Questions on Mathematics
F?t_ S trr @)
1. The roots of the equation
Lear blan 223-3*+82*5:0 ate zp
z, artd z,
Given that
zr: 1 + 2i, firrd zrand z,
(s)
72 =- t:Zl-
II
4
2.
f(x):3cos2x + x-2,
(a)
Show that the equation f(x)
:
-7t
<x <
1T
0 has a root a in the interval 12,31.
(2)
(b)
Use linear interpolation once on the interval f2,31to find an approximation to a.
Give your answer to 3 decimal places.
(3)
(c)
The equation f(x) : 0 has another root B in the interval [-1, 0]. Starting with this interval, use"interval bisection to find an interval of width 0.25 which contains B.
(4)
q) tlz-) -l'9-b [F):9
=
3,:68
Su b3 trtn*c-hanse.-rule-
- -2 <o( < 3-
2
rk4
,
r.?h_
9:1
3'86
3'8S* - 1-+6L =5,trt L.- l'
, S;8!t-1dft+;;r( s*18-6q3&-.
,', g( .* L"bbG
c)-
-i'a-
<
-:'*
B
,
- O,S < B < -6'23
Be GosP'zs)
3.
(t.r) lrl (i)
o=lJ, lrrl \'/z
(a)
'/z )
Describe fully the single transformation represented by the matrix A.
The matrix B represents an enlargement, scale factor
(b) Write
2,with
centre the origin.
down the matrix B.
(1)
(iD
-t = (j,
i),
where k
is
apositive constant.
Triangle Thas an area of 16 square units.
Triangle 7 is transformed onto the triangle matrix M.
T' by the transformation
represented by the
Given that the area of the triangle T' is 224 square units, find the value of ft.
(3)
O)
robrhor
{d
+S
o ant r c lo o{,r-r.o t 5e- C eyl/lrrr*- (o, o )
-T-B
(-;:,)
-iifAeb1ry q4:'LtL
T9 Tl r
e.a[ar3ar.*o.r$
-'. 9 + 2lr = l\
=) 2[r- aS
st' lt
.?- LL z2'S
-l-
4.
The complex number z is given by
,=P*2i
3+ pi
where
(a)
p is in
Express
integer.
z inthe form a+ bi where a
and b
xe
real. Give your answer in its simplest
form in terms ofp.
(4)
(b) Given that arg(z): 6, where tan?
:
1
find the possible values ofp.
5P + (b
(s)
-Pn);
j *P'
b)
$=-
4+?'
AnP'
s-9-::O
( {4 P=l
-?
A-
n
5. (a) Use the standard results for ) r
and
r=1
nl
Lr@' r=t 3)
:
)r=1 r' to show that
)"@
4
+ t)(n + 3)(n
- 2)
1. The roots of the equation
Lear blan 223-3*+82*5:0 ate zp
z, artd z,
Given that
zr: 1 + 2i, firrd zrand z,
(s)
72 =- t:Zl-
II
4
2.
f(x):3cos2x + x-2,
(a)
Show that the equation f(x)
:
-7t
<x <
1T
0 has a root a in the interval 12,31.
(2)
(b)
Use linear interpolation once on the interval f2,31to find an approximation to a.
Give your answer to 3 decimal places.
(3)
(c)
The equation f(x) : 0 has another root B in the interval [-1, 0]. Starting with this interval, use"interval bisection to find an interval of width 0.25 which contains B.
(4)
q) tlz-) -l'9-b [F):9
=
3,:68
Su b3 trtn*c-hanse.-rule-
- -2 <o( < 3-
2
rk4
,
r.?h_
9:1
3'86
3'8S* - 1-+6L =5,trt L.- l'
, S;8!t-1dft+;;r( s*18-6q3&-.
,', g( .* L"bbG
c)-
-i'a-
<
-:'*
B
,
- O,S < B < -6'23
Be GosP'zs)
3.
(t.r) lrl (i)
o=lJ, lrrl \'/z
(a)
'/z )
Describe fully the single transformation represented by the matrix A.
The matrix B represents an enlargement, scale factor
(b) Write
2,with
centre the origin.
down the matrix B.
(1)
(iD
-t = (j,
i),
where k
is
apositive constant.
Triangle Thas an area of 16 square units.
Triangle 7 is transformed onto the triangle matrix M.
T' by the transformation
represented by the
Given that the area of the triangle T' is 224 square units, find the value of ft.
(3)
O)
robrhor
{d
+S
o ant r c lo o{,r-r.o t 5e- C eyl/lrrr*- (o, o )
-T-B
(-;:,)
-iifAeb1ry q4:'LtL
T9 Tl r
e.a[ar3ar.*o.r$
-'. 9 + 2lr = l\
=) 2[r- aS
st' lt
.?- LL z2'S
-l-
4.
The complex number z is given by
,=P*2i
3+ pi
where
(a)
p is in
Express
integer.
z inthe form a+ bi where a
and b
xe
real. Give your answer in its simplest
form in terms ofp.
(4)
(b) Given that arg(z): 6, where tan?
:
1
find the possible values ofp.
5P + (b
(s)
-Pn);
j *P'
b)
$=-
4+?'
AnP'
s-9-::O
( {4 P=l
-?
A-
n
5. (a) Use the standard results for ) r
and
r=1
nl
Lr@' r=t 3)
:
)r=1 r' to show that
)"@
4
+ t)(n + 3)(n
- 2)