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21

200. 10. For all integers n greater than 1, define an = (logn2002)-1
Let b = a2 + a3 + a4 + a5 and c = a10 + a11 + a12 + a13 + a14
Waht is b - c ?
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22

201. 13. How many distinct four - tuples (a, b, c, d) of rational numbers are there with
a log102 + b log103 + c log105 + d log107 = 2005 ?
1. 0 2. 1 3. 17 4. 2004
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23

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24

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1. 9 2. 11 3. 13 4. 15
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1. 320 2. 330 3. 340 4. 350
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= 243 x ¹ y 2544 - xy
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1. 1810 2. 1815 3. 1820 4. 1825
26

205. 25. ก a #$ b '()*)+,-B*ก ./0M53# 5-+$BBM1ก+
log4a - log32b3 = 19
log )-3245- 4b - log32a3 = 8 a
b
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1. -1 2. 0 3. 1 4. 2
27

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28. Let S1 = {(x, y) / log(1 + x2 + y2) £ 1 + log(x + y)} and
S2 = {(x, y) / log(2 + x2 + y2) £ 2 + log(x + y)}
What is the ratio of the area of S2 to the area of S1 ?
1. 99 2. 100 3. 101 4. 102
28

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2 log8(3x - 2y) = log8x + log8y + 1
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270. P(x) Q(x) P(x) - Q(x)
£
$;,= 2 |P(x)| Q(x)
-7

271. -Q(x) P(x) Q(x)
³ £
$;,= 3 |P(x)| |Q(x)|
[P(x) - Q(x)] × [P(x) + Q(x)] 0
8 ก /4กก 3 3

272. 2 -.
:

273. ก 3ก
34

274. .*.0
1. 8 ก

275. 5
:

276. ,.! C:

277. I27

278. #
0

279. ?!1%
1. a ³ 0 8. a + b ³ a + b
2. - a = a 9. a + b ³ a - b
3. a - b = b - a 10. a - b £ a + b
4. a × b = a × b 11. a - b £ a - b
5. a 12.
b
= a
b
, b ¹ 0 a + b = a + b « ab ³ 0
6. a 2 = a2 13. a + b = a - b « ab £ 0
7. n an =
a 4 n 8!4ก
|a| 4 n 8!ก
2. ก
กก8 2 943ก

280. ก ,

281. -ก 2

282. ก 77

283. -ก ³ 0
ก-/8

284. !8,B.

285. ()*

286. +),-. (Remainder theorem)
5

287. 96 P(x)

288. x - c 3

289. DEกก -/ก P(c)
0*91 DEกก 96 2x3 - 4x2 - 3x + 5

290. x - 3

291. DEกก /ก 2(3)3 - 4(3)2 - 3 × 3 + 5 = 14
.*.0
1. 3-5

292. 96 P(x)

293. ax - b 3

294. DEกก -/ก P(b
a )
2. 4ก96/4DE -,

295. ก

296. ก 96/4, 2
DEก/SET/DE HI4, ก 1 I!!/4 (ก 0)
35

297. # ก 96

298. P(x) 96ก n
3- Q(x) 96ก m
#
m n -,

299. 96 S(x) 3- R(x) /4/8

300. P(x) =
ß
Q(x) ×
ß
S(x) +
ß
R(x)
ß
,,2 , B DE
HI4ก R(x) -

301. ก m

302. ()*

303. ก (factor theorem)
4 P(x) ! 96 anxn + an - 1xn - 1 + ... + a1x + a0 #
/ 4 n 8,ก
3- an , an - 1, ....., a1, a0 8 . H4I an ¹ 0 96 P(x) - x - c
, -ก ก,4 P(c) = 0
Viete's formula
5

304. P(x) = an × xn + an - 1xn - 1 + ... + a1x + a0 , n ³ 1 3- an ¹ 0 -9 P(x)
- n !8, (#
!8,/4H28ก
ก7

305. ) ! x1, x2, x3, ..., xn 3-
(1) x1 + x2 + x3 + ..... + xn = - an - 1
an
(Bก!8, n ,)
(2) x1 × x2 × x3 × ..... × xn = (-1)n a0
an
(B!!8, n ,)
(3) ก BกB!!8, k , = (-1)k an - k
an
: BกB!!8, 2 ,
(x1x2 + x1x3 + .... + x1xn) + (x2x3 + x2x4 + .... + x2xn) + .... + xn - 1 × xn
= (-1)2 × an - 2
an
36

306. (Real Number System)
By (

307. ก)
..,
..
ก!

308. !#$
1.

309. ก
2x3 + x2 - 2x - 1
(x2 + 2x - 3)(x2 + 2x + 3)
³ 0
1. (-3, -1] È[-1 2.
2
, 1)È(1,¥) (-3, -1] È[-1
2
,¥)
3. (-¥, -3)È[-1, -1 4.
2] È(1,¥) (-3, -1] È[-1
2
, 1)
2.
x2 - x + 1
x - 1 + x2 - 3x + 1
x - 3 2x - 1
4x - 8
37

310. 3. ก

311. A = {x Î R 2 - x2
1 - x £ x} B = {x Î R x - 2
x - 3
£ 0}
AÇB
4.
ก 1

312. !กก

313. #$%
x + x
+ 1
x - x
£ 1
1. (-¥, 0] È[1, 3) 2. (1, 3] È(5,¥)
3. (-¥, 1)È(7, 11) 4. (-5, 5)
38

314. 5. ก

315. A = {a
b

316. Î R a3 - 5a2b + 6ab2
(a - b)3 ³ 0}
$ A (%ก
R (
+,

317. -.%$)
6. ก

318. A = {x Î R/ x2 - 2x = 3}
B = {x Î R/ 3 - 2x = - x}
% #

319. +1ก%$ %ก'(
1. A - B = Æ 2. AÇB = Æ
3. AÇB +. ก 1 $ 4. AÈB +. ก 4 $
39

320. 7. 3
$ก
(1
x = 2x - 60 - 2x
1. 32 2. 60 3. 92 4. 120
8. ก

321. A

322. ก (2 - x - x2)2 = 2 - x - x2 B

323. ก x2 = -x AÇB
9.
x - 1 × x - 2 = x + 2
40

324. 10.
ก x2 - 3x + 5x + 1 = x2 + 2x + 1
11. 8 x (+: ก
ก x - 2 - k = 5 +(1 5 % $ k +%(%
12.

325. $

326. x (+:ก(+:(+: ก
ก
2x2 - 4
3 ³ 2x2
,-%

327. .%$% #

328. +1
1. [-1, 0.5) 2. [0.5, 1) 3. [1, 1.5) 4. [1.5, 2)
41

329. 13. 8
ก x2 + x - 2 (x + 2) = .%$ (a, b)
$ a + b +%(%ก
(%
14. ก

330. A

331. ก x2 + x - 2 £ x2 - 4x + 3
B = A - {1} 8 a

332. . ก
B A: a - b ³ 0 (ก b Î B $
B C
$% #

333. +1
ก. 4

334. $

335. %-
.

336. $

337. %-
3
a 5
a
% #

338. +1 %ก
1. ก. 8-ก
. 8-ก 2. ก. 8-ก
. 3
3. ก. 3
. 8-ก 4. ก. 3
. 3
42

339. 15. ก ก x2-5x-4
x2+x-2

340. ³ 1
16. ก

341. I =

342. $

343. E S = {x x - 1 - 1 × x - 1 + 1 50}

344. $

345. . ก
SÇI (%ก
%#

346. 1+
1. 13 2. 14 3. 15 4. 16
43

347. 17.

348. $

349. (:+

350. $

351. E
ก -5 £ x2 - 6 (%ก
%#

352. 1+
x £ 1
1. 8 2. 9 3. 10 4. 11
18. 8 A = {x Î R+ 3 x + 2 £ 2x2 + x }
$. ก
A (+:+%

353. ,(+:(%ก
%

354. % #

355. +1
1. 13 - 1 2. 3. 4.
2
13 + 1
2
13 - 1 13 + 1
44

356. 19. S

357. ก 52x + 11 £ 12(5x) - 9
8 a b

358. . ก
S (+:+%ก(+:

359. ,(+:
$ a + b (%ก
%#

360. 1+
1. log 2. 3. 2 4. 515 log520 log530
20. S

361. ก 3x - 2
x - 1 - 1 ³ 0
{x x 0 x Ï S}

362. .$% %#

363. 1+
1. [0, 1] 2. [1 3. 4.
4
, 3
2] [1
2
, 2] [3
4
, 3]
45

364. 21.
ก 3x + 4 + 3x - 5 ³ 7
22.
ก x - 2
x + 3
+ x - 1
x - 2
³ x - 2
x + 3
+ x - 1
x - 2
46

365. 23.
5x + 7 - 2x + 3 = 3x + 4
24.
ก 3 - x + 3 + x = x
1. 6 2. 2 3 - 1 3. 4. 5.
3 3
2
6 + 2 2 2
25.
2x - 4 + x - 1 5x - 1
26.

366. $

367. $ก x (+:

368. ก

369. %

370. +1+(1ก+:$
x = x4 - 1
1. 0 2. 1 3. 2 4. 3 5. 4
27. ก

371. P(x) = x3 + ax2 + bx + 2 I,(+: a b

372. $

373. 8 x - 1
x + 3 % P(x) $= KL 5

374. 1

375. a + 2b +%(%ก
% #

376. +1
1. -11 2. -1 3. 1 4. 9
28. P

377. $

378. NB
$ก m, n

379. $

380. E 8 x + 3
x3 +mx2 + nx + P $ x - 1 x3 +mx2 + nx + P = KL 4 $ m
n +%(%ก
(%#
1. m = 4 n = -4 2. m = 2 n = -2
3. m = -4 n = 4 4. m = -2 n = 2
29. KLกก 22002 + 2202 + 222 $, 210 - 1
30. ก

381. x2553 - ax + 1 $, x2 - 1 = KL r(x) 8 r(2) = 17 % a
31. ก

382. a, b, c Î R P(x) = ax2 + bx + c I, [P(x)]5 - x
+ x3 - 6x2 + 11x - 6

383. $ก
$ 7a + 3b + 2c +%(%
32. Consider the integral expression in x
P = x3 + x2 + ax + 1, where a is a rational number.
At a = A the value of P is a rational number for any x which satisfies the equation
x2 + 2x - 2 = 0, and in this case the value of P is B .
A = ........ , B = ........
47

384. 33. What is the remainder when x200 - 2x199 + x50 - 2x49 + x2 + x + 1 is divided by
(x - 1)(x - 2)?
a. 2x - 1 b. 7 c. 2x + 3 d. 1
e. 6x - 5
34. Let a, b and c be the three roots of x3 - 64x - 14. What is the value of a3 + b3 + c3?
a. -36 b. 12 c. 36 d. 42
e. 64
*************************
!
1. 1 2. (1, 3. {2} 5
3)È(2, 7
3)È(3,¥)
4. 1 5. (-¥, 0] È(1, 2] È[3,¥) 6. 2
7. 3 8. (0, 1] 9. {0, 4} 10. [-1
5
, 0] È[3,¥)
11. k = 7 12. 2 13. 2 14. 3
15. (-¥, -2)È(-2, -1] È[-1 16. 3
3
, 1) È(1, 3]
17. 1 18. 2 19. 2 20. 3
21. R 22. (-¥, -3)È[1, 2)È(2,¥) 23. -4
3
24. 5 25. [2,10) 26. 3 27. 4
28. 2 29. 12 30. -7 31. 2 + 3
1
5 - 2
1
5
32. A Þ - 4,B Þ - 1 33. e 34. d
48