1. Angle and its measurement
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(1) Find the radian measure of the angle between hour-hand and minute hand of a clock at:
(ii) Ten minutes past eleven
(i) Twenty minutes past two
(iii) Thirty minutes past six
(iv) Five minutes past one
Solution : The speed of minute hand is 6 degrees per minute. The speed of the hour hand is 0.5 degrees per minute. The angle between hour hand and minute hand changes at the rate 5.5°.
(i) Twenty minutes past two: 2:20 We consider two positions of hands. First at 2:00 and second at 2:20.
At 2:00 angle between them is 60 degrees This angle gradually decreases, becomes zero and then increases. From 2:00 to 2:20 (i.e. in 20 minute time) the minute hand moves through 120 degrees and hour hand moves through 20 * 0.5 degrees = 10 degrees
The minute hand makes 120 ^ epsilon - 60 degrees = 60 degrees angle with the position of hour hand at 2:00. But hour hand has moved through 10 degrees . The required angles is 60 degrees - 10 degrees = 50 degrees
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(ii) Ten minutes past eleven : ..
Answer/Solucation:
11:10. The angle between any two consecutive numbers of a clock. is age 3.0 IF the minute hand is at 2 and hour hand is. ax 11's angle between them is (2 * 30 degrees) = 90 degrees But 10 past eleven.. i.e. at 11:10, the hour hand is moved 10 minutes towards 12, In 60 minutes, the hourhond Covers.
Page No. 1
3e ^ o therefore in 10 minutes, it Covers ( 30 6 )^ 0 =5... the angle between the two hands of the clock at ten past eleven 90^ 0 -s^ 0 =8s^ 0 = (85 * pi/180) ^ c = I ^ c = (14pi * c)/86
(iii) Thirty minutes past six: 6:30 when the time...1.8.6:30
Answer/Solucation:
The hour hand just little left to the minute one...As. per... method 60 min takes 30 degrees
So: 1 min can take (30 degrees)/60
Again its 6.30. So that
30. min will make So.. angle minute 5...=(15 pi 180 )^ c = pi 12 is ise when the time
18. 6.3.0. Radiam measure > pi 12
(iv) Five minutes past one:
Answer/Solucation:
1 05 At 1 o'clock the angle... between hour and minute hand 30°. In 5. minutes. The angle made by hour d = 5/2 The angle between.. the hour. and minute hand... 30° -30°+ 5/2 -2.50 - (.2/5 * pi/180) ^ 1 = (pi ^ 2)/72
Page No 2
(2) Find the exact time when the angle between hour-hand and the
minute-hand of a clock will be: (i) 44° for the first time after 12 O'clock. (ii) 59 degrees for the first time after 6 O'clock.
Solution: The angle between hour hand and minute hand changes at
the rate 5.5 degrees .
The angle between them will be 11 degrees after 2 minutes.
(i) For 11 degrees angle it takes 2 minute time. How much time will it take for 44 degrees angle? 2×44, 11 =8
At 12:08 the angle between hour-hand and the minute-hand of a clock will be 44 degrees for the first time after 12 O'clock.
(ii) 59 degrees for the first time after 6 O'clock.
Solution : Initially i.e. To get angle sg... e at 6 o'clock angle.is...80⁰ 9 ^ 0 = 121 ^ 0 movement required. between her hand and minute taime. hand.. For 11° angle it takes 2 minutes time... How much time will it take fo.x. 121° angle? 2 x 121 minute. 11 ..=..2*.11...2...
At 6:22 the angle between houx -hand and the minute hand of a clock will be .sgº For the first time after 6 o'clock...
(3) A wire of length 10 cm is bent to form an arc of a circle of radius 4 cm. Find the radian and degree measure of the angle subtended by the arc at the center of the circle. Also find the area of the corresponding sector.
Page No.3
Solution Given r=4, are length We know that s = r * 0 = length of the wire = s = 10
plus/minus 0 = 4theta
theta = S/2 radian: =( 5 2 * 180 Pi )^ 0 =( 450 Pi )^ 0 A= 1 2 r^ 2 theta= 1 2 r*.........= 1 2 *4*...=20
Now, the area of the sector=
sq.cm.
(4) Arrange following angles in ascending order pi ^ c pi^ 0 , 3 ^ c , 100^ ,
Solution : ....To... Canuext angles in degrees. pi^ prime 2
"(..TT. X 180) ?=...180...
Pi.......3!8!29.73^ prime prime 1.3.0 (3.x180)03. ($40.)... 10.0 0=..0^ 0 pi^ c 2 ...( 7t 2 x... 180 11 .)^ 0 =...9 From above Conversion. 3.98! 29, 1.3....<90...<.19:0....< 1.11.880 <180°..... Ascending order is pi^ 2 2 *..100^ o ...3^ c ,.... Pi...
page No.4
(5) ABCDEFGH is a regular octagon inscribed in a circle of
radius 1 unit. O is the center of the circle. Find (i) Radian measure of angle
AOB (ii) 1(chord AB) (iii) l(arcAB) (iv) Area of the region enclosed between
chord AB and arc AB.
Solution:
Draw AK perpendicular to OB.
Given OA = OB = 1 Find AK, OK, BK and then AB.
(i)Regulax. actagan hos all its side equal if we join center H
O with all vertices.. A.B.C. - H we will get OA 08 Radius 1 we will get..8..
Segment.... angle.. of each.. Segment. = 360^ 8 =45^ angle AOB.=..4.5^ .
ii)
Draw Ak. J.D.B.
In right angled ADKA.. Sin ZADK = AK OA
Sin 45° = AK.. :: Akt
In. .A.A.OK.
<KOA <kAo .
Ak = ok.
Delta B..= rho K...+BK +8.K.
In right angled AAK.B.. by p 4 1 theorem..... AB^ 2 ...=.Ak^
2 +Bk...^ 2 AB^ 2 *( 1 sqrt 2 *)^ 2 +(1- 1 sqrt 2 )^ 2 *(b4(1)/(11)
A * B ^ 2 = 1/2 + 1 - 2/(sqrt(2)) + 1/2 AB^ 2 =2- 2 sqrt 2
....................
AB= √2-√₂ unit
AB= 0.745 Units. e (chord AB ) = √²-√₂ = 0.765 Units
(i*r_{i}) e (ancAB J= gamma.0
.... pi 4 - pi/2 Units
... BK = 1-√₂ — (ii)
iv) Area of the resion.. Cendouxd
between chond AB and ore AB - Are are a- DOAB area.
1 2 gamma^ 2 theta- 1 2 x0B gamma .
Page No.5
(6) Prove that the lateral surface area of a right circular cone is rl, where r is the radius of the circular base and is the slant height of the cone..
Solution : Given: A right circular Cone with. is the radius of the Circular base. and is the slant beisht of the cone. To prove Lateral Surface area.. or Curved Surface area Tre Activity: IF a perpendicular. cut is made from a point on..... the Circumferne at the base. to the vertex and the Cone is.. opend out then the radius of a Sectax formed circumference of Circle.. circle
..Since the circumference of the. (base = 2TT.Y.
...lateral Surface area of cane. Area of Sentur. OAB..... - Arc length of Sectox...X Area of
B
= 2T1X X FT. y. 2. 2TTY
Curved surface area or the lateral Surface area of a right Circular Cone is. Tre...
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