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Sixth Term Examination Papers 9475 MATHEMATICS 3 Afternoon WEDNESDAY 27 JUNE 2012 Time: 3 hours
Additional Materials: Answer Booklet
Formulae Booklet
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初中英语语法书All questions attempted will be marked.meanshift
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_____________________________________________________________________________ Thi
s question paper consists of 8 printed pages and 4 blank pages.
BLANK PAGE
Section A:Pure Mathematics
1Given that z=y n
d y
d x
2
,show that
d z
vientiane
d x
=y n−1
d y
d x
n
d y
d x
2
+2y
d2y
d x2
.
(i)U the above result to show that the solution to the equation
d y d x 2
+2y
circle of friendsd2y
d x2
=
√
y(y>0)
that satisfies y=1and d y
d x
=0when x=0is y=
3
8
x2+1
2
3.
(ii)Find the solution to the equation
d y d x 2
−y d
2y
d x2
英语学习网站大全+y2=0
that satisfies y=1and d y
d x
=0when x=0.
2In this question,|x|<1and you may ignore issues of convergence.
(i)Simplify
(1−x)(1+x)(1+x2)(1+x4)···(1+x2n), where n is a positive integer,and deduce that
1 1−x =(1+x)(1+x2)(1+x4)···(1+x2n)+
x2n+1
1−x
.
Deduce further that
ln(1−x)=−
草莓的英文单词
∞
r=0
ln
1+x2r
,
and hence that
1 1−x =
1
1+x
+
2x
1+x2
+
4x3英语四级单词下载
1+x4
+···.
(ii)Show that
1+2x 1+x+x2=
1−2x
1−x+x2
+
2x−4x3
1−x2+x4
+
4x3−8x7
1−x4+x8
+···.
2
3It is given that the two curves
y=4−x2and mx=k−y2,
where m>0,touch exactly once.
(i)In each of the following four cas,sketch the two curves on a single diagram,noting
the coordinates of any interctions with the axes:
trf
(a)k<0;
(b)0<k<16,k/m<2;
(c)k>16,k/m>2;
(d)k>16,k/m<2.
(ii)Now t m=12.
Show that the x-coordinate of any point at which the two curves meet satisfies
x4−8x2+12x+16−k=0.
Let a be the value of x at the point where the curves touch.Show that a satisfies
a3−4a+3=0
杂志社英文and hencefind the three possible values of a.
Derive also the equation
k=−4a2+9a+16.
Which of the four sketches in part(i)ari?
4(i)Show that
∞ n=1n+1
n!
=2e−1
arms
and
∞ n=1(n+1)2
n!
=5e−1.
Sum the ries
∞
n=1
(2n−1)3
n!
.
(ii)Sum the ries
∞
n=0
(n2+1)2−n
(n+1)(n+2)
,giving your answer in terms of natural logarithms.
3
5(i)The point with coordinates(a,b),where a and b are rational numbers,is called:
an integer rational point if both a and b are integers;
a non-integer rational point if neither a nor
b is an integer.
(a)Write down an integer rational point and a non-integer rational point on the circle
x2+y2=1.
(b)Write down an integer rational point on the circle x2+y2=2.Simplify
(cosθ+√
m sinθ)2+(sinθ−
√
m cosθ)2
and hence obtain a non-integer rational point on the circle x2+y2=2.
(ii)The point with coordinates(p+√
2q,r+
√
2s),where p,q,r and s are rational numbers,
is called:
an integer2-rational point if all of p,q,r and s are integers;
a non-integer2-rational point if none of p,q,r and s is an integer.
(a)Write down an integer2-rational point,and obtain a non-integer2-rational point,
on the circle x2+y2=3.
(b)Obtain a non-integer2-rational point on the circle x2+y2=11.
(c)Obtain a non-integer2-rational point on the hyperbola x2−y2=7.
6Let x+i y be a root of the quadratic equation z2+pz+1=0,where p is a real number.Show that x2−y2+px+1=0and(2x+p)y=0.Show further that
either p=−2x or p=−(x2+1)/x with x=0.
Hence show that the t of points in the Argand diagram that can(as p varies)reprent roots of the quadratic equation consists of the real axis with one point missing and a circle.This t of points is called the root locus of the quadratic equation.
Obtain and sketch in the Argand diagram the root locus of the equation
pz2+z+1=0
and the root locus of the equation
pz2+p2z+2=0.
4
