平成 27 年度

入学者選抜学力試験（前期）数学

注 意 事 項

1．試験開始の合図があるまで，この問題冊子と解答冊子を開かないでください．
2．問題は必須問題と選択問題に分かれています．
3．必須問題は 3 問あり，それらは 1 ページにあります．選択問題は問題範囲ごとにそれぞれ 2 問ずつあります．数学 I・数学 II・数学 A・数学 B の問題は 2 ページに，数学 III の問題は 3 ページにあります．

4．解答冊子は，必須問題用と選択問題用の 2 冊に分かれています．それぞれの解答冊
子の表紙の所定欄に氏名と受験番号をはっきりと記入してください．

5．選択問題は解答する問題範囲を選び，選択問題用解答冊子の表紙の解答問題欄の選択欄に○印を記入してください．○印を記入していない場合，または複数の選択欄に○印を記入した場合は，0 点となります．

6．計算用紙は，解答冊子の中に綴じてあります．
7．試験中に問題冊子の印刷不明瞭，ページの落丁・乱丁および解答用紙の汚れ等に気がついた場合は，静かに手を上げて監督員に知らせてください．

8．試験終了後，問題冊子は持ち帰ってください．
9．解答時間は 120 分です．
10．問題ごとに配点が記されています．

必須問題I 以下の問いに答えよ．（配点 60 点）
問 1 a, b を実数とする．また，実数 x に対する 2 つの条件 x(x2 + ax + b) = 0 とx = 0 が，互いに同値であるとする．このとき，a と b がみたす関係を求め，点 (a, b) が存在する領域を座標平面に図示せよ．
問 2 方程式 20 · 15−x + 225x − 21 = 0 を解け．

II 以下の問いに答えよ．（配点 60 点）

問 1 正弦，余弦に関する加法定理
 sin(α + β) = sin α cos β + cos α sin β cos(α + β) = cos α cos β − sin α sin βを用いて等式 sin 3x = 3 sin x − 4 sin3 x を証明せよ．
問 2 関数 y = sin 3x + 3 cos 2x + 6 sin x (0 <= x < 2π) の最大値と最小値，およびそのときの x の値をすべて求めよ．

III 座標平面の原点を O とし，放物線 y = x2 の上を相異なる 2 点 A(a, a2 ), B(b, b2 ) は
̸

AOB が直角になるように動くとする．また，点 A と点 B を通る直線を ℓ とする．

以下の問いに答えよ．（配点 60 点）

問 1 a と b がみたす関係を求めよ．
問 2 直線 ℓ の方程式を y = px + q とする．q の値を求めよ．
問 3 原点 O から直線 ℓ に下ろした垂線を OH とする．点 H の軌跡を求めよ．

必須問題は，このページで終りである．
— 1 —選択問題 (数学 I・数学 II・数学 A・数学 B)
I 数列 {an }, {bn } が以下の漸化式をみたすとする．
a1 = 10, b1 = 24, an+1 = 2an − 8, bn+1 =1
bn + 6 (n = 1, 2, 3, · · ·)2

以下の問いに答えよ．（配点 60 点）

問 1 数列 {an }, {bn } の一般項をそれぞれ求めよ．
問 2 3 辺の長さが，それぞれ a2 , b2 , 6 である三角形は存在しないことを示せ．
問 3 3 辺の長さが，それぞれ an , bn , 6 である三角形が存在するような n の値をすべて求めよ．

II 関数 f (x) = |x2 − 1| に対し，F (a) =∫ a+1

f (x) dx とする．ただし，a > 0 とする．
a以下の問いに答えよ．（配点 60 点）

問 1 関数 y = f (x) のグラフをかけ．
問 2 F (a) を求めよ．
問 3 F (a) の最小値およびそのときの a の値を求めよ．

数学 I・数学 II・数学 A・数学 B の問題は，このページで終りである．
— 2 —選択問題 (数学 III)
I 関数 y = x2 e−x のグラフを曲線 C とする．以下の問いに答えよ．（配点 60 点）
問 1 曲線 C をかけ．ただし，x ≦ 2 の範囲でよい．
問 2 曲線 C が直線 y =1x に接していることを示し，その接点の座標を求めよ．
e

問 3 曲線 C と直線 y =1x で囲まれた図形の面積を求めよ．
e

II n = 1, 2, 3, · · · に対し，x の関数 fn (x) を
fn (x) =n∑(−1)k−1
k=1kxk = x + · · · +(−1)n−1 nxnで定める．ただし，0 <= x < 1 とする．以下の問いに答えよ．（配点 60 点）(12)− fn(12)

問1fn+1

問2

lim f ′ (x) を求めよ．
n→∞ n1<= 1000(n + 1) を満たすような n の最小値を求めよ．

問 3 n が偶数であるとき，不等式 fn (x) <= log(x + 1) を示せ．

数学 III の問題は，このページで終りである．
— 3 —2015 Entrance Exam
平成２７年度

入学者選抜学力試験（前期）English英 語

注意事項

1. 試験開始の合図があるまで，この問題冊子と解答冊子を開かないでください．
2. 問題は1〜10ページにあります．
3. 解答冊子の表紙の所定欄に氏名と受験番号をはっきりと記入してください．
4. 試験中に問題冊子の印刷不明瞭，ページの落丁・乱丁および解答用紙の汚れ等に気づいた場合は，静かに手を上げて監督員に知らせてください．
5. 試験終了後，問題冊子は持ち帰ってください．
6. 解答時間は90分です．
7. 辞書を使用することができます．
8. Part ごとに配点が記されています．
9. Part 1 と Part 2 の読解問題の配点の合計は 120点です．
10. Part 3 と Part 4 の英作文問題の配点の合計は 80点です．

Part 1 Reading Comprehension
Answer questions (1) – (8) based on the following article. （配点80点）
Privacy vs Convenience
We all value our privacy. The thought of strangers entering our home, following ourevery step, and observing what we do twenty-four hours a day would fill most of us withdisgust. Such a society would remind us of George Orwell’s novel Nineteen Eighty-Fourin which ① Winston Smith is under the constant gaze of Big Brother. Surely, in a
democratic society, because people value their privacy so much, they would not tolerate orfacilitate invasions of their privacy by a repressive state or individual such as Big Brother.
However, our daily interaction with the Internet suggests that people value convenienceeven more than privacy.
To access many services on the Internet we need to use a password that we havecreated. If we were really concerned about our privacy we would think of passwords thatno one else (or no machine) could easily guess. So the fact that in 2012 the most popular
password was ‘password’ and the second most used was ‘123456’ [1] indicates that many
people prefer an easy-to-remember password to one that is complicated and safer, but
hard to remember. This is even true for more technically minded people. A leak of 100,000users’ IEEE* website data revealed that the most common passwords were ‘123456’ and‘ieee2012’ [1]. Many companies offer a feature called ② ‘two-factor authentication’ tomake logging into their online services more secure. However, according to the results ofa 2013 survey as many as 79% of Internet users had no knowledge of two-factor
authentication [2].
Many people now use online services known as Social Network Services or SNS. These
SNS are a wonderful and convenient way to keep in touch with family, friends andcontacts. However, it seems that people’s desire to share the details of their lives withfriends is more important than the risk that this data may get into the wrong hands.
Photographs of a wild teenage party may seem harmless now, but in the hands of a future
employer they could have negative consequences.
Even for people who take precautions to protect their privacy, accessing the Internet
can leave a digital trail that reveals their interests, health issues, politics, plans, andfriends. A recent article by the online technology website called Ars Technica
demonstrated that we unknowingly reveal a lot of information about ourselves whenconnecting to the Internet [3]. Steve Henn, the technology correspondent for National
Public Radio (NPR), agreed to have his Internet activities monitored for one week by ArsTechnica. Although he took precautions not to reveal private information, there wasconsiderable leakage from his Internet-connected devices which allowed the monitors topiece together a detailed picture of the reporter’s private life. At the end of the week, Mr.

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Henn realized that anyone tracking his Internet usage “could actually know more about
my own past than I did” [3].
In the futuristic movie Minority Report made in 2002, as Tom Cruise’s character JohnAnderton walks through the streets he is greeted by automatic advertisements thatrecognise him. The existence of ‘cookies’, the small bits of code left by websites on yourhard disk, makes it possible for you to experience the Internet in much the same way thatJohn Anderton experiences life in 2054. Cookies keep a record of sites you have visitedand searches you have made. These cookies are an important component of your digital
trail. This trail allows websites that communicate with these cookies to tailor
advertisements to your own interests. However, according to a recent newspaper report,
more and more people are turning these cookies off [4]. Perhaps people are becoming
more aware that protecting one’s privacy might be more important than convenience.
* IEEE stands for ‘Institute of Electrical and Electronics Engineers’. The IEEE is theworld's largest professional association dedicated to advancing technological innovation.

References[1] S. Gallagher (2012, Nov. 4). “Born to be breached: the worst passwords are still themost common.” Ars Technica [Online]. Available:
http://arstechnica.com/information-technology/2012/11/born-to-be-breached-the-worst-passwords-are-still-the-most-common/. [Accessed: July 3, 2014].
[2] T. Trapp (2013, Nov. 22). “Two-factor authentication: why Twitter and Google need toeducate users,” The Guardian Newspaper [Online]. Available:
http://www.theguardian.com/media-network/media-network-blog/2013/nov/22/two-fac
tor-authentication-twitter-google. [Accessed: July 3, 2014].
[3] S. Gallagher (2014, Jun. 11). “Ars tests Internet surveillance—by spying on an NPR
reporter.” Ars Technica [Online]. Available:
http://arstechnica.com/security/2014/06/what-the-nsa-or-anyone-can-learn-about-you-from-internet-traffic/. [Accessed: July 3, 2014].
[4] J. Barnes (2013, May 24). “Internet users' privacy concerns may mean cookies start tocrumble.” The Guardian Newspaper [Online]. Available:
http://www.theguardian.com/technology/blog/2013/may/24/internet-privacy-cookies-fi
refox. [Accessed: July 3, 2014].

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(1)

下線部①の Winston Smith はどのような人か．以下の選択肢の中から最も適切なものを一つ選び，記号で答えよ．
(ア) a character in a book(イ) the title of a novel(ウ) the author of Big Brother(エ) someone who watches Big Brother(2)

本文の著者は people と privacy について，どのように考えているか．以下の選択肢の中から最も適切なものを一つ選び，記号で答えよ．
(ア) People think that online privacy is more important than offline privacy.
(イ) In democracies, people should value convenience more than privacy.
(ウ) Even in democracies, people value privacy more than convenience.
(エ) The way we use the Internet shows that we don’t value privacy that much.

(3)

下線部②の ‘two-factor authentication’ とは何を指すか．以下の選択肢の中から最も適切なものを一つ選び，記号で答えよ．
(ア) a complicated and difficult-to-remember password that online service
companies want their customers to use(イ) a method of making access to websites more secure(ウ) an easy way to access online services that companies are trying to sell(エ) an online service that only 21% of Internet users use(4)

本文の著者は SNS にはどのような問題があると述べているか．以下の選択肢の中から最も適切なものを一つ選び，記号で答えよ．
(ア) People can waste their time using SNS.
(イ) SNS encourage unruly parties.
(ウ) People may share data that they will regret later in life.
(エ) SNS make employers concerned about users’ privacy.

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(５)

第 4 段落で述べられている Ars Technica の記事が示したこととは何か．以下の選択肢の中から最も適切なものを一つ選び，記号で答えよ．
(ア) It showed that people know how to protect their private data when accessingthe Internet.
(イ) It showed that people unintentionally disclose information about themselves
when accessing the Internet.
(ウ) It showed that Steve Henn leaked some sensitive information from the NPR
website.
(エ) It showed that the technology correspondent for NPR had monitored theInternet activities of Ars Technica reporters.

(６)

一致しないものはどれか. 以下の選択肢の本文の Steve Henn について述べている内容と中から最も適切なものを一つ選び，記号で答えよ．

(ア) He is a journalist.
(イ) He works for National Public Radio.
(ウ) His job is to monitor people’s Internet activities.
(エ) He makes an effort to protect his online privacy.
(オ) He let Ars Technica track his Internet activities for one week.

(7)

映画 Minority Report はいつのことを描いているか．以下の選択肢の中から最も適切なものを一つ選び，記号で答えよ．
(ア) the past(イ) 2002(ウ) the present(エ) 2054(8)

本文によると ‘cookies’ の利便性は何か．以下の選択肢の中から最も適切なものを一つ選び，記号で答えよ．
(ア) Cookies make it possible to see advertisements for products that interest you.
(イ) Cookies allow companies to track everything that you do with your devices.
(ウ) Cookies prevent companies from writing code to your hard disk.
(エ) Cookies make it possible for Tom Cruise to be personally greeted by street
advertisements.

-4-Part 2 Questions about a Japanese Text
Answer questions (1) – (3) based on the following article. （配点 40 点）

著作権保護のため問題文は省略してあります

トーマス・フリードマン、朝日新聞社訳．「（ニューヨーク・タイムズから） IQ ではない グーグルが求める人材５条件」（朝日新聞 朝刊 2014 年 5 月 23 日）より改変

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(1)

Based on the article, mark each of the following statements as True (○) or False(×).

(ア) Having served as the president of a chess club does not demonstrate emergent
leadership.
(イ) When applying to Google, having good grades at college is considered adisadvantage.
(ウ) Fourteen per cent of Google employees have not received any college education.
(エ) Having good programming skills is important for technical positions at Google.
(オ) Google ranks intellectual humility higher than general cognitive ability.

(2)

According to the article, for each of the following terms, choose the best
description from (ア) to (ク).
a)
b)
c)
d)
e)

ソフト面のスキルプライドの高い自分全体的な認識能力
創発的なリーダーシップ
専門性

《選択肢》
(ア) an ability to take a lead when your team faces a problem and continue staying
in the position with a strong sense of responsibility(イ) pride and confidence developed through educational experience at famous
universities(ウ) being able to step in to lead when needed and willing to step back to let otherpeople lead(エ) general cognitive ability, including the ability to put together different kinds ofinformation(オ) personal qualities, such as motivation, leadership and flexibility, that are hardto quantify(カ) extensive knowledge in a discipline which people acquire through repeatedexposure(キ) abilities that can be easily quantified by such things as school grades and testscores(ク) being able to fiercely defend your strongly held opinions

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(3)

Which of the following statements is NOT consistent with Laszlo Bock’s views
described (explicitly or implicitly) in this article? Choose the best answer.

(ア) Having a good academic record is not necessarily a bad thing if it reflects theskills required by the job you are applying for. What is problematic is your belief
that all you need is a college degree because it guarantees the skills for any job.
(イ) Innovations have increasingly become a team effort. Therefore, it makes sense
for a company like Google to recruit its employees who have a set of soft skills
that cannot be easily predicted by their school records.
(ウ) Education has become increasingly important in order to find a securehigh-paying job. Therefore, you should choose a university carefully and do yourbest in order to earn the best grades possible.
(エ) When the ultimate objective of a team is to work cooperatively and solve
problems, who offers an idea is unimportant. What is important is to find thebest idea among various solutions. This is the reason why intellectual humility
becomes so important.

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Part 3 Writing

Write one paragraph (about 100 words) in English explaining the advantages or
disadvantages of the Internet.（配点 40 点）-8-Part 4 Writing
Write one paragraph (about 100 words) in English explaining what skills you have thatmake you a suitable student for Future University Hakodate.（配点 40 点）-9-