FOR OCR
GCE Examinations
Advanced / Advanced Subsidiary
Core Mathematics C1
Paper A
Time: 1 hour 30 minutes
INSTRUCTIONS TO CANDIDATES
• Answer all the questions.
• Give non-exact numerical answers correct to 3 significant figures, unless a different degree
of accuracy is specified in the question or is clearly appropriate.
• You are not permitted to use a calculator in this paper.
INFORMATION FOR CANDIDATES
• The number of marks is given in brackets [ ] at the end of each question or part question.
• The total number of marks for this paper is 72.
• You are reminded of the need for clear presentation in your answers.
Written by Shaun Armstrong
 Solomon Press
These sheets may be copied for use solely by the purchaser’s institute.
Paper A
1.
Find the value of y such that
4y + 3 = 8.
2.
[3]
Express
2
3 5 +7
in the form a + b 5 where a and b are rational.
3.
[3]
A circle has the equation
x2 + y2 − 6y − 7 = 0.
4.
5.
(i)
Find the coordinates of the centre of the circle.
[2]
(ii)
Find the radius of the circle.
[2]
(i)
Express x2 + 6x + 7 in the form (x + a)2 + b.
[3]
(ii)
State the coordinates of the vertex of the curve y = x2 + 6x + 7.
[2]
Solve the simultaneous equations
x+y=2
3x2 − 2x + y2 = 2
C1A page 2
 Solomon Press
[7]
Paper A
6.
y
B
3
y = 3x − x 2
O
A
x
3
The diagram shows the curve with equation y = 3x − x 2 , x ≥ 0.
The curve meets the x-axis at the origin and at the point A and has a maximum
at the point B.
7.
(i)
Find the x-coordinate of A.
[3]
(ii)
Find the coordinates of B.
[5]
(i)
Calculate the discriminant of x2 − 6x + 12.
[2]
(ii)
State the number of real roots of the equation x2 − 6x + 12 = 0 and hence,
explain why x2 − 6x + 12 is always positive.
[3]
(iii) Show that the line y = 8 − 2x is a tangent to the curve y = x2 − 6x + 12.
[4]
f(x) = x3 − 6x2 + 5x + 12.
8.
(a)
Show that
(x + 1)(x − 3)(x − 4) ≡ x3 − 6x2 + 5x + 12.
(b)
(c)
[2]
Sketch the curve y = f(x), showing the coordinates of any points of intersection
with the coordinate axes.
[3]
Showing the coordinates of any points of intersection with the coordinate axes,
sketch on separate diagrams the curves
(i)
y = f(x + 3),
[2]
(ii)
y = f(−x).
[2]
Turn over
 Solomon Press
C1A page 3
Paper A
9.
A curve has the equation y =
x
1
+ 3 − , x ≠ 0.
2
x
The point A on the curve has x-coordinate 2.
(i)
Find the gradient of the curve at A.
(ii)
Show that the tangent to the curve at A has equation
3x − 4y + 8 = 0.
[4]
[3]
The tangent to the curve at the point B is parallel to the tangent at A.
(iii) Find the coordinates of B.
10.
[3]
The straight line l has gradient 3 and passes through the point A (−6, 4).
(i)
Find an equation for l in the form y = mx + c.
[2]
The straight line m has the equation x − 7y + 14 = 0.
Given that m crosses the y-axis at the point B and intersects l at the point C,
(ii)
find the coordinates of B and C,
[4]
(iii) show that ∠BAC = 90°,
[4]
(iv) find the area of triangle ABC.
[4]
C1A page 4
 Solomon Press
Paper B
1.
Find the set of values of the constant k such that the equation
x2 − 6x + k = 0
has real and distinct roots.
2.
[3]
The points A, B and C have coordinates (−3, 0), (5, −2) and (4, 1) respectively.
Find an equation for the straight line which passes through C and is parallel to AB.
Give your answer in the form ax + by = c, where a, b and c are integers.
3.
4.
18
in the form k 3 .
3
(i)
Express
(ii)
Express (1 −
3 )(4 − 2 3 ) in the form a + b 3 where a and b are integers.
[2]
[2]
Solve the inequality
2x2 − 9x + 4 < 0.
5.
[4]
[4]
Given that
(x2 + 2x − 3)(2x2 + kx + 7) ≡ 2x4 + Ax3 + Ax2 + Bx − 21,
find the values of the constants k, A and B.
C1B page 2
 Solomon Press
[7]
Paper B
6.
y
4
y = f(x)
2
−2
O
2
4
x
The diagram shows the graph of y = f(x).
(a)
(b)
Write down the number of solutions that exist for the equation
(i)
f(x) = 1,
[1]
(ii)
f(x) = −x.
[1]
Labelling the axes in a similar way, sketch on separate diagrams the graphs of
(i)
y = f(x − 2),
[3]
(ii)
y = f(2x).
[3]
f(x) = x3 − 9x2.
7.
(i)
Find f ′(x).
[2]
(ii)
Find f ′′(x).
[1]
(iii) Find the coordinates of the stationary points of the curve y = f(x).
[4]
(iv) Determine whether each stationary point is a maximum or a minimum point.
[2]
Turn over
 Solomon Press
C1B page 3
Paper B
f(x) = 9 + 6x − x2.
8.
(i)
(ii)
9.
Find the values of A and B such that
f(x) = A − (x + B)2.
[4]
State the maximum value of f(x).
[1]
(iii) Solve the equation f(x) = 0, giving your answers in the form a + b 2 where
a and b are integers.
[3]
(iv) Sketch the curve y = f(x).
[2]
The circle C has centre (−3, 2) and passes through the point (2, 1).
(i)
Find an equation for C.
[4]
(ii)
Show that the point with coordinates (−4, 7) lies on C.
[1]
(iii) Find an equation for the tangent to C at the point (−4, 7). Give your answer in
the form ax + by + c = 0, where a, b and c are integers.
10.
[5]
A curve has the equation y = ( x − 3)2, x ≥ 0.
(i)
Show that
dy
3
=1−
.
dx
x
[4]
The point P on the curve has x-coordinate 4.
(ii)
Find an equation for the normal to the curve at P in the form y = mx + c.
[5]
(iii) Show that the normal to the curve at P does not intersect the curve again.
[4]
C1B page 4
 Solomon Press
Paper C
1.
Solve the equation
9x = 3x + 2.
2.
[3]
The straight line l has the equation x − 5y = 7.
The straight line m is perpendicular to l and passes through the point (−4, 1).
Find an equation for m in the form y = mx + c.
3.
B
C
A
D
[4]
F
G
E
H
The diagram shows the rectangles ABCD and EFGH which are similar.
Given that AB = (3 − 5 ) cm, AD = 5 cm and EF = (1 + 5 ) cm, find the length
EH in cm, giving your answer in the form a + b 5 where a and b are integers.
[5]
4.
(i)
1
Sketch on the same diagram the curves y = x2 − 4x and y = − .
x
(ii)
State, with a reason, the number of real solutions to the equation
x2 − 4x +
5.
(i)
1
= 0.
x
[2]
Solve the inequality
x2 + 3x > 10.
(ii)
[4]
[3]
Find the set of values of x which satisfy both of the following inequalities:
3x − 2 < x + 3
x2 + 3x > 10
C1C page 2
[3]
 Solomon Press
Paper C
f(x) = 4x2 + 12x + 9.
6.
7.
(i)
Determine the number of real roots that exist for the equation f(x) = 0.
[2]
(ii)
Solve the equation f(x) = 8, giving your answers in the form a + b 2 where
a and b are rational.
[4]
The circle C has centre (−1, 6) and radius 2 5 .
(i)
Find an equation for C.
[2]
The line y = 3x − 1 intersects C at the points A and B.
(ii)
Find the x-coordinates of A and B.
(iii) Show that AB = 2 10 .
[3]
2
f(x) = 2 − x + 3x 3 , x > 0.
8.
(i)
Find f ′(x) and f ′′(x).
[3]
(ii)
Find the coordinates of the turning point of the curve y = f(x).
[4]
(iii) Determine whether the turning point is a maximum or minimum point.
9.
[4]
(i)
(ii)
[2]
Find an equation for the tangent to the curve y = x2 + 2 at the point (1, 3)
in the form y = mx + c.
[4]
Express x2 − 6x + 11 in the form (x + a)2 + b where a and b are integers.
[2]
(iii) Describe fully the transformation that maps the graph of y = x2 + 2 onto the
graph of y = x2 − 6x + 11.
[2]
(iv) Use your answers to parts (i) and (iii) to deduce an equation for the tangent
to the curve y = x2 − 6x + 11 at the point with x-coordinate 4.
[2]
Turn over
 Solomon Press
C1C page 3
Paper C
10.
The curve C has the equation y = f(x) where
f(x) = (x + 2)3.
(i)
(ii)
Sketch the curve C, showing the coordinates of any points of intersection with
the coordinate axes.
[3]
Find f ′(x).
[4]
The straight line l is the tangent to C at the point P (−1, 1).
(iii) Find an equation for l.
[3]
The straight line m is parallel to l and is also a tangent to C.
(iv) Show that m has the equation y = 3x + 8.
C1C page 4
 Solomon Press
[4]
Paper D
1.
Solve the equation
x2 − 4x − 8 = 0,
giving your answers in the form a + b 3 where a and b are integers.
2.
[3]
The curve C has the equation
y = x2 + ax + b,
where a and b are constants.
Given that the minimum point of C has coordinates (−2, 5), find the values of a and b.
3.
(i)
[4]
Solve the simultaneous equations
y = x2 − 6x + 7
y = 2x − 9
(ii)
4.
(i)
[4]
Hence, describe the geometrical relationship between the curve y = x2 − 6x + 7
and the straight line y = 2x − 9.
Evaluate
1
1
1
(36 2 + 16 4 ) 3 .
(ii)
(i)
[3]
Solve the equation
3x
5.
− 12
− 4 = 0.
[3]
Sketch on the same diagram the curve with equation y = (x − 2)2 and the straight
line with equation y = 2x − 1.
Label on your sketch the coordinates of any points where each graph meets the
coordinate axes.
(ii)
[4]
Find the set of values of x for which
(x − 2)2 > 2x − 1.
C1D page 2
[1]
 Solomon Press
[3]
Paper D
6.
(i)
1
Given that y = x 3 , show that the equation
1
2x 3 + 3x
− 13
=7
can be rewritten as
2y2 − 7y + 3 = 0.
(ii)
Hence, solve the equation
1
2x 3 + 3x
7.
[3]
− 13
= 7.
[4]
Given that
y=
x −
4
,
x
(i)
find
dy
,
dx
[3]
(ii)
find
d2 y
,
dx 2
[2]
(iii) show that
4x2
d2 y
dy
+ 4x
− y = 0.
2
dx
dx
[3]
f(x) = 2 + 6x2 − x3.
8.
(i)
Find the coordinates of the stationary points of the curve y = f(x).
[4]
(ii)
Determine whether each stationary point is a maximum or minimum point.
[3]
(iii) Sketch the curve y = f(x).
[2]
(iv) State the set of values of k for which the equation f(x) = k has three solutions.
[1]
Turn over
 Solomon Press
C1D page 3
Paper D
9.
The points P and Q have coordinates (7, 4) and (9, 7) respectively.
(i)
Find an equation for the straight line l which passes through P and Q. Give your
answer in the form ax + by + c = 0, where a, b and c are integers.
[4]
The straight line m has gradient 8 and passes through the origin, O.
(ii)
Write down an equation for m.
[1]
The lines l and m intersect at the point R.
(iii) Show that OP = OR.
10.
[5]
y
l
C
A
O
x
B
The diagram shows the circle C and the straight line l.
The centre of C lies on the x-axis and l intersects C at the points A (2, 4) and B (8, −8).
(i)
Find the gradient of l.
[2]
(ii)
Find the coordinates of the mid-point of AB.
[2]
(iii) Find the coordinates of the centre of C.
[5]
(iv) Show that C has the equation
x2 + y2 − 18x + 16 = 0.
C1D page 4
 Solomon Press
[3]
Paper E
1.
2.
3.
4.
21
in the form k 7 .
7
(i)
Express
(ii)
Express 8
Find
dy
when
dx
(i)
y = x − 2x2,
[2]
(ii)
y=
3
.
x2
[2]
(a)
Express x2 − 10x + 27 in the form (x + p)2 + q.
(b)
Sketch the curve with equation y = x2 − 10x + 27, showing on your sketch
− 13
as an exact fraction in its simplest form.
(i)
the coordinates of the vertex of the curve,
(ii)
the coordinates of any points where the curve meets the coordinate axes.
[2]
[2]
[3]
[3]
The straight line l1 has gradient 2 and passes through the point with coordinates (4, −5).
(i)
Find an equation for l1 in the form y = mx + c.
[2]
The straight line l2 is perpendicular to the line with equation 3x − y = 4 and passes
through the point with coordinates (3, 0).
(ii)
Find an equation for l2.
[3]
(iii) Find the coordinates of the point where l1 and l2 intersect.
C1E page 2
 Solomon Press
[3]
Paper E
5.
Given that the equation
4x2 − kx + k − 3 = 0,
where k is a constant, has real roots,
(i)
show that
k2 − 16k + 48 ≥ 0,
(ii)
[2]
find the set of possible values of k,
[3]
(iii) state the smallest value of k for which the roots are equal and solve the equation
when k takes this value.
6.
[3]
The points P and Q have coordinates (−2, 6) and (4, −1) respectively.
Given that PQ is a diameter of circle C,
(i)
find the coordinates of the centre of C,
(ii)
show that C has the equation
[2]
x2 + y2 − 2x − 5y − 14 = 0.
[5]
The point R has coordinates (2, 7).
(iii) Show that R lies on C and hence, state the size of ∠PRQ in degrees.
7.
(i)
(ii)
[2]
Describe fully the single transformation that maps the graph of y = f(x) onto
the graph of y = f(x − 1).
[2]
Showing the coordinates of any points of intersection with the coordinate axes
1
and the equations of any asymptotes, sketch the graph of y =
.
x −1
(iii) Find the x-coordinates of any points where the graph of y =
graph of y = 2 +
[3]
1
intersects the
x −1
1
. Give your answers in the form a + b 3 , where a and b
x
are rational.
[5]
Turn over
 Solomon Press
C1E page 3
Paper E
8.
y
l
C
A
O
B
x
The diagram shows the curve C with the equation y = x3 + 3x2 − 4x and the
straight line l.
The curve C crosses the x-axis at the origin, O, and at the points A and B.
(i)
Find the coordinates of A and B.
[3]
The line l is the tangent to C at O.
(ii)
Find an equation for l.
[4]
(iii) Find the coordinates of the point where l intersects C again.
9.
3
[4]
1
The curve with equation y = 2x 2 − 8x 2 has a minimum at the point A.
dy
.
dx
(i)
Find
(ii)
Find the x-coordinate of A.
[3]
[3]
The point B on the curve has x-coordinate 2.
(iii) Find an equation for the tangent to the curve at B in the form y = mx + c.
C1E page 4
 Solomon Press
[6]
Paper F
1.
2.
(i)
Calculate the discriminant of 2x2 + 8x + 8.
[2]
(ii)
State the number of real roots of the equation 2x2 + 8x + 8 = 0.
[1]
Find the set of values of x for which
(x − 1)(x − 2) < 20.
3.
(i)
[4]
Solve the equation
3
x 2 = 27.
(ii)
4.
Express ( 2 14 )
− 12
[2]
as an exact fraction in its simplest form.
[2]
Differentiate with respect to x
6 x2 − 1
.
2 x
5.
[5]
y
(−3, 4)
y = f(x)
O
x
(1, −2)
The diagram shows a sketch of the curve with equation y = f(x). The curve has a
maximum at (−3, 4) and a minimum at (1, −2).
Showing the coordinates of any turning points, sketch on separate diagrams the
curves with equations
(i)
y = 2f(x),
[3]
(ii)
y = −f(x).
[3]
C1F page 2
 Solomon Press
Paper F
f(x) = 2x2 − 4x + 1.
6.
(i)
Find the values of the constants a, b and c such that
f(x) = a(x + b)2 + c.
(ii)
7.
[4]
State the equation of the line of symmetry of the curve y = f(x).
[1]
(iii) Solve the equation f(x) = 3, giving your answers in exact form.
[3]
A curve has the equation
y = x3 + ax2 − 15x + b,
where a and b are constants.
Given that the curve is stationary at the point (−1, 12),
8.
(i)
find the values of a and b,
[6]
(ii)
find the coordinates of the other stationary point of the curve.
[3]
The circle C has the equation
x2 + y2 + 10x − 8y + k = 0,
where k is a constant.
Given that the point with coordinates (−6, 5) lies on C,
(i)
find the value of k,
[2]
(ii)
find the coordinates of the centre and the radius of C.
[3]
A straight line which passes through the point A (2, 3) is a tangent to C at the point B.
(iii) Find the length AB in the form k 3 .
[5]
Turn over
 Solomon Press
C1F page 3
Paper F
9.
A curve has the equation y = x +
3
, x ≠ 0.
x
The point P on the curve has x-coordinate 1.
(i)
Show that the gradient of the curve at P is −2.
[3]
(ii)
Find an equation for the normal to the curve at P, giving your answer in the
form y = mx + c.
[3]
(iii) Find the coordinates of the point where the normal to the curve at P intersects
the curve again.
10.
[4]
The straight line l1 has equation 2x + y − 14 = 0 and crosses the x-axis at the point A.
(i)
Find the coordinates of A.
[2]
The straight line l2 is parallel to l1 and passes through the point B (−6, 6).
(ii)
Find an equation for l2 in the form y = mx + c.
[3]
The line l2 crosses the x-axis at the point C.
(iii) Find the coordinates of C.
[1]
The point D lies on l1 and is such that CD is perpendicular to l1.
(iv) Show that D has coordinates (5, 4).
[5]
(v)
[2]
C1F page 4
Find the area of triangle ACD.
 Solomon Press
Paper G
1.
Find the value of y such that
4y + 1 = 82y − 1.
2.
Express
3.
A circle has the equation
[4]
22.5 in the form k 10 .
[4]
x2 + y2 + 8x − 4y + k = 0,
where k is a constant.
(i)
Find the coordinates of the centre of the circle.
[2]
Given that the x-axis is a tangent to the circle,
(ii)
[3]
f(x) = 4x − 3x2 − x3.
4.
5.
find the value of k.
(i)
Fully factorise 4x − 3x2 − x3.
[3]
(ii)
Sketch the curve y = f(x), showing the coordinates of any points of intersection
with the coordinate axes.
[3]
Find in exact form the coordinates of the points where the curve y = x2 − 4x + 2
crosses the x-axis.
[4]
(i)
(ii)
C1G page 2
Find the value of the constant k for which the straight line y = 2x + k is a tangent
[4]
to the curve y = x2 − 4x + 2.
 Solomon Press
Paper G
6.
Some ink is poured onto a piece of cloth forming a stain that then spreads.
The area of the stain, A cm2, after t seconds is given by
A = (p + qt)2,
where p and q are positive constants.
Given that when t = 0, A = 4 and that when t = 5, A = 9,
(i)
find the value of p and show that q = 15 ,
(ii)
find
[5]
dA
in terms of t,
dt
[3]
(iii) find the rate at which the area of the stain is increasing when t = 15.
7.
[2]
The curve C has the equation y = x2 + 2x + 4.
(i)
Express x2 + 2x + 4 in the form (x + p)2 + q and hence state the coordinates
of the minimum point of C.
[4]
The straight line l has the equation x + y = 8.
(ii)
Sketch l and C on the same set of axes.
[3]
(iii) Find the coordinates of the points where l and C intersect.
f(x) ≡
8.
(i)
( x − 4)2
1
2x 2
, x > 0.
Find the values of the constants A, B and C such that
3
1
f(x) = Ax 2 + Bx 2 + Cx
(ii)
[4]
− 12
.
[3]
Show that
f ′(x) =
3 x 2 − 8 x − 16
4x
3
2
.
(iii) Find the coordinates of the stationary point of the curve y = f(x).
[5]
[3]
Turn over
 Solomon Press
C1G page 3
Paper G
9.
y
l1
B
A
O
l2
D
x
C
The diagram shows the parallelogram ABCD.
The points A and B have coordinates (−1, 3) and (3, 4) respectively and lie on the
straight line l1.
(i)
Find an equation for l1, giving your answer in the form ax + by + c = 0, where
a, b and c are integers.
[4]
The points C and D lie on the straight line l2 which has the equation x − 4y − 21 = 0.
(ii)
Show that the distance between l1 and l2 is k 17 , where k is an integer to
be found.
(iii) Find the area of parallelogram ABCD.
C1G page 4
 Solomon Press
[7]
[2]
Paper H
f(x) = ( x + 3)2 + (1 − 3 x )2.
1.
Show that f(x) can be written in the form ax + b where a and b are integers to be
found.
2.
Find in exact form the real solutions of the equation
x4 = 5x2 + 14.
[4]
f(x) = x3 + 4x2 − 3x + 7.
3.
Find the set of values of x for which f(x) is increasing.
4.
5.
[3]
[5]
Express each of the following in the form p + q 2 where p and q are rational.
(i)
(4 − 3 2 )2
[2]
(ii)
1
2+ 2
[3]
Given that the equation
x2 + 4kx − k = 0
has no real roots,
(i)
show that
4k2 + k < 0,
(ii)
6.
[3]
find the set of possible values of k.
[3]
The curve with equation y = x2 + 2x passes through the origin, O.
(i)
Find an equation for the normal to the curve at O.
[4]
(ii)
Find the coordinates of the point where the normal to the curve at O intersects
the curve again.
[3]
C1H page 2
 Solomon Press
Paper H
7.
A circle has centre (5, 2) and passes through the point (7, 3).
(i)
Find the length of the diameter of the circle.
[2]
(ii)
Find an equation for the circle.
[2]
(iii) Show that the line y = 2x − 3 is a tangent to the circle and find the coordinates
of the point of contact.
8.
(i)
(ii)
[5]
Sketch the graphs of y = 2x4 and y = 2 x , x ≥ 0 on the same diagram and
write down the coordinates of the point where they intersect.
[4]
Describe fully the transformation that maps the graph of y = 2 x onto the
graph of y = 2 x − 3 .
[2]
(iii) Find and simplify the equation of the graph obtained when the graph of y = 2x4
is stretched by a factor of 2 in the x-direction, about the y-axis.
9.
[3]
The straight line l1 passes through the point A (−2, 5) and the point B (4, 1).
(i)
Find an equation for l1 in the form ax + by = c, where a, b and c are integers.
[4]
The straight line l2 passes through B and is perpendicular to l1.
(ii)
Find an equation for l2.
[3]
Given that l2 meets the y-axis at the point C,
(iii) show that triangle ABC is isosceles.
[4]
Turn over
 Solomon Press
C1H page 3
Paper H
10.
h
r
The diagram shows an open-topped cylindrical container made from cardboard.
The cylinder is of height h cm and base radius r cm.
Given that the area of card used to make the container is 192π cm2,
(i)
show that the capacity of the container, V cm3, is given by
V = 96πr −
(ii)
1
2
πr3.
Find the value of r for which V is stationary.
[5]
[4]
(iii) Find the corresponding value of V in terms of π.
[2]
(iv) Determine whether this is a maximum or a minimum value of V.
[2]
C1H page 4
 Solomon Press
Paper I
1.
Solve the inequality
x(2x + 1) ≤ 6.
2.
Differentiate with respect to x
3x2 −
3.
[4]
x +
1
.
2x
The straight line l has the equation x − 2y = 12 and meets the coordinate axes at
the points A and B.
Find the distance of the mid-point of AB from the origin, giving your answer in the
form k 5 .
4.
(i)
(ii)
[4]
Hence find the exact roots of the equation
x2 + 6x + 4 = 0.
[2]
8x passes through the point A with x-coordinate 2.
The curve with equation y =
Find an equation for the tangent to the curve at A.
3
[6]
−1
f(x) = x 2 − 8x 2 .
6.
(i)
(ii)
7.
[6]
By completing the square, find in terms of the constant k the roots of the equation
x2 + 2kx + 4 = 0.
5.
[4]
Evaluate f(3), giving your answer in its simplest form with a rational
denominator.
[3]
Solve the equation f(x) = 0, giving your answers in the form k 2 .
[4]
Solve the simultaneous equations
x − 3y + 7 = 0
x2 + 2xy − y2 = 7
C1I page 2
 Solomon Press
[7]
Paper I
8.
y
x2 + y2 − 2x − 18y + 73 = 0
y = 2x − 3
O
x
The diagram shows the circle with equation x2 + y2 − 2x − 18y + 73 = 0 and the
straight line with equation y = 2x − 3.
(i)
Find the coordinates of the centre and the radius of the circle.
[3]
(ii)
Find the coordinates of the point on the line which is closest to the circle.
[6]
f(x) = 2x2 + 3x − 2.
9.
(i)
Solve the equation f(x) = 0.
[2]
(ii)
Sketch the curve with equation y = f(x), showing the coordinates of any
points of intersection with the coordinate axes.
[2]
(iii) Find the coordinates of the points where the curve with equation y = f( 12 x)
crosses the coordinate axes.
[3]
When the graph of y = f(x) is translated by 1 unit in the positive x-direction it maps
onto the graph with equation y = ax2 + bx + c, where a, b and c are constants.
(iv) Find the values of a, b and c.
[3]
Turn over
 Solomon Press
C1I page 3
Paper I
10.
The curve with equation y = (2 − x)(3 − x)2 crosses the x-axis at the point A and
touches the x-axis at the point B.
(i)
Sketch the curve, showing the coordinates of A and B.
(ii)
Show that the tangent to the curve at A has the equation
x + y = 2.
[3]
[6]
Given that the curve is stationary at the points B and C,
(iii) find the exact coordinates of C.
C1I page 4
 Solomon Press
[4]
Paper J
1
2
1.
Evaluate 49 2 + 8 3 .
2.
Solve the equation
[3]
3x −
3.
4.
5
= 2.
x
[4]
Find the set of values of x for which
(i)
6x − 11 > x + 4,
[2]
(ii)
x2 − 6x − 16 < 0.
[3]
(i)
Sketch on the same diagram the graphs of y = (x − 1)2(x − 5) and y = 8 − 2x.
Label on your diagram the coordinates of any points where each graph meets
the coordinate axes.
(ii)
[5]
Explain how your diagram shows that there is only one solution, α, to
the equation
(x − 1)2(x − 5) = 8 − 2x.
[1]
(iii) State the integer, n, such that
n < α < n + 1.
[1]
f(x) = x2 − 10x + 17.
5.
(a)
Express f(x) in the form a(x + b)2 + c.
[3]
(b)
State the coordinates of the minimum point of the curve y = f(x).
[1]
(c)
Deduce the coordinates of the minimum point of each of the following curves:
C1J page 2
(i)
y = f(x) + 4,
[2]
(ii)
y = f(2x).
[2]
 Solomon Press
Paper J
6.
The points P, Q and R have coordinates (−5, 2), (−3, 8) and (9, 4) respectively.
(i)
Show that ∠PQR = 90°.
[4]
Given that P, Q and R all lie on a circle,
(ii)
find the coordinates of the centre of the circle,
[3]
(iii) show that the equation of the circle can be written in the form
x2 + y2 − 4x − 6y = k,
where k is an integer to be found.
7.
The straight line l1 has gradient
(i)
3
2
[3]
and passes through the point A (5, 3).
Find an equation for l1 in the form y = mx + c.
[2]
The straight line l2 has the equation 3x − 4y + 3 = 0 and intersects l1 at the point B.
(ii)
Find the coordinates of B.
[3]
(iii) Find the coordinates of the mid-point of AB.
[2]
(iv) Show that the straight line parallel to l2 which passes through the mid-point
of AB also passes through the origin.
[4]
Turn over
 Solomon Press
C1J page 3
Paper J
8.
y
l
m
B
y = 2 + 3x − x2
A
O
x
The diagram shows the curve with equation y = 2 + 3x − x2 and the straight
lines l and m.
The line l is the tangent to the curve at the point A where the curve crosses the y-axis.
(i)
Find an equation for l.
[5]
The line m is the normal to the curve at the point B.
Given that l and m are parallel,
(ii)
9.
find the coordinates of B.
[6]
The curve C has the equation
1
y = 3 − x 2 − 2x
− 12
, x > 0.
(i)
Find the coordinates of the points where C crosses the x-axis.
[4]
(ii)
Find the exact coordinates of the stationary point of C.
[5]
(iii) Determine the nature of the stationary point.
[2]
(iv) Sketch the curve C.
[2]
C1J page 4
 Solomon Press
Paper K
1.
Express
50 + 3 8 in the form k 2 .
2.
Find the coordinates of the stationary point of the curve with equation
y=x+
3.
[3]
4
.
x2
[5]
y
y = x3 + ax2 + bx + c
−1
O
3
x
The diagram shows the curve with equation y = x3 + ax2 + bx + c, where a, b and c
are constants. The curve crosses the x-axis at the point (−1, 0) and touches the x-axis
at the point (3, 0).
Show that a = −5 and find the values of b and c.
4.
[5]
The curve C has the equation y = (x − a)2 where a is a constant.
Given that
dy
= 2x − 6,
dx
5.
(i)
find the value of a,
[4]
(ii)
describe fully a single transformation that would map C onto the graph of y = x2. [2]
The straight line l1 has the equation 3x − y = 0.
The straight line l2 has the equation x + 2y − 4 = 0.
(i)
(ii)
C1K page 2
Sketch l1 and l2 on the same diagram, showing the coordinates of any points
where each line meets the coordinate axes.
[4]
Find, as exact fractions, the coordinates of the point where l1 and l2 intersect.
[3]
 Solomon Press
Paper K
6.
(a)
(b)
x
Given that y = 2 , find expressions in terms of y for
(i)
2x + 2,
[2]
(ii)
23 − x .
[2]
Show that using the substitution y = 2x, the equation
2x + 2 + 23 − x = 33
can be rewritten as
4y2 − 33y + 8 = 0.
(c)
Hence solve the equation
2x + 2 + 23 − x = 33.
7.
[2]
[4]
The point A has coordinates (4, 6).
Given that OA, where O is the origin, is a diameter of circle C,
(i)
find an equation for C.
[4]
Circle C crosses the x-axis at O and at the point B.
(ii)
Find the coordinates of B.
[2]
(iii) Find an equation for the tangent to C at B, giving your answer in the form
ax + by = c, where a, b and c are integers.
8.
[5]
(i)
Express 3x2 − 12x + 11 in the form a(x + b)2 + c.
[4]
(ii)
Sketch the curve with equation y = 3x2 − 12x + 11, showing the coordinates
of the minimum point of the curve.
[3]
Given that the curve y = 3x2 − 12x + 11 crosses the x-axis at the points A and B,
(iii) find the length AB in the form k 3 .
[5]
Turn over
 Solomon Press
C1K page 3
Paper K
9.
A curve has the equation y = x3 − 5x2 + 7x.
(i)
Show that the curve only crosses the x-axis at one point.
[4]
The point P on the curve has coordinates (3, 3).
(ii)
Find an equation for the normal to the curve at P, giving your answer in the
form ax + by = c, where a, b and c are integers.
[6]
The normal to the curve at P meets the coordinate axes at Q and R.
(iii) Show that triangle OQR, where O is the origin, has area 28 18 .
C1K page 4
 Solomon Press
[3]
Paper L
1.
Solve the inequality
4(x − 2) < 2x + 5.
f(x) = 2 − x − x3.
2.
Show that f(x) is decreasing for all values of x.
3.
(i)
(ii)
[2]
Hence solve the equation
3
x3 + 8 = 9x 2 .
[3]
Given that
y=
5.
[4]
Solve the equation
y2 + 8 = 9y.
4.
[3]
x4 − 3
,
2 x2
dy
,
dx
(i)
find
(ii)
show that
[4]
d2 y
x4 − 9
=
.
dx 2
x4
[2]
Find the pairs of values (x, y) which satisfy the simultaneous equations
3x2 + y2 = 21
5x + y = 7
C1L page 2
[7]
 Solomon Press
Paper L
6.
−1
(i)
Evaluate ( 5 94 ) 2 .
(ii)
Find the value of x such that
1+ x
=
x
[2]
3,
giving your answer in the form a + b 3 where a and b are rational.
7.
[5]
The straight line l passes through the point P (−3, 6) and the point Q (1, −4).
(i)
Find an equation for l in the form ax + by + c = 0, where a, b and c are integers. [4]
The straight line m has the equation 2x + ky + 7 = 0, where k is a constant.
Given that l and m are perpendicular,
8.
(ii)
find the value of k.
(i)
Describe fully a single transformation that maps the graph of y =
graph of y =
(ii)
[4]
1
onto the
x
3
.
x
Sketch the graph of y =
[2]
3
and write down the equations of any asymptotes.
x
(iii) Find the values of the constant c for which the straight line y = c − 3x is a
3
tangent to the curve y = .
x
[3]
[4]
Turn over
 Solomon Press
C1L page 3
Paper L
9.
The circle C has the equation
x2 + y2 − 12x + 8y + 16 = 0.
(i)
Find the coordinates of the centre of C.
[2]
(ii)
Find the radius of C.
[2]
(iii) Sketch C.
[2]
Given that C crosses the x-axis at the points A and B,
(iv) find the length AB, giving your answer in the form k 5 .
10.
[4]
y
y = 2x + 1
Q
y = x2 − 3x + 5
P
O
x
The diagram shows the curve y = x2 − 3x + 5 and the straight line y = 2x + 1.
The curve and line intersect at the points P and Q.
(i)
Using algebra, show that P has coordinates (1, 3) and find the coordinates of Q.
[4]
(ii)
Find an equation for the tangent to the curve at P.
[4]
(iii) Show that the tangent to the curve at Q has the equation y = 5x − 11.
[2]
(iv) Find the coordinates of the point where the tangent to the curve at P intersects
the tangent to the curve at Q.
[3]
C1L page 4
 Solomon Press