111年 - 111 臺北市立西松高級中學教師甄選：數學科#108209

科目：教甄◆數學 | 年份：111年 | 選擇題數：0 | 申論題數：23

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所屬科目：教甄◆數學

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申論題 (23)

Question 1. [Maximum mark: 6]
 The relation sin x +cos  y =0.5 is graphed alongside for 629d90015f56c.jpg . P lies on the y-axis, and the tangents at P and Q are perpendicular. The coordinates of Q is __________.629d8feed898f.jpg
Question 2. [Maximum mark: 6] 
 Jack and John have decided to play a game. One roll of a die is considered as one round of the game. On each round, John agrees to pay Jack $4 if 1 or 2 is rolled, Jack agrees to pay John $2 if 3, 4, 5 or 6 is rolled, and who is paid wins the round. In the end, who earns money wins the game. Now Jack and John roll a die 12 times. Let A denote the number of rounds Jack wins. Find the expected value of A, E[A]= __________.
 (a) Find the equation of the horizontal asymptote to the graph of y=f(x).

(b) Find f (x') for x ∈ 629d914a07fd1.jpg,x≠0. 

(c) Find the expression for 629d917173992.jpg, and state the domain of 629d917a7d6b8.jpg.
(a) Write down the value of n and the value of p. 
(b) State an assumption that is being made for X to be considered as following a binomial distribution. 
(c) Show that the variance of the proportion of marked fish in the sample, Var(X/300), is 0.000425. 
The estate manager feels confident that the proportion of marked fish in the lake will be within 1.5 standard deviations of the proportion of marked fish in the sample and decides these will form the upper and lower bounds of his estimate.
(d) Taking the value for the variance given in (c) as a good approximation for the true variance, find upper and lower bounds for the number of fish in the lake when t=8.
(e) Given this result, comment on the validity of the linear model used initially.
(a) Find an expression for the volume of water V, in m3 , in the left part of the water tank in terms of β?.  
The left part of the tank is now being filled with water at a constant rate of 0.001 m3 per second.
(b) Calculate 629d922e29a7a.jpg when β=629d924349441.jpg (note: sin629d925bcae44.jpg ≈0.5878  ).
(c) Calculate the amount of time it will take for the left part of the tank to be fully filled with water. Give your answer in minutes and correct to the nearest integer.
6.高斯符號 [a] 代表不大於 a 的最大整數， [12.7]= 12 ；[ 2.8 ]=[-3 +0.2]=- 3  。 已知方程式 [x]+[2x]+[4x] = 102 解集合為 A。若 a ∈ A  ，求 [4a+1]  之值? 

7. Fibonacci數列 629d9351c22eb.jpg 的定義如下： 629d93316c576.jpg ，其中 n ∈ N  。 若 m =5k ，其中 k ∈ N  ，試證： 629d93777a8ae.jpg 必為5之倍數。

8.已知△ABC 中 629d94adbd93b.jpg = a ， 629d94baa230d.jpg = b ， 629d94c67416a.jpg = c 。若 tan629d94d2d9b46.jpg (b tan A + a tan B) = (b + a) 。 2 試判斷△ABC 的形狀為何種三角形? 

9.銳角△ABC 中，過 A 作垂直線交 629d94f571d8c.jpg 於 D，在 629d9501e7e8c.jpg 上取一點 P，連接 629d951618720.jpg ， 629d951d85a76.jpg 分別交 629d952b18b22.jpg ， 629d953a0f762.jpg 於 E，F。試證： 629d955b2467f.jpg為 ∠EDF 之角平分線。

10.若 629d95792bffa.jpg ≤x≤ 3 ，求 √3-x +√ 5x-4 之最大值與最小值? 此時對應之 x 值?
11.有一個底面為三角形的直立稜柱，某一彼此相鄰的三個面(即兩個側面和一個 底面)的面積總和為一給定值 P。試求此直立三角柱體積之最大值?並求出此時 柱體的高與底面三角形的形狀? 
(1)取出數字由小到大排列，其數字差距均大於 7。例如(2,10,19,28) 
(2)取出四顆中數字是 2 的倍數恰有兩顆，是 5 的倍數也恰有兩顆。 例如(6,10,19,25) 
(3)取出四顆中數字是 2 的倍數恰有一顆，是 3 的倍數也恰有一顆，且是 5的倍數也恰有一顆。例如(6,13,19,25) 
13.IB 常識題，請以英文作答。
 International mindedness lies at the very heart of the IB mission, which aims to develop students ‘who help create a better and more peaceful world.’ International mindedness is an overarching idea. Please indicate three dimensions of being internationally-minded in DP programs. In addition, please elaborate on ways to facilitate international mindedness in your subject.