①已知x^2y^2-4xy+x^2-2x+5=0求1/xy+1/(x+1)(y+1)+1/(x+2)(y+2)+…+1/(x+2009)(y+2009)的值②计算[（1994^2-2000)(1994+3985)×1995]/(1991×1993×1996×1997)③设x=-2-√3,求x5+6x4+7x3-4x2-8x+1的值④若√(2x-3)+y=4,化简√（x^2-2x+1)+

Question

①已知x^2y^2-4xy+x^2-2x+5=0 求1/xy+1/(x+1)(y+1)+1/(x+2)(y+2)+…+1/(x+2009)(y+2009)的值
②计算[（1994^2-2000)(1994+3985)×1995]/(1991×1993×1996×1997)
③设x=-2-√3,求x5+6x4+7x3-4x2-8x+1的值
④若 √(2x-3)+y=4,化简√（x^2-2x+1)+√(y^2-8y+16)
⑤x=√(3+√2),y=√(3-√2),求x^6+y^6的值
⑥若a>1,p=√2001a-√(2001a-1),q=√(2001a-1)-√2001a,r=√2001a-√(2001a+1),s=√(2001a+1)-√2001a,试比较p,q,r,s的大小,并求出最小的一个

Answer 1

1、
x^2y^2-4xy+x^2-2x+5=0
(x^2y^2-4xy+4)+(x^2-2x+1)=0
(xy-2)^2+(x-1)^2=0
两个平方只有都等于0
所以xy-2=0,x-1=0
x=1,y=2
所以原式=1/1*2+1/2*3+……+1/2010*2011
=1-1/2+1/2-1/3+……+1/2010-1/2011
=1-1/2011
=2010/2011
2、
1994^2-2000
=1994^2-1994-6
=(1994-3)(1994+2)
=1997*1996
1994+3985
=1994+1993+1992
=3*1993
所以原式=1997*1996*1995*3/(1991*1993*1996*1997)
=1995*3/(1991*1993)
3、
x+2=-√3
x^2+4x+4=3
x^2+4x=-1
原式=x^5+4x^4+2x^4+8x^3-x^3-4x^2-8x+1
=x^3(x^2+4x)+2x^2(x^2+4x)-x(x^2+4x)-8x+1
=-x^3-2x^2+x-8x+1
=-x^3-2x^2-7x+1
=-x^3-4x^2+2x^2+8x-15x+1
=-x(x^2+4x)+2(x^2+4x)-15x+1
=x-2-15x+1
=-14x-1
=28+14√3-1
=27+14√3
4、
根号大于等于0
所以√(2x-3)>=0
所以y=4-√(2x-3)=0,x>=3/2
所以x-1>=0.|x-1|=x-1
所以原式=√(x-1)^2+√(y-4)^2=|x-1|+|y-4|=x-1+4-y=x-y+3
5、
x^2+y^2=3+√2+3-√2=6
xy=√(9-2)=√7
(x^2+y^2)^2=6^2
所以x^4+y^4+2(xy)^2=36
x^4+y^4=36-2*7=22
所以x^6+y^6=(x^2+y^2)(x^4-x^2y^2+y^4)
=6*(22-7)
=90
6、
2001a+1>2001a>2001a-1
所以p>0,q√2001a+√(2001a-1)>0
所以1/√(2001a+1)+√2001a