东莞市统考的数学试卷

东莞市统考的数学试卷一、选择题（每题1分，共10分）

1.下列函数中，有最小值的是：

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

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

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

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

2.在直角坐标系中，点\(P(3,4)\)关于\(y\)轴的对称点坐标是：

A.\((3,-4)\)

B.\((-3,4)\)

C.\((3,4)\)

D.\((-3,-4)\)

3.已知等差数列\(\{a_n\}\)的第一项\(a_1=2\)，公差\(d=3\)，则第\(10\)项\(a_{10}\)为：

A.25

B.28

C.31

D.34

4.若\(\frac{1}{x}+\frac{1}{y}=\frac{1}{4}\)，且\(x+y=6\)，则\(xy\)的值为：

A.12

B.24

C.36

D.48

5.在三角形ABC中，已知\(\angleA=90^\circ\)，\(a=3\)，\(b=4\)，则\(c\)的值为：

A.5

B.6

C.7

D.8

6.下列各数中，是等比数列的是：

A.\(1,2,4,8,16,\ldots\)

B.\(2,4,8,16,32,\ldots\)

C.\(3,6,12,24,48,\ldots\)

D.\(1,3,9,27,81,\ldots\)

7.已知\(\sin30^\circ=\frac{1}{2}\)，则\(\cos60^\circ\)的值为：

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

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

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

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

8.下列各数中，是整数的是：

A.\(\sqrt{9}\)

B.\(\sqrt{16}\)

C.\(\sqrt{25}\)

D.\(\sqrt{36}\)

9.已知\(\triangleABC\)中，\(a=5\)，\(b=6\)，\(c=7\)，则\(\sinA\)的值为：

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

B.\(\frac{3}{5}\)

C.\(\frac{4}{5}\)

D.\(\frac{5}{6}\)

10.若\(x^2-5x+6=0\)，则\(x\)的值为：

A.2或3

B.1或4

C.2或4

D.1或3

二、多项选择题（每题4分，共20分）

1.下列哪些函数是奇函数？

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

B.\(f(x)=\cosx\)

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

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

2.在下列各对数中，哪些是同底数？

A.\(\log_24\)和\(\log_28\)

B.\(\log_416\)和\(\log_48\)

C.\(\log_525\)和\(\log_55\)

D.\(\log_327\)和\(\log_39\)

3.下列哪些图形是全等图形？

A.两个等腰三角形，底边和腰长相等

B.两个等边三角形，边长相等

C.两个矩形，对边相等且相邻边垂直

D.两个圆，半径相等

4.下列哪些数是素数？

A.17

B.18

C.19

D.20

5.下列哪些函数是增函数？

A.\(f(x)=2x+3\)

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

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

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

三、填空题（每题4分，共20分）

1.已知等差数列\(\{a_n\}\)的前三项分别为\(a_1,a_2,a_3\)，若\(a_1=2\)，\(a_2=5\)，则\(a_3=\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_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四、计算题（每题10分，共50分）

1.计算下列函数的极值：

\(f(x)=x^3-6x^2+9x+1\)

2.解下列方程：

\(2x^2-5x+3=0\)

3.已知等差数列\(\{a_n\}\)的前\(n\)项和\(S_n=3n^2-2n\)，求该数列的第一项\(a_1\)和公差\(d\)。

4.已知三角形ABC的边长分别为\(a=5\)，\(b=6\)，\(c=7\)，求该三角形面积。

5.已知\(\sin30^\circ=\frac{1}{2}\)，\(\cos60^\circ=\frac{1}{2}\)，求\(\tan45^\circ\)的值。

本专业课理论基础试卷答案及知识点总结如下：

一、选择题答案及知识点详解：

1.B.\(f(x)=-x^2+4x-3\)有最小值，因为这是一个开口向下的二次函数。

2.B.\((-3,4)\)是点\(P(3,4)\)关于\(y\)轴的对称点。

3.B.\(a_{10}=a_1+(n-1)d=2+(10-1)\times3=2+27=29\)。

4.A.\(xy=\frac{(x+y)^2-(x^2+y^2)}{2}=\frac{6^2-(3^2+4^2)}{2}=\frac{36-25}{2}=6\)。

5.A.\(c=\sqrt{a^2+b^2}=\sqrt{3^2+4^2}=\sqrt{9+16}=\sqrt{25}=5\)。

6.D.\(1,3,9,27,81,\ldots\)是等比数列，公比为3。

7.A.\(\cos60^\circ=\frac{1}{2}\)。

8.C.\(\sqrt{25}=5\)是整数。

9.B.\(\sinA=\frac{a}{c}=\frac{5}{7}\)。

10.D.\(x^2-5x+6=(x-2)(x-3)=0\)，解得\(x=2\)或\(x=3\)。

二、多项选择题答案及知识点详解：

1.A.\(f(x)=x^3\)是奇函数，因为\(f(-x)=-f(x)\)。

2.A.\(\log_24=2\)，\(\log_28=3\)，同底数。

3.A.两个等腰三角形，底边和腰长相等，全等。

4.A.17是素数。

5.A.\(f(x)=2x+3\)是增函数，因为斜率为正。

三、填空题答案及知识点详解：

1.\(a_3=a_1+2d=2+2\times3=2+6=8\)。

2.\(a_1=2\)，\(d=3\)，\(a_2=a_1+d=2+3=5\)，\(a_3=a_2+d=5+3=8\)。

3.\(S_n=\frac{n}{2}(a_1+a_n)=\frac{n}{2}(2+(2+(n-1)\times3))=\frac{n}{2}(2+2+3n-3)=\frac{n}{2}(3n+1)\)。

4.\(S=\frac{1}{2}\timesa\timesb\times\sinC=\frac{1}{2}\times5\times6\times\sin90^\circ=\frac{1}{2}