13.某公司趣味競賽共有32位同仁參加，競賽採單淘汰制，每位輸了一場就淘汰。每場沒 有和局，勝者晉級下一輪。公司提供每位參加同仁基本獎1千元，第一輪獲勝者另加獎 金2千元，第二輪獲勝者再加獎金4千元，依此方式，每輪獲勝獎金為前一輪的2倍， 例如第三輪才輸的同仁共可得到7千元，則最後一場獲勝者總共可得_____千元。

14.某品牌手機的電池由A,B兩家廠商製造，其中40%是A廠製造、60%是B廠製造。已 知A廠的電池有90%的機率可以正常使用超過一年，而B廠的電池有75%的機率可以 正常使用超過一年。假設某甲有此品牌的手機且電池已經正常使用超過一年，則某甲手 機的電池是A廠製造的機率為_____。（化為最簡分數）

15.已知a>1，且知當0≤x≤2時，的最小值為，則a=_____。（化為最簡分數）

16.在坐標平面上，任給共線三點A,BC,且A,B在C點的同側，若以C為圓心、半徑為r的圓滿足（線段乘積），則稱A,B互為對圓的反演點。設為x2+y2-2x+6y+2=0，則點A(2,-2)對圓的反演點B的坐標為_____。

17.設ABCD,,,為空間中正四面體的四個頂點，另有一點E與點D分別在△ABC所在平面的兩側，且（向量內積）。則cos∠DAE=_____。（化為最簡根式）

19.若T將直線y=x+1上的點都映射到直線y=5x+13上，試求a、b。（非選擇題，6分）

20.（承19題）設P,Q為平面上兩相異點，令P'=T(P)、Q'=T(Q)，試說明為定值， 並求此值。（非選擇題，6分）

1. You are going to enbance a machine, and there are two possible improvements:either make multiply instructions run four times faster than before, or make memory access instructions run two times faster than before. You repeatedly run a program that takes 200 seconds to execute. Of this time, 50% is used for multiplication, 20% for memory access instructions, and 30% for other tasks. Determine the speedup for the corresponding cases. Copy Table I to your answer sheet and fill your answers.Enhancement cases:C1: Improve only multiplicationC2: Improve only memory accessC3: Improve both multiplication and memory accessTable 1.

2. The dynamic power of a processor is related to the capacitive load (C), operation frequency (F) and voltage (V) with this equation Power=CFV2. Suppose we developed a new, simpler processor that has 80% of the capacitive load of the more complex older processor. Further, we can reduce the voltage of the new processor by 20% and let the new processor run at 20% faster frequency. What's the dynamic power of the new processor with respect to the dynamic power of the old processor ?

3. The following C procedure swap is compiled into the corresponding MIPSassembly code. Assume that the parameters v and k, and variable temp are in registers $a0, sal and $to, respectively. Other registers that can be used are St1 and St2. C procedure:Please determine proper values for the operands (OPA, OPB, OPC, OPD). Copy thefollowing table (Table 3) to your answer sheet and fill in the operand values.

4. Figure 4.1 is a basic multiplication hardware and Figure 4.2 is an improvedversion of the basic one.Figure 4.1: Basic multiplication hardwareFigure 4.2: Improved multiplication hardwareThe following statements are possible observations for the improvement:S1: The multiplicand register, ALU, Multiplier register and the Product register are all 32 bits wide；S2: The Product register always shifts right two bits in each iteration；S3: The original Multiplier register disappears and is now placed in the right half of the Product register initially；S4: The left side of the Product register is initially set to zero.S5: The Multiplicand register is initially set to zero.Please copy the following table (Table 4) to your answer sheet and determine if these statements are true or false.

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