Devoir 1 semestre 1 Mathematique Tronc Commun | الفرض الأول الدورة الأولى الرياضيات جذع مشترك علمي — Transcript

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1. 0:01Peace, mercy, and blessings of God to
2. 0:03our loyal followers, followers of the
3. 0:05School Channel. As always, welcome to
4. 0:08the channel. In this video, we have a
5. 0:11sample of mathematics tests for the
6. 0:13common core level. I will leave the
7. 0:17rest of the tests in the link below the
8. 0:19video on the Telegram group for this
9. 0:21test. I have x = 2400, y = 1350. divide
10. 0:50The first I have is 2...is 1350. Good.
11. 1:07So, naturally, this is divisible by...
12. 1:09gives me 12. I will divide by...it
13. 1:15gives me 600, divide by...it gives me
14. 1:19300, divide by 150, I will divide by 75
15. 1:23, it doesn't divide by...it divides by
16. 1:2725, 25 gives me 5, so I will...y after
17. 1:34this 2400. Everything is clear, 2400.
18. 1:41Meaning 2 ^ 5 times 3 ^ 5...coming here
19. 1:46, so this is divisible by 675. This is
20. 1:52divisible by...the sum of the digits is
21. 1:55a multiple, it gives me 225. It is also
22. 1:58divisible by 3 because 5 is 75. We
23. 2:01reached something like this: 3, 25, 5,
24. 2:065, 1350, 5 Greatest divisor, the
25. 2:23greatest common divisor of x and y. The
26. 2:30greatest common divisor, the smallest
27. 2:35powers. Okay, the common factors that
28. 2:39have the smallest exponent, so here
29. 2:41they share the 3 and the 5, but we will
30. 2:44take them with the smallest exponent, 5
31. 2:47with...we will take here with...we will
32. 2:50take the power 5, we have...smallest,
33. 3:02so we take, as we said, the common and
34. 3:06uncommon factors, but raised to the
35. 3:09largest. We said, look at the x, the
36. 3:13largest. So we have the power, which is
37. 3:16the largest, multiplied by 3 which is
38. 3:18the largest, in 5...21600 This is the
39. 3:25LCM. Determine the number of divisors
40. 3:34of x. ...Here we have 1, we add 1 to it
41. 3:58, and this power of 5 we add 1 to it,
42. 4:01and we calculate these, it is 6 times
43. 4:052/3, it is 36. x = 36 divisors. 36 x*y
44. 4:21belongs to...look...5 in 5 power...I
45. 4:47want to remove the radical, naturally,
46. 4:50a natural number radical, so I need to
47. 4:53combine the same...and apply the power
48. 4:56rule and take out the square of the
49. 4:59number to get rid of it. Okay, I will
50. 5:03add 2 with 2, it will be 6, it is 3, it
51. 5:06is 3 ^ 4*5...is 5. Leave it 5, although
52. 5:13this equals...I will start making the
53. 5:16square, it is...3 2 ^ 5 squared, all to
54. 5:26the...this 3 to n. That's it. So, so
55. 5:47now x*y belongs to n, that is a prime
56. 5:58number, the smallest non-zero natural
57. 6:18number for y to be a cube of a natural
58. 6:20number. Y natural number gives us...
59. 6:34Okay, this isn't it, why? Because...I
60. 6:40have 2 ^...all to the...so how much is
61. 6:47needed to become power 5? How much is
62. 6:50needed to become 5? Understood? We have
63. 6:55y is 2 ^ 3...it becomes...multiply, so
64. 7:04...this 5...So when I multiplied, I
65. 7:22multiplied by 5 and I multiplied by 2
66. 7:24to reach...to become...this m 5 y m*y
67. 7:56it will become 2 ^ 3*5 ^ 5. Okay, why?
68. 8:03Because that m consists of...Everything
69. 8:10is...it is possible now to just write
70. 8:14that...Naturally, the one simplified...
71. 8:23Here it is, we multiplied, it is 20. It
72. 8:31is 3. So the smallest non-zero natural
73. 8:37number is m = 20. 6 The prime numbers
74. 9:21that are less than or equal to 10, so
75. 9:2311. The prime numbers prime less than
76. 9:32or equal to 100 is not not 7 That's 11.
77. 9:58I am checking 139. Not 11. 139 is not
78. 10:00even. The sum of the digits is 13, so
79. 10:03it's not divisible by three. The unit
80. 10:06digit is not five or zero, so it's not
81. 10:09divisible by five. If I divide here
82. 10:12with a calculator, I get a decimal. 11
83. 10:26divides, does not divide 139. So 139 is
84. 10:35a number a prime number. That's it.
85. 10:43Only one divides it, so it is a number.
86. 10:47We are done with the first exercise.
87. 11:04Next numbers: we are going to study the
88. 11:06parity. The following number the
89. 11:11exercise. 2n squared + 9n + 4 The best
90. 11:32shortcut is to show whether it equals
91. 11:372k or 2k + 1. So here, n squared, let's
92. 11:43break this down into n + 8 + 4, which
93. 11:48gives me n + 2 (4n + 2). This case an
94. 12:04even number 2k. So n squared + 9n + 4
95. 12:11is a number, is a number is a number of
96. 12:28things are possible to show it's
97. 12:50written in the form 13 I will reach
98. 12:57what is requested. Even though I have
99. 13:01here what equals 3 times 7n times 5.
100. 13:08Good. To factorize here, I have a
101. 13:12common factor which is 7. Here I
102. 13:15factorize n, here 3. Here I have a plus
103. 13:19, in the case of plus, here 7 remains.
104. 13:24So, times 7 plus 5. So this is 7 n. 7
105. 13:30times 3 is 21, I add 5, that gives me
106. 13:3426. 26 is what? 7 13, therefore 13
107. 13:56divides a that b is a multiple of 4, of
108. 14:064, it is 3 to the power of n + 3 to the
109. 14:13power of n + 1 + 3 to the power of n +
110. 14:173 to the power of n +...So I will show
111. 14:20that I need to extract a 4 times a
112. 14:22number. So here, in the case with a 3.
113. 14:28Here I have. That's it, so it will
114. 14:35remain one. Here it remains one, I
115. 14:43forgot one, it remains one, it remains
116. 14:463, and I added one for you, the one I
117. 14:49remembered is 1 + 3. 4, which is 3 to
118. 14:59the n times 4, times 10 times 4 in the
119. 15:02product, therefore is a multiple of 4
120. 15:16Verify that n + 11/(n + 2) is 1 + 9/(n
121. 15:22+ 2). Then the possible values of n so
122. 15:25that Here, either I reach this or, it
123. 15:31is better from here. So what do we have
124. 15:36? plus over n + 2, and one the
125. 15:43denominators, of course, is n + 2. 11
126. 15:49over n + 2. That's it, so these are
127. 15:54equal. Now the next part is what
128. 15:58concerns us: the values of n so that (n
129. 16:00+ 11) belongs to N/(n + 2) in N so that
130. 16:24so that for I need the division by this
131. 16:32number to have no decimal. , : that is,
132. 16:38this is divisible by n + 2 For it to
133. 16:55belong to N, n + 2 must be a divisor,
134. 17:00must be a divisor of 9. That is I can
135. 17:16add for you I have one and I have...and
136. 17:22I have, that's it. Either it is that n
137. 17:32+ n + 2 = 1, so 1 no, that doesn't work
138. 17:45because -1 does not belong to N If n +
139. 18:002 = 3, then n + 2 = 9, n = 7. So the
140. 18:12possible values of n, so the values
141. 18:20possible for n is = 1, n is = 7 N
142. 18:33carefully and with caution for the
143. 18:46first time a number form of two common
144. 18:58things question that I have n plus 3n
145. 19:123n, show look we must is the number Z,
146. 19:24let's set N cubed + 3n squared + n is
147. 19:30equal to A, that makes therefore a A +
148. 19:402 + 1, A + 2+ the factorization squared
149. 19:54+ A and this is one squared, a notable
150. 19:59identity, right all squared, therefore
151. 20:14squared + n into n, 3 n for perfect.
152. 20:28here we have this entire thing is
153. 20:34difficult, let's name x and to the
154. 20:59power that x + 2x + y is 2 x. so here x
155. 21:18y factoring, factoring by x, right, we
156. 21:25can factor by x here and there remains
157. 21:27x plus, if I factor by x for example,
158. 21:30there remains y plus...but there
159. 21:34remains here I will have to add two to
160. 21:39both sides, okay, so it becomes clear
161. 21:48let's here equals four, okay, to factor
162. 22:02here by two because this one alone I
163. 22:05have nothing to do with it. so I will
164. 22:09factor here by x here by two it gives
165. 22:14me x here there is 4, that's it. so
166. 22:33here, consider what is the product that
167. 22:39can give me four, that makes then the
168. 22:45divisors div in is equal to 1, that
169. 23:12makes and x + y + 2 greater than or
170. 23:16equal to 2, that makes the couples x 4
171. 23:53x y possible x y x + 1 y + 2 are why we
172. 24:08and it could be one 4 and it is
173. 24:10possible that this is 4 and this why,
174. 24:15because it is y + 2 if it is equal to
175. 24:21one here and I put this 4 and here x +
176. 24:27greater than or equal to one, but here
177. 24:30y + 4, x 1, x 1, 0. x y therefore y is
178. 25:20here it is one, okay, what is x equal
179. 25:26to one, y means 0, now either x 0 or x
180. 25:34and 0, they are the possible couples,
181. 25:37actually even if we worked one out, one
182. 25:40of them would come out to be minus,
183. 25:43therefore it does not belong Therefore
184. 25:49the couples, the solution couples 100
185. 26:02and 0, y is minus one, and therefore 3,
186. 26:22let's say a triangle pointle construct
187. 26:31the figure is equal to others, yeah,
188. 26:48okay, this one flips here draw and take
189. 27:12from it how much it shouldn't come here
190. 27:16to be honest, it should come here,
191. 27:18let's put it here, okay, here. it said
192. 27:22it is equal to minus, so minus in the
193. 27:24opposite direction, so minus we
194. 27:31finished the first question minus 3 ab
195. 27:46minus 3 a b b leave that this and I
196. 28:16will insert into it so that I can work
197. 28:19...in my pocket like B is BC B ac minus
198. 28:55B minus here minus AC AB BC in order
199. 29:56the sum AB what we showed the
200. 30:18information, let's show that I minus on
201. 30:20plus on I J we have I minus I I, how
202. 30:52much is it equal to, minus, here it has
203. 30:56started to appear, what I have is okay
204. 31:03and a g therefore ab a lot of time of
205. 31:27ours that I is equal to on minus 2 we
206. 31:39have also I equal I a b ab AB 3 ab
207. 32:34minus 2 AC. Therefore, AB 3 minus 2 AC,
208. 32:50reach 5 on. Here, common denominator,
209. 32:54so here a factor of 3 minus 2. minus 2,
210. 32:59so how much will this be? 3 times 2 is
211. 33:046, minus is 5 over 2 ab, minus 2 vector
212. 33:20show that they are so and show that
213. 33:30these okay here where where the
214. 33:38question is and they gave me, so I'll
215. 33:42find a relation. Okay, so we have, what
216. 33:47does it equal here? It seems a little,
217. 33:52look here I have 5 over, here I have 5
218. 33:56ac, I have a one here, under a five
219. 33:59added here in the relation for we g
220. 34:28over 2 over 5 a must the minus,
221. 34:39multiply here, here the minus is gone,
222. 34:41it was plus and became minus, multiply
223. 34:43by minus 5 here. I multiplied by minus,
224. 34:46that's the coefficient I'm looking for,
225. 34:48minus 5 will give me here 5 over 2 ab
226. 34:55minus 2 and that 5 over 2 ab minus 2 ac
227. 35:13donc 5 g 2 J, let A be the midpoint of
228. 35:41M, so me midpoint midpoint midpoint,
229. 35:51that means is parallel to B C. BC are
230. 36:00the deductions, we show that and BC are
231. 36:04linear. I is the midpoint of ABC, so is
232. 36:16parallel to B and BC. that it is
233. 36:31parallel, it's better we take aya
234. 36:49equals am I have a common factor here
235. 37:23which is over minus ac y it is B ac.
236. 37:43Donc IB, donc IM and BC are coin we, we
237. 37:58deduce, we deduce that I is parallel to
238. 38:04B C ABC. D a parallelogram and M a
239. 38:20point such that AMC, the projection of
240. 38:24point M onto BC parallel to B Show that
241. 38:27CN C a paralle. M is a point of AM BC M
242. 38:51BC M BC in parallel well in on so I
243. 39:23noticed one on, where is it coming from
244. 39:26? It's coming from here, this one on l,
245. 39:30was this 3 on, this with. these
246. 39:50consider the projection parallel to on
247. 39:54BC. We consider we, we consider, see
248. 40:01the projection, the projection uh on BC
249. 40:09parallelly parallely to AB. We have, we
250. 40:18have B is the projection of A, B We
251. 40:25have P and ST and the project and the
252. 40:31project of of A, uh M projection of N
253. 40:40We have N and the project of Me, C it
254. 40:53line PC. We have C and the project of,
255. 40:57C it itself, that's the project of C
256. 41:02and it's project, that is of C of this
257. 41:26AC MC M C MC MC MC M C donc M C AC
258. 41:47minus AC minus here, good. So ac y
259. 41:58minus am is minus am, how much? 3 over
260. 42:064 ac, let's factor here 1 minus 3 over
261. 42:124, here 4 times 4 minus 3 is 1 over 4,
262. 42:15good. So I found that am equals 1 over
263. 42:184, am c equals 1 over vect vector ac
264. 42:34vector B C, so the vector the
265. 42:41projection so CN CNC MC N and MC and we
266. 43:26BC and the project AC MC AC BC, since
267. 43:47since the projection preserves the
268. 43:54coefficient coefficient of collinearity
269. 44:01. , : Therefore, BN, CN, B it is CN
270. 44:31over 4, Cre over MN, MC, N, CN, MC, MC,
271. 45:15AC, MC. Let's return to the figure, M
272. 45:27we showed it here. Okay, everything
273. 45:29will be replaced by its value. AM
274. 45:32equals 1 over 4, and CN is what we just
275. 45:38saw, it equals 1 over, CN equals over 4
276. 45:41that we just saw, 1 over 4 plus 1 over
277. 45:444 Okay, I will factor out 1 over, it
278. 45:51will give me plus C, let's also use
279. 45:54over 4 M, a point such that ABR is
280. 46:09parallel, it will equal over divide
281. 46:15over these. E Let's see let's replace
282. 46:44here first, and you have is over, look,
283. 46:53here it is. So here or here is where I
284. 46:57need to not here. E equals over 4
285. 47:13because the data must be precise for us
286. 47:15to work with them. Okay, I left these
287. 47:18here so we can work on what this equals
288. 47:22. Good, we know AM, AM equals 3 over 4,
289. 47:27this is already known to us. We just
290. 47:32need to see, okay, we will see what it
291. 47:34equals, add to it let's factor. over
292. 47:40here I need to use Chasles and factor
293. 47:45with that, let's go, AM we said is
294. 47:49simple, 3 over 4, okay, 3 over 4, and
295. 47:53we will see, follow me well because it
296. 47:58is inverted, it is not, because it
297. 48:02inverts it EA minus E AB H over, okay,
298. 48:21over 4, okay, we reached 48 minutes EA,
299. 48:29we AB eq AB minus 4 and we will this I
300. 48:54want EA, they inverted it minus 4 AB,
301. 49:03good, here with these, it is inverted,
302. 49:07we factored out 3 over 4, good. So A
303. 49:15minus 3 over 4 A Therefore A minus 3
304. 49:28over 4 A, maybe A 3 AC minus 3 over 4 A
305. 49:41So here I will factor by 3 over 4, it
306. 49:44gives me minus, here I will invert it,
307. 49:47it equals 3 over 4 plus B B eq over 4
308. 49:58BC. Therefore M and BC are collinear,
309. 50:21that is to say it's parallel to it
310. 50:32because they are divided, I mean, the
311. 50:35matter is really difficult. Let's get
312. 50:39into the exercises and tests you have,
313. 50:41they are long, good luck to everyone
314. 50:42and goodbye M.