H
Turnbull High School
Mathematics Department
Mathematics
Higher
Paper 2
Time allowed
1 hour 10 minutes
NATIONAL
QUALIFICATIONS
Practice Prelim B
Read carefully
1
Calculators may be used in this paper.
2
Full credit will be given only where the solution contains appropriate working.
3
Answers obtained by readings from scale drawings will not receive any credit.
FORMULAE LIST
Circle:
The equation x 2 + y 2 + 2 gx + 2 fy + c = 0 represents a circle centre ( − g , − f ) and radius
The equation ( x − a)2 + ( y − b)2 = r 2 represents a circle centre ( a , b) and radius r.
Trigonometric formulae:
sin(A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B msin A sin B
sin 2A = 2 sin A cos A
cos 2A = cos 2 A − sin 2 A
= 2 cos 2 A − 1
= 1 − 2 sin 2 A
g2 + f 2 − c .
ALL questions should be attempted.
Marks
1.
A curve has equation y = x −
16
, x > 0.
x
Find the equation of the tangent at the point where x = 4.
2.
6
Triangles ACD and BCD are right angled at D
with angles p and q and lengths as shown in the
diagram.
(a)
Show that the exact value of sin( p + q) is
(b)
Calculate the exact values of:
84
.
85
4
(i) cos( p + q);
(ii) tan( p + q).
3.
(a)
3
A chord joins the points A(1, 0) and B(5, 4) on
the circle as shown in the diagram.
Show that the equation of the perpendicular
bisector of chord AB is x + y = 5.
(b)
The point C is the centre of the circle. The
tangent at the point A on the circle has
equation x + 3 y = 1.
Find the equation of the radius CA.
(c)
4
4
(i) Determine the coordinates of the point C.
(ii) Find the equation of the circle.
4
Marks
4.
The curves with equations y = x 2
and y = 2 x 2 − 9 intersect at K and L as
shown.
Calculate the area enclosed between
the curves.
8
5.
A zookeeper wants to fence off six individual animal pens.
Each pen is a rectangle measuring x metres by
y metres, as shown in the diagram.
(a)
(i) Express the total length of fencing in terms of x and y.
(ii) Given that the total length of fencing is 360 m, show that the total
16
area, A m2, of the six pens is given by A( x) = 240 x − x 2 .
3
(b)
6.
3
Find the values of x and y which give the maximum area and write down
this maximum area.
6
(a)
Write x 2 − 10 x + 27 in the form ( x + a)2 + b.
2
(b)
Hence show that the function g( x) = x 3 − 5 x 2 + 27 x − 2 is always increasing.
1
3
4
Marks
7.
Two functions f and g, are defined
by f ( x) = k sin 2 x and g( x) = sin x
where k > 1.
The diagram shows the graphs of
y = f ( x) and y = g( x) intersecting at
0, A, B, C and D.
Show that, at A and C, cos x =
1
.
2k
6
8.
The diagram shows a sketch of a
parabola passing through ( −1, 0),
(0, p) and ( p , 0).
(a)
(b)
Show that the equation of the
parabola is y = p + ( p − 1)x − x 2 .
3
For what value of p will the
line y = x + p be a tangent to
3
this curve?
End of Question Paper