﻿ Class 11 NCERT Math Solution
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Exercise - 3.1

Question-1 :-  Find the radian measures corresponding to the following degree measures:
(i) 25°  (ii) – 47°30′  (iii) 240°  (iv) 520°.

Solution :-
```(i) 25°
We know that 180° = π radian.
Hence,
25° = π/180 x 25 radian

(ii) – 47°30′
We know that 180° = π radian.
Hence,
-47° 30′ = -47 + 1/2 degree
= -31π/120
Therefore, -47° 30′ = -31π/120 radian

(iii) 240°
We know that 180° = π radian.
Hence,
240° = π/180 x 240 radian

(iv) 520°
We know that 180° = π radian.
Hence,
520° = π/180 x 520 radian
```

Question-2 :-  Find the degree measures corresponding to the following radian measures (Use π = 22/7)
(i) 11/16  (ii) -4  (iii) 5π/3  (iv) 7π/6.

Solution :-
```(i) 11/16
We know that π radian = 180°.
Hence,
11/16 radians = 180/π x 11/16 degree
= (45 x 11 x 7)/ (22 x 4) degree
= 315/8 degree
= 39° + 3/8 degree
= 39° + (3 x 60)/8 minute    [1° = 60']
= 39° + 180/8 minute
= 39° + 22' + 1/2 minute     [1' = 60"]
= 39° + 22' + 30"
11/16 radians = 39° 22' 30".

(ii) -4
-4 radians = 180/π x (-4) degree
= [180 x 7 x (-4)]/22 degree
= -2520/11 degree
= -229° + 1/11 degree
= -229° + (1 x 60)/11 minute    [1° = 60']
= -229° + 60/11 minute
= -229° + 5' + 5/11 minute     [1' = 60"]
= -229° + 5' + 27"
-4 radians = -229° 5' 27" approximately.

(iii) 5π/3
5π/3 radians = 180/π x 5π/3 degree
= 300°

(iv) 7π/6
7π/6 radians = 180/π x 7π/6 degree
= 210°
```

Question-3 :-  A wheel makes 360 revolutions in one minute. Through how many radians does it turn in one second?

Solution :-
```  No. of revolution made by wheel in 1 minute = 360°
No. of revolution made by wheel in 1 second = 360/60 = 6
In one complete revolution, the wheels turns an angle of 2π radian.
Hence, in 6 complete revolutions, it will turn  an angle of 6 x 2π radian.
Thus, in one second, the wheels turns an angle of 12π radian.
```

Question-4 :-  Find the degree measure of the angle subtended at the centre of a circle of radius 100 cm by an arc of length 22 cm (Use π = 22/7).

Solution :-
```  We know that in a circle of radius r unit, length of arc l and angle θ.
Then, l = r.θ
Given :
radius of circle (r) = 100 cm
length of arc (l) = 22 cm
angle (θ) = ?
θ = l/r = 22/100 radian
= 180/π x 22/100 degree
= (180 x 22 x 7)/ (22 x 100) degree
= 126/10 degree
= 12° + 3/5 degree
= 12° + (3 x 60)/5 minute    [1° = 60']
= 39° + 36'
= 39° 36'
The required angle is 39° 36'.
```

Question-5 :-  In a circle of diameter 40 cm, the length of a chord is 20 cm. Find the length of minor arc of the chord.

Solution :-
```  Given:
Diameter of circle = 40 cm
radius of circle = 40/2 = 20 cm
length of chord = 20 cm
length of minor arc of chord = ?

Let AB be a chord(length = 20 cm) of the circle. In ∆AOB,
OA = OB = radius of circle = 20 cm
Also, AB = 20 cm
Thus, ∆AOB is a an equilateral triangle.
So, θ = 60° = π/3 radian
We know that in a circle of radius r unit, length of arc l and angle θ.
Then, l = r.θ
length of arc (AB) = 20 x π/3 = 20π/3 cm
Length of minor arc of chord is 20π/3 cm.
```

Question-6 :-  If in two circles, arcs of the same length subtend angles 60° and 75° at the centre, find the ratio of their radii.

Solution :-
```  Let the radii of two circles are r₁ and r₂.
Let an arc of length l subtend an angle of 60° at the centre of the circle of radius r₁
Let an arc of length l subtend an angle of 75° at the centre of the circle of radius r₂

We know that in a circle of radius r unit, length of arc l and angle θ.
Then, l = r.θ
l = r₁.θ₁ and l = r₂.θ₂
l = π/3 . r₁ and l = 5π/12 . r₂
π/3 . r₁ = 5π/12 . r₂
r₁/r₂ = 5π/12 x 3/π
r₁/r₂ = 5/4
r₁ : r₂ = 5 : 4
Ratio of radii is 5 : 4
```

Question-7 :-  Find the angle in radian through which a pendulum swings if its length is 75 cm and the tip describes an arc of length
(i) 10 cm  (ii) 15 cm  (iii) 21 cm

Solution :-
```  We know that in a circle of radius r unit, length of arc l and angle θ.
Then, l = r.θ
Given:

(i) 10 cm
length of arc (l) = 10 cm
radius of circle (r) i.e.,Pendulum = 75 cm
angle (θ) = ?
θ = l/r = 10/75 = 2/15 radian

(ii) 15 cm
length of arc (l) = 15 cm
radius of circle (r) i.e.,Pendulum = 75 cm
angle (θ) = ?
θ = l/r = 15/75 = 1/5 radian

(iii) 21 cm
length of arc (l) = 21 cm
radius of circle (r) i.e.,Pendulum = 75 cm
angle (θ) = ?
θ = l/r = 21/75 = 7/25 radian
```
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