertices (x1,y1), (x2,y2) and (x3,y3) is the numerical value of ................

33.

.............. is the point of intersection of the axes of coordinates.

34.

The centroid of triangle divides the median in the ratio ...........

35.

Coordinate geometry is ........ tool for studying geometry.

Case Study Questions

3 x 4 = 12

36.

Radio towers are used for transmitting a range of communication services including radio and television. The tower will either act as an antenna, itself or support one or more antennas on its structure.On a similar concept, a radio station tower was built in two Sections A and B. Tower is supported by wires from a point O.
Distance between the base of the tower and point O is 36 cm. From point O, the angle of elevation of the top of the Section B is 30° and the angle of elevation of the top of Section A is 45°.

Based on the above information, answer the following questions :
(i) Find the length of the wire from the point O be the top of Section B.
(ii) Find the distance AB.
(iii) Find the height of the Section A from the base of the tower.
Or
Find the area of \(\Delta\)OPB.

37.

Students of a school are standing in rows and columns in their playground for a drill practice. A, B, C and D are the positions of four students as shown in figure.

Based on the above information, answer the following questions.
(i) Is it possible to place Jaspal in the drill in such a way that he is equidistant from each of the four students A, B, C and D? If so, what should be his position?
(ii) Determine the distance between the points C and D.

38.

Ayush starts walking from his house to office. Instead of going to the office directly, he goes to a bank first, from there to his daughter's school and then reaches the office.
(i) What is the extra distance travelled by Ayush in reaching his office? [assume that all distances covered are in straight lines, if the house is situated at (2, 4), bank at(5, 8), school at(13, 14)and office at (13, 26) and coordinates are in kilometres],
(ii) How much far from bank to the office?

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Answers Key:

Fill up / 1 Marks

35 x 1 = 35

()
\(5 \sqrt{3} \mathrm{~m}\)

()
38 m

()
\(\frac{100}{\sqrt{3}} \mathrm{~m}\)

()
\(\tan ^{-1}\left(\frac{1}{2}\right)\)

()
17.m

()
42m

()
remains unchanged

()
line of sight

()
30\(\unicode{xb0} \)

10.

()
1:1

11.

()
\(10\sqrt{3}m\)

12.

()
173.2m

13.

()
decreasing

14.

()
60⁰

15.

()
\(\sqrt{st}\)

16.

()
depression

17.

()
elevation

18.

()
45⁰

19.

()
75m

20.

()
10m

21.

()
unchanged

22.

()
8m

23.

()
0

24.

()
CA

25.

()
CB

26.

()
BC

27.

()
centroid

28.

()
concurrent

29.

()
5 units

30.

()
positive, negative

31.

()
coordinates

32.

()
\({1\over2}[x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)]\)

33.

()
origin

34.

()
2:1

35.

()
an algebraical

Case Study Questions

3 x 4 = 12

36.

(i) In right angled \(\Delta\)OPB,
\(\begin{aligned} & \cos 30^{\circ}=\frac{O P}{O B}=\frac{36}{O B} \\ \end{aligned}\)
\(\begin{aligned} \Rightarrow \quad O B & =\frac{36}{\cos 30^{\circ}}=\frac{36}{\frac{\sqrt{3}}{2}}=\frac{72}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} \\ \end{aligned}\)
\(\begin{aligned} \Rightarrow \quad & O B=24 \sqrt{3} \mathrm{~cm} \end{aligned}\)
In right angled \(\Delta\)OPB,
\(\begin{aligned} \tan 30^{\circ} & =\frac{B P}{O P} \\ \end{aligned} \)
\(\begin{aligned} \Rightarrow \quad \frac{1}{\sqrt{3}} & =\frac{B P}{36} \\ \end{aligned}\)
\(\begin{aligned} \Rightarrow \quad B P & =\frac{36}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} \\ \end{aligned}\)
\(\begin{aligned} =12 \sqrt{3} \mathrm{~cm} \end{aligned}\)
In right angled \(\Delta\)APO,
\(\begin{aligned} & \tan 45^{\circ}=\frac{A P}{O P} \\ \end{aligned}\)
\(\Rightarrow\) OP = AP [\(\because\) tan 45° = 1]
\(\Rightarrow\) AP = 36 cm ....(i)
\(\therefore\) Distance AB = AP - BP
= 36 - 12\(\sqrt3\)
=12(3-\(\sqrt3\)) cm
(iii) Height of the section A from the base of the tower
= AP = 36 cm [using Eq (i)]
Or
In right angled \(\Delta\)OPB,
\(\begin{aligned} \tan 30^{\circ} & =\frac{B P}{O P} \\ \end{aligned}\)
\(\Rightarrow\) BP = OP tan 30^\(\circ\)
\(=36 \times \frac{1}{\sqrt{3}}=12 \sqrt{3} \mathrm{~cm}\)
\(\therefore\) Area of \(\Delta\)OPB = \(\begin{aligned} & =\frac{1}{2} \times O P \times B P \\ \end{aligned}\)
\(\begin{aligned} & =\frac{1}{2} \times 36 \times 12 \sqrt{3} \\ \end{aligned}\)
\(\begin{aligned} & =216 \sqrt{3} \mathrm{~cm}^2 \end{aligned}\)

37.

(i) Yes, Jaspal should be placed at the point (7, 5).
(ii) 4\(\sqrt2\) units

38.

(i)
Extra distance travelled by Ayush = Distance between house and office via Bank and School-Direct Distance between house and office.
Now, using distance formula,
Extra distance
\(=\sqrt{(5-2)^{2}+(8-4)^{2}}+\sqrt{(13-5)^{2}+(14-8)^{2}}+\sqrt{(13-13)^{2}+(26-14)^{2}}-\sqrt{(13-2)^{2}+(26-4)^{2}}\) = 2.4km
(ii) \(2 \sqrt{97} \mathrm{~km}\)