Angle and its Measurement — MH State Board 11 Textbook Solutions

1.1 Directed Angles

14 q

Solved Examples

Worked · 14

Convert the following degree measures in the radian measures.
1.1 SolvedEx.1(i) $70^\circ$
1.1 SolvedEx.1(ii) $-120^\circ$
1.1 SolvedEx.1(iii) $\left(\frac{1}{4}\right)^\circ$
Convert the following radian measures in the degree measures.
1.1 SolvedEx.2(i) $\left(\frac{7\pi}{3}\right)^{c}$
1.1 SolvedEx.2(ii) $\left(\frac{-\pi}{18}\right)^{c}$
1.1 SolvedEx.2(iii) $\left(\frac{4}{7}\right)^{c}$
Express the following angles in degree, minute and second.
1.1 SolvedEx.3(i) $74.87^\circ$
1.1 SolvedEx.3(ii) $-30.6947^\circ$
1.1 SolvedEx.4 $The measures of the angles of the triangle are in A. P. The smallest angle is 40. Find the angles of the triangle in degree and in radian.$
1.1 SolvedEx.5 $\frac{7\pi^{c}}{30}$
1.1 SolvedEx.6 $\frac{2\pi}{9}$
1.1 SolvedEx.7 $\left(\frac{4\pi}{5}\right)^{c}$
Find the angle between hour hand and minute hand of a clock at
1.1 SolvedEx.8(i) $Quarter past five$
1.1 SolvedEx.8(ii) $Quarter to twelve$

Exercise 1.1

48 q

Determine which of the following pairs of angles are co-terminal.
Ex 1.1 Q1A(i) $210^\circ,\ -150^\circ$
Ex 1.1 Q1A(ii) $360^\circ,\ -30^\circ$
Ex 1.1 Q1A(iii) $-180^\circ,\ 540^\circ$
Ex 1.1 Q1A(iv) $-405^\circ,\ 675^\circ$
Ex 1.1 Q1A(v) $860^\circ,\ 580^\circ$
Ex 1.1 Q1A(vi) $900^\circ,\ -900^\circ$
Draw the angles of the following measures and determine their quadrants.
Ex 1.1 Q1B(i) $-140^\circ$
Ex 1.1 Q1B(ii) $250^\circ$
Ex 1.1 Q1B(iii) $420^\circ$
Ex 1.1 Q1B(iv) $750^\circ$
Ex 1.1 Q1B(v) $945^\circ$
Ex 1.1 Q1B(vi) $1120^\circ$
Ex 1.1 Q1B(vii) $-80^\circ$
Ex 1.1 Q1B(viii) $-330^\circ$
Ex 1.1 Q1B(ix) $-500^\circ$
Ex 1.1 Q1B(x) $-820^\circ$
Convert the following angles in to radian.
Ex 1.1 Q2(i) $85^\circ$
Ex 1.1 Q2(ii) $250^\circ$
Ex 1.1 Q2(iii) $-132^\circ$
Ex 1.1 Q2(iv) $65^\circ 30'$
Ex 1.1 Q2(v) $75^\circ 30'$
Ex 1.1 Q2(vi) $40^\circ 48'$
Convert the following angles in degree.
Ex 1.1 Q3(i) $\frac{7\pi^{c}}{12}$
Ex 1.1 Q3(ii) $\frac{-5\pi^{c}}{3}$
Ex 1.1 Q3(iii) $5^{c}$
Ex 1.1 Q3(iv) $\frac{11\pi^{c}}{18}$
Ex 1.1 Q3(v) $\left(\frac{-1}{4}\right)^{c}$
Express the following angles in degree, minute and second.
Ex 1.1 Q4(i) $(183.7)^\circ$
Ex 1.1 Q4(ii) $(245.33)^\circ$
Ex 1.1 Q4(iii) $\left(\frac{1}{5}\right)^{c}$
Ex 1.1 Q5 $\triangle$
Ex 1.1 Q6 $\frac{5\pi^{c}}{9}$
Ex 1.1 Q7 $4:5$
Ex 1.1 Q8 $5\pi^{c}$
Ex 1.1 Q9 $3:7:8$
Ex 1.1 Q10 $The measures of the angles of a triangle are in A.P. and the greatest is 5 times the smallest (least). Find the angles in degree and radian.$
Ex 1.1 Q11 $40^\circ$
Ex 1.1 Q12 $\frac{2\pi^{c}}{5}$
Find the degree and radian measure of exterior and interior angle of a regular
Ex 1.1 Q13(i) $Pentagon$
Ex 1.1 Q13(ii) $Hexagon$
Ex 1.1 Q13(iii) $Heptagon$
Ex 1.1 Q13(iv) $Octagon$
Find the angle between hour-hand and minute-hand in a clock at
Ex 1.1 Q14(i) $ten past eleven$
Ex 1.1 Q14(ii) $twenty past seven$
Ex 1.1 Q14(iii) $thirty five past one$
Ex 1.1 Q14(iv) $quarter to six$
Ex 1.1 Q14(v) $2:20$
Ex 1.1 Q14(vi) $10:10$

1.2 Arc Length and Area of a Sector

6 q

Solved Examples

Worked · 6

1.2 SolvedEx.1 $54^\circ$
1.2 SolvedEx.2 $30^\circ$
1.2 SolvedEx.3 $225\pi$
1.2 SolvedEx.4 $The perimeter of a sector is equal to half of the circumference of a circle. Find the measure of the angle of the sector at the centre in radian.$
1.2 SolvedEx.5 $36^\circ$
1.2 SolvedEx.6 $ABCDEFGH is a regular octagon inscribed in a circle of radius 9 cm. Find the length of minor arc AB.$

Exercise 1.2

10 q

Ex 1.2 Q1 $108^\circ$
Ex 1.2 Q2 $The radius of a circle is 9 cm. Find the length of an arc of this circle which cuts off a chord of length, equal to length of radius.$
Ex 1.2 Q3 $Find the angle in degree subtended at the centre of a circle by an arc whose length is 15 cm, if the radius of the circle is 25 cm.$
Ex 1.2 Q4 $18^\circ$
Ex 1.2 Q5 $60^\circ$
Ex 1.2 Q6 $25\pi$
Ex 1.2 Q7 $m\angle AOB = 45^\circ$
Ex 1.2 Q8 $m\angle POQ = 30^\circ$
Ex 1.2 Q9 $25\pi$
Ex 1.2 Q10 $64\pi$

Miscellaneous Exercise 1

21 q

(I) Select the correct option

Practice · 10

Misc I Q1 $\left(\dfrac{22\pi}{15}\right)^{c}$
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Misc I Q2 $156^\circ$
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Misc I Q3 $72^\circ$
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Misc I Q4 $12^\circ$
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Misc I Q5 $Angle between hands of a clock when it shows the time 9.45 is$
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Misc I Q6 $20 meter of wire is available for fancing off a flower-bed in the form of a circular sector of radius 5 meters, then the maximum area (in sq. m.) of the flower-bed is$
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Misc I Q7 $If the angles of a triangle are in the ratio 1:2:3, then the smallest angle in radian is$
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Misc I Q8 $A semicircle is divided into two sectors whose angles are in the ratio 4:5. Find the ratio of their areas?$
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Misc I Q9 $Find the measure of the angle between hour-hand and the minute hand of a clock at twenty minute past two.$
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Misc I Q10 $9\pi$
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(II)

Practice · 11

Misc II Q1 $\left(\dfrac{3\pi}{4}\right)^{c}$
Misc II Q2 $7\sqrt{2}$
Misc II Q3 $\Delta$
Misc II Q4 $60^\circ$
Misc II Q5 $A wire of length 10 cm is bent so as to form an arc of a circle of radius 4 cm. What is the angle subtended at the centre in degrees ?$
Misc II Q6 $65^\circ$
Misc II Q7 $81\pi$
Misc II Q8 $5^\circ 30'$
Misc II Q9 $A train is running on a circular track of radius 1 km at the rate of 36 km per hour. Find the angle to the nearest minute, through which it will turn in 30 seconds.$
Misc II Q10 $In a circle of diameter 40 cm, the length of a chord is 20 cm. Find the length of minor arc of the chord.$
Misc II Q11 $The angles of a quadrilateral are in A.P. and the greatest angle is double the least. Find angles of the quadrilateral in radian.$