C3 Chapter 4 Numerical Methods – Exam Questions
[June 2009] 1.
Figure 1 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.
(b) Show that α = 2.359 correct to 3 decimal places.
(3)
(3)
f(x) = x3 + 2x2 − 3x − 11
[Jan 2010] 2.
(a) Show that f(x) = 0 can be rearranged as
x=
 3x  11 

,
 x2 
(2)
x  –2.
The equation f(x) = 0 has one positive root α.
The iterative formula xn + 1 =
 3xn  11 

 is used to find an approximation to α .
 xn  2 
(b) Taking x1 = 0, find, to 3 decimal places, the values of x 2 , x3 and x 4 .
(c) Show that α = 2.057 correct to 3 decimal places.
(3)
(3)
f(x) = x3 + 3x2 + 4x – 12
[June 2012] 2.
(a) Show that the equation f(x) = 0 can be written as
x=
 4(3  x) 

 ,
 (3  x) 
x  –3.
(3)
C3 Chapter 4 Numerical Methods – Exam Questions
[June 2009] 1.
Figure 1 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.
(b) Show that α = 2.359 correct to 3 decimal places.
(3)
(3)
f(x) = x3 + 2x2 − 3x − 11
[Jan 2010] 2.
(a) Show that f(x) = 0 can be rearranged as
x=
 3x  11 

,
 x2 
(2)
x  –2.
The equation f(x) = 0 has one positive root α.
The iterative formula xn + 1 =
 3xn  11 

 is used to find an approximation to α .
 xn  2 
(b) Taking x1 = 0, find, to 3 decimal places, the values of x 2 , x3 and x 4 .
(c) Show that α = 2.057 correct to 3 decimal places.
(3)
(3)
f(x) = x3 + 3x2 + 4x – 12
[June 2012] 2.
(a) Show that the equation f(x) = 0 can be written as
x=
 4(3  x) 

 ,
 (3  x) 
x  –3.
(3)
The equation x3 + 3x2 + 4x – 12 = 0 has a single root which is between 1 and 2.
(b) Use the iteration formula
xn + 1 =
 4(3  x n ) 

 , n  0,
 (3  xn ) 
with x 0 = 1 to find, to 2 decimal places, the value of x1 , x 2 and x3 .
(3)
The root of f(x) = 0 is  .
(c) By choosing a suitable interval, prove that  = 1.272 to 3 decimal places.
f (x) = 2 sin (x2) + x − 2,
[June 2011] 2.
(3)
0  x < 2.
(a) Show that f (x) = 0 has a root  between x = 0.75 and x = 0.85.
(2)
The equation f (x) = 0 can be written as x = arcsin (1  0.5 x)2 .
1
(b) Use the iterative formula
xn + 1 = arcsin (1  0.5 x n )2 ,
1
x 0 = 0.8,
to find the values of x1 , x 2 and x3 , giving your answers to 5 decimal places.
(c) Show that  = 0.80157 is correct to 5 decimal places.
[June 2008] 7.
(3)
(3)
f(x) = 3x3 – 2x – 6.
(a) Show that f (x) = 0 has a root, α, between x = 1.4 and x = 1.45.
(b) Show that the equation f (x) = 0 can be written as
x=
(2)
 2 2
   , x  0.
 x 3
(3)
(c) Starting with x 0 = 1.43, use the iteration
xn + 1 =
 2 2
  
 xn 3 
to calculate the values of x1 , x 2 and x3 , giving your answers to 4 decimal places.
(d) By choosing a suitable interval, show that α = 1.435 is correct to 3 decimal places.
(3)
(3)
(b) Use the iteration formula
xn + 1 =
 4(3  x n ) 

 , n  0,
 (3  xn ) 
with x 0 = 1 to find, to 2 decimal places, the value of x1 , x 2 and x3 .
(3)
The root of f(x) = 0 is  .
(c) By choosing a suitable interval, prove that  = 1.272 to 3 decimal places.
f (x) = 2 sin (x2) + x − 2,
[June 2011] 2.
(3)
0  x < 2.
(a) Show that f (x) = 0 has a root  between x = 0.75 and x = 0.85.
(2)
The equation f (x) = 0 can be written as x = arcsin (1  0.5 x)2 .
1
(b) Use the iterative formula
xn + 1 = arcsin (1  0.5 x n )2 ,
1
x 0 = 0.8,
to find the values of x1 , x 2 and x3 , giving your answers to 5 decimal places.
(c) Show that  = 0.80157 is correct to 5 decimal places.
[June 2008] 7.
(3)
(3)
f(x) = 3x3 – 2x – 6.
(a) Show that f (x) = 0 has a root, α, between x = 1.4 and x = 1.45.
(b) Show that the equation f (x) = 0 can be written as
x=
(2)
 2 2
   , x  0.
 x 3
(3)
(c) Starting with x 0 = 1.43, use the iteration
xn + 1 =
 2 2
  
 xn 3 
to calculate the values of x1 , x 2 and x3 , giving your answers to 4 decimal places.
(d) By choosing a suitable interval, show that α = 1.435 is correct to 3 decimal places.
(3)
(3)