问题 解答题

已知:关于x的方程mx2-3(m-1)x+2m-3=0。

(1)求证:m取任何实数时,方程总有实数根;

(2)若二次函数y1=mx2-3(m-1)x+2m-3的图象关于y轴对称,

①求二次函数y的解析式;

②已知一次函数y2=2x-2,证明:在实数范围内,对于x的同一个值,这两个函数所对应的函数值y1≥y2均成立;

(3)在(2)条件下,若二次函数y3=ax2+bx+c的图象经过点(-5,0),且在实数范围内,对于x的同一个值,这三个函数所对应的函数值y1≥y3≥y2 均成立,求二次函数y3=ax2+bx+c的解析式。

答案

解:(1)分两种情况:

当m=0时,原方程可化为3x-3=0,即x=1;

∴m=0时,原方程有实数根;

当m≠0时,原方程为关于x的一元二次方程,

∵△=[-3(m-1)]2-4m(2m-3)=m2-6m+9=(m-3)2≥0,

∴方程有两个实数根;

综上可知:m取任何实数时,方程总有实数根;

(2)①∵关于x的二次函数y1=mx2-3(m-1)x+2m-3的图象关于y轴对称;

∴3(m-1)=0,即m=1;

∴抛物线的解析式为:y1=x2-1;

②∵y1-y2=x2-1-(2x-2)=(x-1)2≥0,

∴y1≥y2(当且仅当x=1时,等号成立);

(3)由②知,当x=1时,y1=y2=0,即y1、y2的图象都经过(1,0);

∵对应x的同一个值,y1≥y3≥y2成立,

∴y3=ax2+bx+c的图象必经过(1,0),

又∵y3=ax2+bx+c经过(-5,0),

∴y3=a(x-1)(x+5)=ax2+4ax-5a;

设y=y3-y2=ax2+4ax-5a-(2x-2)=ax2+(4a-2)x+(2-5a);

对于x的同一个值,这三个函数对应的函数值y1≥y3≥y2成立,

∴y3-y2≥0,

∴y=ax2+(4a-2)x+(2-5a)≥0;

根据y1、y2的图象知:a>0,

∴y最小=≥0

∴(4a-2)2-4a(2-5a)≤0,

∴(3a-1)2≤0,

而(3a-1)2≥0,只有3a-1=0,解得a=

∴抛物线的解析式为:y3=x2+x-

判断题
阅读理解

The Great Fire of London started in the very early hours of September 2, 1666. In four days it destroyed more than three-quarters of the old city, where most of the houses were wooden and close together. One hundred thousand people became homeless, but only a few lost their lives.

The fire started on Sunday morning in the house of the King's baker in Pudding Lane. The baker, with his wife and family, was able to get out through a window in the roof. A strong wind blew the fire from the bakery into a small hotel next door. Then it spread quickly into the Thames Street. That was the beginnings.

By eight o'clock three hundred houses were on fire. On Monday nearly a kilometer of the city was burning along the River Thames. Tuesday was the worst day. The fire destroyed many well-known buildings, old St Paul's and the Guildhall among them.

Samuel Pepys, the famous writer, wrote about the fire. People threw their things into the river. Many poor people stayed in their houses until the last moment. Birds fell out of the air because of the heat.

The fire stopped only when the King finally ordered people to destroy hundreds of buildings in the path of the fire. With nothing left to burn, the fire became weak and finally died out.

After the fire, Christopher Wren, the architect, wanted a city with wider streets and fine new houses of stone. In fact, the streets are still narrow; but he did build more than fifty churches, among which was new St Paul's.

The fire caused great pain and loss, but after it London was a better place: a city for the future and not just of the past.

1. The fire began in ___________________________.

A. a hotel         B. the palace          C. Pudding Lane          D. Thames Street

2. The underlined word "family" in the second paragraph means “________________________________”.

A. home             B. children              C. wife and husband    D. wife and children

3. It seems that the writer of the text was most sorry for the fact that ________________________________.

A. some people lost their lives              

B. the birds in the sky were killed by the fire

C. many famous buildings were destroyed     

D. the King’s bakery was burned down

4. Why did the writer cite(引用) Samuel Pepys?

A. Because Pepys was among those putting out the fire.

B. Because Pepys also wrote about the fire.

C. To show that poor people suffered most

D. To give the reader a clearer picture of the fire.

5. How was the fire put out according to the text?

A. The king and his soldiers came to help.

B. All the wooden houses in the city were destroyed.

C. People managed to get enough water from the river.

D. Houses standing in the direction of the fire were pulled down.

6. Which of the following were reasons for the rapid spread of the big fire?

(a) There was a strong wind.

(b) The streets were very narrow.

(c) Many houses were made of wood.

(d) There was not enough water in the city.

(e) People did not discover the fire earlier.

A. (a) and (b)                                        B. (a), (b) and (c)         

C. (a), (b), (c) and (d)                            D. (a), (b), (c), (d) and (e)