佛山市一模高三數(shù)學(xué)試卷一、選擇題（每題1分，共10分）

1.下列函數(shù)中，在實(shí)數(shù)范圍內(nèi)有最大值的是（）

A.\(f(x)=x^2-4x+4\)

B.\(f(x)=-x^2+4x-4\)

C.\(f(x)=x^3-3x^2+4x-4\)

D.\(f(x)=\frac{1}{x^2-2x+1}\)

2.若復(fù)數(shù)\(z\)滿足\(|z-1|=|z+1|\)，則復(fù)數(shù)\(z\)的軌跡是（）

A.直線\(x=0\)

B.圓\(x^2+y^2=1\)

C.雙曲線\(x^2-y^2=1\)

D.焦點(diǎn)在\(x\)軸上的橢圓

3.已知等差數(shù)列\(zhòng)(\{a_n\}\)的前\(n\)項(xiàng)和為\(S_n=3n^2-n\)，則該數(shù)列的公差為（）

A.5

B.4

C.3

D.2

4.若\(\sin\alpha+\cos\alpha=\frac{\sqrt{2}}{2}\)，則\(\sin2\alpha\)的值為（）

A.\(\frac{1}{2}\)

B.\(\frac{\sqrt{2}}{2}\)

C.\(\frac{3}{2}\)

D.\(\frac{\sqrt{3}}{2}\)

5.下列函數(shù)中，在\(x=0\)處有極值的是（）

A.\(f(x)=x^3\)

B.\(f(x)=x^2\)

C.\(f(x)=x^4\)

D.\(f(x)=\frac{1}{x^2}\)

6.已知\(\triangleABC\)中，\(\sinA=\frac{3}{5}\)，\(\cosB=\frac{4}{5}\)，則\(\sinC\)的值為（）

A.\(\frac{7}{25}\)

B.\(\frac{24}{25}\)

C.\(\frac{12}{25}\)

D.\(\frac{3}{25}\)

7.下列數(shù)列中，是等比數(shù)列的是（）

A.\(\{2,4,8,16,\ldots\}\)

B.\(\{1,3,6,10,\ldots\}\)

C.\(\{1,2,4,8,\ldots\}\)

D.\(\{1,3,9,27,\ldots\}\)

8.若\(\log_2x+\log_2y=3\)，則\(xy\)的值為（）

A.8

B.16

C.32

D.64

9.下列函數(shù)中，在\(x=1\)處有拐點(diǎn)的是（）

A.\(f(x)=x^2\)

B.\(f(x)=x^3\)

C.\(f(x)=e^x\)

D.\(f(x)=\lnx\)

10.若\(\tan\alpha=\frac{1}{3}\)，則\(\sin\alpha\)的值為（）

A.\(\frac{3}{\sqrt{10}}\)

B.\(\frac{1}{\sqrt{10}}\)

C.\(\frac{3}{\sqrt{2}}\)

D.\(\frac{1}{\sqrt{2}}\)

二、多項(xiàng)選擇題（每題4分，共20分）

1.下列各式中，屬于三角函數(shù)的有（）

A.\(\sin^2x+\cos^2x=1\)

B.\(\frac{1}{\sinx}=\cscx\)

C.\(\tan^2x+1=\sec^2x\)

D.\(\log_2(\sinx)\)

E.\(\sqrt{\frac{1}{1+\tan^2x}}=\cosx\)

2.已知數(shù)列\(zhòng)(\{a_n\}\)的通項(xiàng)公式為\(a_n=3^n-2^n\)，則下列說(shuō)法正確的是（）

A.\(\{a_n\}\)是等差數(shù)列

B.\(\{a_n\}\)是等比數(shù)列

C.\(\{a_n\}\)的前\(n\)項(xiàng)和\(S_n=3^n-2^n\)

D.\(\{a_n\}\)的前\(n\)項(xiàng)和\(S_n=3^n+2^n-1\)

E.\(\{a_n\}\)的前\(n\)項(xiàng)和\(S_n=3^n-2^n+1\)

3.下列各函數(shù)中，其圖像關(guān)于原點(diǎn)對(duì)稱的有（）

A.\(f(x)=x^2\)

B.\(f(x)=\frac{1}{x}\)

C.\(f(x)=\sinx\)

D.\(f(x)=e^x\)

E.\(f(x)=\lnx\)

4.若\(\log_2x\)和\(\log_3x\)是方程\(ax^2-2x+b=0\)的兩個(gè)實(shí)數(shù)根，則下列說(shuō)法正確的是（）

A.\(a>0\)

B.\(a<0\)

C.\(b>0\)

D.\(b<0\)

E.\(ab>0\)

5.下列各曲線中，屬于二次函數(shù)圖像的是（）

A.\(y=x^2-4x+4\)

B.\(y=-x^2+4x-4\)

C.\(y=\frac{1}{x^2-2x+1}\)

D.\(y=x^2+2x+1\)

E.\(y=\frac{1}{x}+x\)

三、填空題（每題4分，共20分）

1.已知\(\sin\alpha=\frac{3}{5}\)，且\(\alpha\)的終邊在第二象限，則\(\cos\alpha=\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\

四、計(jì)算題（每題10分，共50分）

1.計(jì)算下列三角函數(shù)的值：

\(\sin75^\circ\)和\(\cos75^\circ\)。

2.解下列方程：

\(3x^2-5x+2=0\)。

3.已知等差數(shù)列\(zhòng)(\{a_n\}\)的前\(n\)項(xiàng)和為\(S_n=4n^2-3n\)，求該數(shù)列的首項(xiàng)\(a_1\)和公差\(d\)。

4.已知\(\log_2x\)和\(\log_3x\)是方程\(ax^2-2x+b=0\)的兩個(gè)實(shí)數(shù)根，且\(a>0\)，\(b<0\)，求\(a\)和\(b\)的值。

5.設(shè)函數(shù)\(f(x)=x^3-6x^2+9x-1\)，求\(f(x)\)的極值點(diǎn)。

6.解下列不等式：

\(\frac{x^2-4x+3}{x-1}>0\)。

7.已知\(\triangleABC\)中，\(\sinA=\frac{3}{5}\)，\(\cosB=\frac{4}{5}\)，\(\sinC=\frac{7}{25}\)，求\(\cosA\)和\(\cosB\)的值。

8.設(shè)\(f(x)=\lnx+\frac{1}{x}\)，求\(f(x)\)的最小值。

本專業(yè)課理論基礎(chǔ)試卷答案及知識(shí)點(diǎn)總結(jié)如下：

一、選擇題

1.答案：B

解題過(guò)程：\(f(x)=-x^2+4x-4\)是一個(gè)開(kāi)口向下的拋物線，其頂點(diǎn)坐標(biāo)為\((2,0)\)，在實(shí)數(shù)范圍內(nèi)有最大值0。

2.答案：A

解題過(guò)程：由\(|z-1|=|z+1|\)可得\(z\)到點(diǎn)1和點(diǎn)-1的距離相等，因此\(z\)的軌跡是\(x=0\)。

3.答案：D

解題過(guò)程：\(S_n=3n^2-n\)的差分\(S_{n+1}-S_n=6n-1\)為等差數(shù)列的通項(xiàng)，公差\(d=6\)。

4.答案：B

解題過(guò)程：\(\sin2\alpha=2\sin\alpha\cos\alpha=2\cdot\frac{3}{5}\cdot\frac{4}{5}=\frac{12}{25}\)。

5.答案：A

解題過(guò)程：\(f(x)=x^3\)在\(x=0\)處有極大值。

6.答案：D

解題過(guò)程：由正弦定理，\(\sinC=\sin(180^\circ-A-B)=\sin(A+B)=\sinA\cosB+\cosA\sinB=\frac{3}{5}\cdot\frac{4}{5}+\frac{4}{5}\cdot\frac{3}{5}=\frac{24}{25}\)。

7.答案：C

解題過(guò)程：等比數(shù)列的定義是相鄰兩項(xiàng)之比相等，因此\(\{1,2,4,8,\ldots\}\)是等比數(shù)列。

8.答案：C

解題過(guò)程：\(\log_2x+\log_2y=3\)可化簡(jiǎn)為\(\log_2(xy)=3\)，即\(xy=2^3=8\)。

9.答案：B

解題過(guò)程：\(f(x)=x^3\)在\(x=0\)處有極小值。

10.答案：B

解題過(guò)程：\(\sin\alpha=\frac{1}{\sqrt{10}}\)。

二、多項(xiàng)選擇題

1.答案：ABCE

解題過(guò)程：\(\sin^2x+\cos^2x=1\)，\(\frac{1}{\sinx}=\cscx\)，\(\tan^2x+1=\sec^2x\)，\(\sqrt{\frac{1}{1+\tan^2x}}=\cosx\)都是三角函數(shù)的基本關(guān)系。

2.答案：ACE

解題過(guò)程：\(\{a_n\}\)是等差數(shù)列，前\(n\)項(xiàng)和\(S_n=3^n-2^n\)。

3.答案：BCE

解題過(guò)程：\(\sinx\)，\(\cosx\)，\(\tanx\)是三角函數(shù)，而\(\frac{1}{x}\)和\(\lnx\)不是。

4.答案：AD

解題過(guò)程：\(\log_2x\)和\(\log_3x\)是方程的實(shí)數(shù)根，因此\(a>0\)，且\(\log_2x\)和\(\log_3x\)互為倒數(shù)，所以\(ab>0\)。

5.答案：ABD

解題過(guò)程：\(y=x^2-4x+4\)，\(y=-x^2+4x-4\)，\(y=x^2+2x+1\)是二次函數(shù)圖像。

三、填空題

1.答案：\(-\frac{4}{5}\)

解題過(guò)程：由\(\sin\alpha=\frac{3}{5}\)和\(\alpha\)的終邊在第二象限，可得\(\cos\alpha=-\sqrt{1-\sin