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C3 Algebra
1.
Express
2 x 2  3x
6
 2
(2 x  3) ( x  2) x  x  2
as a single fraction in its simplest form.
(7)
2.
(a)
Simplify
3x 2  x  2
x 2 1
(3)
(b)
Hence, or otherwise, express
3x 2  x  2
x 1
2

1
as a single fraction in its simplest form.
x( x  1)
(3)
f( x)  1 
3.
(a)
Show that f( x) 
x 2  x 1
(x  2 ) 2
3
3

, x   2.
x  2 ( x  2) 2
, x   2.
(4)
(b)
Show that x2 + x +1 > 0 for all values of x.
(3)
(c)
Show that f (x) > 0 for all values of x, x ≠ –2.
(1)
f x  
4.
(a)
Show that f x  
2x  3
9  2x
 2
,
x  2 2 x  3x  2
x
1
2
4x  6
.
2 x 1
(7)
(b)
Hence, or otherwise, find f′(x) in its simplest form.
(3)
5.
Given that
2 x 4  3x 2  x  1
dx  e
 (ax2  bx  c)  2
,
( x 2  1)
( x  1)
find the values of the constants a, b, c, d and e.
(4)
f ( x) 
6.
(a)
2x  2
x 1

x  2x  3 x  3
2
Express f (x) as a single fraction in its simplest form.
(4)
1
(b)
Hence show that f' ( x) 
2
( x  3) 2
(3)
7.
The function f is defined by
f( x) 1 –
(a)
2
x–8

, x
( x  4) ( x – 2)( x  4)
Show that f( x) 
, x ≠ –4, x ≠ 2
x–3
x–2
(5)
The function g is defined by
g ( x) 
(b)
ex  3
ex  2
Differentiate g(x) to show that g' ( x) 
,
x
, x ≠ 1n 2
ex
,
(e x – 2) 2
(3)
(c)
Find the exact values of x for which g′(x) = 1
(4)
8.
Express
x 1
1

2
3x  3 3x  1
as a single fraction in its simplest form.
(4)
9.
(a)
Simplify fully
2x 2  9x  5
x 2  2 x  15
(3)
Given that
ln(2x2 + 9x – 5) = 1 + ln(x2 + 2x – 15), x ≠ – 5,
(b)
find x in terms of e.
(4)
1. Edexcel C3 January 2006 Question 2
2. Edexcel C3 June 2006 Question 1
3. Edexcel C3 January 2007 Question 2
4. Edexcel c3 June 2007 Question 2
5. Edexcel C3 January 2008 Question 1
6. Edexcel C3 June 2008 Question
7. Edexcel C3 January 2009 Question 2
8. Edexcel C3 June 2009 Question 7
9. Edexcel C3 January 2010 Question 1
10. Edexcel C3 June 2010 Question 8
2
C3 Exponential and Logs
1.
A heated metal ball is dropped into a liquid. As the ball cools, its temperature, T °C,
t minutes after it enters the liquid, is given by
T = 400 e–0.05t + 25, t  0.
(a)
Find the temperature of the ball as it enters the liquid.
(1)
(b)
Find the value of t for which T = 300, giving your answer to 3 significant figures.
(4)
(c)
Find the rate at which the temperature of the ball is decreasing at the instant when t = 50. Give your
answer in °C per minute to 3 significant figures.
(3)
(d)
From the equation for temperature T in terms of t, given above, explain why the temperature of the
ball can never fall to 20 °C.
(1)
(Total 9 marks)
2.
Find the exact solutions to the equations
(a)
ln x + ln 3 = ln 6,
(2)
(b)
ex + 3e–x = 4.
(4)
(Total 6 marks)
3.
The amount of a certain type of drug in the bloodstream t hours after it has been taken is given by the
formula
x  De
1
 t
8 ,
where x is the amount of the drug in the bloodstream in milligrams and D is the dose given in milligrams.
A dose of 10 mg of the drug is given.
(a)
Find the amount of the drug in the bloodstream 5 hours after the dose is given.
Give your answer in mg to 3 decimal places.
(2)
A second dose of 10 mg is given after 5 hours.
(b)
Show that the amount of the drug in the bloodstream 1 hour after the second dose is 13.549 mg to 3
decimal places.
(2)
No more doses of the drug are given. At time T hours after the second dose is given, the amount of the drug
3
in the bloodstream is 3 mg.
(c)
Find the value of T.
(3)
(Total 7 marks)
4.
The point P lies on the curve with equation
y = 4e2x+1.
The y -coordinate of P is 8.
(a)
Find, in terms of ln 2, the x -coordinate of P.
(2)
(b)
Find the equation of the tangent to the curve at the point P in the form y = ax + b, where a and b are
exact constants to be found.
(4)
(Total 6 marks)
5.
The functions f and g are defined by
f : x  3x  ln x, x  0, x
2
g : x  ex , x
(a)
Write down the range of g.
(1)
(b)
Show that the composite function fg is defined by
2
fg : x  x 2  3 x , x 
(2)
(c)
Write down the range of fg.
(1)
(d)
Solve the equation
2
d
fg ( x)  x( xe x  2).
dx
(6)
(Total 10 marks)
f(x) = 3xex – 1
6.
The curve with equation y = f (x) has a turning point P.
(a)
Find the exact coordinates of P.
(5)
The equation f (x) = 0 has a root between x = 0.25 and x = 0.3
(b)
Use the iterative formula
1
x n 1  e  xn
3
4
with x0 = 0.25 to find, to 4 decimal places, the values of x1, x2 and x3.
(3)
(c)
By choosing a suitable interval, show that a root of f(x) = 0 is x = 0.2576 correct to 4 decimal places.
(3)
(Total 11 marks)
7.
Rabbits were introduced onto an island. The number of rabbits, P, t years after they were introduced is
modelled by the equation
1
P = 80 e 3
(a)
t
t
,t≥0
Write down the number of rabbits that were introduced to the island.
(1)
(b)
Find the number of years it would take for the number of rabbits to first exceed 1000.
(2)
(c)
Find
dp
.
dt
(2)
(d)
Find P when
dp
 50.
dt
(3)
(Total 8 marks)
8.
(i)
Find the exact solutions to the equations
(a)
ln (3x – 7) = 5
(3)
(b)
3xe7x + 2 = 15
(5)
(ii)
The functions f and g are defined by
(a)
f (x) = e2x + 3,
x
g(x) = ln (x – 1),
x
,x>1
Find f–1 and state its domain.
(4)
(b)
Find fg and state its range.
(3)
(Total 15 marks)
5
9.
(a)
Simplify fully
2x 2  9x  5
x 2  2 x  15
(3)
Given that
ln(2x2 + 9x – 5) = 1 + ln(x2 + 2x – 15), x ≠ – 5,
(b)
find x in terms of e.
(4)
(Total 7 marks)
6
C3 Functions
1.
The functions f and g are defined by
(a)
f : x  2x + ln 2,
x
g : x  e2x,
x
Prove that the composite function gf is
gf : x  4e4x, x 
(4)
(b)
In the space provided below, sketch the curve with equation y = gf(x), and show the coordinates of
the point where the curve cuts the y-axis.
(1)
(c)
Write down the range of gf.
(1)
(d)
Find the value of x for which
d
gf x 3 , giving your answer to 3 significant figures.
dx
(4)
(Total 10 marks)
2.
For the constant k, where k > 1, the functions f and g are defined by
x > –k,
f: x  ln (x + k),
g: x  |2x – k |,
(a)
x ∈.
On separate axes, sketch the graph of f and the graph of g.
On each sketch state, in terms of k, the coordinates of points where the graph meets the coordinate axes.
(5)
(b)
Write down the range of f.
(1)
k
Find fg  in terms of k, giving your answer in its simplest form.
 4
(c)
(2)
The curve C has equation y = f(x). The tangent to C at the point with x-coordinate 3 is parallel to the line
with equation 9y = 2x + 1.
(d)
Find the value of k.
(4)
(Total 12 marks)
3.
The function f is defined by
f : x  ln(4 – 2x), x < 2 and x 
(a)
.
Show that the inverse function of f is defined by
7
1
f 1 : x  2  e x
2
and write down the domain of f–1.
(4)
(b)
Write down the range of f–1.
(1)
(c)
Sketch the graph of y = f–1(x). State the coordinates of the points of intersection with the x and y axes.
(4)
The graph of y = x + 2 crosses the graph of y = f–1(x) at x = k.
The iterative formula
1
x n 1   e xn , x 0   0.3
2
is used to find an approximate value for k.
(d)
Calculate the values of x1 and x2, giving your answers to 4 decimal places.
(2)
(e)
Find the value of k to 3 decimal places.
(2)
(Total 13 marks)
4.
The functions f and g are defined by
f : x  ln 2 x  1,
g:x
(a)
2
,
x3
1
x  , x  .
2
x  , x  3.
Find the exact value of fg(4).
(2)
(b)
Find the inverse function f–1(x), stating its domain.
(4)
(c)
Sketch the graph of y = g(x). Indicate clearly the equation of the vertical asymptote and the
coordinates of the point at which the graph crosses the y-axis.
(3)
(d)
Find the exact values of x for which
2
 3.
x 3
(3)
(Total 12 marks)
5.
The functions f and g are defined by
f : x  1  2x 3 ,
3
g : x   4, x  0
x
(a)
x 
x 
Find the inverse function f–1.
(2)
8
(b)
Show that the composite function gf is
gf : x 
8 x3  1
1  2 x3
(4)
(c)
Solve gf(x) = 0.
(2)
(d)
Use calculus to find the coordinates of the stationary point on the graph of y = gf(x).
(5)
(Total 13 marks)
6.
The function f is defined by
f :x
(a)
Show that f ( x) 
2( x  1)
1

,
x  2x  3 x  3
x3
2
1
,
x 1
x3
(4)
(b)
Find the range of f.
(2)
(c)
Find f–1 (x). State the domain of this inverse function.
(3)
The function g is defined by
g : x  2 x 2  3, x   .
(d)
Solve fg( x) 
1
.
8
(3)
(Total 12 marks)
7.
The functions f and g are defined by
f : x  3x  ln x, x  0, x
2
g : x  ex , x
(a)
Write down the range of g.
(1)
(b)
Show that the composite function fg is defined by
2
fg : x  x 2  3 x , x 
(2)
(c)
Write down the range of fg.
(1)
(d)
Solve the equation
2
d
fg ( x)  x( xe x  2).
dx
(6)
(Total 10 marks)
9
8.
(i)
Find the exact solutions to the equations
(a)
ln (3x – 7) = 5
(3)
(b)
3xe7x + 2 = 15
(5)
(ii)
The functions f and g are defined by
(a)
f (x) = e2x + 3,
x
g(x) = ln (x – 1),
x
,x>1
Find f–1 and state its domain.
(4)
(b)
Find fg and state its range.
(3)
(Total 15 marks)
9.
The function f is defined by
f : x  2x  5 ,
(a)
x
Sketch the graph with equation y = f(x), showing the coordinates of the points where the graph cuts
or meets the axes.
(2)
(b)
Solve f = (x) = 15 + x.
(3)
The function g is defined by
g : x  x 2  4 x  1, x  , 0 ≤ x ≤ 5
(c)
Find fg(2).
(2)
(d)
Find the range of g.
(3)
(Total 10 marks)
10
C3 Transformations
1.
y
M (2, 4)
O
–5
5x
The figure above shows the graph of y = f(x), –5  x  5.
The point M (2, 4) is the maximum turning point of the graph.
Sketch, on separate diagrams, the graphs of
(a)
y = f(x) + 3,
(2)
(b)
y= |f(x)|,
(2)
(c)
y = f(|x|).
(3)
Show on each graph the coordinates of any maximum turning points.
(Total 7 marks)
2.
The functions f and g are defined by
(a)
f : x  2x + ln 2,
x
g : x  e2x,
x
Prove that the composite function gf is
gf : x  4e4x, x 
(4)
(b)
In the space provided below, sketch the curve with equation y = gf(x), and show the coordinates of
the point where the curve cuts the y-axis.
(1)
(c)
Write down the range of gf.
(1)
(d)
Find the value of x for which
d
gf x 3 , giving your answer to 3 significant figures.
dx
(4)
(Total 10 marks)
11
3.
y
y = f(x)
O
Q
(3, 0)
x
(0, –2)
P
The figure above shows part of the curve with equation y = f(x), x  , where f is an increasing function of
x. The curve passes through the points P(0, –2) and Q(3, 0) as shown.
In separate diagrams, sketch the curve with equation
(a)
y = |f(x)|,
(3)
(b)
y = f –1(x),
(3)
(c)
y=
1
2
f(3x).
(3)
Indicate clearly on each sketch the coordinates of the points at which the curve crosses or
meets the axes.
(Total 9 marks)
4.
For the constant k, where k > 1, the functions f and g are defined by
f: x  ln (x + k),
g: x  |2x – k |,
(a)
x > –k,
x ∈.
On separate axes, sketch the graph of f and the graph of g.
On each sketch state, in terms of k, the coordinates of points where the graph meets the coordinate axes.
(5)
(b)
Write down the range of f.
(1)
(c)
k
Find fg  in terms of k, giving your answer in its simplest form.
 4
(2)
The curve C has equation y = f(x). The tangent to C at the point with x-coordinate 3 is parallel to the line
with equation 9y = 2x + 1.
(d)
Find the value of k.
(4)
(Total 12 marks)
12
5.
y
A (5, 4)
O
x
B (– 5, – 4)
The diagram above shows a sketch of the curve with equation y = f(x).
The curve passes through the origin O and the points A(5, 4) and B(–5, –4).
In separate diagrams, sketch the graph with equation
(a)
y = |f(x)|,
(3)
(b)
y = f(|x|),
(3)
(c)
y = 2f(x + 1).
(4)
On each sketch, show the coordinates of the points corresponding to A and B.
(Total 10 marks)
6.
y
P
Q
–3
R
x
The diagram above shows the graph of y  f ( x), x   .
The graph consists of two line segments that meet at the point P.
The graph cuts the y-axis at the point Q and the x-axis at the points (–3, 0) and R.
Sketch, on separate diagrams, the graphs of
(a)
y = |f(x)|
(2)
13
(b)
y = f (–x).
(2)
Given that f(x) = 2 – |x + 1|,
(c)
find the coordinates of the points P, Q and R,
(3)
(d)
solve f ( x) 
1
x.
2
(5)
(Total 12 marks)
7.
The figure above shows the graph of y = f (x), 1 < x < 9.
The points T(3, 5) and S(7, 2) are turning points on the graph.
Sketch, on separate diagrams, the graphs of
(a)
y = 2f(x) – 4,
(3)
(b)
y  f ( x) .
(3)
Indicate on each diagram the coordinates of any turning points on your sketch.
(Total 6 marks)
8.
14
The figure above shows a sketch of part of the curve with equation y = f(x), x 
The curve meets the coordinate axes at the points A(0, 1 – k) and 1 B  12 1n k , 0 ,
where k is a constant and k >1, as shown in the diagram above.
On separate diagrams, sketch the curve with equation
(a)
y = f ( x) ,
(3)
(b)
y = f–1(x)
(2)
Show on each sketch the coordinates, in terms of k, of each point at which the curve meets or cuts the axes.
Given that f(x) = e2x – k,
(c)
state the range of f,
(1)
(d)
find f–1(x),
(3)
(e)
write down the domain of f–1.
(1)
(Total 10 marks)
9.
Sketch the graph of y = ln x , stating the coordinates of any points of intersection with the axes.
(Total 3 marks)
10.
The diagram above shows a sketch of the graph of y = f(x).
The graph intersects the y-axis at the point (0, 1) and the point A(2, 3) is the maximum turning point.
15
Sketch, on separate axes, the graphs of
(i)
y = f(-x) + 1,
(ii)
y = f(x + 2) + 3,
(iii)
y = 2f(2x).
On each sketch, show the coordinates of the point at which your graph intersects the y-axis and the
coordinates of the point to which A is transformed.
(Total 9 marks)
11.
The diagram above shows a sketch of the curve with the equation y = f(x), x 
.
The curve has a turning point at A(3, – 4) and also passes through the point (0, 5).
(a)
Write down the coordinates of the point to which A is transformed on the curve with equation
(i)
y  f ( x) ,
(ii)
y = 2f  12 x .
(4)
(b)
Sketch the curve with equation
y  fx 
(3)
On your sketch show the coordinates of all turning points and the coordinates of the point at which the
curve cuts the y-axis. The curve with equation y = f(x) is a translation of the curve with equation y = x2.
(c)
Find f(x).
(2)
(d)
Explain why the function f does not have an inverse.
(1)
(Total 10 marks)
16
C3 Numerical Methods
1.
f(x) = 2 x3– x – 4.
(a)
Show that the equation f(x) = 0 can be written as
2 1
x   
 x 2
(3)
The equation 2x3 – x – 4 = 0 has a root between 1.35 and 1.4.
(b)
Use the iteration formula
 2 1
xn 1     ,
 x 2
with x0 = 1.35, to find, to 2 decimal places, the values of x1, x2 and x3.
(3)
The only real root of f(x) = 0 is α.
(c)
By choosing a suitable interval, prove that α = 1.392, to 3 decimal places.
(3)
(Total 9 marks)
f(x) = – x3 + 3x2 – 1.
2.
(a)
Show that the equation f(x) = 0 can be rewritten as
 1 
 .
x  
 3 x 
(2)
(b)
Starting with x1 = 0.6, use the iteration
 1
x n 1  
 3  xn



to calculate the values of x2, x3 and x4, giving all your answers to 4 decimal places.
(2)
(c)
Show that x = 0.653 is a root of f(x) = 0 correct to 3 decimal places.
(3)
(Total 7 marks)
f(x) = ln(x + 2) – x + 1, x > –2, x 
3.
(a)
.
Show that there is a root of f(x) = 0 in the interval 2 < x < 3.
(2)
(b)
Use the iterative formula
xn+1 = 1n(xn + 2) + 1, x0 = 2.5
17
to calculate the values of x1, x2 and x3 giving your answers to 5 decimal places.
(3)
(c)
Show that x = 2.505 is a root of f(x) = 0 correct to 3 decimal places.
(2)
(Total 7 marks)
f(x) = 3x3 – 2x – 6
4.
(a)
Show that f(x) = 0 has a root, α, between x = 1.4 and x = 1.45
(2)
(b)
Show that the equation f (x) = 0 can be written as
 2 2
x    , x  0 .
 x 3
(3)
(c)
Starting with x0 = 1.43, use the iteration
 2 2
xn 1    
 xn 3 
to calculate the values of x1, x2 and x3, giving your answers to 4 decimal places.
(3)
(d)
By choosing a suitable interval, show that α = 1.435 is correct to 3 decimal places.
(3)
(Total 11 marks)
f(x) = 3xex – 1
5.
The curve with equation y = f (x) has a turning point P.
(a)
Find the exact coordinates of P.
(5)
The equation f (x) = 0 has a root between x = 0.25 and x = 0.3
(b)
Use the iterative formula
1
x n 1  e  xn
3
with x0 = 0.25 to find, to 4 decimal places, the values of x1, x2 and x3.
(3)
(c)
By choosing a suitable interval, show that a root of f(x) = 0 is x = 0.2576 correct to 4 decimal places.
(3)
(Total 11 marks)
6.
18
The diagram above shows part of the curve with equation y = –x3 + 2x2 + 2, which intersects the x-axis at
the point A where x = α.
To find an approximation to α, the iterative formula
xn 1 
2
2
( xn ) 2
is used.
(a)
Taking x0 = 2.5, find the values of x1, x2, x3 and x4.
Give your answers to 3 decimal places where appropriate.
(3)
(b)
Show that α = 2.359 correct to 3 decimal places.
(3)
(Total 6 marks)
f(x) = x3 + 2x2 – 3x – 11
7.
(a)
Show that f(x) = 0 can be rearranged as
 3x  11 
x 
,
 x2 
x  2.
The equation f(x) = 0 has one positive root α.
(2)
 3x  11 
 is used to find an approximation to α.
The iterative formula x n 1   n
 xn  2 
(b)
Taking x1 = 0, find, to 3 decimal places, the values of x2, x3 and x4.
(3)
(c)
Show that α = 2.057 correct to 3 decimal places.
(3)
(Total 8 marks)
f(x) = 4 cosec x – 4x + 1, where x is in radians.
8.
(a)
Show that there is a root α of f (x) = 0 in the interval [1.2, 1.3].
(2)
(b)
Show that the equation f(x) = 0 can be written in the form
19
x
1
1

sin x 4
(2)
(c)
Use the iterative formula
xn 1 
1
1
 , x0  1.25,
sin xn 4
to calculate the values of x1, x2 and x3, giving your answers to 4 decimal places.
(3)
(d)
By considering the change of sign of f(x) in a suitable interval, verify that α = 1.291 correct to 3
decimal places.
(2)
(Total 9 marks)
20