AS Level Mathematics 9709 | Circular Measures 5 | Past Paper Questions | Raja Shafqat — Transcript

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1. 0:08Assalamu Alaikum beta, finally today we
2. 0:11will start practice for circular
3. 0:12measurement. Is it clear, beta? We only
4. 0:16covered the basics in the last four or
5. 0:18five classes. Today we will finally
6. 0:21start our chapter and do further
7. 0:22practice on it. Look, what new things
8. 0:25did we study? The only new thing we
9. 0:28have studied so far in this entire
10. 0:30topic is what the formula for arc
11. 0:32length is. The formula for arc length
12. 0:35is r*theta. And we studied one more
13. 0:37thing, that the area of a sector
14. 0:39formula is 1/2 rs. The formula for the
15. 0:43area of a sector is 1/2 rs or the
16. 0:45formula for the area of a sector is 1/2
17. 0:47r² theta. Technically, this is the
18. 0:50only new thing. Otherwise, we haven't
19. 0:52studied anything new. Is it clear, beta
20. 0:54? We only learned these two formulas as
21. 0:56new. s = r theta and area = 1/2 rs and
22. 0:591/2 r² theta. Now, let's start. Now,
23. 1:0399%of the work here is O-Level Math.
24. 1:06These are just two extra formulas.
25. 1:08Clear? Okay, now look at the wording of
26. 1:10the question. In the diagram, AC is an
27. 1:12arc of a circle with center O and
28. 1:14radius six. So, if any student has a
29. 1:18doubt, I will clarify this question a
30. 1:20bit more to explain what this question
31. 1:22technically meant. Look, what was this
32. 1:25question technically like? Look,
33. 1:27technically this meant that there is a
34. 1:29circle like this, and this is the
35. 1:30center of the circle. And the line
36. 1:33given to you, line AC, what is line AC?
37. 1:37Line AC is tangent to the circle. And
38. 1:40we know that according to the property,
39. 1:41the radius is always perpendicular to
40. 1:43the tangent line. So, this line,
41. 1:45basically, what is this line? This line
42. 1:48is tangent to the circle. Now look at
43. 1:50the question, what is the radius of
44. 1:52this circle or this sector? The radius
45. 1:54of this sector is 6 cm, so obviously OA
46. 1:56will also be a radius. Because what is
47. 1:59the definition of radius? The gap from
48. 2:01the center to the circumference is what
49. 2:02we call the radius. The gap from the
50. 2:04center to the circumference, that
51. 2:06length is what we call the radius. Is
52. 2:08it clear, beta? And this point is
53. 2:10already given to you. This point is
54. 2:12your B. So, this point is basically
55. 2:14your B. So, your required region is
56. 2:16this required region. If you check your
57. 2:20diagram, then this is the required
58. 2:22region of your question which I have
59. 2:23highlighted. It is also obvious from
60. 2:26your actual diagram. This is its
61. 2:27required region. Is it clear, beta? So,
62. 2:30the first thing we know is what this
63. 2:32larger shape is. This larger shape is a
64. 2:34triangle. So, if I cut this sector out
65. 2:37of this right-angled triangle, which
66. 2:39area will be found? The shaded area
67. 2:41will be found. Is that clear, kid? So
68. 2:43now, what is the first extra effort we
69. 2:45need to make? Let's start slowly. This
70. 2:47is the diagram you have. Well, first of
71. 2:49all, we don't have the area of the
72. 2:51triangle. So, we know what type of
73. 2:53triangle it is, right? The triangle is
74. 2:55a right-angled triangle. The triangle
75. 2:57is a right-angled triangle. So, the
76. 3:00point you have here is point O. This is
77. 3:03point C. This is point B. And the angle
78. 3:05you have here was π/3, this angle is
79. 3:08π/3. You are given this angle as π/3,
80. 3:11and what is the base of this triangle?
81. 3:13It is six. By the way, if any student
82. 3:16remembers, what is π/3 in degrees?
83. 3:18180/3, so in degrees, this technically
84. 3:21means 60 °. Is that clear, kid? I
85. 3:24explained earlier as well what pi means
86. 3:26. Pi means 180 °. So when I simplify
87. 3:30this, what answer will be derived?
88. 3:31Technically, in degrees, it is 60 °.
89. 3:34Now, what is missing for us? With
90. 3:36respect to this angle, what is this?
91. 3:38This is the opposite. So, first of all,
92. 3:40we don't have the height. So, we
93. 3:42applied it, we applied tan. Because we
94. 3:45know, what is the formula for tan in a
95. 3:47right-angled triangle? The formula for
96. 3:50tan is, tan is always opposite divided
97. 3:52by adjacent. Opposite divided by
98. 3:55adjacent. Opposite divided by adjacent.
99. 3:58Is this clear? Opposite divided by
100. 4:01adjacent. Now, you have to use a
101. 4:02calculator. And when you check the
102. 4:05answer for tan π/3 on your calculator,
103. 4:07your calculator will show √3. This
104. 4:09job is obviously for the calculator. Is
105. 4:11that clear, kid? We know that all
106. 4:13trigonometric ratios are inside the
107. 4:15calculator. So, from here, I finally
108. 4:17found out, what is the derived answer
109. 4:20for BC? The answer for BC is derived as
110. 4:236 times √3. This is the answer you
111. 4:25found for BC. So, what was BC
112. 4:27technically? BC was the height of the
113. 4:30triangle for you. Height of a triangle.
114. 4:32So, what is the height derived for this
115. 4:34triangle? The height of this triangle
116. 4:36is derived as 6√3, 6√3. So now,
117. 4:39what is your final target? Your target
118. 4:41is the shaded area. Is that clear, kid?
119. 4:43Your target in the question is the
120. 4:45shaded area. So, if we are asked to
121. 4:47find the shaded area. Then, what will
122. 4:49the shaded area be equal to? The shaded
123. 4:51area will be equal to. Area of the
124. 4:55right-angled triangle minus area of the
125. 4:57sector minus area of the sector. You
126. 5:01have this right-angled triangle here.
127. 5:03What is the base of the right-angled
128. 5:04triangle? The base of the right-angled
129. 5:06triangle is 6. What height did we find
130. 5:08for the right-angled triangle? We just
131. 5:10found the height of the right-angled
132. 5:11triangle. 6√3, this is the height of
133. 5:14the right-angled triangle, 6√3. This
134. 5:16is the height of the right-angled
135. 5:18triangle. And now, when you talk about
136. 5:20the sector, what is the radius of the
137. 5:22sector? The radius of the sector is 6,
138. 5:24right here. And what is this angle of
139. 5:26the sector? This angle of the sector is
140. 5:29π/3, so how will your target be found?
141. 5:32How will you find this shaded area?
142. 5:34Area of the whole triangle minus area
143. 5:36of the sector. Area of the whole
144. 5:39triangle minus area of the sector. So
145. 5:41what is the area of the triangle? Since
146. 5:43the triangle is right-angled, we will
147. 5:45say 1/2*height*base. So what height of
148. 5:47the triangle was found? 6√3, this is
149. 5:49the height of the triangle. So what did
150. 5:51the area of the triangle become? 1/2*
151. 5:53height*base. 1/2*height*base, and we
152. 5:57already know the radius of the sector.
153. 6:001/2*6²*θ, θ means the angle, 1/2
154. 6:04r²θ. Now, when this was further
155. 6:07simplified, it became 2*3*6 = 18, so
156. 6:10what answer was derived here? 18√3,
157. 6:1318√3, minus sign copied as it is. Now
158. 6:18this three, let's see, 3*2 = 6, and six
159. 6:20gets cancelled. So what answer was
160. 6:22derived from here? From here, your
161. 6:24answer was derived as 6π. What answer
162. 6:27was derived from here? From here, your
163. 6:29answer was derived as 6π. This is the
164. 6:31final required answer derived for your
165. 6:33question. Now look, what was the
166. 6:35statement in your question, 'find the
167. 6:37area of the shaded region', and in what
168. 6:39form was the answer? In terms of π.
169. 6:41And what does 18√3-6π mean? That the
170. 6:45answer you required was needed in exact
171. 6:47form. What does exact form mean?
172. 6:50Without decimals. So this is the most
173. 6:53average question with which we started.
174. 6:55Meaning, what was this simply? If we
175. 6:57talk about it, this was simply a full
176. 6:59diagram like this. There was a circle,
177. 7:01and this portion of it was what? It was
178. 7:03a sector. This line was a tangent. That
179. 7:05is why this angle is what? 90 °. So
180. 7:08what did we basically do? When we
181. 7:10subtract the sector from this entire
182. 7:12right-angled triangle, our required
183. 7:14area, meaning the shaded area, will be
184. 7:16found. Is that clear, son? So, for us,
185. 7:19this is 99.99%O-Level Math. What is the
186. 7:22only new formula? The arc length
187. 7:25formula is r*theta, and the area
188. 7:27formula is 1/2 rs or 1/2 r² theta.
189. 7:30This is the formula for the area of the
190. 7:31sector. Otherwise, we have already
191. 7:33covered all the other basics. Is that
192. 7:35clear, son? Come on, this was our first
193. 7:37question. So, in what form was the
194. 7:39answer required? The answer was
195. 7:41required in exact form. The answer was
196. 7:43required in exact form. That is why I
197. 7:46did not show decimals anywhere in the
198. 7:48answers to this question. Is that clear
199. 7:50, son? So, this was the first example
200. 7:52of circular measurement. Let's try more
201. 7:54questions on this. Next question.
202. 7:57Please read the question first so that
203. 7:59I can guide you. Hurry up, son. Look at
204. 8:01what the condition is. In the diagram,
205. 8:03OCD is an isosceles triangle. The
206. 8:05examiner is saying that this whole
207. 8:07triangle, this whole triangle is
208. 8:08isosceles. He is right, what is this
209. 8:10side? It is 10. This side is also 10.
210. 8:13And what is this angle? It is 0.8. RAD
211. 8:15means the angle is already given in
212. 8:17radians form. The angle is already
213. 8:19given in radians form. Is that clear,
214. 8:21son? The angle is already given. What
215. 8:23is your target? Shaded area and shaded
216. 8:25perimeter. If any student wants to see
217. 8:27this diagram in more detail, I will
218. 8:29show you that basically, the examiner
219. 8:31is saying something like this. If I
220. 8:34focus a bit more on this question,
221. 8:36basically the examiner is giving you a
222. 8:38situation like this. The situation is
223. 8:41something like this. If you connect
224. 8:43this with your question, this is your
225. 8:45O. This radius of yours is 6 cm. This
226. 8:48radius of yours is 6 cm. And this angle
227. 8:51of yours, this angle is 0.8 radians.
228. 8:540.8 radians. This point is your B. This
229. 8:57point is your A. This point is your C.
230. 8:59This point is your D. And what is this
231. 9:01length given? This length is given as 4
232. 9:03cm and this length is also given as 4
233. 9:05cm. Is that clear, son? So, I purposely
234. 9:08plotted the complete diagram. The
235. 9:10examiner has no interest in this
236. 9:11particular portion. Is that clear, son?
237. 9:14Your original question is this, and
238. 9:15what has the examiner done with all
239. 9:17this area? The examiner has shaded all
240. 9:19of this area. So, basically, this whole
241. 9:21region, this whole region is yours that
242. 9:23is shaded. Okay, son? This region is
243. 9:26already given in the actual diagram
244. 9:27above. I am just highlighting this for
245. 9:30you so that you can get a bit of an
246. 9:31idea. Clear, son? This entire area is
247. 9:35your required region that the examiner
248. 9:36has highlighted for you. This is all
249. 9:39your required region. Meaning the
250. 9:40shaded region is our required region.
251. 9:42Clear, son? This is our required region
252. 9:45. The one I have deliberately
253. 9:46highlighted. Clear, son? Now let’s
254. 9:48start our question: what is the basic
255. 9:50target in our question? First of all,
256. 9:52the examiner says find the area of the
257. 9:54shaded region. So, your number one
258. 9:56target is the shaded area. Okay, son?
259. 9:58Our target in the question, first and
260. 10:00foremost, is the shaded area. So,
261. 10:03observe this question carefully. What
262. 10:05will we do for the shaded area? This
263. 10:07entire shape is an isosceles triangle.
264. 10:11So, we can read it as: how will the
265. 10:13shaded area be found? What will you
266. 10:15have to do for the shaded area? From
267. 10:17the area of the whole isosceles
268. 10:19triangle OCD, you will have to......
269. 10:21take this sector out. So, what is this
270. 10:24sector you have? The name of this
271. 10:26sector is sector OB. So, what will we
272. 10:29have to do with this sector? Finally,
273. 10:31we will have to subtract this sector.
274. 10:33Clear, son? Read the question first.
275. 10:35Now tell me, what will be the area of
276. 10:37the triangle? This side of the triangle
277. 10:39is 10. This side is 10. This angle is
278. 10:41already given. So, we have already
279. 10:42studied in the basics what its area
280. 10:44will be. Its area will be 1/2 ab sinC,
281. 10:47so this will be 1/2*10*10*sin (0.8);
282. 10:501/2 ab sinC, clear son? 1/2 ab sinC.
283. 10:53Now, what will be subtracted from that?
284. 10:57The area of the sector; so we already
285. 10:59know what the area of a sector is.
286. 11:01It’s always 1/2 r² theta, 1/2 r²
287. 11:04theta; so what will be the area of this
288. 11:06sector? 1/2*radius squared*0.8*0.8. Now
289. 11:12finally, if you want to make it a bit
290. 11:15easier, we took help from a calculator:
291. 11:185*10*10*sin (0.8), we just put
292. 11:21everything into the calculator-5*6
293. 11:24squared*0.8. So what answer was finally
294. 11:28derived? The answer finally derived was
295. 11:3124.467. 80 80454. But obviously, the
296. 11:38answer we have to provide must be to
297. 11:39three significant figures. So what will
298. 11:42the answer be to three significant
299. 11:43figures? 21.5 cm². This is the final
300. 11:48answer to your question. Because we
301. 11:50know that in A-Level Maths, the final
302. 11:52answer is required to be to three
303. 11:54significant figures. So this was a very
304. 11:56average question. We simply took this
305. 11:59sector out from the entire larger shape
306. 12:01, meaning the isosceles triangle. So
307. 12:04our shaded area is found. Clear, dear?
308. 12:07This was the first step of our question
309. 12:09. Clear? Now look, what is the second
310. 12:12part of this question? What does the
311. 12:14examiner require in the second part?
312. 12:16The perimeter of the shaded region. Do
313. 12:18we know what the word perimeter means?
314. 12:20The word perimeter means the sum of all
315. 12:22sides. What does the word perimeter
316. 12:25mean? The word perimeter means the sum
317. 12:27of all sides. So now look, what are you
318. 12:29interested in? You are interested in
319. 12:33the shaded perimeter. Okay? We are
320. 12:37basically interested in the shaded
321. 12:38perimeter. Okay? Now look, which region
322. 12:41is the shaded perimeter? Our interest
323. 12:44lies in this shaded area, the shaded
324. 12:46perimeter. Okay? In the second part,
325. 12:49the examiner is telling us what they
326. 12:50want. They want the shaded perimeter.
327. 12:52They want the shaded perimeter. So look
328. 12:54, we have copied the same diagram from
329. 12:56above. What is our interest in? Our
330. 12:59interest is in the shaded perimeter. So
331. 13:00we know, look, what does shaded
332. 13:02perimeter mean? What does the word
333. 13:04perimeter itself mean? The word
334. 13:07perimeter means adding up all the
335. 13:08boundaries. Clear? What does the word
336. 13:11perimeter even mean? Sum of all sides.
337. 13:13Sum of all lengths. Clear? What
338. 13:15definition of perimeter have we always
339. 13:17heard in math? Sum of all sides, sum of
340. 13:19all lengths. Now what did we do? We
341. 13:22have to add all these sides. Okay? Look
342. 13:25, first of all, we already know this
343. 13:27side. It is four. We already know this
344. 13:29side is four. What is this side? This
345. 13:31is the arc length. Whose arc length is
346. 13:33it? It is the arc length of the sector.
347. 13:36It is the arc length of the sector.
348. 13:37Clear, dear? This is the arc length of
349. 13:39your sector. This was your sector. The
350. 13:42name of this region of the sector was
351. 13:44ABCD. So first of all, what is your
352. 13:46target? The shaded perimeter. The
353. 13:48examiner says what they are interested
354. 13:50in. They are interested in the shaded
355. 13:52perimeter. Shaded perimeter. So
356. 13:55perimeter means the sum of all sides.
357. 13:57Now look, what does shaded perimeter
358. 13:59mean? The shaded perimeter will include
359. 14:01arc length AB. Look, what is this
360. 14:04boundary? Arc length plus length AC. So
361. 14:07first you told the examiner that this
362. 14:09is arc length AB. It is arc length AB.
363. 14:12Plus length AC, this length here is
364. 14:14length AC. What is this length? This
365. 14:16length is length BD. Here, this length
366. 14:20is BD, plus what is this? This is also
367. 14:22a length you have. Look, this is a side
368. 14:24of a triangle. Side of a triangle means
369. 14:27this is also a side you have, side CD,
370. 14:29so in your perimeter you have an arc
371. 14:31length. This side you have is four.
372. 14:35This side you have is four. And this is
373. 14:38a side of a triangle you have. The side
374. 14:40of the triangle is CD. Now your target
375. 14:43is to add all of these one by one. We
376. 14:45know what the formula for arc length is
377. 14:47. We have already studied the arc
378. 14:49length formula. s = r theta, and we
379. 14:52have also studied the area of a sector
380. 14:54formula, which is 1/2 rs or the area of
381. 14:57a sector formula is 1/2 r² theta. That
382. 15:00is the only new thing. Otherwise, there
383. 15:02is nothing new. Now we will calculate
384. 15:04this one by one. What is the formula
385. 15:06for arc length? The formula for arc
386. 15:08length is radius times angle. This is
387. 15:11your sector. So what is your radius? It
388. 15:13is six, and your angle is 0.8. So what
389. 15:16is the result of the arc length? 6*0.8
390. 15:20This is the arc length you have. Now
391. 15:22look, what is length AC? Length AC is
392. 15:25four. What is length BD? Length BD is
393. 15:28also four. So this is four, and this is
394. 15:30also four. Done. This is found. Last
395. 15:33step. Now see, this side is 10. This
396. 15:35side is 10. Both these sides are 10 and
397. 15:3710, and this angle is already given as
398. 15:390.8. So we studied the cosine rule.
399. 15:41According to the cosine rule, this side
400. 15:43is 10. This side is 10. So if you
401. 15:45summarize this, this is 10 and this is
402. 15:4710. And the angle between them is 0.8.
403. 15:50Your target is this side here. So you
404. 15:52have two sides and an angle. So
405. 15:54basically, what will you apply?
406. 15:56Basically, you will apply the cosine
407. 15:58rule. Basically, you will apply the
408. 16:01cosine rule. So according to the cosine
409. 16:03rule, what is the answer? a² + b²-2*a
410. 16:09*b*cos of 0.8 0.8, simply which rule
411. 16:17did we apply? We applied the cosine
412. 16:20rule. If I simplify this, what is 6*0.8
413. 16:25? It is 4.8. + 4. + 4. Done. Now we
414. 16:30took help from the calculator for this.
415. 16:3210 squared + 10²-2*10*10*cos of 0.8,
416. 16:35so from the calculator, the answer
417. 16:39derived is 7.788. 7.788, here it is.
418. 16:45Now add up all these lengths: 4 + 4 +
419. 16:484.8. When you add them all up, the
420. 16:51result is 20.58. We know that our
421. 16:56required answer must always be to three
422. 16:58significant figures. So, the shaded
423. 17:03perimeter you find will be 20.6 cm.
424. 17:0720.6 cm. Clear, son? There is nothing
425. 17:11new in this whole topic anymore. We
426. 17:14will now apply the basic concepts we
427. 17:16studied previously, one by one, to
428. 17:18different questions. The only new
429. 17:22things were the arc length formula r*
430. 17:24theta, the area formula 1/2 radius*arc
431. 17:26length, and the area formula 1/2 r²
432. 17:28theta. Clear, son? Just simply plug in
433. 17:32the values. Also, whenever it is not
434. 17:34mentioned, we know that in all of
435. 17:36A-level maths, the required answer is
436. 17:38to three significant figures. Clear,
437. 17:40son? Either the examiner will ask you
438. 17:43to show the answer in exact form. Exact
439. 17:45form means without a decimal. Otherwise
440. 17:48, your answer will be to 3sf. If the
441. 17:50answer should be something else, they
442. 17:52will mention it, like one decimal place
443. 17:54or two decimal places, so the examiner
444. 17:56will guide us. Clear, son? So, this was
445. 17:59also an average circular measurement
446. 18:00question. Let's try more. Here, now
447. 18:03read the question properly; let me
448. 18:05guide you. Look, this is a very clear
449. 18:08question. The examiner is telling you
450. 18:10that this is a sector; by the way, this
451. 18:12is a sector, and the examiner says this
452. 18:14sector is named OPQ. What is its angle?
453. 18:18Theta. Its perimeter is already given.
454. 18:21The perimeter of the sector is already
455. 18:23given. The examiner is asking you to
456. 18:25show theta in terms of R. The examiner
457. 18:27is demanding this from you. Number one.
458. 18:31Number one. First of all, look at what
459. 18:33the word perimeter means. If you listen
460. 18:37to their question, the word perimeter
461. 18:39means the sum of all sides. Clear, son?
462. 18:41The word perimeter means the sum of all
463. 18:43sides. Look, this is basically your
464. 18:45sector. Let's take this; I have copied
465. 18:47the sector here. This is your sector.
466. 18:49The examiner is telling you that its
467. 18:51perimeter is given. Now you have to
468. 18:53show something. Number one, first of
469. 18:55all, see if the radius is r, then
470. 18:57obviously this will also be r. If this
471. 19:00is r, then this will also be r. And we
472. 19:02call this length the arc length. And we
473. 19:04know what the arc length formula is. r*
474. 19:07theta. r*theta. This is the arc length
475. 19:11formula. Now we start slowly. First of
476. 19:14all, look at what hint you are given.
477. 19:17First, the examiner is telling you that
478. 19:19the perimeter of this sector, what is
479. 19:21the perimeter given? 20 cm, this hint
480. 19:24is already given. Perimeter means the
481. 19:27sum of all sides. r + r + r theta. So,
482. 19:30what is its answer given? Its answer is
483. 19:33given as 20. Who is he making the
484. 19:34subject? Theta. Make it a little easier
485. 19:37. This is finally it. When simplified
486. 19:40further, it became 2r + r theta = 20,
487. 19:42copy r theta as it is, so what will it
488. 19:45be here? 20-2r, there it is. Your
489. 19:49target is theta. So obviously, r will
490. 19:51go there and divide both. So r will
491. 19:54divide 20, and what does 2r/r mean? It
492. 19:57became two. So, it was finally a small
493. 19:59and easy question, the result you had
494. 20:01to show. The result you had to show.
495. 20:05Clear, son? This was a very average
496. 20:07question. The examiner only asked you
497. 20:09to show its theta. Clear, son? Show
498. 20:12theta. The perimeter was already given.
499. 20:15The perimeter was already given. Clear?
500. 20:18Okay, what do you need in the next part
501. 20:20? Read the area. And another
502. 20:23interesting thing is that your area
503. 20:24answer is in terms of r. Meaning, theta
504. 20:27is not required in the area answer.
505. 20:31Theta is not required. You have already
506. 20:33found theta previously. This means
507. 20:36theta should be replaced so that theta
508. 20:38is not involved in your required answer
509. 20:39. So we know that we learned today,
510. 20:43what is the formula for the area of a
511. 20:44sector? The area of a sector formula is
512. 20:471/2 r² theta. This is the formula for
513. 20:49the area of a sector. There is another
514. 20:52one, 1/2 rs, why did we use this?
515. 20:54Because we do not require theta. We
516. 20:57will put in theta, so theta will be
517. 20:59ignored or replaced. Theta was not
518. 21:02required in the answer. Now, when you
519. 21:05simplify this a bit more, see, 1/2
520. 21:07copied as is, r² copied as is. Now
521. 21:10tell me, what is the answer for theta?
522. 21:13The answer for theta was 20/r, the
523. 21:15answer for theta was 20/r-2, this is
524. 21:18the theta answer. Now you multiply the
525. 21:21bracket. So it will multiply with both,
526. 21:26r²/2*20/r and-r²/2*2, there you go,
527. 21:29it will multiply with both. This became
528. 21:332, 10, r and r got cut. So what is the
529. 21:36area answer derived from here? The area
530. 21:38became 10*r, from here 2 and 2 got
531. 21:40cancelled out. What answer was derived
532. 21:42from this? r². So finally, what area
533. 21:45was found from here? Finally, the area
534. 21:48found was tanr-r², and this was your
535. 21:50target, that the answer you had to show
536. 21:52had to be in terms of r. Meaning, theta
537. 21:56should not be involved anywhere in your
538. 21:58answer. Clear, kid? So this was also a
539. 22:01simple question. Look at part three of
540. 22:03this same question to see what the
541. 22:05examiner requires in part three. Now
542. 22:07the examiner is guiding you. Look, it's
543. 22:10the same sector. Again, it's exactly
544. 22:12the same sector. Clear? Now see what
545. 22:15the examiner is guiding you with. Okay?
546. 22:19It's the same sector. Now let's see
547. 22:21what the examiner is guiding us with.
548. 22:22It's the same sector. Done. Okay. Now
549. 22:25the examiner is telling you that the
550. 22:26radius is eight. So, what hint did he
551. 22:28give you? That the numeric value of the
552. 22:30radius is eight. So obviously, what did
553. 22:32this radius become? It became eight. He
554. 22:34stated the numeric value of the radius
555. 22:36is eight. Now tell me, what was your
556. 22:38theta equal to? Theta. Your theta was
557. 22:41equal to 20/r-2. Your theta was equal
558. 22:45to 20/r-2. Now tell me, what is your r?
559. 22:50Your r is 8, so what will be the answer
560. 22:52for theta? The answer for theta will be
561. 22:5520/8-2. The answer for theta will be 20
562. 22:58/8-2, here it is. So its answer came
563. 23:01out to be 4/8, which is 0.5 radians. So
564. 23:04, what did we find out first from here?
565. 23:06We found the numeric value of this
566. 23:08theta. So from here, I found out that
567. 23:10the numeric value of theta came out to
568. 23:13be 0.5? RAD means radians, RAD means
569. 23:16radians. Now let's read the question, "
570. 23:19Find the length of the chord", "Find
571. 23:21the length of the chord", chord. Now we
572. 23:23already know what the length of a chord
573. 23:25means, right? Obviously, what is a
574. 23:27chord? It is a line, by the way. So now
575. 23:29if you summarize and look at it again,
576. 23:31basically a triangle is being formed.
577. 23:33Clear, kid? So basically, what does the
578. 23:35examiner want to do? If I summarize
579. 23:38this, it means this. It means that
580. 23:41originally you had a sector like this,
581. 23:43originally. And what did the examiner
582. 23:47do? The examiner joined these two
583. 23:49points. The examiner joined these two
584. 23:53points to form a chord. By joining
585. 23:55these two points, the examiner formed a
586. 23:58chord. So this point was your O. This
587. 24:00point was your P. This point was your
588. 24:02Q. The examiner joined these to form a
589. 24:05chord. Now, you already know this side.
590. 24:07You already know this side. And you
591. 24:09already know this angle too. What is
592. 24:10this angle? Your 0.5 radians. What is
593. 24:14your angle? Your angle is 0.5 radians.
594. 24:18This is your angle. Now, what do you
595. 24:20simply need to apply? Side given, side
596. 24:23given, angle between sides given. So,
597. 24:25what will we apply? We will simply
598. 24:28apply the cosine rule. So, what did we
599. 24:30simply apply? We just have to apply the
600. 24:33cosine rule. So, what does the examiner
601. 24:35require? Length. The side of the
602. 24:37triangle is required. This means the
603. 24:39length of chord PQ is required. So, for
604. 24:41chord PQ, what will we apply? The
605. 24:44cosine rule. We will have to apply the
606. 24:46cosine rule. Now look, what are the
607. 24:49sides? You have this side as eight. You
608. 24:51have this side as eight. So, you
609. 24:53applied the cosine rule: 8 squared. 8
610. 24:55squared, 2*8*8*cos of 0.5, cos of 0.5.
611. 25:03Now, if I summarize this. Inside the
612. 25:06root, it became 8² + 8²-2*8*8*cos 0.5
613. 25:11. So, from the calculator, what length
614. 25:14of chord PQ was found here? The length
615. 25:17found will be 3.958. 95846, here it is.
616. 25:22We know that we have to show the answer
617. 25:24to three significant figures. So, what
618. 25:27will your answer for chord PQ be to
619. 25:29three significant figures? 3.96. cm,
620. 25:333.96 cm. So, what was your target? You
621. 25:40required the length of the chord. You
622. 25:43required the length of the chord. Clear
623. 25:45, kid? So, we didn't see anything new
624. 25:47in this either. We just know that in
625. 25:49this whole topic, so far we have only
626. 25:50learned these two new formulas. What is
627. 25:52the formula for arc length? S = Rθ,
628. 25:54and what is the formula for the area of
629. 25:57a sector? 1/2 Rθ, the area of sector
630. 26:00formula is 1/2 RS, this is the area of
631. 26:03sector formula 1/2 R²θ. Theta.
632. 26:07Otherwise, there is nothing new. We
633. 26:09have already studied all these things
634. 26:11in the basics. Clear, kid? We have
635. 26:13already studied them in the basics. Now
636. 26:15, let's try some more questions one by
637. 26:17one. Look, let's try one more question.
638. 26:21Now, first look at this question and
639. 26:23read the language to see what the
640. 26:24examiner wants to do. Okay, kid? Read
641. 26:26the language first. Look, first of all,
642. 26:29the examiner is saying in the diagram,
643. 26:31AB is an arc. The examiner is saying
644. 26:33this is an arc. Is that clear? If this
645. 26:35is an arc, then this is a radius, so
646. 26:37this is also a radius. So, let me make
647. 26:39this even easier for you to see what
648. 26:41the examiner wants to do. Basically,
649. 26:43let me simplify this question a bit
650. 26:45more to show you what the original
651. 26:47situation was. The situation is
652. 26:48something like this. Basically, the
653. 26:50situation is like this. Meaning, you
654. 26:53essentially have a sector here. If we
655. 26:55drop a line from here, your question is
656. 26:57now converted into this model. This was
657. 27:00your O. Your radius is r cm. He is
658. 27:02calling this point X. This angle is 90
659. 27:05°. What is this angle you have? 90 °
660. 27:07and what is he calling this angle? He
661. 27:09is calling it theta. This angle of
662. 27:11yours is theta. This is B for you. And
663. 27:13this is A for you. Now notice what this
664. 27:15whole thing is. This whole thing is a
665. 27:17sector. What is the radius of this
666. 27:19sector? The radius is r. The radius of
667. 27:21the sector is r. So that means the gap
668. 27:24from O to B should also be the radius.
669. 27:28The gap from O to B, what should that
670. 27:30gap be? It should obviously be the
671. 27:32radius. Because any gap from the center
672. 27:34of a sector to the circumference is
673. 27:36called the radius. If this is r, then
674. 27:38this is also r. This entire thing is
675. 27:40also r. Now look at what this region is
676. 27:42. This is your entire required region
677. 27:46which the examiner has highlighted. In
678. 27:49your question, this is clearly given in
679. 27:51the diagram above. This is clearly
680. 27:53given in the question above. Meaning
681. 27:55the situation is of this type. Now
682. 27:57let's read this question one by one. So
683. 27:59I showed you the actual diagram, that
684. 28:01the original diagram is something like
685. 28:03this. Meaning, the examiner erased all
686. 28:06of this portion and copied this diagram
687. 28:08above. Clear, kid? Technically, this is
688. 28:11what it means. Now, shall we start?
689. 28:13First of all, what does the examiner
690. 28:15want? The shaded area. So what is this
691. 28:17whole shape? A sector. If this triangle
692. 28:19is removed from the sector, then the
693. 28:21shaded area will be found. So what is
694. 28:24your first task? Your first task is
695. 28:27that you will look. Let me start with
696. 28:30part one. The very first task is to
697. 28:32look at this triangle first. The
698. 28:34triangle that is given to you, what is
699. 28:36given in it? First, focus on this
700. 28:38triangle. You have a triangle here.
701. 28:41Here you go. This is a right-angled
702. 28:42triangle. Okay? Now look at this
703. 28:44right-angled triangle. This is your O.
704. 28:46This is your X. This is your A. This is
705. 28:49your r. And this angle, the name of
706. 28:50this angle is theta. This angle is your
707. 28:53theta. Let's see, we don't know the
708. 28:55height or the base of this triangle.
709. 28:56With respect to this angle, this is
710. 28:58opposite. With respect to this angle.
711. 29:00This is adjacent and the side opposite
712. 29:02to 90 is called the hypotenuse. Meaning
713. 29:04in simple language, this is a
714. 29:06right-angled triangle. Is it clear, son
715. 29:08? This is a right-angled triangle. Now
716. 29:10let's start slowly. Number one, first I
717. 29:13applied cos, so with respect to cos,
718. 29:15what will be its answer? Base over
719. 29:17hypotenuse, here is base over
720. 29:19hypotenuse. So from here, what did the
721. 29:21answer for ox become? The answer for ox
722. 29:24became r*cos theta. r*cos theta. This
723. 29:30is the answer you got for cos. Number
724. 29:33two. Now if you apply sin to this, then
725. 29:35we know what the answer for sin is?
726. 29:38Opposite divided by hypotenuse. So if I
727. 29:40apply sin to this, what will the answer
728. 29:42for sin be? ax/ax/r, here is ax,
729. 29:45opposite over hypotenuse, here it is.
730. 29:48So from here I found out that what is
731. 29:50the answer for ax? The answer for ax
732. 29:53became r*sin theta. This is your answer
733. 29:57for ax. So we got the triangle's height
734. 29:59and the triangle's base as well.
735. 30:01Obviously, this is the height of the
736. 30:02triangle. Obviously, this is a
737. 30:04right-angled triangle. So we will say
738. 30:06this is the height of the triangle and
739. 30:08this is the base of the triangle. Clear
740. 30:10, son? Height and base. So now if I ask
741. 30:12you, what will be the area of the
742. 30:14triangle? Area of the triangle. So for
743. 30:16the area of the triangle, what will you
744. 30:18say? You will say the formula for the
745. 30:20area of the triangle is 1/2*height*base
746. 30:22. This is the formula for the area of
747. 30:24the triangle. So now tell me, what is
748. 30:25the height of your triangle? The height
749. 30:27is ax. The base is ox, height is ax,
750. 30:32base is ox, so 1/2*height*base, here is
751. 30:351/2*height*base, so now you tell me
752. 30:37what is ax? ax is r sin theta, so tell
753. 30:42me what is ox? ox is r cos theta. Then
754. 30:46if I ask you, man, what is the area of
755. 30:48this entire triangle? Then the area of
756. 30:54the triangle will be 1/2*r², r*r
757. 30:56became r squared, multiplied by sin
758. 30:59theta*cos theta. So whose area was
759. 31:05found out? The area of the whole
760. 31:07triangle was found out. The area of the
761. 31:10whole right-angled triangle was found
762. 31:11out. So I have the area of this
763. 31:13right-angled triangle, meaning I first
764. 31:15used the right-angled triangle to find
765. 31:17the height and base, then the area of
766. 31:19the triangle is 1/2*height*base, and I
767. 31:21got the area of the triangle. What is
768. 31:23your final target? Your final target is
769. 31:26the shaded area. Your target was the
770. 31:29shaded area. This is your target. This
771. 31:33is your target, the shaded area. So how
772. 31:35will we get the shaded area? This
773. 31:37entire shape is a sector. We have to
774. 31:39remove the triangle from it. So we will
775. 31:41say that if we require its area, what
776. 31:43word should we use for it? We will say
777. 31:45area of the whole sector OAB minus area
778. 31:50of the triangle, the area of the
779. 31:52triangle that we just found above. What
780. 31:56is the name of the triangle? OAX, so
781. 31:58what is the area of the sector? We
782. 32:00already know 1/2*R*r, here it is. So,
783. 32:04we already know that the formula for
784. 32:06the area of a sector is 1/2 r² theta.
785. 32:09This is the formula for the area of a
786. 32:10sector. So, now if I ask you what the
787. 32:13formula for the area of a sector is,
788. 32:16you will say 1/2*r²*θ; 1/2 r² theta
789. 32:19is the formula. 1/2 r² theta, here you
790. 32:22go. 1/2 r² theta, what do you subtract
791. 32:25from this? Subtract the area of the
792. 32:28triangle from this: 1/2 r² sin theta*
793. 32:31cos theta, here you go. Now simplify
794. 32:34this so that your answer matches the
795. 32:36required answer. 1/2 is common in this,
796. 32:39r² is common in both. So what will you
797. 32:41do in both? You will take half common,
798. 32:44and take r² common. Took 1/2 and r²
799. 32:46common; when 1/2 and r² were taken
800. 32:48common, what remained inside? Theta,
801. 32:50and what remained here? sin theta*cos
802. 32:53theta. This is your target achieved. So
803. 32:58there was nothing extraordinary in this
804. 32:59question, clear? So this was just a
805. 33:03normal routine question; the question
806. 33:04was just in terms of an unknown
807. 33:06variable, otherwise it was in terms of
808. 33:08theta, otherwise there was nothing new.
809. 33:11This is the result you had to show in
810. 33:13the question. Clear, dear? So, it
811. 33:15became area of the whole sector minus
812. 33:17area of triangle. Clear, dear? Okay,
813. 33:19now answer me one thing. We know that
814. 33:22this whole thing is the radius. Done.
815. 33:23We know that this is also the entire
816. 33:25radius. Done. And we also already know
817. 33:28the answer for Ox, what is the answer
818. 33:29for Ox? r cos theta. The answer for Ox
819. 33:34is r cos theta. So, I know that this
820. 33:37entire gap from here to there, this
821. 33:39full length, is r. So, if I ask you
822. 33:42what xb will be? xb, if I ask you what
823. 33:45xb would be? Then what will be the
824. 33:48answer for xb? xb will be ob-ox, here
825. 33:51it is. So, who is ob? OB is the full r.
826. 33:54So, how will its answer come? Subtract
827. 33:57r cos theta from R. Clear, kid? Now,
828. 34:01read the second part of this question,
829. 34:03what is the examiner's target in the
830. 34:05second part? In the second part, r is
831. 34:07given, theta is given. Perimeter of the
832. 34:09shaded region. The restriction is that
833. 34:13the shaded area's answer is required in
834. 34:15terms of 3 and in terms of 3 and pi.
835. 34:18Now look at your shaded area. Shaded
836. 34:20area, this is your shaded area. Who is
837. 34:22this? Length ax. Now, let's first copy
838. 34:25this. Slowly. Your target is the
839. 34:27perimeter, and it's also giving a hint,
840. 34:29perimeter of what? Perimeter of axb,
841. 34:32that is your target. You need the
842. 34:34perimeter of axb. So, what will be its
843. 34:37answer? ax, what is ax? ax is a line +
844. 34:41xb, ax + xb, here it is. Plus, what is
845. 34:45this? I will call this the arc length
846. 34:48of a sector. Arc length of a sector.
847. 34:51Arc length of a sector. So s means arc
848. 34:54length, sab means arc length ab. Now
849. 34:57you tell me, what is ax? I already know
850. 35:00the answer for ax. Here it is, ax, what
851. 35:02was the answer for ax? The answer for
852. 35:06ax was r*sin theta. The answer for ax
853. 35:10was r*sin theta. This was your answer
854. 35:13for ax. This was your answer for ax. r
855. 35:16sin theta. Now put it here. So what
856. 35:19will be the answer for ax? r sin theta.
857. 35:21Done. Here you go. What was XB? Here is
858. 35:24XB. XB was R-R cos theta. Here you go.
859. 35:29This is done too. Plus, what was this
860. 35:32thing? This thing was the arc length
861. 35:34you have. The formula for arc length is
862. 35:36. The formula for arc length is radius
863. 35:39times angle. The radius is r and the
864. 35:41angle is theta. So the formula for arc
865. 35:43length is r*theta. So what did we do?
866. 35:46We put all these values into the
867. 35:48question. Clear, kid? We put all these
868. 35:50values into the question. Now tell me,
869. 35:52what r is given to you in the question?
870. 35:54r is given as 12. Here you go. r is
871. 35:57given as 12 and what theta is given to
872. 35:59you? Theta is given as pi/6, now put
873. 36:02these values in. Then what did this
874. 36:04question become? 12*sin pi/6, here you
875. 36:08go. We will simply plug in r and theta.
876. 36:12Plus what is r? r is 12, here it is.
877. 36:16-12 cosθ, so that becomes cos times π
878. 36:20/6, here it is, here it is + r*θ, r*θ
879. 36:24, r is 12 and theta is π/6. There you
880. 36:28go, everything has been plugged in. Now
881. 36:30when you use the calculator, the answer
882. 36:33for sinπ/6 from the calculator is half
883. 36:35. I will show it too, so this is 12*1/2
884. 36:40And 12 copied as it is, here is 12
885. 36:42copied as it is, here it is, and what
886. 36:45is the answer for cosπ/6 from the
887. 36:47calculator? 3/2, this task is obviously
888. 36:50for the calculator. We don't have any
889. 36:52issues. In this, you just have to use
890. 36:54the calculator. And what did this
891. 36:56become? 12*π/6 means 2π, now you make
892. 36:59this a bit easier. When you simplify it
893. 37:03further, it becomes six. 12 and 6
894. 37:06became 18. So what is the answer? 18
895. 37:09plus 2π copied as it is, here it is.
896. 37:14And 2 6, so this became -6 3, your
897. 37:16target has been achieved. In what
898. 37:20specific form did you need the answer?
899. 37:22Read the question. Your target, your
900. 37:25target was in exact form. What is the
901. 37:29meaning of exact form? Exact form means
902. 37:31without decimals. Exact form means
903. 37:35without decimals. It was clearly
904. 37:37written in the question, right? Look,
905. 37:38if you read the language of the
906. 37:40question, what is the examiner guiding?
907. 37:43The examiner is saying leave your
908. 37:45answer in terms of 3 and π, meaning 3
909. 37:46and pi should be involved in your
910. 37:48answer. Clear, kid? 3 and pi should be
911. 37:52involved in the answer. So we simply
912. 37:54just did the calculation. This final
913. 37:56answer is derived. So we haven't
914. 37:58studied anything extraordinary yet that
915. 38:00is very tricky or very technical. Let's
916. 38:04practice more questions of this type.
917. 38:06Here you go. There is another question.
918. 38:08Read it a little properly first, kid.
919. 38:10Then I will guide you further. Okay?
920. 38:12Look, what hint is given in this
921. 38:13question now? First you read the
922. 38:15question a little, then I will guide
923. 38:16you. Look, first of all, this is a
924. 38:18somewhat good question. Interesting
925. 38:20question. First you read the language
926. 38:22to see what the examiner is saying.
927. 38:23Number one, this is a larger circle.
928. 38:25Its name is C1. The radius of C1 is
929. 38:27eight, so then this will also be eight.
930. 38:30Obviously. Its radius is two, so then
931. 38:32this will also be two. Obviously. These
932. 38:34are two circles. Clear? First, read the
933. 38:36question thoroughly. Okay? Good. So,
934. 38:38this is circle C1. This is circle C2.
935. 38:41These two lines are parallel. Or let's
936. 38:43just say that the radius is always
937. 38:45perpendicular to the tangent line. The
938. 38:47radius is always perpendicular to the
939. 38:49tangent line. This line here is tangent
940. 38:51to both of these circles. This line is
941. 38:55tangent to both of these circles. Is
942. 38:57that clear, kid? So, if a student is
943. 38:59well-read, in mathematical terms, they
944. 39:01call this something too. What is this
945. 39:03line common to? This line is a common
946. 39:05tangent. This line is a common tangent.
947. 39:07What is this line forming with both
948. 39:09these circles? It is forming a 90 °
949. 39:11angle. Clear, kid? So please look at
950. 39:14the question properly first so that we
951. 39:16understand what our question is. Clear,
952. 39:18kid? Now, let's start slowly on what
953. 39:21our basic target is in this whole topic
954. 39:23. Okay? Now look, first of all, if we
955. 39:26have read the question properly, I will
956. 39:28show you the diagram to see what the
957. 39:30examiner wants to do. Let me cut this
958. 39:33diagram and show it to you here.
959. 39:35Meaning, the situation is something
960. 39:37like this. The situation is something
961. 39:40like this. Look, it's like this. Okay?
962. 39:42What did I do? I copied that part, the
963. 39:45middle part. Like this, your region is
964. 39:47shaded here. And like this, your region
965. 39:49is shaded here. There it is. It’s
966. 39:51like that, right, kid? Technically,
967. 39:52your question is something like this,
968. 39:54and this whole area in the middle is
969. 39:56highlighted. This whole area in the
970. 39:57middle is highlighted. Meaning, this
971. 39:59entire area is shaded. Please read the
972. 40:02question again. I haven't told you
973. 40:03anything new. I just copied that same
974. 40:06diagram. Okay, kid? I copied the same
975. 40:09diagram here. This is your region.
976. 40:11Clear? This was your region. This is
977. 40:13your diagram. So, what did I do? I
978. 40:15picked up this diagram from there and
979. 40:17copied it here. Now let's start it
980. 40:19slowly. Look, this point you have, this
981. 40:22point is P. Look, now let's read it
982. 40:24slowly. This is your point P. This
983. 40:28radius is 8 cm. This is your R. And
984. 40:30obviously, what was this line? 90 °.
985. 40:33What was this? Obviously, the line was
986. 40:3590 °. This was also obviously what? 90
987. 40:37°. Okay? Good. Now, what else do you
988. 40:40know? This point was your T. This point
989. 40:43was your Q. This was your 2 cm. This
990. 40:45was also your 2 cm. And if this is the
991. 40:48radius, then PTB was the radius. So
992. 40:50what is PTB? 8 cm, meaning I have
993. 40:52copied this diagram from there to down
994. 40:54here. Clear, son? Now let's start the
995. 40:57work even more slowly. Number one,
996. 40:59first of all, you need RS. RS, this is
997. 41:02your RS. Clear, son? Need RS. So look,
998. 41:05the first thing is, if I drop a line
999. 41:07from here. Note it. These two sides are
1000. 41:11your parallel sides. It is common sense
1001. 41:14that if two lines are perpendicular to
1002. 41:16the x-axis. Both are forming 90 °, so
1003. 41:18then what are they to each other? Both
1004. 41:20have become parallel to each other,
1005. 41:22right son? Both have become what to
1006. 41:23each other? Parallel. Okay, now look,
1007. 41:26your target is RS. Now look at the
1008. 41:30question slowly, that if you have this
1009. 41:32gap of 2 cm. Look, watch carefully, son
1010. 41:35. Now what will we do? If this gap is 2
1011. 41:37cm, then this will also be 2 cm. Look
1012. 41:40at the working of the question properly
1013. 41:42. If this is 2 cm, then this too will
1014. 41:44be 2 cm. Clear? Okay, now tell me, if
1015. 41:46this gap is two, then this is also two.
1016. 41:48What is this entire length? Eight. If
1017. 41:51we take this two out of eight, then
1018. 41:53this small length, this length. What
1019. 41:56will this length be found to be? This
1020. 41:58small length, what shall we call this
1021. 42:00small length? This small length, we
1022. 42:02will call this small length six. This
1023. 42:04small length of yours, what will this
1024. 42:06small length become? It will become six
1025. 42:08. There it is. What should we call this
1026. 42:11small length? We will call this small
1027. 42:13length six. Because the entire length
1028. 42:14was eight. So what did this small
1029. 42:16length come out to be? It came to 6 cm.
1030. 42:18Did it? Clear, son? Now see, this is
1031. 42:21six. And what did you have here? You
1032. 42:22had this as 8 cm. This you already know
1033. 42:25. This was 8 cm. Clear, son? So I
1034. 42:29haven't done any extraordinary work yet
1035. 42:30. Come on, number one, he needs RS. So
1036. 42:33now, first of all, look at this. Okay,
1037. 42:36let's name this point point X, name it
1038. 42:38anything, here, I named it point X. So
1039. 42:40obviously, this is also a right-angled
1040. 42:42triangle. Look again, this point...this
1041. 42:44is also a right-angled triangle.
1042. 42:46Obviously, we named this point point X,
1043. 42:48there it is, this point, it became
1044. 42:50point X. Now you see, you have this
1045. 42:52right-angled triangle, you have this
1046. 42:55right-angled triangle. Now look, this
1047. 42:58point is P, this point is X, and this
1048. 43:01point is Q. This point is your q. Here
1049. 43:03you go. Now tell me, what will be this
1050. 43:05entire length? 10 cm. 10 cm. This gap
1051. 43:09here was two. So obviously, this was
1052. 43:11also two. So this entire length came
1053. 43:14out to be 10 cm; here is 8 and here is
1054. 43:162. 8 + 2 becomes 10. This became 10.
1055. 43:19Okay, what is this height? You have a
1056. 43:21height of 6 cm. This height here is 6
1057. 43:25cm, and your target is x. So you
1058. 43:28applied Pythagoras. Because this is a
1059. 43:31right-angled triangle. So we applied
1060. 43:33Pythagoras. Is that clear, child? We
1061. 43:35applied Pythagoras. So what will
1062. 43:37Pythagoras be? 10²-6², so that
1063. 43:40becomes 100-36 = 64, which is 8. So
1064. 43:44from this, I found out, what is the
1065. 43:46answer? 8 cm. So whose answer is this?
1066. 43:50It is the answer for x. Then you answer
1067. 43:52one thing for me. This line here, x,
1068. 43:55this line and that line are both
1069. 43:57parallel to each other. Obviously. So
1070. 44:00technically, this shape is a rectangle.
1071. 44:02If you note, what is this shape? This
1072. 44:04shape is technically a rectangle. So
1073. 44:06this means, technically. This means,
1074. 44:11technically, that you have x and RS
1075. 44:12both being the same. Out of which the
1076. 44:15length of both came out to be what? 8
1077. 44:17cm. This target of yours is achieved.
1078. 44:19So, first of all, what were you
1079. 44:21required to do? First of all, you were
1080. 44:24required to find RS. To show this. So
1081. 44:26this part one of yours is solved. In
1082. 44:28part one, this was your target. The
1083. 44:31target's name was RS. The target's name
1084. 44:34was RS. Clear, child? This was your
1085. 44:37target. Now look at the second part of
1086. 44:39this again. In the second part, it
1087. 44:42mentions angle RPQ. Which one is angle
1088. 44:45RPQ? If you note, this angle is angle
1089. 44:48RPQ. Look again. Angle RPQ. This angle
1090. 44:52is your target. This angle is your
1091. 44:54target. Let's name this angle alpha. We
1092. 44:58will read this as alpha. So now look at
1093. 45:00the triangle; you already have the
1094. 45:02triangle. Look, you already have the
1095. 45:04sides of the triangle. This is it; it's
1096. 45:06the same triangle. This is your target;
1097. 45:08this is the angle. The name of this
1098. 45:10angle is alpha. So what is this
1099. 45:12triangle? p x q is right here. So its
1100. 45:15side is already six. This side is
1101. 45:17already 10. So what is your target? By
1102. 45:20the way, your target is only alpha. And
1103. 45:22you even know x. This is also eight.
1104. 45:24Your target is alpha. So with respect
1105. 45:26to this angle. For this, it is opposite
1106. 45:27. This is the hypotenuse. So you can
1107. 45:29apply any ratio. Suppose I applied sine
1108. 45:32. I applied sine. Actually, I can apply
1109. 45:36anything because everything is given.
1110. 45:38So what is the answer for sine?
1111. 45:39Opposite divided by hypotenuse.
1112. 45:42Opposite divided by hypotenuse.
1113. 45:44Opposite divided by hypotenuse here.
1114. 45:46Clear? Opposite over hypotenuse. So
1115. 45:49when I found the answer for sine
1116. 45:51inverse of 8/10, it became 0.927. Okay?
1117. 45:54Okay? Now what was your target? You had
1118. 45:56to find this angle to 4sf. So the
1119. 45:59answer came to 0.9273. There you go,
1120. 46:020.9273; this work was just about
1121. 46:04putting it into the calculator. There
1122. 46:07is no logical issue in this. Your angle
1123. 46:10has been found. Your next target in the
1124. 46:12step was that you required this angle.
1125. 46:15The angle whose name was alpha. This
1126. 46:17angle was your target. The name of this
1127. 46:19angle was alpha. Clear, kid? So now
1128. 46:21this part of yours is also solved. One
1129. 46:23is done, two is done. Last and final
1130. 46:26step. Now you need the shaded area. Now
1131. 46:29what is your target? Now your target is
1132. 46:31the shaded area. This is now your
1133. 46:34target in this whole question, which is
1134. 46:35called the shaded area. Okay, kid? Now
1135. 46:38let's copy our diagram here, this same
1136. 46:40diagram. Now your target is the shaded
1137. 46:43area. Now let's see what information we
1138. 46:45have. Okay? What information is there
1139. 46:48and what is missing. Now, what is your
1140. 46:50target in the third part? Your target
1141. 46:52in the third part is the shaded area.
1142. 46:54The target is the shaded area. We have
1143. 46:56already found the answer for alpha. The
1144. 46:57answer for alpha is 0.9273. This is
1145. 47:01your answer for alpha. You know that
1146. 47:03these two sides are already parallel.
1147. 47:06These two sides are already parallel. I
1148. 47:08will highlight this. These two sides
1149. 47:10are already parallel. Obviously. And
1150. 47:13you also already know what the answer
1151. 47:15for RS was. If you remember, we found
1152. 47:17the answer for RS in part one. In part
1153. 47:20one. In part one, we found the answer
1154. 47:22for RS. The answer for RS was 8 cm. 8
1155. 47:26cm. Okay, now you just need to put
1156. 47:29extra effort into one thing. You know
1157. 47:32this entire shape is a quadrilateral.
1158. 47:35In a quadrilateral, the sum of angles
1159. 47:38is 360, which means 2 pi. You just need
1160. 47:42a little bit of common sense. Look,
1161. 47:43this became 90. This became 90. 90 and
1162. 47:4690 became 180. Now this one is gone. So
1163. 47:49that means what should its sum and its
1164. 47:50sum be? π means 180, so if I ask you
1165. 47:53that I only need this much angle, just
1166. 47:56this much. Let's name this angle beta.
1167. 48:01I have named this angle beta. So you
1168. 48:03see that this 90 and this 90 make 180.
1169. 48:06So that means its sum and this sum
1170. 48:08should also be 180. So then what will
1171. 48:10beta be equal to? This small angle beta
1172. 48:13will be equal to 180-0.9273. Now I know
1173. 48:17that 180 means pi, by the way. So if I
1174. 48:20make it easy, it becomes π-0.9273. So
1175. 48:24the answer came out to be 2.214, so
1176. 48:26from here the answer for beta is 2.214.
1177. 48:313, this is the answer for beta and this
1178. 48:33is in terms of radians. What does RAD
1179. 48:36mean? RAD means radians. Your answer
1180. 48:40for beta has also been found from here.
1181. 48:42And this answer for beta is in terms of
1182. 48:44radians. Clear, kid? Beta is also found
1183. 48:46. Let me explain again. This entire
1184. 48:48shape is a quadrilateral. It is a
1185. 48:50trapezium. If anyone's basic math is
1186. 48:52strong, this shape is a trapezium.
1187. 48:53Trapezium. Its sum, this 90 and this is
1188. 48:56also 90, meaning the sum is 180. So
1189. 48:58then what should this sum and this sum
1190. 49:00be? It should be 180. So now if I tell
1191. 49:02you that my final target is the shaded
1192. 49:04area. Now let's solve the question. So
1193. 49:06finally, how will the shaded area be
1194. 49:08found? This entire shape is a trapezium
1195. 49:10. What is the name of the trapezium? PQ
1196. 49:12. PRSQ. So what should I name it? We
1197. 49:17will say that this entire shape is a
1198. 49:19trapezium and what is the name of the
1199. 49:21trapezium? The trapezium's name is PRSQ
1200. 49:24, P R S Q, this is your trapezium. What
1201. 49:28will you subtract from this entire
1202. 49:30trapezium? What do you need to subtract
1203. 49:32from this entire trapezium? Number one,
1204. 49:35you have to subtract this sector and
1205. 49:36this sector. Both these sectors need to
1206. 49:39be subtracted. So what will we subtract
1207. 49:41from this? What is the name of this
1208. 49:43larger sector? PRT. Sector PRT, the
1209. 49:47name of this sector is PRT, and we will
1210. 49:50also subtract this other sector. The
1211. 49:52name of this sector is TQS. The name of
1212. 49:56this sector is sector TQS. T Q S. Now
1213. 50:02let's start. If we remember the area of
1214. 50:04a trapezium, the area is 1/2*sum of
1215. 50:07parallel sides*height. 1/2*by sum of
1216. 50:10parallel sides multiplied by the height
1217. 50:12. This is the area of a trapezium. So
1218. 50:15now tell me, what are the parallel
1219. 50:17sides of the trapezium? You have the
1220. 50:19parallel sides of the trapezium now. So
1221. 50:21what is your target now? 1/2*sum of
1222. 50:23parallel sides*height. So this side is
1223. 50:25two. This side of yours is two. This
1224. 50:27side of yours is eight. These are the
1225. 50:29parallel sides of the trapezium.
1226. 50:32Multiply by the height. The definition
1227. 50:34of height is the line that connects two
1228. 50:36parallel lines. Forming a 90-degree
1229. 50:37angle with both. So, who is the height?
1230. 50:39It is RS. Who is the height? It is
1231. 50:41eight. Height is RS, height is eight.
1232. 50:43Here you go. The area of the trapezium
1233. 50:45is here. This is the area of the whole
1234. 50:47trapezium. Now, we have to subtract the
1235. 50:49area of two sectors from this. We have
1236. 50:51to subtract the area of two sectors. Of
1237. 50:53the two sectors. Do we remember the
1238. 50:55formula for the area of a sector? The
1239. 50:57formula for the area of a sector is 1/2
1240. 51:00rs, and there is another option: 1/2
1241. 51:02r² theta. This is the only new thing,
1242. 51:05otherwise, there is nothing new. What
1243. 51:07will be the area of the sector? Okay,
1244. 51:08now look, sector PRT, here is your
1245. 51:11sector PRT. Here it is: 1/2*radius*
1246. 51:13radius*angle, so 1/2 r² theta. So what
1247. 51:16will be its answer? 1/2*8²*theta
1248. 51:200.9273. There you go, this sector is
1249. 51:23done. Number two, this is also a sector
1250. 51:27. We already know its angle. Done. This
1251. 51:30is known. This is known. So 1/2 r²
1252. 51:32theta, so we will say 1/2 r² beta, so
1253. 51:36it will be 1/2*2²*beta. What is the
1254. 51:39answer for beta? 2.2143 This is the
1255. 51:43answer for beta. Clear, dear? Okay, now
1256. 51:46we just have to simplify this. Now if I
1257. 51:48simplify this, well, this becomes 10,
1258. 51:5210, 80, 80/2 = 40. There you go, and
1259. 51:55now we simplified it: 0.5*8²*0.9273;
1260. 51:59calculating this gave 29.6736. Here it
1261. 52:06is, and what is the area of the next
1262. 52:08one? 0.5*2² multiplied by 2.2143. Its
1263. 52:12area is 4.4286. 4.4286. Now let's
1264. 52:18calculate this finally. Let's finally
1265. 52:21calculate this. So our target is the
1266. 52:23shaded area, so it will be 40-29.6736-
1267. 52:294.4286 Obviously, we will have to do a
1268. 52:33bit of calculation. So what is the
1269. 52:35answer to this? 5.8 978. We usually
1270. 52:39have to give the answer to 3 SF. So
1271. 52:43when I give the answer to 3 SF, it will
1272. 52:45be 5.90. 5.90, and what will be its
1273. 52:49unit? Centimeters squared. This was
1274. 52:53your ultimate target. This was your
1275. 52:56ultimate target. Clear, dear? So we
1276. 52:58didn't study anything extraordinary.
1277. 53:00But this was a good and slightly
1278. 53:02different question. Clear, dear? That
1279. 53:04was a slightly good and different
1280. 53:05question. This was the difference in
1281. 53:06this question. Clear? So the first rule
1282. 53:09is that circular measurement questions
1283. 53:11are mostly different. So first, observe
1284. 53:13it properly to see what statements are
1285. 53:15given. What is required? There should
1286. 53:18be no error in calculation, and
1287. 53:19obviously, a calculator will be used. I
1288. 53:22am also using a calculator for
1289. 53:23everything, obviously. Let's try more
1290. 53:26questions of this type. Here is another
1291. 53:28question. Read the question properly
1292. 53:31first so that I can guide you. Look,
1293. 53:33the only new thing is what formula we
1294. 53:35studied for arc length? r*theta and
1295. 53:38what formula did we study for the area
1296. 53:40of a sector? 1/2 rs and another formula
1297. 53:43we studied 1/2 r² theta, otherwise
1298. 53:45there is nothing new. Otherwise, there
1299. 53:48is more common sense in this topic.
1300. 53:50Otherwise, there is more common sense
1301. 53:52in this topic. Clear? You just read the
1302. 53:54question for a minute. Well done. Then
1303. 53:57I will guide you on what to do. Okay?
1304. 54:03Come on. Number one. It says first that
1305. 54:05there is a sector. Here is the sector.
1306. 54:07Here is the sector. Here is the sector.
1307. 54:09Here is its arc length. Radius, radius,
1308. 54:12arc length. Okay? We read this one.
1309. 54:15This is its center. This is the radius.
1310. 54:17This is the radius. This is the arc
1311. 54:19length. We read this. Number two. Then
1312. 54:21this whole shape. Is a rectangle. The
1313. 54:23examiner says it himself in the
1314. 54:25question. So what are all the corners
1315. 54:26of a rectangle? 90 °. 90 °. So look,
1316. 54:30if I show this question by zooming in a
1317. 54:32bit. So I will copy this diagram and
1318. 54:35then explain it to you in detail. Here,
1319. 54:37I have copied the exact same diagram
1320. 54:40here. Here you go. To explain it to you
1321. 54:42, come on, I have copied the diagram.
1322. 54:44Here you go. Okay? Now we do our
1323. 54:47working on this diagram and see what is
1324. 54:49missing and what is given. Clear, beta?
1325. 54:52What is missing and what is given.
1326. 54:54Let's begin. Number one. The first
1327. 54:58thing is that we already know that the
1328. 55:00corners of every rectangle are always
1329. 55:0290 °. This is very basic. A class
1330. 55:06three or four kid knows that rectangle
1331. 55:08corners are always 90 °. Rectangle
1332. 55:11corners are always 90 °. Rectangle
1333. 55:14corners are always 90 °. So the first
1334. 55:16thing is the corner is always 90. Okay,
1335. 55:18if anyone remembers, if any child
1336. 55:20remembers, then I can also tell you,
1337. 55:22look, what was the meaning of 90 °?
1338. 55:24First of all, by now we probably
1339. 55:26remember that 90 ° is called π/2.
1340. 55:29Still, if anyone doesn't remember, I
1341. 55:31told you what the method is to convert
1342. 55:33degrees to radians. For those who
1343. 55:35remember, we studied that if you want
1344. 55:37to convert degrees to radians, what
1345. 55:39should you multiply it by? Multiply by
1346. 55:42π/180. Multiply by π/180. So it
1347. 55:45became 90/2. That is why it became 90/2
1348. 55:48. So that is why what do we call 90 °?
1349. 55:50We call it π/2. 90 ° is called π/2.
1350. 55:54So now if I tell any student, "Man, I
1351. 55:56need this small angle." Let me
1352. 55:58highlight it. This angle is my target.
1353. 56:01Here, this angle is my target. If I
1354. 56:04require this small angle, then how will
1355. 56:06this small angle be calculated? Very
1356. 56:09easy. So what will we do for this small
1357. 56:11angle? What is the name of this angle?
1358. 56:14The name of this angle is angle COA.
1359. 56:17Let's say I need angle COA. So what
1360. 56:20will we do? From 90 °, from 90 °—
1361. 56:24meaning from π/2—subtract π/3. If a
1362. 56:28student is a bit educated, what is the
1363. 56:30meaning of π/3? π/3 means 180/3,
1364. 56:32which is basically 60 °. So if a
1365. 56:35student is sensible, this is 60 ° and
1366. 56:37this whole thing is 90 °. So if you
1367. 56:39take 60 out of 90, what will this be?
1368. 56:42It will be 30 °. So I am working in
1369. 56:44the form of radians. 3π-2π, so what
1370. 56:46answer is derived from this? The answer
1371. 56:49derived will be π/6. The answer
1372. 56:53derived will be π/6, there it is. So
1373. 56:55what did your answer turn out to be?
1374. 56:57This small angle will be π/6, here you
1375. 57:00go. So first of all, what are the
1376. 57:03corners of a rectangle? Always 90 °,
1377. 57:05because what is this shape? A large
1378. 57:07shape. It's a rectangle, by the way.
1379. 57:09It's a rectangle. Okay, now you note
1380. 57:11one thing. Let's copy this triangle a
1381. 57:13little bit. This triangle, here you go.
1382. 57:15If I copy this small triangle over here
1383. 57:17. I have copied this same triangle here
1384. 57:19. Here you go. I am showing this to
1385. 57:21explain it to you. I have copied this
1386. 57:23same triangle here. Here you go. Now,
1387. 57:26what was the name of this triangle?
1388. 57:28This point was A. Alright, this point
1389. 57:30was your A, this point was your O, this
1390. 57:33point was your C. And this small angle
1391. 57:35here was your π/6. If any student is
1392. 57:39curious, what does π/6 mean? 180/6
1393. 57:42means if it were in degrees, we would
1394. 57:44call it 30 °, but we have studied that
1395. 57:46degrees are not in my syllabus. Radians
1396. 57:49are in my syllabus, radians. So now I
1397. 57:52need to check what I have. This side is
1398. 57:54already 12 cm. With respect to this
1399. 57:57angle. This is opposite. Opposite the
1400. 58:0090, this is the hypotenuse. And what
1401. 58:02remains, this is the adjacent. So I
1402. 58:04neither know the height, nor do I know
1403. 58:06the base of the triangle; I don't know
1404. 58:08the height, nor the base. Clear, kid? I
1405. 58:10don't know anything. First of all, I
1406. 58:12applied, um, I applied 'sin'. Here you
1407. 58:15go, first of all, I applied 'sin'. Here
1408. 58:18you go, I applied 'sin'. Here you go,
1409. 58:20applied 'sin'. So what is the result of
1410. 58:23'sin'? The result of 'sin' is opposite
1411. 58:25divided by hypotenuse. Here you go, so
1412. 58:27with respect to this angle, this is
1413. 58:29opposite, this is hypotenuse. So what
1414. 58:30did the result come to? The result is
1415. 58:32AC. AC/12, here you go. Now look, what
1416. 58:38is sin π/6? You can call it sin π/6
1417. 58:41or sin 30. But obviously, we will work
1418. 58:44in radians. So sin 30 is half, by the
1419. 58:46way. So from here, I found out what the
1420. 58:49result of AC is. The result of AC is 6
1421. 58:52cm. Here you go. From here, I found out
1422. 58:54the answer for AC. Clear? Here is your
1423. 58:57AC. So, from here, I found out what the
1424. 58:59result of AC is. It is 6 cm. Here you
1425. 59:02go, it is 6 cm. Now tell me one thing.
1426. 59:06This whole side is 12. This is six. So
1427. 59:08if I ask you what this would be? Then
1428. 59:10common sense says you will say this is
1429. 59:12also six. 12-6 is 6, by the way. 12-6
1430. 59:15is 6, by the way. Okay. Now I will also
1431. 59:20find this out. This OC. I applied cos,
1432. 59:24okay? I applied cos, the same one here.
1433. 59:28Similarly, I applied cos. Here is cos,
1434. 59:31this is a very basic thing.
1435. 59:32Right-angled triangle. So what is the
1436. 59:34formula for cos? Base over hypotenuse,
1437. 59:36here it is. What is cos π/6? cosπ/6
1438. 59:40is √3/2, so take this, √3/2*12. So,
1439. 59:45from here, I found out what the answer
1440. 59:47for OC is? The answer for OC is 6√3
1441. 59:51cm, there you go. So, what is the
1442. 59:53answer for OC? The answer for OC is
1443. 59:566√3. The answer for OC is 6√3,
1444. 1:00:02clear child? What is your final target?
1445. 1:00:04Read it again. Your target is the
1446. 1:00:06shaded area in this form. What does
1447. 1:00:09this form indirectly mean? That the
1448. 1:00:11examiner wants the answer in the form
1449. 1:00:13of exact values. Meaning, the examiner
1450. 1:00:15wants the answer without decimals.
1451. 1:00:18Meaning, there should be no decimals
1452. 1:00:20involved in your answer. The answer
1453. 1:00:22should be without decimals. Now come
1454. 1:00:24and see, what is your final target?
1455. 1:00:27Your final target is the shaded area.
1456. 1:00:29Clear child? Your final target is the
1457. 1:00:31shaded area. This is your target. The
1458. 1:00:34shaded area is your target. Here it is.
1459. 1:00:37There are two or three options. Solve
1460. 1:00:38this. There are two or three options.
1461. 1:00:42Okay? One option is to remove this from
1462. 1:00:44the entire rectangle. Remove this too.
1463. 1:00:46You get the shaded area. So if I ask
1464. 1:00:48you, what would be the area of this
1465. 1:00:50whole rectangle? Then you can say that
1466. 1:00:52the name of this entire shape is a
1467. 1:00:54rectangle. The name of the rectangle is
1468. 1:00:56OCDB, OCDB. From this entire rectangle,
1469. 1:01:00remove this triangle. The name of the
1470. 1:01:02triangle is OCA. Triangle OCA. Remove
1471. 1:01:08this one, and remove this one as well.
1472. 1:01:10What is this? This is your sector. This
1473. 1:01:14is your sector. The name of the sector
1474. 1:01:16is OB. This is your sector. The name of
1475. 1:01:19the sector is OAB. Alright, now see,
1476. 1:01:21what is the area of a rectangle? This
1477. 1:01:23side and this side. This side is 6√3
1478. 1:01:26and this side is 12, so the area of a
1479. 1:01:28rectangle is length*breadth 6√3*12,
1480. 1:01:34there you go, now subtract this from it
1481. 1:01:36. What is this thing? This is a
1482. 1:01:38right-angled triangle, its height is 6
1483. 1:01:40and its base is 6√3. So if I say what
1484. 1:01:43its area is, you will say 1/2*height
1485. 1:01:46multiplied by base. There you go, its
1486. 1:01:48area is also found, this is calculated.
1487. 1:01:50What is this thing? This is a sector.
1488. 1:01:52The area of a sector, what is the area?
1489. 1:01:541/2 r² theta. The radius of the sector
1490. 1:01:57is 12 and the angle is? π/3, so what
1491. 1:01:59will its area be? 1/2*radius squared
1492. 1:02:04multiplied by π/3, now we will make
1493. 1:02:07this easier. What is 12*6? 72, there
1494. 1:02:13you go. Made it even easier. Now what
1495. 1:02:15happened here? 2*3 = 3*6 = 18, so this
1496. 1:02:19became 18 3, there you go. Nothing
1497. 1:02:22extraordinary here. 144 6, so this will
1498. 1:02:25become 24π. 24π, there you go. Clear,
1499. 1:02:30dear? Now make it a bit easier. Now
1500. 1:02:33what is this answer? 72 3-18 3-24π,
1501. 1:02:38here it is. Now look, what is 72-18? 54
1502. 1:02:44, so finally what is the answer?
1503. 1:02:49Finally the answer is 54 3-24π, this
1504. 1:02:51is your final answer and you see this
1505. 1:02:54answer is absolutely correct. What is
1506. 1:02:58the reason? Your required answer, did
1507. 1:03:00you see what form it was? The required
1508. 1:03:04answer was a 3-bπ-bπ, so if you match
1509. 1:03:06it, your answer is derived exactly in
1510. 1:03:09this form. Your answer is derived
1511. 1:03:13exactly in this form. You see, what is
1512. 1:03:16your A? If you match it, your A is 54,
1513. 1:03:20and if you match it, what is your B?
1514. 1:03:23Your B is 24. Clear, dear? If you match
1515. 1:03:27it, your A is 54 and what is your B?
1516. 1:03:29Your B is 24. Clear, dear? So what did
1517. 1:03:33we simply do today? Today we did a bit
1518. 1:03:35more practice, today for the first time
1519. 1:03:37we practiced circular measurement. In
1520. 1:03:39the next class, God willing, we will
1521. 1:03:41practice this topic more. Okay, dear?
1522. 1:03:44Goodbye.