◎特別奨学生試験（2019年12月15日実施）

਺ ֶ 2

<஫ҙ> ࣍ͷ Ξ ͔Β ϩ ʹ͋ͯ͸·Δ਺ࣈ·ͨ͸ූ߸ΛɼϚʔΫγʔτղ౴

༻ࢴͷ֘౰͢Δղ౴ཝʹϚʔΫͤΑɻͨͩ͠ɼ෼਺͸ط໿෼਺Ͱදͤɻ

1 7998×8321 = 66 Ξ Π ΢ Τ 58

2 xʹؔ͢Δํఔࣜ x⁴+ Φ x³− Χ x²−35x−30 = 0ͷղ͸ɼx=−2ɼ3ͱɼ x= Ω Ϋ ±

έ

2 Ͱ͋Δɻ

3 ฏ໘্ʹ4఺A, B, C, D͕͋ΓɼAB = BC = CD = 1ɼ−→AB·−→BC =−→CB·−→CD = 1 2 Ͱ͋Δͱ͖ɼ−→AD =a−→AB +b−→BCͱͯ͠ɼ(a, b) =

ί , 0·ͨ͸

0, α Ͱ͋Γɼ

−→AC·−→AD = γ ·ͨ͸ ε

η Ͱ͋Δɻ

4 √

1 + sinx= 1 + cos

x− π

ι

=

λ cos

x

ν − π π

（注）この問題は，「数学②」の問題である。

工学部は必須科目。

解答用紙は，数学のマークシート１枚。

（解答用紙の選択欄に「数学②」を必ず記入・マークすること。）

数　　　学　②

◎特別奨学生試験（2019年12月15日実施）

〔数　　　学〕

数　　　学　②

（工・経営情報・国際関係・人文・応用生物・生命健康科・現代教育学部）

5 lim

h→0

f(x+ 3h)−f(x)

h = logx ͔ͭf(1) = 1ͷͱ͖ɼ f(x) = ς

τ x

�

logx− φ

�

+ χ

ψ Ͱ͋Δɻ

6 αΠίϩΛnճ౤͛ͯɼ1ͷ໨͕kճग़Δ֬཰pk͸nCk

⎛

⎝ ω ϊ

⎞

⎠

k⎛

⎝ ϋ ϊ

⎞

⎠

n−k

Ͱ͋Δɻ͜͜Ͱɼk= 0, 1, 2, . . . , nͰ͋ΓɼnCk͸nݸ͔ΒkݸΛબͿ૊߹ͤͷ

਺Ͱ͋Δɻ͜ͷͱ͖ɼk= 0, 1, 2, . . . , n−1ʹରͯ͠ɼ pk+1

pk

= n−k

ώ �

k+ ϑ �

͕੒Γཱͭɻ͕ͨͬͯ͠ɼpk+1

pk 1ͱͳΔͷ͸ɼk

⎡

⎣n− ϔ ϗ

⎤

⎦ͷͱ͖Ͱ͋Γɼ

pk+1

pk

<1ͱͳΔͷ͸ɼk

⎡

⎣ n+ Ϛ ϗ

⎤

⎦ͷͱ͖Ͱ͋Δɻ͜͜Ͱɼ[x]͸ɼxΛ͑͜

ͳ͍࠷େͷ੔਺Ͱ͋Δɻ

7

�n k=1

k= n�

n+ ϛ �

Ϝ Ͱ͋Δ͜ͱΛ༻͍Δͱɼ 1 + 1

1 + 2 + 1

1 + 2 + 3 +· · ·+ 1

1 + 2 + 3 +· · ·+ 100 = ϝ Ϟ Ϡ Ϣ Ϥ ϥ

8 f(x) = (x−[x])^[x]([x] + 1−x)ͱ͓͘ɻ͜͜Ͱɼ[x]͸ɼxΛ͑͜ͳ͍࠷େͷ੔਺

Ͱ͋Δɻࣗવ਺nʹର͠ɼ� n+1 n

f(x)dx=

� n+1 n

(x−n)ⁿ(n+1−x)dx= 1 n+ Ϧ − 1

n+ ϧ ͱͳΔͷͰ,ࣗવ਺mʹର͠

� m+1 1

f(x)dx= m

Ϩ m+ ϩ ͕੒Γཱͭɻ

分 修正日：2019年11月19日 午前11時38分

特奨－No.4

計算用紙が次ページにあります。

　 2 　

― 2 ―

਺ ֶ 1

<஫ҙ> ࣍ͷ Ξ ͔Β Ϯ ʹ͋ͯ͸·Δ਺ࣈ·ͨ͸ූ߸ΛɼϚʔΫγʔτղ

౴༻ࢴͷ֘౰͢Δղ౴ཝʹϚʔΫͤΑɻͨͩ͠ɼ෼਺͸ط໿෼਺Ͱද

ͤɻ·ͨɼࠜ߸ΛؚΉܗͰղ౴͢Δ৔߹ɼࠜ߸ͷதʹݱΕΔࣗવ਺͕

࠷খͱͳΔܗͰ౴͑Αɻ 1 √

2√ 3√

5

1

√2 +√ 3−√

5 − 1

√2 +√ 3 +√

5

= Ξ

2 ͍ͣΕ΋0Ͱͳ͍࣮਺a, b, c͕ɼ (a+ 2b)(b+ 2c)(c+ 2a)

abc = 5ɼ (a−b)(b−c)(c−a)

abc = 3

Λຬͨ͢ͱ͖ɼx= a

bɼy= b

cɼz= c

aͱஔ͘ͱɼa b + b

c + c

a =x+y+z= Π

΢ Ͱ͋Γɼ b

a + c b + a

c = Τ Φ

Χ Ͱ͋Δɻ

3 ࡾ֯ܗABCʹ͓͍ͯɼsin∠A : sin∠B : sin∠C = 4 : 5 : 6ͷͱ͖ɼ cos∠A : cos∠B : cos∠C = Ω Ϋ : έ : ί ͱͳΔɻ

4 A, B, C, D, Eͷ5ਓ͕1ճ͚ͩ͡ΌΜ͚ΜΛ͢Δɻ͜ͷͱ͖ɼA͚͕ͩউͭ֬཰͸

α

γ ε Ͱ͋Γɼ͍͋͜ͷ֬཰͸ η ι

λ ν Ͱ͋Δɻ·ͨɼ2ਓҎ্͕উͭ֬཰͸

π ς

τ φ Ͱ͋Δɻ

5 aɼbɼcΛఆ਺ͱ͢Δɻ์෺ઢy=ax²+bx+cΛx࣠ํ޲ʹ2ɼy࣠ํ޲ʹ2͚ͩฏ 3

（注）この問題は，「数学①」の問題である。

解答用紙は，数学のマークシート１枚。

（解答用紙の選択欄に「数学①」を必ず記入・マークすること。）

数　　　学　① 数　　　学　①

（経営情報・国際関係・人文・応用生物・生命健康科・現代教育学部）

6 ਖ਼ീ֯ܗABCDEFGH͕͋Δɻ͜ͷਖ਼ീ֯ܗͷ4௖఺Λ݁ΜͰͰ͖Δ࢛֯ܗ͸ ϋ ώ ௨Γ͋Δɻ·ͨɼͦͷΑ͏ͳ࢛֯ܗͷ͏ͪ୆ܗ͸ ϑ ϔ ௨Γ͋Δɻͨͩ͠ɼ௕ํ

ܗ΍ਖ਼ํܗ΋୆ܗͱΈͳ͢ɻ

7 ઈର஋͕50ҎԼͷ੔਺ͷू·ΓΛશମू߹Uͱ͢Δɻू߹U ͷ෦෼ू߹Ͱ2ͷഒ

਺ͷू߹ΛAɼ3ͷഒ਺ͷू߹ΛBɼ4ͷഒ਺ͷू߹ΛCͱ͢Δɻ͜ͷͱ͖A∪B ͷཁૉͷݸ਺͸ ϗ Ϛ Ͱ͋Γɼ(A∩B)∪Cͷཁૉͷݸ਺͸ ϛ Ϝ Ͱ͋Δɻ

8 ӈਤͷΑ͏ʹɼ2ͭͷԁOͱԁO͕઀͍ͯͯ͠ɼͦΕ Βͷ൒ܘ͸ͦΕͧΕ10ͱ7Ͱ͋Δɻ௚ઢ�͸ԁOͱ ԁOͷڞ௨઀ઢͰɼA, B͸઀఺Ͱ͋Δɻઢ෼ABͷ

௕͞͸ ϝ

Ϟ Ϡ Ͱ͋Δɻ

10

O 7

O B A

�

9 ӈਤͷΑ͏ʹ1ลͷ௕͕͞3ͷਖ਼࢛໘ମOABCͷล OAɼOB্ʹͦΕͧΕɼ఺D, EΛOD = 1, OE = 2 ͱͳΔΑ͏ʹͱΔɻ͜ͷͱ͖ɼ௖఺O͔Βࡾ֯ܗABC ʹԼͨ͠ਨઢͷ௕͞͸

Ϣ Ͱ͋Γɼ࢛໘ମOCDE ͷମੵ͸

Ϥ

ϥ Ͱ͋Δɻ·ͨɼࡾ֯ܗCDEͷ໘ੵ

͸ Ϧ ϧ

Ϩ Ͱ͋Δ͔ΒɼO͔ΒC, D, EΛ௨Δฏ ໘΁Լͨ͠ਨઢͷ௕͞͸ ϩ

ϫ

Ϯ Ͱ͋Δɻ

A

C D

O

E

B

分 修正日：2019年11月19日 午前11時39分

特奨－No.4

計算用紙が次ページにあります。

　 6 　

― 4 ―

（注）この問題は，「英語」の問題である。

解答用紙は，英語のマークシート１枚。

英　　　　　語

（ 解答番号 1 ～ 40 ）

〔１〕

Many of us may have a clear mental image of a so-called “cave man,” a grunting, heavy- browed pre-human wearing rough furs and carrying a crude club. This image is based mostly on popular representations of Neanderthals, an extinct species of ancient humans that lived between 400,000 and 40,000 years ago. Neanderthal skulls and jaws were first discovered in what is now Belgium in 1829, and in Gibraltar in 1848, but they were not recognized as being distinct from modern humans until a more-complete skeleton was unearthed in the Neanderthal valley of Germany in 1856. In the intervening years, the remains of more than 400 Neanderthals have been found. Over time, scientists have developed a greater understanding of these people and their relationship to modern humans, and we even have some hints about their culture and belief systems.

It is incorrect to think that Neanderthals were ancestors of modern humans. Instead, it is believed that we shared a common ancestor, Homo erectus, and split off between 800,000 and 400,000 years ago. Neanderthals were shorter and stockier than modern humans, and probably much stronger physically. They had a very distinctly-shaped skull, with a low forehead and large nose, and actually had significantly larger brains than modern humans do. Many of these differences are thought to be because they lived mainly in the colder, forested climates of northern Europe, unlike the warm open savannahs where modern humans developed. Shorter limbs and a thicker body helped them conserve heat, while a large nose warmed cold air before it was breathed into the lungs. Short, strong legs also made them suited for short runs and jumps, useful for hunting and living in heavy forests.

We know for sure that Neanderthals made tools out of stone, knew how to use fire, and hunted for much of their food, but little else for certain. They appear to have lived in small family groups, and while there is little clear evidence, they may have had some degree of culture. They may have cared for their elderly and injured with fairly sophisticated techniques;

〔英　　　語〕

（工・経営情報・国際関係・人文・ 応用生物・生命健康科・現代教育学部）

分 修正日：2019年11月19日 午前11時22分 特奨－No.2

　 2 　

widespread. There is certainly nothing like the complex cave paintings and symbolic carvings associated with early modern humans who lived at about the same time.

What we do know is that the Neanderthals died out around 40,000 years ago, very soon after modern humans arrived in the areas where the Neanderthals had been living. Whether the Neanderthals disappeared because the climate in Europe was rapidly warming, because modern humans killed them off or drove them out, or some combination of those factors, is not well known. But we do know, from analysis of the genes of modern humans, that most people have some Neanderthal DNA̶as much as 2% in some cases. So even though these mysterious relatives of modern humans are no longer with us, we are still carrying their legacy with us today.

〔設問〕

それぞれ下のア～エのうちから一つずつ選べ。

Neanderthals were a kind of human who １ . ア　only lived in caves

イ　lived hundreds of thousands of years ago ウ　had a clear mental image

エ　had very small brains

The first Neanderthal remains discovered were ２ . ア　only the bones of the head and jaw

イ　found in the Neanderthal valley in Germany ウ　dug up in Gibraltar in 1848

エ　immediately recognized as different from humans today In the time since the first Neanderthal bones were discovered, ３ .

ア　we have learned very little about them

イ　scientists have discovered much about their culture ウ　we have learned everything there is to know about them エ　remains of hundreds of individuals have been found

― 6 ―

Neanderthals were ４ modern humans.

ア　unrelated to イ　a direct ancestor of

ウ　descended from a common ancestor to エ　descendants of

Neanderthals were probably ５ than modern humans.

ア　shorter but stronger イ　shorter and weaker ウ　taller and stronger エ　taller but weaker

The physical differences between Neanderthals and modern humans are thought to be caused by _６ .

ア　the Neanderthals’ belief systems イ　modern humans’ hunting habits ウ　the difference in the size of their brains エ　the different climates where they lived Scientists are certain that Neanderthals _７ .

ア　buried their dead

イ　made complex cave paintings ウ　made stone tools

エ　had an interesting culture

Neanderthals ８ may be evidence that they took care of their elderly.

ア　living in small family groups イ　dying long after they lost their teeth ウ　having some degree of culture エ　being buried after death

９ may be one of the reasons the Neanderthals disappeared.

ア　The size of their brains イ　Sustaining serious injuries

分 修正日：2019年11月19日 午前11時22分 特奨－No.2

　 4 　 DNA analysis shows that 10 .

ア　pure Neanderthals are still alive

イ　modern humans drove Neanderthals away ウ　Neanderthals died out about 40,000 years ago エ　most people have some Neanderthal DNA

― 8 ―

〔２〕

My son has just graduated from college, so I don’t think he’s old 11 to get married.

ア　good イ　well ウ　much エ　enough

I’ll go back to Japan 12 completion of my studies here in the U.S.

ア　to イ　upon ウ　with エ　of

Please call me when you 13 in Los Angeles.

ア　arrive イ　will arrive ウ　arrived エ　will have arrived This explanation is lengthy and ₁₄ from clear.

ア　heavy イ　light ウ　far エ　close

In Japan, you’re not allowed 15 alcohol if you’re under 20.

ア　to drink イ　drinking ウ　drink エ　to drinking

If you have any questions, feel 16 to contact us.

ア　happy イ　easy ウ　free エ　helpful

If we don’t have anything to eat at home, 17 don’t we go out to that new restaurant?

ア　what _イwhere _ウwho _エwhy

18 that I’m eighteen, I can get a driver’s license.

ア　Once イ　Now ウ　Before エ　After

The workshop is for anyone 19 is interested in haiku.

ア　who イ　which ウ　whose エ　what

I hurried to my house only 20 that my parents had already gotten home.

ア　find イ　finding ウ　to find エ　found

分 修正日：2019年11月19日 午前11時22分 特奨－No.2

　 6 　

〔３〕

Runa and David are in a pet shop.

Runa: Oh, what an adorable little puppy!

David: Yeah, he’s really cute.

Runa: You mean she’s really cute. This is a sweet little girl. I really want to 21 with me.

David: As much as I like dogs, you know we can’t have one right now.

Runa: Why not? Our back yard has plenty of space and she’s just a small little thing.

David: That’s true, but 22 to keep her outside. She’s an indoor dog.

Runa: Well, our house is big enough, too.

David: The problem isn’t 23 . The problem is having enough money. Look at how much she costs!

Runa: Yeah, she’s expensive, but I can work part time to afford it.

David: And 24 of her while you are working?

Runa: You’re right. I can’t leave her alone all day. I’m not giving up, though. I’ll think of something.

David: Well, if you 25 , then we can get a dog. Good luck.

ア　come up with a good plan イ　who will take care ウ　we aren_’t going エ　always want a pet オ　having enough space カ　having too much space キ　take her home ク　leave her home

― 10 ―

Kenta and Emilia are talking about part-time jobs.

Kenta: What are you looking at there?

Emilia: Oh, just an app I downloaded for 26 .

Kenta: I thought you worked at the convenience store.

Emilia: Yeah, but I think I want to quit. The hours ₂₇ . Kenta: Hmm. So what kind of job 28 ?

Emilia: I was thinking about applying 29 . The pay is good and the hours are flexible.

Kenta: Yeah, that looks good. You know, Jim works there. You should ask him about it.

Emilia: That’s a good idea. I’ll 30 . Kenta: Good luck!

ア　send him a text message イ　are pretty bad

ウ　do you have now エ　sound very long

オ　work at the convenience store カ　finding part-time jobs

キ　at this coffee shop ク　are you looking for

分 修正日：2019年11月19日 午前11時22分 特奨－No.2

　 8 　

〔４〕

My mother

31 earned a living as a nurse.

ア　saved people イ　risked her life

ウ　became famous _エmade money

We are all glad to hear that you ₃₂ got over your illness.

ア　overcame イ　overtook ウ　raised エ　worsened

The lecturer ₃₃ brought up a new theory about human development.

ア　represented イ　denied ウ　introduced エ　decided

Every time I try to talk to the director, her secretary ₃₄ gets in my way.

ア　stops me イ　irritates me ウ　offers me エ　receives me

I live alone in Tokyo, but my mother comes to visit ₃₅ from time to time.

ア　rarely イ　occasionally ウ　hourly エ　timely

― 12 ―

〔５〕

36 Party members are ^{A B} their candidate.

1. thoughts 2. supporting 3. second

4. about 5. having

ア　A-4 B-5 イ　A-5 B-2 ウ　A-3 B-4

エ　A-5 B-1 _オA-1 B-2

37 Masaki ^{A B} his graduation thesis.

1. a lot of 2. for 3. needed to

4. research 5. do

ア　A-3 B-1 イ　A-1 B-3 ウ　A-2 B-1

エ　A-5 B-4 オ　A-5 B-3

38 Pressure at work has been ^{A B} what to do.

1. I 2. know 3. rapidly that

4. don’t 5. increasing so

ア　A-3 B-1 イ　A-3 B-4 ウ　A-4 B-3

エ　A-4 B-1 オ　A-5 B-3

分 修正日：2019年11月19日 午前11時22分 特奨－No.2

　 10 　

39 I will definitely ^{A B} week.

1. my 2. by 3. submit

4. next 5. assignment

ア　A-5 B-2 イ　A-5 B-1 ウ　A-3 B-4

エ　A-4 B-1 オ　A-1 B-2

40 After the operation, Susan ^{A B} .

1. until 2. her little sister 3. cared for

4. recovered 5. she was completely

ア　A-2 B-5 イ　A-1 B-5 ウ　A-1 B-2

エ　A-3 B-5 オ　A-4 B-1

― 14 ―

2020ಛ঑޻using pr2015ˏTakamaruฏ੒31೥11݄22೔18࣌ 7෼

෺ɹɹɹɹɹཧ

(ղ౴൪߸ 1 ʙ 35 )

I

બ΂ɻ

ɹҎԼͷ໰୊Ͱ͸ɼॏྗՃ଎౓ͷେ͖͞Λgͱ͠ɼۭؾ఍߅͸ແࢹͰ͖Δ΋ͷͱ͢Δɻ (1) ਤ1ʹࣔ͢Α͏ʹɼ࣭ྔmͷখ͞ͳ෺ମAΛॳ଎౓V Ͱচ͔ΒԖ௚ํ޲ʹ౤্͛͛Δ

ͱɼߴ͞H ʹୡͨ͠ޙɼݩͷচ·Ͱ໭ͬͯ͘Δɻ౤্͛͛ͨॠؒͷӡಈΤωϧΪʔ͸

1 Ͱ͋ΔɻҐஔΤωϧΪʔͷج४Λচ໘ͱ͢Δɻ࠷ߴ఺ͰͷӡಈΤωϧΪʔ͸ 2 ͱ ͳΓɼͦͷͱ͖ͷҐஔΤωϧΪʔ͸ 3 ͱͳΔɻྗֶతΤωϧΪʔอଘଇΑΓV ͱH ͱͷؔ܎͸V = 4 Ͱ͋Δ͜ͱ͕Θ͔Δɻ

V

H

ਤ1

෺ମA

（注）この問題は，「物理②」の問題である。

解答用紙は，理科のマークシート１枚。

（解答用紙の選択欄に「物理②」を必ず記入・マークすること。）

物　　　理　②

〔理　科（物理，化学，生物）〕

物　　　理　②（工・応用生物・生命健康科・現代教育学部）

(2) ࣍ʹɼਤ2ʹࣔ͢Α͏ʹɼ௕͞Rͷܰͯ͘ۂ͕Βͣ৳ॖ͠ͳ͍๮ͷઌ୺ʹ࣭ྔmͷখ

͞ͳ෺ମAΛऔΓ෇͚ɼ๮ͷ΋͏Ұํͷઌ୺Λճస࣠ʹऔΓ෇͚ԁӡಈͰ͖ΔΑ͏ʹ͠

ͨɻԁӡಈͷ࠷Լ఺ʹ͓͍ͯɼ෺ମAʹେ͖͞V0ͷॳ଎౓Λਫฏํ޲ʹ༩͑ͨͱ͢Δɻ

͜ͷͱ͖ɼ࠷ߴ఺ʹୡͨ͠෺ମAͷ଎͕͞0ͱͳͬͨͱ͢ΔͱྗֶతΤωϧΪʔอଘଇ

͔ΒɼV0ͱRͷؔ܎͸ɼV0= 5 ͱͳΔɻ(1)ͷV = 4 ʹࣔؔ͢܎ͱൺ΂Δͱɼӡ ಈͷܦ࿏ʹΑΒͣɼӡಈͷ࢝ΊͱऴΘΓͷྗֶతΤωϧΪʔͷ૯࿨͕ෆมͰ͋Δ͜ͱ͕

͔֬ΊΒΕΔɻ

R

V0

ਤ2

෺ମA

分 修正日：2020年6月4日 午後2時59分

特奨－No.3

　 2 　

― 16 ―

(3) ࣍ʹɼਤ3ʹࣔ͢Α͏ʹɼઌ΄Ͳͷ๮ͷ୅ΘΓʹ௕͞Rͷܰͯ͘৳ॖ͠ͳ͍ࢳΛ༻͍ͯ

෺ମAΛԁӡಈͤ͞ଓ͚Δ͜ͱΛߟ͑Δɻͭ·ΓɼԁӡಈΛଓ͚ΔͨΊʹ͸ɼࢳͷுྗ

T͕T >0ͷؔ܎Λҡ࣋͢Δඞཁ͕͋Δɻ͜ͷ͜ͱΛ೦಄ʹஔ͍ͯɼԁӡಈͷ࠷ߴ఺Ͱ ͷ଎౓ͷେ͖͞ΛVtopͱ͢Δͱɼ࠷ߴ఺Ͱͷ޲৺Ճ଎౓ͷେ͖͞͸ 6 Ͱ͋Γɼ͜ͷ ͱ͖ͷԖ௚ํ޲ͷӡಈํఔࣜ͸ 7 Ͱ͋Δɻ࣍ʹɼ࠷Լ఺Ͱͷਫฏํ޲ͷॳ଎౓ͷେ͖

͞Λv0ͱ͢Δɻ 7 ͱྗֶతΤωϧΪʔอଘଇ͔ΒԁӡಈΛଓ͚ΔͨΊʹ͸ɼv0ͱR ͱͷؔ܎͸ 8 Λຬͨ͢ඞཁ͕͋Δɻ

R

v0

Vtop

ਤ3

෺ମA

R θc

x

y

V0

Vc

ਤ4

෺ମA

(4) ਤ4ʹ͓͍ͯɼ࠷Լ఺Ͱͷਫฏํ޲ͷॳ଎౓ͷେ͖͞v0͕(2)Ͱఆٛ͢ΔV0ͱ౳͍͠

(v0 =V0)৔߹Λߟ͑Δɻ͜ͷͱ͖ɼԁӡಈͷ్தͰࢳ͕ͨΔΜͰ͠·͏(ࢳͷுྗT

͕ɼT = 0ͱͳΔ)ɻਤ4ʹࣔ͢Α͏ʹ࠷ߴ఺͔Β࣌ܭճΓʹճస֯θΛఆٛ͢Δɻ࠷Լ

఺Λ࠲ඪݪ఺ͱ͠ɼਫฏํ޲ӈ޲͖ʹx࣠ΛɼԖ௚্ํʹy࣠Λߟ͑Δͱɼԁӡಈͷ్

தͰࢳ͕ͨΔΉ৔ॴͷθͷ஋Λθcͱ͓͚͹ɼͦͷ࠲ඪ(Xc ,Yc)͸ɼθcͱRΛ༻͍ͯ( 9 , 10 )ͱͳΔɻ͜ͷ͜ͱʹ஫ҙͯ͠ɼ࠲ඪ(Xc ,Yc)Ͱͷࢳ͕ͨΔΉ௚લͷ෺ମA ͷ଎͞VcΛ࢖ͬͯɼճసத৺ํ޲ͷӡಈํఔࣜΛද͢ͱɼ 11 Ͱ͋Γɼͦͷͱ͖ͷྗ

ֶతΤωϧΪʔอଘଇ͸ɼ 12 Ͱ͋Δɻ 5 ɼ 11 ɼ 12 Λ૊Έ߹ΘͤΔͱ 13 ͷؔ

܎ࣜΛಘΔɻ·ͨɼ(Xc ,Yc) = ( 14 , 15 )Ͱ΋͋Δɻ

1 ɼ 2 ͷղ౴܈

(Ξ) 0 (Π) mV² (΢) 1

2mV² (Τ) 3 2mV²

3 ͷղ౴܈

(Ξ) 0 (Π) 1

2mgH (΢) mgH (Τ) 1

2mgH 3

分 修正日：2020年6月4日 午後2時59分

特奨－No.3

　 4 　

― 18 ―

4 ͷղ౴܈

(Ξ) gH (Π) 2gH (΢) 3gH (Τ) 4gH

(Φ)

gH (Χ)

2gH (Ω)

3gH (Ϋ)

4gH

5 ͷղ౴܈

(Ξ) gR (Π) 2gR (΢) 3gR (Τ) 4gR

(Φ)

gR (Χ)

2gR (Ω)

3gR (Ϋ)

4gR

6 ͷղ౴܈

(Ξ) V_top²

R ^(Π)

Vtop

R ^(΢)

V_top² R²

7 ͷղ౴܈

(Ξ) mVtop

R =T−mg (Π) mV_top²

R² =T+mg (΢) mV_top²

R² =T−mg (Τ) mV_top²

R =T+mg (Φ) mV_top²

R =T−mg (Χ) mVtop

R =T+mg

8 ͷղ౴܈

(Ξ) v₀²>5gR (Π) v²₀>2gR (΢) v²₀>4gR

9 ɼ 10 ͷղ౴܈

(Ξ) Rsinθc (Π) Rcosθc (΢) Rtanθc

(Τ) R(1 + sinθc) (Φ) R(1 + cosθc) (Χ) R(1 + tanθc)

(Ω) R(1−sinθc) (Ϋ) R(1−cosθc) (έ) R(1−tanθc)

11 ͷղ౴܈

(Ξ) mV_c²

R = 0 (Π) mV_c²

R =mgsinθc

(΢) mV_c²

R =mgcosθc (Τ) mV_c²

R =mgtanθc

(Φ) mV_c²

R =−mgsinθc (Χ) mV_c²

R =−mgcosθc

(Ω) mV_c²

R =−mgtanθc (Ϋ) mV_c² R =mg

12 ͷղ౴܈

(Ξ) 1

2mV₀²=1

2mV_c²+mgR (Π) 1

2mV₀²=1

2mV_c²+mgR(1 + sinθc)

(΢) 1

2mV₀²=1

2mV_c²+mgR(1 + cosθc) (Τ) 1

2mV₀²=1

2mV_c²+mgR(1 + tanθc)

(Φ) 1

2mV₀²=1

2mV_c²−mgR (Χ) 1

2mV₀²=1

2mV_c²+mgR(1−sinθc) (Ω) 1

2mV₀²=1

2mV_c²+mgR(1−cosθc) (Ϋ) 1

2mV₀²=1

2mV_c²+mgR(1−tanθc)

13 ͷղ౴܈

(Ξ) sinθc=2

3 ^(Π) cosθc= 2

3 ^(΢) tanθc=2 3

(Τ) sinθc=1

2 ^(Φ) cosθc= 1

2 ^(Χ) tanθc=1 2

14 ɼ 15 ͷղ౴܈

(Ξ)

√2

3 R (Π) 2

3R

(΢)

√5

3 R (Τ) 5

3R

(Φ)

√3

2 R (Χ) 2

√3R

分 修正日：2020年6月4日 午後2時59分

特奨－No.3

　 6 　

― 20 ―

II

બ΂ɻ

ɹిؾճ࿏ͷجຊతͳૉࢠʹ͍ͭͯߟ͑Α͏ɻ఍߅͸εΠονͷ։ดͳͲʹΑΓిѹΛՃ͑Δ ͱిྲྀ͕ଈ࠲ʹมԽ͢ΔɻҰํɼίΠϧ͸ిѹΛՃ͑ͯ΋ిྲྀͷมԽ͸ଈ࠲ʹ͸ى͜Βͳ͍ɻ

͜ͷ͜ͱʹ͍ͭͯߟ͑ͯΈΔɻ

ɹίΠϧʹిྲྀΛྲྀ͢ͱɼίΠϧΛ؏͘ 16 ͕ੜ͡ΔɻίΠϧΛྲྀΕΔిྲྀΛ૿Ճͤ͞Δͱɼ 16 ΋૿Ճ͢Δɻͦͷͱ͖ɼίΠϧͷ྆୺ʹ͸ɼిྲྀͷ૿ՃΛ๦͛Δ޲͖ʹ 17 ͕ੜ͡Δɻ͜

ͷΑ͏ͳݱ৅ΛίΠϧͷ 18 ͱ͍͏ɻ࣌ؒΔt[s]ͷؒʹίΠϧΛྲྀΕΔిྲྀ͕ΔI[A]͚ͩม Խ͢Δͱ͖ɼίΠϧͷࣗݾΠϯμΫλϯεL[ 19 ]Λ༻͍ΔͱɼίΠϧͷ྆୺ʹੜ͡Δిѹ VL͸ిྲྀͷ૿ՃΛଅਐͤ͞Δ޲͖Λਖ਼ͱͯ͠ɼVL= 20 ͱͳΔɻ͜ͷ͜ͱ͔ΒɼίΠϧʹ͸

ྲྀΕΔిྲྀΛҰఆʹอͱ͏ͱ͢Δੑ࣭͕͋ΓɼεΠονͷ։ดͳͲͷ௚ޙͷిྲྀ͸௚લͷిྲྀ

ͱ౳͍͠ɻ

ɹਤͷΑ͏ͳճ࿏Λߟ͑ΔɻS͸εΠονɼE͸಺෦఍߅ͷແࢹͰ͖ΔىిྗE[V]ͷి஑ɼ

͜ͷి஑͔ΒྲྀΕग़ΔిྲྀΛI[A]ɼR͸఍߅஋͕R[Ω]ͷిٿͰ͋Γɼ࣮ݧதʹ͸఍߅஋͸

มԽ͠ͳ͍΋ͷͱ͢ΔɻL͸ࣗݾΠϯμΫλϯεL[ 19 ]ͷίΠϧͰɼr͸ͦͷίΠϧͷרઢ ͷ఍߅஋r[Ω]Ͱ͋ΓɼRʹൺ΂ͯे෼ʹখ͍͞΋ͷͱ͢Δɻ͜͜ͰɼίΠϧΛྲྀΕΔిྲྀΛ IL[A]ɼిٿRΛྲྀΕΔిྲྀΛIR[A]ɼిٿRʹ͔͔ΔిѹΛV[V](અ఺bʹର͢Δઅ఺aͷ

ిҐ)ͱ͢Δɻ֤ిྲྀͷਖ਼ෛ͸ɼਤதʹࣔ͢໼ҹͷ޲͖Λਖ਼ͱ͢Δɻ

ɹεΠονΛ࣌ࠁt=t1[s]Ͱดͨ͡௚ޙ͸I=IR= 21 [A]Ͱ͋Γɼे෼ʹ͕࣌ؒܦͬͨ࣌

ࠁt=t2[s]ͷͱ͖ʹ͸I=IL+IR= 22 [A]ͱͳΔɻ࣌ࠁt=t2[s]ͷͱ͖ɼίΠϧʹ஝͑

ΒΕΔΤωϧΪʔU͸ 23 [J]ͱͳΔɻ

ɹ࣌ࠁt=t3(> t2) [s]ͷͱ͖ʹεΠονΛ։͘ͱI= 0 AͱͳΓɼిྲྀIL͸ిٿRʹྲྀΕࠐ ΉͷͰɼV = 24 [V]ͱͳΔɻిٿR͸εΠονΛ։͘ͱ 25 ɻ͜ͷҰ࿈ͷ࣮ݧʹ͓͚Δి

ѹV ͷมԽ͸ 26 ͷΑ͏ʹͳΔɻ

L r

R

a b

IL

IR

I

16 ɼ 17 ɼ 18 ͷղ౴܈

(Ξ) ిք(ి৔) (Π) ిՙ (΢) ࣗݾ༠ಋ

(Τ) ಁ࣓཰ (Φ) ࣓ଋ (Χ) ༠ి෼ۃ

(Ω) ͏ͣిྲྀ (Ϋ) ిؾྗઢ (έ) ༠ಋىిྗ

19 ͷղ౴܈

(Ξ) F(ϑΝϥυ) (Π) H(ϔϯϦʔ) (΢) Wb(΢Σʔό)

(Τ) C(Ϋʔϩϯ) (Φ) Hz(ϔϧπ) (Χ) T(ςεϥ)

20 ͷղ౴܈

(Ξ) LΔI

Δt ^(Π) LΔt

ΔI ^(΢) LΔIΔt (Τ) L 1 ΔIΔt

(Φ) −LΔI

Δt ^(Χ) −LΔt

ΔI ^(Ω) −LΔIΔt (Ϋ) −L 1 ΔIΔt

21 ɼ 22 ͷղ౴܈

(Ξ) E

R+r ^(Π) E

R−r ^(΢) E R +E

r ^(Τ) E R−E

r ^(Φ) E

r −E R (Χ) E

R ^(Ω)

E

r ^(Ϋ) ER (έ) Er (ί) E(R+r)

23 ͷղ౴܈

(Ξ) LE

2R ^(Π) LE²

2R ^(΢) LE²

2R^{2 (Τ)} LE

2(R+r) ^(Φ)

LE² 2(R+r)²

(Χ) LE

2r ^(Ω) LE²

2r ^(Ϋ) LE²

2r^{2 (έ)}

LE²

2(R+r) ^(ί)

LE 2(R+r)²

24 ͷղ౴܈

(Ξ) R

rE (Π) r

RE (΢) R+r

r E (Τ) R+r

R E (Φ) E (Χ) −R

rE (Ω) −r

RE (Ϋ) −R+r

r E (έ) −R+r

R E (ί) −E

分 修正日：2020年6月4日 午後2時59分

特奨－No.3

　 8 　

― 22 ―

25 ͷղ౴܈

(Ξ) ด͍ͯ͡Δͱ͖ͱൺ΂ͯ໌Δ͘ޫΔ (Π) ด͍ͯ͡Δͱ͖ͱಉ͡໌Δ͞ͰޫΔ (΢) ด͍ͯ͡Δͱ͖ͱൺ΂ͯ҉͘ޫΔ

(Τ) ফ͑Δ

(Φ) Ұॠ໌Δً͖͘ɼͦͷޙด͍ͯ͡Δͱ͖ͱಉ͡໌Δ͞ʹͳΔ (Χ) Ұॠ໌Δً͖͘ɼͦͷޙด͍ͯ͡Δͱ͖ͱΑΓ҉͘ޫΓଓ͚Δ (Ω) Ұॠ໌Δً͖͘ɼͦͷޙফ͑Δ

26 ͷղ౴܈

(Ξ)

t[s]

V[V]

t1 t3

0 E

−E

(Π)

t[s]

V[V]

t1 t3

0 E

−E .......

...

.... ... .... ........

...................

(΢)

t[s]

V[V]

t1 t3

0 E

−E

......... ....... ............. . ...

.....

(Τ)

t[s]

V[V]

t1 t3

0 E

−E .......

...

.... ... .... ........

. ...

...

...

...

...

...

...

..

...

(Φ)

t[s]

V[V]

t1 t3

0 E

−E

...................

(Χ)

t[s]

V[V]

t1 t3

0 E

−E ......... ....... .............

....

...............

(Ω)

t[s]

V[V]

t1 t3

0 E

−E

....

.....

(Ϋ)

t[s]

V[V]

t1 t3

0 E

−E

. ...

...

...

...

...

...

...

.....

(έ)

t[s]

V[V]

t1 t3

0 E

−E ......... ....... .............

. ...

...

...

...

...

...

...

.....

t3

...

...

...

.........

t3

..................

t3

..................

III

બ΂ɻ

ɹ෺࣭Λߏ੒͢Δݪࢠ΍෼ࢠ͸ɼ௨ৗɼܹ͘͠ཚࡶʹӡಈ͍ͯ͠Δɻ͜ͷݪࢠ΍෼ࢠͷӡಈΛ 27 ͱ͍͏ɻ 28 ͸ɼ͜ͷ 27 ͷܹ͠͞Λ͋ΒΘ͢෺ཧྔͰ͋Δɻ೔ৗͷݴ༿Ͱ͸ɼ 28 Λ Լ͛Δ͜ͱΛʮྫྷ΍͢ʯɼ 28 Λ্͛Δ͜ͱΛʮ೤͢Δ(ԹΊΔ)ʯͳͲͱදݱ͢Δɻ ɹ෺࣭Λ೤͢Δͱ෺࣭಺෦ͷ 27 ͷΤωϧΪʔ͕େ͖͘ͳΔɻ௨ৗͷ෺࣭Λɼݻମͷঢ়ଶ͔

Β೤͍ͯ͘͠ͱɼӷମͷঢ়ଶʹͳΓɼ͞Βʹؾମͷঢ়ଶ΁ͱมԽ͢Δɻ͜Ε͸֎෦͔Β෺ମʹ 27 ͷΤωϧΪʔ͕ྲྀೖ͠ɼݪࢠɾ෼ࢠͷ 27 ͕ܹ͘͠ͳΔ͔ΒͰ͋Δɻٯʹɼ෺࣭Λྫྷ΍

͢ͱ෺࣭಺෦ͷ 27 ͷΤωϧΪʔ͕খ͘͞ͳΔɻ௨ৗͷ෺࣭Λɼؾମͷঢ়ଶ͔Βྫྷ΍͍ͯ͠

͘ͱɼӷମͷঢ়ଶʹͳΓɼ͞Βʹݻମͷঢ়ଶ΁ͱมԽ͢Δɻ͜Ε͸෺࣭಺෦ͷ 27 ͷΤωϧ Ϊʔ͕֎෦΁ྲྀग़͢Δ͔ΒͰ͋Δɻ͜ͷΑ͏ʹɼ෺࣭ͱ֎෦ͱͷؒΛग़ೖΓͯ͠ 28 ΍ঢ়ଶ ͷมԽΛҾ͖ى͜͢ 27 ͷΤωϧΪʔΛ೤ͱ͍͍ɼͦͷྔΛ 29 ͱ͍͏ɻͦͷ୯Ґ͸ɼ࢓ࣄ

΍ӡಈΤωϧΪʔͳͲͱಉ͡δϡʔϧ(J) Ͱ͋Δɻ·ͨɼݻମɾӷମɾؾମͷ3ͭͷঢ়ଶΛ 30 ͱ͍͏ɻ

ɹ 28 ͷҟͳΔ෺ମΛ઀৮͓ͤͯ͘͞ͱɼ 28 ͷߴ͍෺ମ͔Βग़ͨ೤͕ɼ 28 ͷ௿͍෺ମʹ

ೖΓɼ 28 ͷߴ͍෺ମͷ 28 ͕Լ͕Γɼ 28 ͷ௿͍෺ମͷ 28 ͸্͕Δɻͦͯ͠ɼ࠷ऴతʹ

͸྆෺ମͷ 28 ͕౳͘͠ͳΓɼ྆෺ମؒͰͷΤωϧΪʔͷҠಈ͕ͳ͘ͳΔɻ͜ͷΑ͏ͳঢ়ଶ Λ 31 ͱ͍͏ɻ͜ͷΑ͏ʹɼ 28 ͷߴ͍෺ମͱ 28 ͷ௿͍෺ମΛ઀৮ͤͨ͞Γࠞͥ߹ͤͨΓ

ͨ͠৔߹ɼ 28 ͷߴ͍෺ମ͔Βग़ͨ 29 ͸ɼ 28 ͷ௿͍෺ମʹೖͬͨ 29 ʹ౳͍͠ɻ͜ͷ

ؔ܎Λ 32 ͱ͍͏ɻ

ɹͯ͜͜͞Ͱɼ 32 Λద༻ͯ͠ɼҎԼͷ໰୊Λղ͍ͯΈΑ͏ɻ

ɹʮ೤༰ྔͷແࢹͰ͖Δ༰ثʹ140 gͷਫΛೖΕɼ͠͹Β͔ͯ͘͠ΒਫԹΛଌఆͨ͠ͱ͜Ζɼ 15.0^◦CͰ͋ͬͨɻൺ೤ͷΘ͔Βͳ͍࣭ྔ100 gͷۚଐΛ80.0^◦C·ͰԹΊͯɼ͜ͷਫͷதʹೖ

Εͯɼ͠͹Β͘͢ΔͱɼਫԹ͕19.0^◦Cʹͳͬͨͱ͢Δɻ͜ͷͱ͖ͷۚଐͷൺ೤͸ԿJ/(g·K)

͔ΛٻΊΑɻͨͩ͠ɼਫͷൺ೤Λ4.2 J/(g·K)ͱ͠ɼۚଐͱਫͱͷؒͷΈͰɼ 29 ͷ΍Γͱ Γ͕ߦΘΕΔ΋ͷͱ͢Δɻʯ

ɹ͜ͷ໰୊Ͱ͸ɼۚଐ͕ 28 ͷߴ͍෺ମʹରԠ͢Δɻۚଐͷൺ೤Λc[J/(g·K)]ͱ͢Δͱɼ 28 ͷߴ͍෺ମ͔Βग़ͨ 29 ͸ɼ 33 g·KʹcΛ͔͚Δ͜ͱͰٻΊΒΕΔɻҰํɼ͜ͷ໰୊Ͱ͸ɼ ਫ͕ 28 ͷ௿͍෺ମʹରԠ͠ɼ 28 ͷ௿͍෺ମʹೖͬͨ 29 ͸ɼ 34 JͰ͋Δɻ͕ͨͬ͠

ͯɼ 32 ΑΓɼc= 35 J/(g·K)ͱٻΊΒΕΔɻͳ͓ɼܭࢉ஋ͷ༗ޮ਺ࣈ͸2ܻͱ͢Δɻ

分 修正日：2020年6月4日 午後2時59分

特奨－No.3

　 10 　― 24 ―

27 ɼ 28 ɼ 29 ɼ 30 ɼ 31 ɼ 32 ͷղ౴܈

(Ξ) ॏྗ (Π) Թ౓ (΢) ೤ྔ

(Τ) ೤ฏߧ (Φ) ి৔ (Χ) ෺࣭ͷࡾଶ (Ω) ೤ӡಈ (Ϋ) ைࣚ (έ) ೤ྔͷอଘ

33 ɼ 34 ͷղ౴܈

(Ξ) 2.4×10² (Π) 2.4×10³ (΢) 2.4×10⁴

(Τ) 6.1×10² (Φ) 6.1×10³ (Χ) 6.1×10⁴

(Ω) 6.5×10² (Ϋ) 6.5×10³ (έ) 6.5×10⁴

35 ͷղ౴܈

(Ξ) 3.9×10⁻¹ (Π) 3.9×10⁰ (΢) 3.9×10¹

(Τ) 2.6×10⁻¹ (Φ) 2.6×10⁰ (Χ) 2.6×10¹

2020ಛ঑ݱ୅Ԡੜusing pr2015ˏTakamaruฏ੒31೥11݄20೔16࣌ 48෼

෺ɹɹɹɹɹཧ

(ղ౴൪߸ 1 ʙ 29 )

I

બ΂ɻ

ɹຎࡲͷ͋Δਫฏͳচͷ্ʹ࣭ྔmͷശA͕੩ࢭ͓ͯ͠ΓɼுྗT1ͰਫฏʹҾͬுͬͨͱ

͜Ζɼਤ1ͷΑ͏ʹҰఆͷՃ଎౓a1Ͱಈ͖ͩͨ͠ɻಈ͖ͩͨ࣌͠ࠁΛt= 0ͱ͠ɼശAͷਐ ߦํ޲ʢਤ1Ͱࠨ͔Βӈͷํ޲ʣΛਖ਼ͱ͢ΔɻҎԼͰ͸ɼॏྗՃ଎౓ͷେ͖͞Λgͱ͢Δɻ ɹ͜ͷͱ͖ɼശAͱচͷؒͷಈຎࡲ܎਺Λμ_Aͱ͢ΔͱശAʹ͔͔ΔຎࡲྗFAͷେ͖͞͸ɼ

1 Ͱ͋Δɻ͜ͷͱ͖ശAͷӡಈํఔࣜ͸ 2 ͱͳΔɻശA͸࣌ࠁt=t1·Ͱͷؒʹ͸ 3

͚ͩҠಈ͠ɼͦͷؒʹຎࡲྗ͕ͨ͠࢓ࣄͷେ͖͞͸ 4 Ͱ͋Δɻ

ɹ࣍ʹɼ੩ࢭ͍ͯ͠Δ࣭ྔmͷശAʹɼ࣭ྔ͕ಉ͕ͩ͡ɼচͱͷຎࡲ܎਺͕ҟͳΔശBΛ௕

͞Lͷ͍ܰࢳͰͨΔ·ͳ͍Α͏ʹͭͳ͗ɼுྗT2ͰਫฏʹҾͬுͬͨͱ͜Ζɼਤ2ͷΑ͏ʹ ࢳ͕ͨΔΉ͜ͱͳ͘ɼ͢΂ͯͷശ͕ҰఆͷՃ଎౓a2Ͱಈ͖ͩͨ͠ɻಈ͖ͩͨ࣌͠ࠁΛt= 0 ͱ͠ɼശAͱശBͷਐߦํ޲ʢਤ2Ͱࠨ͔Βӈͷํ޲ʣΛਖ਼ͱ͢Δɻ

ɹശAͱচͷؒͷಈຎࡲ܎਺Λμ_AɼശBͱচͱͷؒͷಈຎࡲ܎਺Λμ_BɼശAͱശBΛ݁

Ϳࢳʹ͔͔Δுྗͷେ͖͞ΛTABͱ͢ΔͱɼശAͷӡಈํఔࣜ͸ 5 ɼശBͷӡಈํఔࣜ

͸ 6 ͱͳΔɻ͜ͷͱ͖ɼࢳͷுྗͷେ͖͞TAB͸ 7 ͱͳΓɼ࣌ࠁt=t2ͷͱ͖ͷശA

͓ΑͼശBͷ଎͞v2͸ 8 ͱͳΔɻ

ɹ࣌ࠁt=t2ͷͱ͖ʹҾͬுΔͷΛ΍Ίͨͱ͜ΖɼശAͱശBͷؒͷࢳ͕ͨΔΈͳ͕Βຎࡲ

ྗʹΑͬͯݮ଎ͯ͠ശA΋ശB΋੩ࢭͨ͠ɻ͜ͷ͜ͱ͔ΒശAͱശBͷಈຎࡲ܎਺ʹ͸ 9 ͷؔ܎͕͋ΓɼҾͬுΔͷΛ΍Ί͔ͯΒ੩ࢭ͢Δ·ͰͷؒʹശAͱശB͕઀৮͠ͳ͍ͨΊʹ

͸ɼ࣌ࠁt=t2ͷͱ͖ͷശA͓ΑͼശBͷ଎͞v2Λ༻͍ͯ͋ΒΘ͢ͱL > 10 Ͱ͋Δඞ ཁ͕͋Δɻ

修正日：2019年11月21日 午後2時22分 特奨－No.3

（解答用紙の選択欄に「物理①」を必ず記入・マークすること。）

物　　　理　①

　 77 　― 26 ―

物　　　理　①（生命健康科・現代教育学部）

࣭ྔm ശA

FA

T1

ਤ1

ࢳͷ௕͞L ࣭ྔm

TAB TAB

ശA

FA

T2

࣭ྔm ശB

FB

ਤ2

1 ͷղ౴܈

(Ξ) μ_Amg (Π) mg (΢) μ_Amg

2 ^(Τ)

μ_Am g

2 ͷղ౴܈

(Ξ) ma1=T1−μ_Amg (Π) ma1=T1+μ_Amg

(΢) ma1=T1−mg (Τ) ma1=T1+mg

(Φ) ma1=T1−μ_Amg

2 ^(Χ) ma1=T1+μ_Amg 2

(Ω) ma1=T1−μ_Am

g ^(Ϋ) ma1=T1+μ_Am g

3 ͷղ౴܈

(Ξ) (T1+μ_Amg)t12

2m ^(Π)

(T1−μ_Amg)t12

2m ^(΢)

(T1+mg)t12

2m ^(Τ)

(T1−mg)t12

2m

(Φ) (T1+μ_Amg)t12

4m ^(Χ)

(T1−μ_Amg)t12

4m ^(Ω)

(T1+μ_Amg)t1

4m ^(Ϋ)

(T1−μ_Amg)t1

4m

4 ͷղ౴܈

(Ξ) μ_A(T1+mg)gt12 (Π) μ_A(T1−μ_Amg)gt1

4 ^(΢)

μ_A(T1+μ_Amg)gt1

4

(Τ) μ_A(T1−μ_Amg)gt12

2 ^(Φ)

μ_A(T1+μ_Amg)gt12

2 ^(Χ) μ_A(T1−mg)gt12

5 ɼ 6 ͷղ౴܈

(Ξ) ma2=T2−μ_Am+TAB (Π) ma2=TAB−μ_Bm (΢) ma2=T2−μ_Amg+ 2TAB (Τ) ma2= 2TAB−μ_Bmg (Φ) ma2=T2−μ_Amg−TAB (Χ) ma2=TAB−μ_Bmg (Ω) ma2=T2+μ_Amg−TAB (Ϋ) ma2=TAB+μ_Bmg

7 ͷղ౴܈

(Ξ) T2−(μ_A+μ_B)mg

2 ^(Π) T2−(μ_A−μ_B)g (΢) T2−(μ_A+μ_B)mg (Τ) T2−(μ_A−μ_B)mg

2 ^(Φ)

T2−(μ_A+μ_B)mg

4 ^(Χ) T2−(μ_A−μ_B)mg

8 ͷղ౴܈

(Ξ) {T2−(μ_A+μ_B)mg}t2

2m ^(Π)

{T2+ (μ_A−μ_B)mg}t2

2m ^(΢)

{T2−2mg}t2

4m

(Τ) {T2−2mg}t22

4m ^(Φ)

{T2−(μ_A+μ_B)mg}t22

2m ^(Χ)

{T2+ (μ_A+μ_B)mg}t22

2m

修正日：2019年11月21日 午後2時22分 特奨－No.3

　 79 　― 28 ―

9 ͷղ౴܈

(Ξ) μ_A< μ_B (Π) μ_A=μ_B (΢) μ_A> μ_B

10 ͷղ౴܈

(Ξ) (μ_A+μ_B)v2

2μ_Aμ_Bg ^(Π) 2μ_Aμ_B(μ_A−μ_B)gv2 (΢) 2μ_Aμ_B(μ_A+μ_B)gv22

(Τ) (μ_A−μ_B)v22

2μ_Aμ_Bg ^(Φ)

(μ_A+μ_B)v22

2μ_Aμ_Bg ^(Χ)

(μ_A−μ_B)v2

2μ_Aμ_Bg

II

બ΂ɻ

ɹ఍߅ͷ྆୺ͷిѹV ͱྲྀΕΔిྲྀI͓Αͼ఍߅஋Rͱͷؒʹ͸ɼ 11 ͷ๏ଇͱݺ͹ΕΔ V= 12 ͷؔ܎͕͋Δɻ

ɹਤͷճ࿏໢ʹ͍ͭͯɼͦΕͧΕͷ఍߅ͷ྆୺ʹՃΘΔిѹͱྲྀΕΔిྲྀΛٻΊΔɻ୺ࢠbc

ؒͷ߹੒఍߅Rbc͸఍߅R2ɼR3ͱR4ͷฒྻ઀ଓͳͷͰɼRbc= 13 ͷؔ܎͕͋Δɻ࣍ʹɼ

୺ࢠadؒͷ߹੒఍߅Rad͸ɼ఍߅R1ɼRbcͱR5ͷ௚ྻ઀ଓͱΈͳ͢͜ͱ͕Ͱ͖ΔͷͰɼ Rad= 14 ͷؔ܎͕͋Δɻ͜ͷճ࿏ͷిѹVadͷిݯ͔ΒྲྀΕग़ΔిྲྀIa͸ɼIa= 15 Ͱ͋

Δɻ

ɹ͍·ɼ఍߅R1ɼR2ɼR3ɼR4ɼR5 ʹྲྀΕΔిྲྀΛͦΕͧΕI1ɼI2ɼI3ɼI4ɼI5ͱ͢Δɻ ɹ͜͜ͰɼVad= 18 VɼR1= 600 ΩɼR2= 900 ΩɼR3= 1.2 kΩɼR4= 1.8 kΩɼR5= 200 Ω Ͱ͋Δͱ͢ΔɻI1= 16 mAͰ͋Γɼ఍߅R1ʹՃΘΔిѹV1͸ɼV1= 17 VͱͳΔɻͦ͠

ͯɼI3= 18 mAͰ͋Γɼ఍߅R3ʹՃΘΔిѹV3͸ɼV3= 19 VͱͳΔɻ

ɹ·ͨɼిྲྀI1ɼI2ɼI3ɼI4ɼI5ͷؒʹ͸ 20 ͷؔ܎͕͋Δɻ͜ͷؔ܎͸ճ࿏໢ͷb఺ʹ͓

͍ͯྲྀΕࠐΉిྲྀͷ૯࿨͸ྲྀΕग़͢ిྲྀͷ૯࿨ʹ౳͍͜͠ͱΛҙຯ͍ͯ͠Δɻ͜ͷ͜ͱΛΩ ϧώϗοϑͷୈ̍๏ଇͱ͍͏ɻ

a R1 b c d

R2

R3

R4

R5

Vad

ਤ

11 ͷղ౴܈

(Ξ) ΦʔϜ (Π) ϑοΫ (΢) ΤωϧΪʔอଘ (Τ) ൓ࣹ (Φ) Ϩϯπ (Χ) ϑϨϛϯάͷࠨख

修正日：2019年11月21日 午後2時22分 特奨－No.3

　 81 　― 30 ―

12 ͷղ౴܈

(Ξ) R

I ^(Π)

R²

I ^(΢)

I

R ^(Τ)

I² R

(Φ) R²I (Χ) RI (Ω) RI² (Ϋ) 1

RI

13 ͷղ౴܈

(Ξ) R2+R3+R4 (Π) R2R3+R3R4+R4R2

R2R3R4

(΢) 1 R2

+ 1 R3

+ 1 R4

(Τ) R2R3R4

R2+R3+R4

(Φ) R2+R3+R4

R2R3R4

(Χ) R2R3R4

R2R3+R3R4+R4R2

14 ͷղ౴܈

(Ξ) R1+Rbc+R5 (Π) R1Rbc+RbcR5+R5R1

R1RbcR5

(΢) 1 R1

+ 1 Rbc

+ 1 R5

(Τ) R1RbcR5

R1+Rbc+R5

(Φ) R1+Rbc+R5

R1RbcR5

(Χ) R1RbcR5

R1Rbc+RbcR5+R5R1

15 ͷղ౴܈

(Ξ) VadRad (Π) Rad

Vad

(΢) Vad

Rad

(Τ) 1

VadRad

(Φ) V_ad²Rad (Χ) VadR_ad² (Ω) V_ad² Rad

(Ϋ) R²_ad Vad

16 ɼ 18 ͷղ౴܈

(Ξ) 3.3 (Π) 5.0 (΢) 6.7 (Τ) 10 (Φ) 13

17 ɼ 19 ͷղ౴܈

(Ξ) 1.5 (Π) 3.0 (΢) 4.5 (Τ) 5.0 (Φ) 6.0

(Χ) 7.2 (Ω) 9.0 (Ϋ) 10.8 (έ) 12 (ί) 18

20 ͷղ౴܈

(Ξ) I1=I2I3I4=I5 (Π) I1I5=I2I3I4 (΢) I1=I2I3I4+I5

(Τ) I1=I2+I3+I4=I5 (Φ) I1+I5=I2+I3+I4 (Χ) I1=I2+I3+I4+I5

(Ω) I1=I2=I3=I4=I5 (Ϋ) I1I5=I2+I3+I4 (έ) I1+I5=I2I3I4

修正日：2019年11月21日 午後2時22分 特奨－No.3

　 83 　― 32 ―

III

બ΂ɻ

ɹ෺࣭Λߏ੒͢Δݪࢠ΍෼ࢠ͸ɼ௨ৗɼܹ͘͠ཚࡶʹӡಈ͍ͯ͠Δɻ͜ͷݪࢠ΍෼ࢠͷӡಈΛ 21 ͱ͍͏ɻ 22 ͸ɼ͜ͷ 21 ͷܹ͠͞Λ͋ΒΘ͢෺ཧྔͰ͋Δɻ೔ৗͷݴ༿Ͱ͸ɼ 22 Λ Լ͛Δ͜ͱΛʮྫྷ΍͢ʯɼ 22 Λ্͛Δ͜ͱΛʮ೤͢Δ(ԹΊΔ)ʯͳͲͱදݱ͢Δɻ ɹ෺࣭Λ೤͢Δͱ෺࣭಺෦ͷ 21 ͷΤωϧΪʔ͕େ͖͘ͳΔɻ௨ৗͷ෺࣭Λɼݻମͷঢ়ଶ͔

Β೤͍ͯ͘͠ͱɼӷମͷঢ়ଶʹͳΓɼ͞Βʹؾମͷঢ়ଶ΁ͱมԽ͢Δɻ͜Ε͸֎෦͔Β෺ମʹ 21 ͷΤωϧΪʔ͕ྲྀೖ͠ɼݪࢠɾ෼ࢠͷ 21 ͕ܹ͘͠ͳΔ͔ΒͰ͋Δɻٯʹɼ෺࣭Λྫྷ΍

͢ͱ෺࣭಺෦ͷ 21 ͷΤωϧΪʔ͕খ͘͞ͳΔɻ௨ৗͷ෺࣭Λɼؾମͷঢ়ଶ͔Βྫྷ΍͍ͯ͠

͘ͱɼӷମͷঢ়ଶʹͳΓɼ͞Βʹݻମͷঢ়ଶ΁ͱมԽ͢Δɻ͜Ε͸෺࣭಺෦ͷ 21 ͷΤωϧ Ϊʔ͕֎෦΁ྲྀग़͢Δ͔ΒͰ͋Δɻ͜ͷΑ͏ʹɼ෺࣭ͱ֎෦ͱͷؒΛग़ೖΓͯ͠ 22 ΍ঢ়ଶ ͷมԽΛҾ͖ى͜͢ 21 ͷΤωϧΪʔΛ೤ͱ͍͍ɼͦͷྔΛ 23 ͱ͍͏ɻͦͷ୯Ґ͸ɼ࢓ࣄ

΍ӡಈΤωϧΪʔͳͲͱಉ͡δϡʔϧ(J) Ͱ͋Δɻ·ͨɼݻମɾӷମɾؾମͷ3ͭͷঢ়ଶΛ 24 ͱ͍͏ɻ

ɹ 22 ͷҟͳΔ෺ମΛ઀৮͓ͤͯ͘͞ͱɼ 22 ͷߴ͍෺ମ͔Βग़ͨ೤͕ɼ 22 ͷ௿͍෺ମʹ

ೖΓɼ 22 ͷߴ͍෺ମͷ 22 ͕Լ͕Γɼ 22 ͷ௿͍෺ମͷ 22 ͸্͕Δɻͦͯ͠ɼ࠷ऴతʹ

͸྆෺ମͷ 22 ͕౳͘͠ͳΓɼ྆෺ମؒͰͷΤωϧΪʔͷҠಈ͕ͳ͘ͳΔɻ͜ͷΑ͏ͳঢ়ଶ Λ 25 ͱ͍͏ɻ͜ͷΑ͏ʹɼ 22 ͷߴ͍෺ମͱ 22 ͷ௿͍෺ମΛ઀৮ͤͨ͞Γࠞͥ߹ͤͨΓ

ͨ͠৔߹ɼ 22 ͷߴ͍෺ମ͔Βग़ͨ 23 ͸ɼ 22 ͷ௿͍෺ମʹೖͬͨ 23 ʹ౳͍͠ɻ͜ͷ

ؔ܎Λ 26 ͱ͍͏ɻ

ɹͯ͜͜͞Ͱɼ 26 Λద༻ͯ͠ɼҎԼͷ໰୊Λղ͍ͯΈΑ͏ɻ

ɹʮ೤༰ྔͷແࢹͰ͖Δ༰ثʹ140 gͷਫΛೖΕɼ͠͹Β͔ͯ͘͠ΒਫԹΛଌఆͨ͠ͱ͜Ζɼ 15.0^◦CͰ͋ͬͨɻൺ೤ͷΘ͔Βͳ͍࣭ྔ100 gͷۚଐΛ80.0^◦C·ͰԹΊͯɼ͜ͷਫͷதʹೖ

Εͯɼ͠͹Β͘͢ΔͱɼਫԹ͕19.0^◦Cʹͳͬͨͱ͢Δɻ͜ͷͱ͖ͷۚଐͷൺ೤͸ԿJ/(g·K)

͔ΛٻΊΑɻͨͩ͠ɼਫͷൺ೤Λ4.2 J/(g·K)ͱ͠ɼۚଐͱਫͱͷؒͷΈͰɼ 23 ͷ΍Γͱ Γ͕ߦΘΕΔ΋ͷͱ͢Δɻʯ

ɹ͜ͷ໰୊Ͱ͸ɼۚଐ͕ 22 ͷߴ͍෺ମʹରԠ͢Δɻۚଐͷൺ೤Λc[J/(g·K)]ͱ͢Δͱɼ 22 ͷߴ͍෺ମ͔Βग़ͨ 23 ͸ɼ 27 g·KʹcΛ͔͚Δ͜ͱͰٻΊΒΕΔɻҰํɼ͜ͷ໰୊Ͱ͸ɼ ਫ͕ 22 ͷ௿͍෺ମʹରԠ͠ɼ 22 ͷ௿͍෺ମʹೖͬͨ 23 ͸ɼ 28 JͰ͋Δɻ͕ͨͬ͠

21 ɼ 22 ɼ 23 ɼ 24 ɼ 25 ɼ 26 ͷղ౴܈

(Ξ) ॏྗ (Π) Թ౓ (΢) ೤ྔ

(Τ) ೤ฏߧ (Φ) ి৔ (Χ) ෺࣭ͷࡾଶ (Ω) ೤ӡಈ (Ϋ) ைࣚ (έ) ೤ྔͷอଘ

27 ɼ 28 ͷղ౴܈

(Ξ) 2.4×10² (Π) 2.4×10³ (΢) 2.4×10⁴

(Τ) 6.1×10² (Φ) 6.1×10³ (Χ) 6.1×10⁴

(Ω) 6.5×10² (Ϋ) 6.5×10³ (έ) 6.5×10⁴

29 ͷղ౴܈

(Ξ) 3.9×10⁻¹ (Π) 3.9×10⁰ (΢) 3.9×10¹

(Τ) 2.6×10⁻¹ (Φ) 2.6×10⁰ (Χ) 2.6×10¹

修正日：2019年11月21日 午後2時22分 特奨－No.3

　 85 　― 34 ―

（注）この問題は，「化学②」の問題である。

解答用紙は，理科のマークシート１枚。

（解答用紙の選択欄に 「化学②」 を必ず記入・マークすること。）

化　　　学　②

（ 解答番号 1 ～ 32 ）

Ⅰ

族 １ ２ 13 14 15 16 17 18

元素 ⒜ Be B ⒝ N ⒞ ⒟ ⒠

問１　次の文章の空欄Ａには元素⒜～⒠のうちのいずれかが，空欄ＢおよびＣにはそれぞれ数字 が入る。空欄Ａ～Ｃに入れるのに最も適当な組み合わせを，下の解答群のア～カのうちから一 つ選べ。 １

　ケイ素の同族元素である Ａ は原子核に Ｂ 個の陽子をもつ。いずれの元素も最外殻に Ｃ 個の電子をもつ。

１ の解答群

Ａ Ｂ Ｃ

ア b ４ ４

イ b ６ ４

ウ b ６ ６

エ c ６ ６

オ c ６ ４

カ c ８ ６

化　　　学　②（工・応用生物・生命健康科・現代教育学部）