2009-2010,2008-2009学年澳门大学入学试试题(数学B).pdf

Pa st Examination Pape rs 2008 2009 2009 2010 84848484 2009 2010 學年 入學考試試題 ADMISSION EXAMINATION PAPER 第一部份第一部份第一部份第一部份 Part I Write your answers in the space provided for each question Answers written elsewhere will not be marked 將答題步驟及答案寫在每題所提供的空間內將答題步驟及答案寫在每題所提供的空間內將答題步驟及答案寫在每題所提供的空間內將答題步驟及答案寫在每題所提供的空間內 寫在該空間以外的答案不會被寫在該空間以外的答案不會被寫在該空間以外的答案不會被寫在該空間以外的答案不會被 評分評分評分評分 Part I 1 a 3 marks Solve the following equation for x 3 分 對以下方程求解x xx 2122 b 3 marks Let 23 2 xxf For what values of x is xf a real number 3 分 設 23 2 xxf 求所有使 xf為實數 的 x 值 2 4 marks Express 1 1 25 2 2 xx xx as sum of partial fractions 4 分 將 1 1 25 2 2 xx xx 化為部分分式和 3 6 marks The following data show the selling prices of 9 models of mobile phones 6 分 以下數據所示為9款不同型號手提電話的售價 2 130 3 250 4 760 1 800 2 770 6 050 2 340 1 980 1 560 Find the a mean b median and c standard deviation of the set of data listed above 求該組數據的 a 平均數 b 中位數 c 標準差 4 The marked price of a television is 7 500 A customer may choose one of the following three methods of payment 一部電視機的標價是 7 500 有三種付款方法可供顧客選擇 Syllabu s 2010 2011 85858585 Method 1 Pay by one single payment with a 5 discount 方法一 以現金一次付清所有款項 可享標價的九五折優惠 Method 2 First pay 20 of the marked price as initial payment Then pay the remainder together with a 7 simple interest by 12 monthly equal installments 方法二 先以現金支付標價的20 餘額要另加7 單利息 可分12 個月每月支付相同的款項清還 Method 3 At first no initial payment is needed Afterwards pay the marked price together with an 8 simple interest by 12 monthly equal installments 方法三 開始時不用付款 所有欠款要另加8 單利息 可分12個 月每月支付相同的款項清還 a 4 marks What is the percentage increase of the total amount paid by using Method 2 instead of Method 1 4 分 以方法二付款 比以方法一付款 所需款項增加的百份比是多 少 b 4 marks What is the difference between the amount of a monthly installment when using Method 2 and that when using Method 3 4 分 方法二和方法三每月支付的分期款項相差多少 5 A squad of soldiers are marching from a barracks to a mountain area They carry a load of 4 800 kg and it is shared equally by all soldiers In the middle of the journey 22 soldiers left and brought away 320 kg of load The rest of the soldiers continue their journey and the load carried by each remaining soldier now is 3 kg heavier than before 一隊軍人奉召由軍營出發前往一個山區 每人背負相同重量的物資 全隊攜 帶了4 800 kg 在途中 22人離開大隊並帶走共重320 kg 的物資 其餘軍 人繼續前往該山區 每人須背負的物資比出發時重了3 kg a 4 marks Let the number of soldiers setting out from the barracks be x In the context of this question establish a quadratic equation of x 4 分 設該隊軍人由軍營出發時有x人 根據題意 試設立一條有關 x 的二次方程式 b 4 marks After 22 soldiers have left how many soldiers remain in the squad What is the load that each remaining soldier must carry 4 分 當22人在途中離隊後 還剩下軍人多少名 餘下各人所攜物資 有多重 6 Box A contains 1 black ball and 3 white balls Box B contains 3 black balls and 2 white balls A ball is drawn out at random from A and put into B Then a ball is drawn out at random from B Find the probabilities of the following Pa st Examination Pape rs 2008 2009 2009 2010 86868686 events A盒載有一個黑球和三個白球 B盒載有三個黑球和兩個白球 若從A 盒中隨機抽取一球放入B盒內 然後又從B盒內隨機抽取一球 求以下 事件的概率 a 4 marks both balls drawn out are white in color 4 分 抽出的兩個球皆為白色 b 4 marks the second ball drawn out is white in color 4 分 第二個抽出的球為白色 第二部份第二部份第二部份第二部份 Part II This part contains 20 multiple choice questions Please select the desired answers on the pink answer sheet appeared on the last page of Part I Please follow the instruction given on that page 本部份有本部份有本部份有本部份有20條選擇題條選擇題條選擇題條選擇題 請在本試卷第一部份請在本試卷第一部份請在本試卷第一部份請在本試卷第一部份尾尾尾尾頁的頁的頁的頁的粉紅色粉紅色粉紅色粉紅色答案紙答案紙答案紙答案紙上上上上選擇答選擇答選擇答選擇答 案案案案 請依照該頁內的指示填寫答案請依照該頁內的指示填寫答案請依照該頁內的指示填寫答案請依照該頁內的指示填寫答案 1 If x x xf 1 1 then a faf 1 若 x x xf 1 1 則 a faf 1 A 0 B 1 C a 1 2 D a a 1 2 E a a 1 4 2 If the equation 052 2 kxx has no real roots find the range of the values of k 若方程 052 2 kxx 沒有實根 求k值的範圍 A 8 25 k Syllabu s 2010 2011 87878787 B 8 25 k D 8 25 k E 8 25 k 3 If 1 axx and bx 2 2 are identical find a and b 若 1 axx 與bx 2 2 相同 求a 及b A a 2 and b 0 a 2 及b 0 B a 3 and b 1 a 3 及 b 1 C a 3 and b 0 a 3 及 b 0 D a 3 and b 1 a 3 及 b 1 E a 3 and b 1 a 3 及 b 1 4 The figure below shows the graph of cbxaxy 2 The coordinates of the vertex are 3 5 With reference to the figure below which of the following must be correct 下圖所示為cbxaxy 2 的圖像 其頂點的坐標是 3 5 參看下圖 以下何者必為正確 I The y intercept of the graph 0 圖像的y軸截距 0 II 04 2 acb III The equation of the axis of symmetry of the graph is x 3 o x y 3 5 cbxaxy 2 Pa st Examination Pape rs 2008 2009 2009 2010 88888888 圖像的對稱軸的方程是x 3 A II only 只有II B I and II only 只有I及II C I and III only 只有I及III D II and III only 只有II及III E I II and III I II及III 5 What is the sum of all the different five digit integers formed by using the digits 1 2 3 5 9 without repetition 用1 2 3 5 9 組成數位不同的所有五位數總和為多少 A 5 333 280 B 4 666 640 C 3 999 960 D 3 555 520 E 2 666 640 6 Mr Chan has deposited a sum in a bank The interest rate is 6 per annum compounded monthly If the interest is 5 720 after 2 years find the original sum 陳先生把一筆款項存入銀行 年利率為6 每月計算複利息一次 若 兩年後他可得利息 5 720 求原來那筆存款 A 44 983 B 45 220 C 45 574 D 46 278 E 48 200 7 If a is greater than b by 20 and a is greater than c by 50 find a b c 若a比b大20 a比c大50 求a b c A 18 15 10 B 15 12 10 C 10 8 5 D 8 6 3 E 6 5 4 8 Solve 2 107 1 3 5 3 25 1 2 43 xx xx Syllabu s 2010 2011 89898989 解 2 107 1 3 5 3 25 1 2 43 xx xx A 2 x B 2 x C 22k Find the quotient 若多項式754 2 xx除以kx 的餘數是15 k 其中0 k 求商式 A 4x 7 B 4x 5 C 4x 5 D 4x 3 E 4x 3 19 An arithmetic series is such that its first term is a and its third term is b Let the sum of its first n terms be denoted by n S Given that 754 SSS are consecutive terms of a geometric series then 有一等差數列其首項為a及第三項為b 設 n S為其首n項之和 若 754 SSS 為一等比數列的相鄰項 則 A 22 2528ba B 22 716ba C 22 1011ba D 22 137ba E ba2528 20 In the figure ABC is an equilateral triangle inscribed in a circle as shown The chord MN bisects two sides of ABC at D E respectively If 2 AB the length of EN is 圖中ABC 為圓內接全等三角形 弦MN把ABC 兩邊等分於點 D E 若2 AB EN 的長度為 Syllabu s 2010 2011 93939393 A 5 15 B 2 15 C 5 10 D 4 12 E 4 6 Pa st Examination Pape rs 2008 2009 2009 2010 94949494 2009 2010 學年 參考答案 MODEL ANSWER Part I 1 a 2 2 122 2122 xx xx 0 4 2 082 2 xx xx 4or 2 xx rejected Thus there is only one real root namely 2 b xfis a real number if 02 2 x and 023 2 x 02 2 x 0 2 2 xx 2or 2 xx 23 2 x29 2 x 11 2 x 1111 x Combining and we see that 112or 211 xx 2 Let 11 1 1 25 22 2 x CBx x A xx xx Then 1 1 25 22 xCBxxAxx for all1 x In particular CAx 20 BCACBAx 42 281 852542 CACBAx Solving the above equations gives 2 1 BA and 3 C Thus 1 32 1 1 1 1 25 22 2 x x xxx xx Syllabu s 2010 2011 95959595 3 a 2 960 b 2 340 c 91 1423 9 18247600 4 a Total amount paid by using Method 1 125 7 95 0 7500 Total amount paid by using Method 2 920 7 07 01 2 01 75002 07500 Percentage increase of the total amount paid 16 11 100 7125 71257920 b The month installment when using Method 2 535 12 07 01 2 01 7500 The month installment when using Method 3 675 12 08 01 7500 Thus monthly installment of the latter method is 140 higher than the former 5 a 32048003 4800 22 x x or 01056002543 2 xx b The two roots of the quadratic equation formulated in a are 150 and 3 2 234 rejected So the number of soldiers remaining in the squad 12822150 Each remaining soldier must carry 353 150 4800 kg 6 a 8 3 6 3 4 3 or 0 375 b 24 11 6 2 4 1 6 3 4 3 or 0 458 Pa st Examination Pape rs 2008 2009 2009 2010 96969696 Part II 1 A 2 D 3 E 4 D 5 A 6 A 7 E 8 E 9 D 10 C 11 D 12 B 13 E 14 E 15 B 16 E 17 A 18 A 19 D 20 B Syllabu s 2010 2011 97979797 2008 2009 學年 入學考試試題 ADMISSION EXAMINATION PAPER 第一部份第一部份第一部份第一部份 Part I Write your answers in the space provided for each question Answers written elsewhere will not be marked 將答題步驟及答案寫在每題所提供的空間內將答題步驟及答案寫在每題所提供的空間內將答題步驟及答案寫在每題所提供的空間內將答題步驟及答案寫在每題所提供的空間內 寫在該空間以外的答案不會被寫在該空間以外的答案不會被寫在該空間以外的答案不會被寫在該空間以外的答案不會被 評分評分評分評分 1 Solve the equations for x 對以下方程求解x i 2 marks 12212 654 xxx 2 分 12212 654 xxx ii 3 marks 11000 log52 x x 3 分 11000 log52 x x 2 5 marks Find the values of k if the sum of square of the roots of the following equation is equal to 25 5 分 若下面方程式的根的平方和為25 求k的值 0 1 2 2 2 kxkx 3 Let AxAxxxf 4 23 When f x is divided by 2x 1 the remainder is 5 設AxAxxxf 4 23 當f x 除以2x 1時 其餘數為5 a 3 marks Find the value of A 3 分 求A的值 b 2 marks Find the remainder when f 2x 1 is divided by 2x 1 2 分 求當 f 2x 1 除以2x 1時的餘數 Pa st Examination Pape rs 2008 2009 2009 2010 98989898 4 By selling a shirt at 84 a man makes a profit of 40 Now the cost of the shirt increases by 25 若某商人以 84 出售一件恤衫 盈利百分率為 40 現恤衫的成本增加 了 25 a 3 marks if the man sells the shirt at the same price as before find the new profit percentage 3 分 若恤衫的售價不變 求新的盈利百分率 b 3 marks what should the new selling price be so that he still makes a profit of 40 3 分 若要保持盈利百分率為 40 問每件恤衫的新售價是 多少 5 The manufacturing cost of a video game console is C It is known that C is the sum of two parts one part is fixed and the other part varies inversely as the number N of game consoles produced When 3 000 game consoles are produced the cost of each game console is 328 When 6 000 game consoles are produced the cost of each game console is 224 5 生産一部遊戲機的成本是 C 已知C是兩部分的和 而其中一部分固定不變 另一部分則隨遊戲機的生産量N而反變 當生産3000部時 每部的成本是 328 當生産 6000 部時 每部的成本是 224 5 a 3 marks If the cost of each game console is 293 5 how many game consoles are produced 3 分 若每部的成本是 293 5 共生產了多少部遊戲機 b 3 marks If the least number of game consoles to be produced is 2 300 what will be the highest cost of each game console 3 分 如果最少須生産2300部 每部的成本最高是多少 6 6 marks One hour after a train had departed Station A and heading for Station B a fault occured which required the train to reduce its speed to 3 4 of its normal time table speed As a result the train reached Station B three hours late If the fault had occurred after traveling for another 60 kilometers the train would have been late for two hours and forty minutes Find the distance between Stations A and B 6 分 有一列火車從甲站開往乙站一個小時後在途中發生故障 需要 以正常速度的3 4行駛 故比平時晚了三個小時到達乙站 若列車多行60公 里才發生故障 則列車將會比平時晚了兩小時四十分鐘才到達乙站 求甲 乙兩站之間的距離 Syllabu s 2010 2011 99999999 7 In throwing two dice the outcome is called a double when the numbers on the two dice are equal 在投擲兩顆均稱的骰子時 若兩顆骰子的點數相同 我們便稱之爲 圍 骰 a 2 marks Find the probability of getting a double when throwing two dice 2 分 求投擲兩顆骰子一次而取得 圍骰 的概率 b 5 marks In a game two dice are thrown three times in succession Find the probability of getting only one double in the three throws 5 分 求連續投擲兩顆骰子三次 只得一次 圍骰 的概率 第第第第二部份二部份二部份二部份 Part II Please select the answers on the answer sheet on the last page of Part I 請在本試卷第一部份請在本試卷第一部份請在本試卷第一部份請在本試卷第一部份尾尾尾尾頁的答案紙頁的答案紙頁的答案紙頁的答案紙上上上上選擇答案選擇答案選擇答案選擇答案 1 Which of the following is an identity are identities 以下何者是恆等式 I 2 1 x x II ba ab ba 11 III mn nm 1 1 A I II and III I II及III B II and III only 只有II及III C I only 只有I D II only 只有II E III only 只有III Pa st Examination Pape rs 2008 2009 2009 2010 100100100100 2 Simplify 3 2 35 log loglog x xx 化簡 3 2 35 log loglog x xx A 4 3 B 6 7 C xlog 4 3 D xlog 6 7 E xlog 6 7 3 A medal is made of 12g of pure gold and a large amount of pure silver If the ratio of the weights of gold and silver is 3 37 find the weight of the medal 一個獎牌由12 g 純金和大量純銀鑄造而成 如果金與銀重量的比是3 37 求該獎牌的重量 A 148g B 154g C 160g D 166g E 170g 4 For all real numbers a and b the expression 2 abba is equal to 對所有實數 a b 2 abba 等於 A the average of a and b a和b 的平均值 B the average of a and b a 和 b的平均值 C the maximum of a and b a和b 中的極大值 D the minimum of a and b a和b 中的極小值 E ba Syllabu s 2010 2011 101101101101 5 In a summer camp there are x male and y female campers On each day every male camper and every female camper have to run 3 kilometers and 2 kilometers respectively so that the campers have run a total of 255 kilometers It is known that the number of male campers is less than that of female campers by 25 According to the above information we have 在某夏令營中 有x名男營友和y名女營友 每名男營友和每名女營友 每天須分別跑步 3 公里和 2 公里 全體營友共跑步 255 公里 已知男營 友的人數比女營友少 25 依題意 可得 A 251 25532 yx yx B 251 25532 xy yx C 25 25523 xy yx D 251 25523 xy yx E 251 25523 yx yx 6 Let R be the region in the xy plane satisfying the inequalities 0 x 3 x 0 yx and 2 yx The minimum value of yx56 on this region R is 設R為在xy 平面上的區域 且滿足不等式 0 x 3 x 0 yx 及 2 yx 在區域R上 yx56 的極小值是 A 9 B 10 C 11 D 12 E 33 7 When 666 2 is divided by 13 the remainder is 666 2除以13的餘數為 i 1 ii 3 iii 4 iv 7 v 12 Pa st Examination Pape rs 2008 2009 2009 2010 102102102102 8 Find the area of the shaded region in the figure below 求以下圖中陰影區域的面積 A 32 5 square units 32 5平方單位 B 31 5 square units 31 5平方單位 C 30 5 square units 30 5平方單位 D 29 5 square units 29 5平方單位 E 28 5 square units 28 5平方單位 9 If a set of data consists of 1 7 1 7 1 7 2 1 2 1 and 3 9 then 2 2 is the I median II mean III range of the set of data 若一組數據為1 7 1 7 1 7 2 1 2 1及3 9 則2 2是該組數據的 I 中位數 II 平均數 III 極差 分佈域 A I only 只有I B II only 只有II C III only 只有III D II and III only 只有II及III E I II and III I II及III y x 5 5 2 4 3 2 O Syllabu s 2010 2011 103103103103 10 In the figure the circle centered at A is tangent to the parabola 2 xy at M and N If the radius of the circle AM is 3 units and O is the origin what is the distance AO 圖中以A為圓心的圓與拋物 線 2 xy 相切於點M和N 若圓形的半徑AM為3公分 且O 原點 那麽距離AO是 多少 A 4 37 cm 公分 B 2 105 cm 公分 C 5 42 cm 公分 D 2 17 cm 公分 E 2 39 cm 公分 11 Let 42244225 1 34 1 23 1 12 1 KKS Which of the following items is are correct 設 42244225 1 34 1 23 1 12 1 KKS 以下項目有 那些是正確的 I S is divisible by 13 S 被13整除 II S is not an integer S不是整數 III 4030 S IV S is divisible by 8 S 被8整除 A I only 只有I B II only 只有 II C III only 只有 III D IV only 只有 IV E II and III only 只有 II和III Pa st Examination Pape rs 2008 2009 2009 2010 104104104104 12 The inequality 6 5 xx has solution 等式 6 5 xx的解是 A 12 x B 11 x C 54 11 xx D 54 11 xx E 63 21 xx 13 Let xxxf8 2 and 74 2 xxxg Find the minimum value of xgxf 設xxxf8 2 及74 2 xxxg 求 xgxf 的極小值 A 5 B 9 C 25 D 27 E no minimum value 沒有極小值 14 What is the number of positive integers less than or equal to 70 000 which are not multiple of 3 or 8 有多少個小於或等於 70 000 的正整數不是 3 或 的倍數 A 40 833 B 37 917 C 32 083 D 29 167 E 19 829 15 Given the points 2 2 A 10 0 B and 6 2 C in the xy plane For how many different points D in the plane are A B C and D the vertices of a parallelogram 給定在xy 平面上的點 2 2 A 10 0 B和 6 2 C 在平面上有多 少個不同的點 D 使 A B C 和 D 為一個平行四邊形的端點 A None 零個 B One 一個 C Two 兩個 D Three 三個 E Four 四個 Syllabu s 2010 2011 105105105105 16 The figure below shows a bridge with a circular arch Its height is 4 m and its span is 48 m Find the radius of this circular arch 下圖所示為一道圓弧形的拱橋 橋高4m跨度為48m 求該圓弧的半徑 A 72 m B 74 m C 76 m D 78 m E 80 m 17 If 3600 how many roots does the equation 0cossin2cos have 若 3600 方程0cossin2cos 有多少個根 A 1 B 2 C 3 D 4 E 5 18 If 2 w x 6 y z are six consecutive terms of a geometric sequence then yz 若2 w x 6 y z為一等比數列中的六個連續項 則yz A 48 B 72 C 108 D 144 E 288 48m 4m Pa st Examination Pape rs 2008 2009 2009 2010 106106106106 19 The coefficient of 8 x in the expansion of 1222 21 1 xx is 在 1222 21 1 xx 的展開式中 8 x 的系數是 A 6160 B 8712 C 17424 D 1496 E 1760 20 Suppose the graph of xfconsists of two half lines as shown If 6 xfxfxg which of the following is true 設 xf的圖形如附圖所示由兩條射 線組成 若 6 xfxfxg 以下哪一項正確 A xg has minimum value at 6 x xg在 6 x 有極小值 B xg has minimum value at 2 x xg在 2 x 有極小值 C xghas minimum value at 0 x xg在 0 x 有極小值 D xghas maximum value at 3 x xg在 3 x 有極大值 E xghas maximum value at 6 x xg在 6 x 有極大值 Syllabu s 2010 2011 107107107107 2008 2009 學年 參考答案 MODEL ANSWER Part I 1 i Taking logarithm to the base 10 on both sides gives 0511 0or 6log25log4log2 5log24log6log 5log24log6log 6log25log4log2 6log 12 5log 2 4log 12 x x xxx ii Taking logarithm to the base 10 on both sides gives the equivalent equation 1or 5 3log 0 1 log3log5 03log2 log5 0log log52 3 2 x xx xx x