Download FP1 Revision Worksheet Number 1

FP1 Revision Worksheet Number 1
1.
Fill in the gaps in the following table:
Angle in
degrees
Angle in
radians
sine
cosine
tangent
1
3
1
2

2
1
2.
(a)
(b)
For the equation 7 x3  23x  4  34 x 2
Show there is a root between 0 and 1
Use interval bisection to find the interval of width 0.25 that contains this root
3.
Try the following improper integrals. Either find the value, or explain why a
value cannot be found.
5
2 

0  6  x 2 dx
(a)

 6 
dx
7 

1
(b)
  x
4.
Solve the following equation to find z in the form a+bi:
zz*  23  15i  5 z
5.
Calculate the value of the following:
8
r
2
 4r  3
r 1
6.
Find the general solution of the following equation:
 1

cos  3 x   
4 2

7.
Write down the matrix that represents a transformation by 60 anticlockwise
about the origin, and find the image of the point (2,-1) under this
transformation.
8.
The equation 2 x 2  3x  1  0 has roots  ,  . Find an equation with integer
1
1
coefficients that has roots 2 and 2 .


FP1 Revision Worksheet Number 2
1.
Fill in the gaps in the following table:
Description
of
transforma
tion
2.
 1, 0 


 0, 1
1, 0 


 0, 1
Matrix
Rotation
by 30
clockwise
about the
origin
 4, 0 


 0, 4 
Reflection
in the line
1
y
x
3
Give the general solution of the following trig equation:
 1

sin  x   
6
2

3. Use Euler’s step-by-step method, with a step length of 0.1 to estimate the value of
dy
1
 2
y when x=0.3, given that y=1 when x=0 and
, giving your answer to
dx  x  4 
3 significant figures.
4.
5.
Solve the following quadratic equation:
5x2  8x  5  0
Sketch the following curve:
1 2x
y
x 1
Hence or otherwise solve the following inequality:
1 2x
2
x 1
6.
Find the following sum:
10
 3r
3
r 5
7. Sketch the following curve writing down the equations of any asymptotes or
points where the curve cuts the co-ordinate axes:
2
y 2
x  2x  3
8. By differentiating FROM FIRST PRINCIPLES find the gradient of the curve
y  x 2  3x at the point where x  2
FP1 Revision Worksheet Number 3
We want to plot a straight line Y  mX  c to verify the following laws. Fill in
the gaps in the table
1.
Law
y  ax 2 
Y
X
m
c
b
x
y  ab x
y  bx a
1
 ax  bx 2
y
2.
Show that the equation x 4  5 x  2  0 has a root between 0 and 1. Use linear
interpolation once to get an approximation for this root.
3.
Sketch the curve y 
4.
Find the general solution of the trig equation:


tan  5 x    3
6

5.
(a)
(b)
(c)
(d)
6.
x2  x  1
stating the equations of asymptotes and the
x 2  3x  2
coordinates of the points where the curve crosses the axes.
1, 0 
 0, 4 
A
 and B  
 . Find:
 2, 2 
 3, 1
AB
BA
3A-2B
2A+AB
Sketch the following curve, giving its name, the equations of any asymptotes
and the points at which it crosses the co-ordinate axes:
x2
 y2  1
4
Now describe the transformations that change the above curve into the curve
with equation:
( x  1) 2
2
  y  1  1
4
Sketch this new curve.
FP1 Revision Worksheet Number 4
1.
Fill in the gaps in the following table:
 1, 0 


 0,1 
Matrix
Description
of
transforma
tion
 2, 0 


 0,1 
Rotation
by 45
anticlockwi
se about
the origin
 1, 0 


 0, 4 
Reflection
in the line
yx
2.
Show that the equation x 5  2  x has a root between x  1 and x  2 .
Starting with an approximation of x  1.5 and using the Newton-Raphson
method once, obtain a better approximation to this root.
3.
Sketch the curve y 
4.
Solve the following equation finding z in the form a+bi:
x2
stating the equations of any asymptotes and points
x2  4
where the curve crosses the axes.
4 z  2  2 z *  12i
5.
6.
2 x 2  2 x  1
For the curve y 
, show that the curve does not exist for
x2  x
2  x  2 . Hence find the coordinates of the turning points of the curve.
Prove the following:
n
 6r
2
 2r  1  n(2n 2  4n  3)
r 1
7.
(a)
(b)
(c)
(d)
8.
On the same diagram sketch clearly the curves:
y2  x
y2  x  4
( y  4)2  x
y2  4x
Give the general solution of the following trig equation:


cos   x   0
4

FP1 Revision Worksheet Number 5
1.
By differentiating from first principles, find the gradient function for the
curve:
y  2 x2  7
2.
Find the general solution of the following trig equation:
1
sin 2 x 
2
3.
4.
(a)
5x  4
Sketch the curve y 
and hence or otherwise, solve the inequality
4 x
5x  4
0
4 x
Either give a value for the following integrals or a reason why the integral
does not exist:
5
1
0 x3 dx
2
(b)
4
 x dx
5

5.
Find a matrix that does the composite transformation of first rotating by
60 clockwise about the origin and then reflecting in the line y   tan 22.5  x
6.
Evaluate the following sum:
7
  r  1
2
r 2
7.
Use Euler’s method with a step length of 0.05 to estimate y when x=2.1 given
dy 20

that y=10 when x=2 and
dx x
8.
Show that the equation x3  3x  1  0 has a root between 1 and 2. Use interval
bisection to find an interval of width 0.125 which contains the root.
9.
(a)
(b)
The quadratic equation x 2  2 x  3  0 has roots  and  . Find the following:
 

(c)
 2
2
1

1
(d)

10.
Sketch the following curve, and write down its name and the co-ordinates of
the centre, and any points where it crosses the co-ordinate axes:

 x  4
25
2
 y  2

100
2
1
FP1 Revision Sheet Answers
Sheet 1: Q1 30,  6 , 1 2 , 3 2
45,  4 , 1 2 , 3
Q2 (0,0.25)
Q3 (a) undefined (b) 1
Q5 84
Q8
Q6

36
90,1,0, 
 2 n3 , 367  2 n3
180,  ,0,0
Q4 2+3i or -7+3i
1, 3
Q7  2 2 
 3,1 
 2 2 
x2  5x  4  0
 23 , 12 
 12 , 23 
Sheet 2: Q1 reflection in x axis, 
 , rotation by 180, 
 , enlargement
 1 , 3 
 3 , 1 
2 2
 2 2
scale factor 4 centre the origin
Q2 512  2n , 1112  2n
Q3 1.07
Q4 453i
Q5 Asymptotes x=-1, y=-2 Points (0,1) (0.5,0)
1  x  41
Q6 8775
Q7 Asymptotes x=-1, x=3, y=0 Does not cross x-axis; crosses y axis

2
at  0, 3 
Q8 1
Sheet 3: Q1 Possible answers are
yx, x3 , a, b
1
log y, log x, a, log b
Q2 0.5
yx , x, b, a
log y, x,  log b, log a
Q3 Asymptotes are x=1,x=2,y=1
Doesn’t cross x-axis. Crosses y-axis at (0,0.5)

Q4 x  30
 n5
 0, 4   8,8   3, 8   2, 4 
Q5 
 '
,
 '

 6, 10   5, 2   12,8   2, 6 
Q6
cross y-axis but crosses x-axis at (2,0) and(-2,0)
Asymptotes are y  
Hyperbola
Does not
x
2
Shifts it to the right by 1 and down by 1
 12 , 21 
 0,1 
Sheet 4: Q1 reflection in y-axis,  1 1  , stretch by factor 2 in x direction,   ,
 , 
 1, 0 
 2 2
stretch by factor 4 in y direction
Q2 -1.33
Q3 x=2, x=-2,
y=0, (-2,0), (0,-0.5)
Q4 1-2i
Q5 (-0.5,2) is only turning point
Q8 34  2n , 4  2n
Sheet 5: Q1 4x
Q2
Q3 x=4, y=-5 (0,-1)
45  360n ,135  360n , 45  360n , 215  360n
 4 5 ,0
x  4 ,x  4
5
Q4 (a) undefined (b)
 1 3 , 1 3 
2 2
2 2

Q5 
Q6 91
Q7 11.0
Q8 (1.5,1.625)
 1 3 , 3 1 
2 2 2 2
2
Q9 2, 3,10,
Q10 ellipse with centre (4,-2) which crosses axes at (0,4),
3
(0,-8), 4  2 6, 0 , 4  2 6, 0



1
16