平成 5 年度次世代の科学技術を担う人材育成事業 福岡県 高校生科学技術コンテスト 総合問題 数学 解答解説 受験番号 氏 名 所属校名 福岡県教育委員会 3FG- 数学 -KK_H-4.indd 3/04/08 9:36
第 問 ( 総合問題 ) [ 出題のねらい ] 理科は一つなのである! [ 解答 ] 問 D C A 問 問 3 E D B C F A F 問 4 問 5 問 6 問 7 問 8 A C (B D) ( ) 問 9 問 0 6 問 Si 4 O 問 問 3 SiO 問 4.0 0 n mol 問 5.4 0 n mol 問 6 6.3 [ 解説 ] 問 D E C B A B D C A 問 F 問 3 E D B C A G F A A F C C A E D B C F A F (G ) 問 4 - - 3FG- 総合 -KK_00-005.indd 3/05/7 4:37
問 5 アメリカワイオミング州で採取された K-T 境界を含む岩石 中央の白い粘土層は下位の白亜紀層 上位の古第三紀層に比べて約千倍のイリジウムを含んでいる (Wikipedia より ) 問 6 977 K-T K-P 980 K-T 0% ( ) 問 7 5 A B C D 0 A 3 0 B 3 0 C 6 6 7 0 D 3 0 7 0 B D 0 B D B D B&D B&D B&D C 6.5 A 3 A.5 3-3 - 3FG- 総合 -KK_00-005.indd 3 3/05/7 4:37
A B&D 3 SiO 3 B D A C Si O 問 8 4 B&D A C (B D) 問 Si 4 O (+4) 4+(-) =-6 6 Si 4 O 問 9 DNA DNA DNA 問 0 Si O 6 Si O= 6= 3 (+4) +(-) 3=- 問 Si O 図の読解能力をみる問題 G 3 問 0 問 問 3-4 - 3FG- 総合 -KK_00-005.indd 4 3/05/7 4:37
40K 問 3 O 3 Si O SiO 問 4 K 0.0 40 K x nmol 0 x=8.3 0 5 nmol. 0 00 =99.6 ].0 0 nmol 0.8 0.6 0.4 0. 0 0.5 5 37.5 問 5 40 K y nmol ( ) 40 Ar 40 K y y= + Ar Ca 00 =x+ Ar 00 =99.6+4.5 ].4 0 n mol 問 6 t t N =d n T N 0 t. 5 40 K 40 =d n K. 0. 4 t. 5 =d n. 0 t ] =d n. 4. 5 = t ] 6.3 別解. 0 ] 0.7. 4 0.7 3FG- 総合 -KK_00-005.indd 5 3/05/7 4:37-5 -
第 問 ( 専門問題 ) [ 出題のねらい ] [ 解答 ] A^p 0h B^p+ q ph C^q p+ qh D^0 qh O E^p+ q 0h F^p+ q p+ qh G^0 p+ qh OEFG y C(q, p+q) G(0, p+q) F(p+q, p+q) D(0, q) B(p+q, p) O A(p, 0) E(p+q, 0) x OEFG p+q OEFG ^p+ q-h H^p qh y G C F D H(p, q) B O A E x OAHD ^p-h^q-h AD q y =- x+ q p x p q x q x = n n p qx = np qx p p q x p x 0 x p p q x q y =- x+ q p y AD A D ioad ^p-h^q-h ieba ifcb igdc ^p-h^q-h ABCD ^ p+ q-h -4# ^p-h^q-h = p + q + + pq-p-q-^pq-p- q+ h = p + q - OEFG ^ p+ q+ h - 4# ^p+ h^q+ h = p + q - - 6-3FG- 数学 -KK_006-00.indd 6 3/05/8 4:7
i b S S = i+ b- S b S = p + q b = 4 i i = S- b+ i = p + q - $ 4 + i = p + q - 第 3 問 ( 専門問題 ) [ 出題のねらい ] n n=4 0 3 3 4 3 33 [ 解答 ] 問 D^x y zh = x + y + z -xy-yz-zx = " ^x - xy+ y h+ ^y - yz+ z h = " ^x- yh + ^y- zh + ^z-xh, 問 n = 4 + ^z - zx+ x h, 3 0 3 0 3 0 0 問 3V S V S =D^n-a n -b n h + D^n-b n n -h+ D^n n - n -dh -" D^n-a n -b n -h + D^n-b n - n -dh, = " ^- a+ bh + ^- bh + a, + " ^- bh + + ^b-h, + " + ^- + dh + ^-dh, - " ^- a+ bh + ^- b+ h + ^- + ah, - " ^- b+ h + ^- + dh + ^- d+ bh, = ^a + b h+ ^b + h+ ^ + d h - 7-3FG- 数学 -KK_006-00.indd 7 3/05/8 4:7
- ^- + ah - ^- b+ h - ^- d+ bh = a+ b+ bd 問 4 a b d 4 p q r s 0 + p + p+ ^ + qh= + p+ q + p+ q+ ^ + rh= 3+ p+ q+ r 3+ p+ q+ r+ ^ + sh= 4+ p+ q+ r+ s a+ b+ bd = ^a+ b h^+ dh-ad " a b, " d, ^a+ bh^+ dh ad " a b, = " + p + p+ q, " d, = " 3+ p+ q+ r 4+ p+ q+ r+ s, ad a = + p+ q b = + p = 3+ p+ q+ r d = 4+ p+ q+ r+ s a+ b+ bd = ^+ p+ qh^3+ p+ q+ rh + ^+ ph^3+ p+ q+ rh + ^+ ph^4+ p+ q+ r+ sh =^p q r s h+ 3 ^p q r s h p q r s 0 ^p q r s h p=q=r=s=0 0 a+ b+ bd ^a b dh= ^ 3 4h 3 " a b, = " + p 3+ p+ q+ r, " d, = " + p+ q 4+ p+ q+ r+ s, ad a = 3+ p+ q+ r b = + p = + p+ q d = 4+ p+ q+ r+ s a+ b+ bd = ^3+ p+ q+ rh^+ p+ qh+ ^+ ph^+ p+ qh + ^+ ph^4+ p+ q+ r+ sh = ^p q r s h+ ^p q r s h p q r s 0 ^p q r s h p=q=r=s=0 0 a+ b+ bd ^a b dh= ^3 4h " a b, = " + p 4+ p+ q+ r+ s, " d, = " + p+ q 3+ p+ q+ r, ad a = 4+ p+ q+ r+ s b = + p = + p+ q d = 3+ p+ q+ r a+ b+ bd = ^4+ p+ q+ r+ sh^+ p+ qh + ^+ ph^+ p+ qh+ ^+ ph^3+ p+ q+ rh =^p q r s h+ 3 ^p q r s h p q r s 0 ^p q r s h p=q=r=s=0 0 a+ b+ bd ^a b dh= ^4 3h 3 ^a b dh= ^3 4h {a b} { d} ^a b dh= ^4 3h - 8-3FG- 数学 -KK_006-00.indd 8 3/06/03 3:
a b d a+ b+ bd " a b d, = " 3 4, a+ bd " 3 4, a+ bd = $ + 3$ 4 = 4 a+ bd = $ 3+ $ 4 = a+ bd = $ 4+ $ 3 = 0 b b " a d, = " 4, " b, = " 3, a+ b+ bd = 7 b b " a d, = " 3 4, " b, = ", a+ b+ bd = 3 b b " a d, = " 3 4, " b, = ", a+ b+ bd = a+ b+ bd 問 5 S n ^a, n -3 n - n n - n -4, a + 0 n- = S n^a 0, n -3 n - n - n - 4, a n-h+ S n+ ^a 0, a i a j n a k a l, a n-h F S n^a 0, a i a j a k a l, an-h+ n=4 0 n=5 n=4 3 0 4 3 4 0 n=6 3 5 4 0 n=7 3 5 6 4 0 n=8 3 5 7 6 4 0 n=9 3 5 7 8 6 4 0 n=0 3 5 7 9 8 6 4 0 S n = S n^a 0 a a a n-h S 5 F S 4+ S 6 F S 5+ S 0 F S 9+ S 0 F S 4 + # 6 = 0+ # 6 = 9 S 0 9 h 第 4 問 ( 専門問題 ) [ 出題のねらい ] ABC a = bos C+ os B b = os A+ aos C = aos B+ b os A a b [ 解答 ] 問 x y z a = bz+ y * b = x+ az = ay+ bx #a a = abz+ ay #b b = bx+ abz a - b = ay-bx # = ay+ bx - a -b - =-bx b 0 b a x = + - b x"y y"z z"x a"b b" "a b a x = + - b x y a b b a a b y = + - a a b z = + - ab - 9-3FG- 数学 -KK_006-00.indd 9 3/06/03 3:
- x b a = - d + - n b ^bh - ^b + -a h = ^bh " ^b a b b a b =- + - h +," ^ + - h -, ^bh " ^b a b a =- + h -," ^ - h -, ^bh " ^b a b a b a b a =- + h +," ^ + h -," ^ - h +," ^ - h -, ^bh ^a b a b b a a b = + + h^ + - h^ + - h^ + - h ^bh a+ b+ 0 a+ b- 0 b+ -a 0 + a-b 0 - x 0 x x F 0 x y z x y z b a x = + - b + a - b + = ay a 0 a b y = + - a + a -b a = bz+ $ a a + b - bz = a b 0 a b z = + - ab a 0 b 0 0 a b BC = a CA = b AB = ABC b a os A = + - b os A E 0 A 80 os A x y z 問 x y z x y z a = bz+ y * b = x+ az = ay+ bx a b+ * b + a a+ b a b a b b a = ^x+ az hz+ y ^-z ha = ^y+ zxh z - z 0 y+ zx a = - z a b = x+ ^bz+ yhz ^- z hb = ^x+ yzh z - z 0 b x + = yz z - a 0 b 0 0 - z 0 x+ yz 0 y+ zx 0-0 - 3FG- 数学 -KK_006-00.indd 0 3/05/8 4:8
y+ zx x+ yz = $ y + $ x - z - z x + xyz+ y = -z 0 x + xyz+ y = -z x + y + z + xyz- = 0 y+ zx x+ yz a+ b - = + - - z - z y+ zx+ x+ yz-^-z h a+ b+ = -z ^+ zh^x+ yh-^ + zh^-zh a + b + = ^+ zh^-zh x+ y+ z- a+ b+ = -z a+ b- 0 -z 0 0 x+ y+ z a b a 0 b 0 a + b b a x + yz z y + + - = + - zx z - - x+ yz+ -z -y-zx b + - a = -z x-y-z^x-yh+ ^+ zh^-zh b+ -a = ^+ zh^-zh ^x-yh^- zh+ ^+ zh^-zh b+ -a = ^+ zh^-zh x- y+ z+ b+ -a = + z - y 0 x- y+ z+ -y 0 b+ -a 0 b+ a + a b x+ yz -y- z+ yz x+ yz = ^-yh^-zh 0 y+ zx 0 x y z x + y + z + xyz- = 0 x+ y+ z - - 3FG- 数学 -KK_006-00.indd 3/06/03 3:
3FG- 数学 -KK_H-4.indd 4 3/04/08 9:36