Relations :
A relation R on set A is said to be -
1. Reflexive: If (x, x) ∊ R for each element x ∊ A, ie, if xRx for each element x ∊ A.
2. Symmetric: If (x, y) ∊ R ⇒ (y, x) ∊ R for all x, y ∊ A, ie., if xRy ⇒ yRx for all x, y ∊ A.
3. Transitive: If (x, y) ∊ R and (y, z) ∊ R then (x, z) ∊ R for all x, y, z ∊ A, ie, if xRy and yRz ⇒ xRz.
Equivalence Relation Any relation R on a set A is said to be an equivalence relation if R is reflexive, symmetric and transitive.
Equivalence Class: Let R be an equivalence relation on a non-empty set A. For all a ∊ A,, the equivalence class of ‘a’ is defined as the set of all such elements of A which are related to ‘a’ under R. It is denoted by [a].
ie., [a] = equivalence class of ‘a’ = {x ∊ A,: (x, a) ∊ R}
For example , Let A= {1.2.3}
Que. 1
A vertical stick 10 cm long casts a shadow 8 cm long. At the same time, a tower casts a shadow 30 m long. Determine the height of the tower.
Que. 2
A person standing on the bank of a river observes that the angle subtended by a tree on the opposite bank is 600. When he retreats 20m from the bank, he finds the angle to be 300. Find the height of the tree and the breadth of the river.
Que. 3
A person standing on the bank of a river observes that the angle subtended by a tree on the opposite bank is 600. When he retreats 20m from the bank, he finds the angle to be 300. Find the height of the tree and the breadth of the river.
Que. 4
A boy is standing on ground and flying a kite with 150m of string at an elevation of 300. Another boy is standing on the roof of a 25m high building and flying a kite at an elevation of 450. Find the length of string required by the second boy so that the two kites just meet, if both the boys are on opposite side of the kites.
Que 5.
An aeroplane flying horizontally 1000m above the ground, is observed at an angle of elevation 600 from a point on the ground. After a flight of 10 seconds, the angle of elevation at the point of observation changes to 300. Find the speed of the plane in m/s.
Que 6.
An aeroplane when flying at a height of 4000 m from the ground passes vertically above another aeroplane at an instant when the angles of the elevation of the two planes from the same point on the ground are 600 and 450 respectively. Find the vertical distance between the aeroplanes at that instant.
Que 7.
An aeroplane at an altitude of 200 m observes the angles of depression of opposite points on the two banks of a river to be 450 and 600. Find the width of the river.
Que 8.
The shadow of a flag staff is three times as long as the shadow of the flag staff when the sun rays meet the ground at an angle of 600. Find the angle between the sun rays and the ground at the time of longer shadow.
Que 9.
A vertically straight tree, 15m high is broken by the wind in such a way that it top just touches the ground and makes an angle of 600 with the ground, at what height from the ground did the tree break?
Que 10.
A man in a boat rowing away from lighthouse 100 m high takes 2 minutes to changes the angle of elevation of the top of lighthouse from 600 to 450. Find the speed of the boat.
Que 11
As observed from the top of a light house, 100m above sea level, the angle of depression of ship, sailing directly towards it, changes from 300 to 450. Determine the distance travelled by the ship during the period of observation.
Que 12
A man standing on the deck of ship, which is 10m above the water level, observes the angle of elevation of the top of a hill as 600 and the angle of depression of the base of the hill as 300. Calculate the distance of the hill from the ship and the height of the hill.
Que 13.
The angles of elevation of the top of a tower from two points at a distance of ‘a’ m and ‘b’ m from the base of the tower and in the same straight line with it are complementary, then prove that the height of the tower is ab
Que 14.
A tower stands vertically on the ground. From a point on the ground, which is 15 m away from the foot of the tower, the angle of elevation of the top of the tower is found to be 60°. Find the height of the tower.
Que 15.
An electrician has to repair an electric fault on a pole of height 5 m. She needs to reach a point 1.3m below the top of the pole to undertake the repair work. What should be the length of the ladder that she should use which, when inclined at an angle of 60° to the horizontal, would enable her to reach the required position? Also, how far from the foot of the pole should she place the foot of the ladder? (You may take 3 = 1.73)
Que 16.
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?
Que 17.
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 P is 45°. Find the length of the flagstaff and the distance of the building from the point P. (You may take 3 = 1.73)
Que 18.
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.
Que 19.
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.
Que 20.
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.
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