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KC Sinha Mathematics Solution Class 11 Chapter 4 Angles and their measures Exercise 4.1

PAGE-4.11

 Exercise 4.1

TYPE-I

Question 1

Find the degree measure corresponding to the following radian measures :

(i) \left(\frac{4 \pi}{3}\right)^{c}

Sol :

1^{c}=\left\{\frac{180^{\circ}}{\pi}\right\}

Now, \frac{4 \pi}{3} \times \frac{180^{\circ}}{\pi}

=4×60=240°


(ii) \left(\frac{7 \pi}{6}\right)^{c}

Sol :

=\frac{7 \pi}{6} \times \frac{180^{\circ}}{\pi}

=7×30=210°


(iii) \left(\frac{5 \pi}{3}\right)^{c}

Sol :

=\frac{5 \pi}{3} \times \frac{180^{\circ}}{\pi}

=5×60=300°


(iv) \left(-\frac{5 \pi}{24}\right)^{c}

Sol :

=-\frac{5 \pi}{24} \times \frac{180^{\circ}}{\pi}

=-\frac{5}{2} \times 15^{\circ}=\frac{-75^{\circ}}{2}

=-37.5°


∴1°=60'

(0.5×60)'=30'


=-37° 30'


(v) \left(-\frac{2 \pi}{3}\right)^{c}

Sol :

=-\frac{2 \pi}{3} \times \frac{180^{\circ}}{\pi}

=-2×60=-120°


(vi) \left(\frac{33 \pi}{320}\right)^{c}

Sol :


(vii) 6^{c}

Sol :


(viii) (-4)^{c}

Sol :


(ix) \left(\frac{11}{16}\right)^{c}

Sol :


(x) (2.64)^{c}

Sol :


Question 2

Express the following angles in radian measure

(i) 105°

(ii) 25°

(iii) 240°

(iv) -56°

(v) 520°

(vi) 7^{\circ} 30^{\prime}

(vii) 40^{\circ} 20^{\prime}

(viii) 42^{\circ} 57^{\prime} 16^{\prime \prime}

(ix) -47^{\circ} 30^{\prime}


Question 3

Two angles of a triangle are 72^{\circ} 53^{\prime} 51^{\prime \prime} and 41^{\circ} 22^{\prime} 50^{\prime \prime} respectively. Find the third angle in radians.

Sol :


Question 4

Find the angle between the hour-hand and minute-hand in circular measure at 4 O'clock.

Sol :


Question 5

The angles of a triangle are in A.P. and the number of degrees in the least is to the number of radians in the greatest is 36 : 𝝿, find the angles in degrees.

Sol :


TYPE-II

Question 6

Find the degree measure of the angle subtended at the centre of a circle of radius 100 cm by an arc of length 22 \mathrm{~cm}\left(\right. use \left.\pi=\frac{22}{7}\right).

Sol :



Question 7

(i) Find the radius of the circle in which a central angle of 45° makes an arc of 187 cm \left(\right. use \left.\pi=\frac{22}{7}\right)

(ii) Find the radius of the circle in which a central angle of 60^{\circ} intercepts an arc of length 37.4 cm use \left.\pi=\frac{22}{7}\right).

Sol :



Question 8

In a circle of diameter 40 cm, the length of a chord is 20 cm. Find the length of minor are of the chord.

Sol :


Question 9

At what distance does a man 5 \frac{1}{2} \mathrm{ft} in height, subtend an angle of 15^{\prime \prime} ?

Sol :



Question 10

In two circles, arcs of equal length subtend angles of 60° and 75° at their centres. show that their radii are in the ratio 5: 4



Question 11

(i) If the arcs of same length in two circles subtend angles of 75^{\circ} and 120 at their respective centres, find the ratio of their radii.

(ii) If arcs of the same lengths in two circles subtend angles of 65^{\circ} and 110 at the centre, find the ratio of their radii.

Sol :


Question 12

Find the angle in radian through which a pendulum swings if its length is 75 cm and the tip describes an are of length :

(i) 10 cm

(ii) 21 cm

(iii) 15 cm \left(\right. use \left.\pi=\frac{22}{7}\right)



Question 13

The minute hand of a watch is 1.5 cm long. How far does its tip move in

(i) 50 minutes (ii) 40 minutes ? (use \pi=3.14 )


Question 14

(i) A wheel makes 30 revolutions per minute. Find the radian measure of the angle described by one of the spokes of the wheel in \frac{1}{2} second.

(ii) A wheel makes 360 revolutions in one minute. Through how many radians does it turn in one second?

Sol :



Question 15

Assuming the average distance of the earth from the sun to be 149700000 km and the angle subtended by the sun at the eye of a person on the earth to be 32^{\prime}. find the sun's diameter.



Question 16

A truck is travelling on a circular road of radius 1500 metres at 66 km/hr. Through what angle (in radians) does it turn in 10 seconds ?



Question 17

If the angular diameter of the moon be 30', how far from the eye should a coin of diameter 2.2 cm be kept to hide the moon ?

Sol :














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