OP Malhotra Class 11 Maths Solutions Chapter 3 Angles and Arc Lengths Chapter Test

Question 1. Find the radian measure of
(i) 25°
(ii) 240°
Answer:
We know that \( \pi \) radians = 180°. So, 1° = \( \frac{\pi}{180} \) radians.
(i) To convert 25° to radians, we multiply by \( \frac{\pi}{180} \).
\( 25^\circ = 25 \times \frac{\pi}{180} \) rad
\( \implies 25^\circ = \frac{5\pi}{36} \) rad
(ii) To convert 240° to radians, we multiply by \( \frac{\pi}{180} \).
\( 240^\circ = 240 \times \frac{\pi}{180} \) rad
\( \implies 240^\circ = \frac{4\pi}{3} \) rad
In simple words: To change degrees to radians, just multiply the degree value by \( \frac{\pi}{180} \). This ratio helps us switch between the two ways of measuring angles.

🎯 Exam Tip: Always remember the conversion factor: \( 1^\circ = \frac{\pi}{180} \) radians. Simplifying the fraction to its lowest terms is important for full marks.

Question 2. Find the degree measure of (i) \( \frac { 5\pi }{ 3 } \) (ii) -4.
Answer:
We know that \( \pi \) radians = 180°. So, 1 radian = \( \frac{180^\circ}{\pi} \).
(i) To convert \( \frac{5\pi}{3} \) radians to degrees, we multiply by \( \frac{180^\circ}{\pi} \).
\( \frac{5\pi}{3} = \frac{5\pi}{3} \times \frac{180^\circ}{\pi} \)
\( \implies \frac{5\pi}{3} = 5 \times 60^\circ \)
\( \implies \frac{5\pi}{3} = 300^\circ \)
(ii) To convert -4 radians to degrees, we multiply by \( \frac{180^\circ}{\pi} \). We use \( \pi = \frac{22}{7} \).
\( -4 = -4 \times \frac{180^\circ}{\pi} \)
\( \implies -4 = -4 \times \frac{180^\circ}{22/7} \)
\( \implies -4 = -4 \times \frac{180^\circ \times 7}{22} \)
\( \implies -4 = -\frac{720 \times 7}{22} \)
\( \implies -4 = -\frac{360 \times 7}{11} \)
\( \implies -4 = -\frac{2520}{11} \)
\( \implies -4 \approx -229^\circ 5' 27'' \)
In simple words: To change radians to degrees, multiply the radian value by \( \frac{180^\circ}{\pi} \). Remember that a negative angle simply means turning in the opposite direction.

🎯 Exam Tip: When converting radians to degrees, if \( \pi \) is present in the radian measure, it will often cancel out. If not, use the approximation \( \pi \approx \frac{22}{7} \) or 3.14159 as specified or implied.

Question 3. If an angle measures D degrees or C radians, show that \( \frac{D}{90}=\frac{2 C}{\pi} \).
Answer:
We know the standard relationship between degrees (D) and radians (C) for an angle:
\( \frac{D}{180} = \frac{C}{\pi} \)
Now, we want to show \( \frac{D}{90}=\frac{2 C}{\pi} \). To get \( \frac{D}{90} \) on the left side, we can multiply both sides of our known relationship by 2.
\( \frac{D}{180} \times 2 = \frac{C}{\pi} \times 2 \)
\( \implies \frac{D}{90} = \frac{2C}{\pi} \)
Thus, the relationship is shown.
In simple words: Degrees and radians are just two ways to measure angles. There's a set rule to change one into the other, and if you use that rule, you can easily prove other connections between them.

🎯 Exam Tip: Always start with the fundamental conversion formula \( \frac{D}{180} = \frac{C}{\pi} \). From this basic relationship, you can derive many other forms by simple algebraic manipulation like multiplying or dividing both sides by a constant.

Question 4. One angle of a triangle in 54° and another angle is \( \frac { \pi }{ 4 } \) radians. Find the third angle is centesimal unit.
Answer:
First, we need to convert the radian angle to degrees so all angles are in the same unit. We know that \( \pi \) radians = 180°. So, 1 radian = \( \frac{180^\circ}{\pi} \).
\( \frac{\pi}{4} \) rad = \( \frac{\pi}{4} \times \frac{180^\circ}{\pi} \)
\( \implies \frac{\pi}{4} \) rad = 45°
Given angles of the triangle are 54° and 45°. The sum of angles in a triangle is always 180°.
Let the third angle be \( \theta \).
\( 54^\circ + 45^\circ + \theta = 180^\circ \)
\( \implies 99^\circ + \theta = 180^\circ \)
\( \implies \theta = 180^\circ - 99^\circ \)
\( \implies \theta = 81^\circ \)
Now, we need to convert this degree measure to the centesimal unit (grades). We know that 180° = 200 grades.
So, 1° = \( \frac{200}{180} \) grades = \( \frac{10}{9} \) grades.
\( 81^\circ = 81 \times \frac{10}{9} \) grades
\( \implies 81^\circ = 9 \times 10 \) grades
\( \implies 81^\circ = 90 \) grades
Therefore, the third angle in centesimal unit is 90 grades.
In simple words: First, change all angles to degrees. Then, use the rule that a triangle's angles add up to 180 degrees to find the missing angle. Finally, convert that degree measure to grades using the conversion factor that 180 degrees equals 200 grades.

🎯 Exam Tip: Remember to convert all angle measures to a consistent unit (usually degrees) before performing calculations. Also, be careful to convert the final answer to the specific unit requested, in this case, centesimal units (grades).

Question 5. Express in circular measure and also in degrees the angle of a regular octagon.
Answer:
For a regular polygon with 'n' sides, the formula for each interior angle is given by \( \frac{(n-2) \times 180^\circ}{n} \).
For a regular octagon, the number of sides \( n = 8 \).
So, each interior angle in degrees is:
\( \text{Angle} = \frac{(8-2) \times 180^\circ}{8} \)
\( \implies \text{Angle} = \frac{6 \times 180^\circ}{8} \)
\( \implies \text{Angle} = \frac{1080^\circ}{8} \)
\( \implies \text{Angle} = 135^\circ \)
To express this in circular measure (radians), we use the conversion factor 1° = \( \frac{\pi}{180} \) radians.
\( 135^\circ = 135 \times \frac{\pi}{180} \) radians
\( \implies 135^\circ = \frac{3 \times 45 \times \pi}{4 \times 45} \) radians
\( \implies 135^\circ = \frac{3\pi}{4} \) radians
So, the interior angle of a regular octagon is 135° or \( \frac{3\pi}{4} \) radians.
In simple words: A regular octagon has 8 equal sides and 8 equal angles. You can find the measure of one of its inside angles using a special formula, first in degrees, and then change that degree value into radians.

🎯 Exam Tip: Memorize the formula for the interior angle of a regular polygon. Always show both the degree and radian measures when asked for both, ensuring clear conversion steps.

Question 6. If in two circles, arcs of the same length subtend angles of 60° and 75° at the centre, find the ratio of their radii.
Answer:
Let the arc length for both circles be \( l \). Let the radii of the two circles be \( r_1 \) and \( r_2 \).
The angles subtended at the centre are given as \( \theta_1 = 60^\circ \) and \( \theta_2 = 75^\circ \).
First, we need to convert these angles to radians, as the arc length formula \( l = r\theta \) requires \( \theta \) in radians.
For the first circle:
\( \theta_1 = 60^\circ = 60 \times \frac{\pi}{180} \) rad \( = \frac{\pi}{3} \) rad
For the second circle:
\( \theta_2 = 75^\circ = 75 \times \frac{\pi}{180} \) rad \( = \frac{5\pi}{12} \) rad
Using the arc length formula \( l = r\theta \):
For the first circle: \( l = r_1\theta_1 \implies l = r_1 \left(\frac{\pi}{3}\right) \)
For the second circle: \( l = r_2\theta_2 \implies l = r_2 \left(\frac{5\pi}{12}\right) \)
Since the arc lengths are the same, we can set the two expressions for \( l \) equal to each other:
\( r_1 \left(\frac{\pi}{3}\right) = r_2 \left(\frac{5\pi}{12}\right) \)
We need to find the ratio \( r_1 : r_2 \), which is \( \frac{r_1}{r_2} \).
\( \frac{r_1}{r_2} = \frac{5\pi/12}{\pi/3} \)
\( \implies \frac{r_1}{r_2} = \frac{5\pi}{12} \times \frac{3}{\pi} \)
\( \implies \frac{r_1}{r_2} = \frac{5 \times 3}{12} \)
\( \implies \frac{r_1}{r_2} = \frac{15}{12} \)
\( \implies \frac{r_1}{r_2} = \frac{5}{4} \)
Thus, the ratio of their radii is \( r_1 : r_2 = 5 : 4 \).
In simple words: When two circles have the same arc length but different angles, their radii will be different. To find the ratio of their radii, first change the angles to radians, then use the arc length formula and compare the two circles.

🎯 Exam Tip: Always convert angles to radians before using the arc length formula \( l = r\theta \). Carefully set up the equation when quantities like arc length are equal between two different scenarios.

Question 7. In a circle of diameter 60 cm the length of a chord is 30 cm. Find the length of the minor arc subtended by the chord.
Answer:
Given the diameter of the circle is 60 cm, so the radius \( r = \frac{60}{2} = 30 \) cm.
The length of the chord is also given as 30 cm.
Let O be the center of the circle and A, B be the endpoints of the chord. Then OA and OB are radii, so \( OA = OB = 30 \) cm. The chord \( AB = 30 \) cm.
Since \( OA = OB = AB = 30 \) cm, triangle OAB is an equilateral triangle. An equilateral triangle has all angles equal to 60°.
Therefore, the central angle \( \theta = \angle AOB = 60^\circ \).
To find the arc length, we must convert the angle to radians:
\( \theta = 60^\circ = 60 \times \frac{\pi}{180} \) radians \( = \frac{\pi}{3} \) radians.
The length of the minor arc (let's call it \( l \)) is given by the formula \( l = r\theta \).
\( l = 30 \times \frac{\pi}{3} \)
\( \implies l = 10\pi \) cm.
The length of the minor arc subtended by the chord is \( 10\pi \) cm. (If \( \pi \approx 3.14 \), then \( 10 \times 3.14 = 31.4 \) cm).
The major arc is the circumference minus the minor arc. The circumference of the circle is \( 2\pi r = 2\pi (30) = 60\pi \) cm.
Major arc \( = 60\pi - 10\pi = 50\pi \) cm.

In simple words: When a chord in a circle is the same length as the radius, it forms a special triangle at the center where all sides are equal. This means the angle at the center is 60 degrees. Convert this angle to radians and use the arc length formula to find the length of the curved part.

🎯 Exam Tip: Recognize special triangles like equilateral triangles formed by chords and radii, as this directly gives you the central angle. Always convert the central angle to radians before calculating arc length.

Question 8. Find the angle in radian through which a pendulum swings and its length is 75 cm and the tip describes an arc of length 21 cm.
Answer:
In this problem, the length of the pendulum acts as the radius of a circular path, and the distance the tip describes is the arc length.
Given: Length of pendulum \( r = 75 \) cm.
Length of arc described by the tip \( l = 21 \) cm.
We use the formula relating arc length, radius, and central angle: \( l = r\theta \), where \( \theta \) must be in radians.
Rearranging the formula to find the angle \( \theta \):
\( \theta = \frac{l}{r} \)
Substitute the given values:
\( \theta = \frac{21 \text{ cm}}{75 \text{ cm}} \)
Simplify the fraction:
\( \theta = \frac{7}{25} \) radians
The required angle through which the pendulum swings is \( \frac{7}{25} \) radians. This angle is quite small, indicating a narrow swing.

In simple words: Imagine a pendulum swinging. The length of the string is like the radius of a circle, and the path its tip travels is an arc. If you know how long the string is and how far the tip moved, you can find the angle it swung through using a simple division, and that angle will be in radians.

🎯 Exam Tip: For pendulum problems, the length of the pendulum is the radius, and the distance covered by the tip is the arc length. Always ensure your angle calculation results in radians, as the formula \( \theta = \frac{l}{r} \) intrinsically yields radians.

Question 9. Find the area of the sector of a circle of radius 14 cm having central angle \( \frac { \pi }{ 4 } \). (Take \( \pi = \frac { 22 }{ 7 } \)).
Answer:
Given: Radius of the circle \( r = 14 \) cm.
Central angle \( \theta = \frac{\pi}{4} \) radians.
We are asked to take \( \pi = \frac{22}{7} \).
The formula for the area of a sector is \( A = \frac{1}{2}r^2\theta \), where \( \theta \) is in radians.
Substitute the given values into the formula:
\( A = \frac{1}{2} \times (14)^2 \times \frac{\pi}{4} \)
\( \implies A = \frac{1}{2} \times 196 \times \frac{\pi}{4} \)
\( \implies A = 98 \times \frac{\pi}{4} \)
\( \implies A = \frac{49}{2} \pi \)
Now, substitute the value of \( \pi = \frac{22}{7} \):
\( A = \frac{49}{2} \times \frac{22}{7} \)
\( \implies A = \frac{7 \times 7}{2} \times \frac{2 \times 11}{7} \)
\( \implies A = 7 \times 11 \)
\( \implies A = 77 \) cm\( ^2 \)
The area of the sector is 77 cm\( ^2 \). This area represents a wedge-shaped portion of the circle.
In simple words: To find the area of a slice of a circle, like a piece of pizza, use a special formula that involves the circle's radius and the angle of the slice, measured in radians. Remember to use the given value of pi for calculations.

🎯 Exam Tip: Always use the central angle in radians when calculating the area of a sector using \( A = \frac{1}{2}r^2\theta \). Be careful with arithmetic and simplification, especially when a specific value for \( \pi \) is provided.

Question 10. A horse is tied to a post by a rope. If the horse moves along a circular path always keeping the rope tight and describes 88 metres when it has traced out 72° at the centre, find the length of the rope.
Answer:
In this scenario, the length of the rope is the radius \( r \) of the circular path, the distance the horse moves is the arc length \( l \), and the angle traced at the centre is \( \theta \).
Given: Arc length \( l = 88 \) metres.
Central angle \( \theta = 72^\circ \).
We need to find the length of the rope, which is \( r \).
First, convert the angle from degrees to radians, as the formula \( l = r\theta \) requires \( \theta \) in radians.
\( \theta = 72^\circ = 72 \times \frac{\pi}{180} \) radians
\( \implies \theta = \frac{2 \times 36 \times \pi}{5 \times 36} \) radians
\( \implies \theta = \frac{2\pi}{5} \) radians
Now, use the arc length formula \( l = r\theta \):
\( 88 = r \times \frac{2\pi}{5} \)
Rearrange to solve for \( r \):
\( r = \frac{88 \times 5}{2\pi} \)
\( \implies r = \frac{440}{2\pi} \)
\( \implies r = \frac{220}{\pi} \)
Using the approximation \( \pi = \frac{22}{7} \):
\( r = \frac{220}{22/7} \)
\( \implies r = 220 \times \frac{7}{22} \)
\( \implies r = 10 \times 7 \)
\( \implies r = 70 \) metres
The length of the rope is 70 metres. This means the horse is tethered quite far from the post.
In simple words: Imagine a horse tied to a pole, walking in a circle. The rope is the radius, the path it walks is the arc, and the angle it turns is the central angle. To find the rope's length, convert the angle to radians, then use the arc length formula and rearrange it to solve for the radius.

🎯 Exam Tip: Always ensure the central angle is in radians when using the arc length formula \( l = r\theta \). If \( \pi \) appears in the calculation, use its approximate value (like \( \frac{22}{7} \)) at the final step to get a numerical answer, unless asked to leave it in terms of \( \pi \).