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ratt 2004+corrigé.pdf


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‫ ‬
‫ ‬

‫ ‬


‫ ‪2004 :‬‬
‫) (‬

‫ا ا ول‬

‫‪0.5‬‬
‫‪0.5‬‬

‫‪0.25‬‬

‫‪0.5‬‬

‫ ‬
‫‪ 3‬‬
‫ ‪:‬‬
‫‪7 :‬‬
‫ ‬

‫)
ن و (‬

‫‪u3‬‬
‫‪. n un +1 = 2 n‬‬
‫ ا ا د ) ‪ ( un‬ا ‪ u0 = 1 :‬و‬
‫‪3un + 1‬‬
‫‪ (1‬أ‪ -‬أن ‪. n un > 0‬‬

‫ب‪ -‬أن ا ) ‪. & '( ( un‬‬

‫ج‪ -‬ا‪ * ' +‬أن‬

‫) ‪( un‬‬

‫ ر ‪.‬‬

‫‪1‬‬
‫‪ (2‬أ‪ -‬أن ‪. n un +1 ≤ un‬‬
‫‪3‬‬
‫‪n‬‬

‫‪0.75‬‬

‫‪1‬‬
‫ب‪ -‬ا‪ * ' +‬أن‪ 12 n un ≤   :‬ا‪. lim un ./0‬‬
‫∞‪x →+‬‬
‫‪3‬‬

‫ا ا ‬

‫) ‪ 3‬ﻁ و (‬

‫ ا <; ء ا '‪9/‬ب إ ‪16' 1 7‬‬

‫‪0.5‬‬

‫‪1‬‬

‫‪0.75‬‬
‫‪0.5‬‬

‫‪0.25‬‬
‫‪0.5‬‬

‫ ‬

‫ ‪( O, i , j , k ) 5‬‬

‫ا ' ﻁ ) ‪ A (1, 2, −2‬و )‪ B ( 0,3, −3‬و ) ‪ C (1,1, −2‬وا ‪9 /‬ى ) ‪ ( P‬ا @ي د ﻩ‪. x + y − 3 = 0 :‬‬

‫‪ (1‬أ‪ -‬ا‪ / ./0‬ا '
)‪ A Ω ( 0,1, −1‬ا ‪9 /‬ى ) ‪. ( P‬‬

‫ب‪ -‬ا‪ * ' +‬أن د د ر( < ) ‪ ( S‬ا آ‪C‬ه )‪Ω ( 0,1, −1‬‬

‫وا ‪9 / +‬ى ) ‪( P‬‬

‫ه ‪x 2 + y 2 + z 2 − 2 y + 2 z = 0 :‬‬
‫ ‬
‫‪ (2‬أ‪ 0 -‬د ‪ 12 AB ∧ AC‬ا‪ * ' +‬أن ا ' ﻁ ‪ A‬و ‪ B‬و ‪. / E C‬‬
‫ب‪ -‬أن‪ x − z − 3 = 0 :‬د د ر( ‪9 /‬ى ) ‪. ( ABC‬‬

‫‪ (3‬أ‪ F G( -‬ا < ‬

‫) ‪(S‬‬

‫ ‪9 / +‬ى ) ‪. ( ABC‬‬

‫ب‪ -‬ا‪ ./0‬ا ‪ ΩC /‬وا‪ (
* ' +‬س ) ‪ ( S‬وا ‪9 /‬ى ) ‪. ( ABC‬‬

‫ا ا ‬

‫) ‪ 3‬ﻁ (‬

‫ ‪ A9 J‬ا‪ AI‬اد ا ا د ا ‪2 z 2 − 2iz − 1 = 0 :‬‬

‫)‪(E‬‬

‫‪0.75‬‬

‫‪ (1‬أ‪ 0 -‬ا د ) ‪ z1 ) . ( E‬و ‪ z2‬ه ‪ L0‬ا د ‪.( Re ( z1 ) > 0 K G‬‬

‫‪0.5‬‬

‫ب‪ -‬اآ ‪ .‬ا ‪ z1 G‬و ‪ 7 A z2‬ا ‪ P‬ا ‪. O O‬‬
‫ ‬
‫‪ (2‬ا ‪9 /‬ى ا ي ا '‪9/‬ب إ ‪( O, e1 , e2 ) 16' 1 7‬‬

‫‪0.75‬‬

‫‪0.5‬‬
‫‪0.5‬‬

‫‪1 1‬‬
‫‪1 1‬‬
‫ ا ' ﻁ ‪ A‬و ‪ B‬و ‪ S‬ا أ ‪ 7 A Q& G‬ا ‪9‬ا ه ‪ a = + i :‬و ‪ b = − + i‬و ‪. s = i‬‬
‫‪2 2‬‬
‫‪2 2‬‬
‫‪a−s‬‬
‫‪.‬‬
‫أ‪ -‬اآ ‪ 7 A .‬ا ‪ P‬ا ‪ O O‬ا د ا ي‪:‬‬
‫‪b−s‬‬
‫ب‪ -‬ا‪ * ' +‬أن ا ‪ / SAB K O‬وي ا ‪ & /‬و& ‪ 1R‬ا ‪C‬او ‪. S‬‬
‫ج‪ -‬أن ا ‪.S OASB A‬‬

‫‪1‬‬