Download Advanced Higher Revision

Advanced Higher Revision
Unit 1 Outcome 1
1.
Expand
(u  3v) 4 by the binomial theorem.
2.
Expand
(2 x  3 y)5 by the binomial theorem.
3.
D E
 2 2
8
5
2
Express  x   in the form Ax  Bx  Cx 

.
x
x x4

4.

2 
Find the y term in the expansion of  3 y  2 
y 

5.
Expand
6.
Find the term independent of
4
5
2
(2  x)4 and use your expansion to find
x
a)
(2.1)4
b)
(1.9)4 .
in the expansions
 3

 2x 
2
x

6
a) 
1

 4x 
x

4
b) 
 n  1  n 

     36
 2  1
7.
Solve for n:
8.
Prove that
9.
Express
10.
Obtain partial fractions for
 n   n   n  1
    

 3  2  3 
x
in partial fractions.
( x  2)( x  3)
a)
10
x2  2
;
; b)
2
2
x ( x  1)
( x  1)( x  3)
b)
g ( x)  ( x  1)sin 2 x
c)
x 3  4x  1
.
( x  2)( x  1)
Unit 1 Outcome 2
1. Differentiate with respect to x:
a) f ( x)  exp(sin 2 x)
2. Given that f ( x)  x tan 2 x , show that
3. Show that the function
equation
y
c) h( x ) 
ln 3 x
.
cos 3 x
f ''( x)  4sec2 2x 1  2xtan2x  .
sin(kx)
where x  0 and k is a non-zero constant, satisfies the differential
x
d 2 y 2 dy

 k 2 y  0.
2
dx
x dx

4. The turning effect, T, of a power boat, is given by the formula T  4 cos x sin x , 0  x 
2
where x is
2
the angle (in radians) between the rudder and the central line of the boat. Find the size of x which
maximises the turning effect.
Unit 1 Outcome 3
1. Find
a)
x
 x  3dx
2x  4
b)
 2 x  3dx
b)
x3
dx
x 1
c)

3
0
6x
dx
2 x2  3
d)
sin xcos 4 xdx
d)
x
3x
dx
4
2
2. Integrate by substitution.
a)
3.

e
x
x
dx
a) Show that

ln 2
ln
3
2

e x  e x
9
dx  ln
x
x
e e
5
c)
b) Find


3
0

2
1
e2 x
dx
(e2 x  1)2
12 x3  6 x
 x 4  x 2  1dx
4. A car is travelling along a straight road. Its acceleration at time t seconds is
a(t )  20  6t  4t 2 measured in metres per second per second. The car started from rest at time t  0
from a point O on the road. Find the speed of the car and its distance from O after t seconds.
5. By using the substitution
1
x  sin  , evaluate
3
6. If a(t )  1  t , and when t  2, v  1 & s  4
1
6
0

1
1
2 2
dx .
(1  9 x )
2
find expressions for v (t ) & s (t ).
3
7. A particle P moves in a straight line and passes a fixed point O with a velocity of V m/s. Its acceleration,
a m/s2 is given by a  16  4t for 0  t  3 & a  t  1 for t  3 , when t is the time in seconds after
passing O. Given that the velocity of P when t  3 is 38m/s find the velocity of v (t ) when
a) t  2 b)
t  4.
8. Find the volume of the solid of revolution which is created when one revolution around the
area bounded between the
x
axis and
x
axis of the
y  x  1 between x  1 and x  1.
2
9. Find the volume of the solid of revolution created by the area lying in the first quadrant, bounded by the
curve
y  x 2 , the y axis, and the line y  4 .
Unit 1 Outcome 4
1. For each of the following functions, state whether they are even, odd, or neither.
a)
f ( x)  4  3 x 2
b)
f ( x)  3 x 2  x
2. Sketch the function f ( x)  x 
3. Sketch the graph y 
c) f ( x)  x 
1
x
d)
f ( x)  x3  x
1
showing all important points and lines. Use this result to sketch f ( x) .
x
x2
, marking on any asymptotes, stationary points, and points where axes are
x2 1
crossed.
4. Sketch the function f ( x) 
x 1
, marking on all important features as above. Use this result to
x( x  1)
f 1 ( x).
sketch f ( x  2) and
Unit 1 Outcome 5
x 
1. Use Gaussian Elimination to solve the equations
 4
y
x  2y 
z
 5
3 y  2 z  6.
2. For what value of
a and b will the following system of equations have
2x 
x
y
 3y
 3z
a) No solution
b) Infinitely many solutions
c) A unique solution?
 5
 2z  8
7 x  4 y  az
 b
3. Which of these systems would you consider ill conditioned?
a)
2x 
3x
7y
 5
b)
 10 y  6
2x 
3x
7y
 5
 10 y  6
Unit 2 Outcome 1
1. Find the inverse of the following functions
a) y 
x
3
b) y  4  x
c) y  2 x  4
d) y 
1
.
x2
2. Differentiate
1 2
a) sin x
b)
ex
sin 1 2 x
1
c) ln x tan x
2 x2
d)
cos 1 (1  x)
3. Find the Cartesian Equation for each of the following parametric forms.
x  3t  2 , y  4  t 2
a)
4. Find
b) x 
1
2
, y t 2
1 t
d2y
dy
and
in terms of t when
dx
dx 2
x  t  2, y  t2  4
a)
b) x  t  4t , y 
2
1
t.
5. Find the Cartesian Equation for the parametric form
x  sin   2cos ,
y  2sin   cos  .
6. Find
dy
in terms of x and y if:
dx
a)
7. Find an expression for
2 y 3  3 y  xy  x 2
b)
6 x 2  3xy  2 y  4 .
dy
given that
dx
a)
y  2x

b) y
x2
x 1
c)
y  2xx
8. If r, the radius of a circle increases at the rate of 2cm/s, find an expression in terms of r for the rate at
which the area of the circle is increasing.
9. Air is being pumped into a spherical balloon at a rate of 54cm 3/s. Find the rate at which the radius is
increasing when the volume of the balloon is 36 cm3.
x  1  cos t  sin t , y  1  cos t  sin t. .
2
2
If v and a are the velocity and acceleration at any instant, prove that v  a  4 .
10. A particle is moving in a plane under the equations of motion
Unit 2 Outcome 2
1. Express
3
in partial fractions, and hence evaluate
2
2x  5x  2

4
2
3
dx to 3.d.p.
2 x  5x  2
2
2. Find the following integrals.
a)
1
 25  16 x
2
dx (Hint: let u  4x .)
b)
3. Find the following indefinite integrals in terms of
a)
 x( x  2) dx ; u  x  2
9
b)

1
2  9 x2
dx
c)
x
2
1
dx
 4 x  29
x using the suggested substitution.
1
 1  16 x
2
dx ; u  4x
c)

1
2
dx ; x  sin u
3
4  9x
2
4. Use integration by parts to evaluate


0
x sin xdx.
5. Find;
a)

4
2
( x  1) ln(2 x)dx
b)


4
0
x cos xdx
c)
e
x
cos 2 xdx
d)

2
0
x3e x dx
8x
x3  11x  3
dx
dx .
6. Find
a)  2
b) 
( x  4)5
9  x2
dy
1
 2 given that x  1 when y  2.
7. Solve
dx x y
dy
 (2 x 2  4 x  1)( y  3) if when x  0 , y  7.
8. Solve ( x  2)
dx
9. a) Express
1
in partial fractions.
x  x2
b) The spread of disease in a large population can be modelled by means of a differential equation. The
proportion
x of the population infected with the disease after t days satisfies
i) Given that
x
dx 1
1
 x  x 2 for t  0 .
dt 2
2
1
when t  0 , find x in terms of t .
500
ii) Verify that about 6% of the population was infected after seven days.
iii) How long will it take for 25% of the population to become infected?
t  0 , just one individual with a contagious
dn
 k ( N  n)n where n(t ) is
disease. Assume that the spread of the disease is governed by the equation
dt
the number of infected individuals after a time t days and k is a constant.
10. A large population of
a) Find
N
individuals contains, at time
n explicitly as a function of t .
b) Given that measurements reveal that half the population is infected after 100 days, show that
k  ln( N 1) /100 N.
c) What value does
11. a) Express
n(t ) approach as t tends to infinity.
1
in partial fractions.
x  x2
b) The spread of disease in a large population can be modelled by means of a differential equation. The
proportion
x of the population infected with the disease after t days satisfies
i) Given that
x
1
when t  0 , find x in terms of t .
500
ii) Verify that about 6% of the population was infected after seven days.
iii) How long will it take for 25% of the population to become infected?
dx 1
1
 x  x 2 for t  0 .
dt 2
2
Unit 2 Outcome 3
1. Given that
z  2  3i , plot on an Argand diagram the points that represent the complex numbers z , z and
2
z , where z is the complex conjugate of z.
2. Express in the form
a  ib
a)
2  3i
1 i
3. Find the modulus and principle argument of


4. If z  2  cos

3
 i sin
a)
b)
1 i
2i
5
4  3i
b)
c)
1  6i
.
1  6i
(3  i ) 2
. Hence express in polar form.
1 i

6
 find z .
3
sin 3  3sin   4sin 3 
5. Use De Moivre’s Theorem to show that
Hence obtain an expression for
cos 3  4 cos3   3cos 
tan 3 in terms of tan  .
6. Find the locus of P when
a)
7. Given that
z 3
b)
z 2  4
c)
2  i is a root of the equation z 3  11z  20  0
8. Show that 1  i is a root of the equation
Hence find all roots.
z 1  3i  5
11. Given

4
.
find the remaining roots.
x 4  3x 2  6 x  10  0
9. By Using De Moivre’s Theorem or otherwise, find the roots of the equation
10. Given that
d) arg z 
z4  4  0
z  2  i is one root of z 4  z 3  z 2  9 z  30  0 find all the remaining roots.
z  2 3  2i find i) z 2 ,
ii)
z10 .
12. Solve the following equations, leaving your answers in polar form. Illustrate the solutions on an Argand
diagram.
a) z  2  2i
4
b)
z 5  3  3 3i .
Unit 2 Outcome 4
3
and S  40.
5
2
2. Find the sum of the first seven terms of a geometric series which has u8 
and u5  18 .
3
1. Find the first term of a geometric progression which has a common ratio of
3. An arithmetic sequence has a common difference of 7, and the 25th term equals 300. What is the first
term?
4. The fourth term of an arithmetic sequence is -3 and the tenth term is -15.
a) Identify the sequence
b) For what value of n is
5. By writing
un  35 ?
(k  1)2  k 2  2k  1 show that
 (k  1)
n
k 1
n
Hence show that
k 
k 1
2

n
 k 2  2 k  n.
k 1
n(n  1)
.
2
Extend this idea by considering
(k  1)3  k 3 to show that
n
k
2
k 1

n(n  1)(2n  1)
.
6
Unit 2 Outcome 5
1. Prove that
k (k 2  5) id divisible by 6. Hence prove that, if n is even, then n2 (n2  20) is divisible by 48.
2. Prove by contradiction that given a, b 
3. Prove by induction that a)
ab is even then at least one of a or b is even.
2n  n, n  ;
b)
2n  n2 , n  4, n  .
n
4. Prove by induction that
, if
r
n
 r (r  1)  n  1 .
r 1
n
5. Prove by induction that
r
2
(r  1) 
r 1
n
(n  1)(n  2)(3n  1).
12
Unit 3 Outcome 1
2
3
 4 
 
 
 
1. Find a.(b  c) if a  1 , b  1 and c  1 .
 
 
 
4
0
 3
 
 
 
2. Find the symmetric form of the equation of a line through the points A(2, 7, 4) and B(1, 2, 4) .
3. Find the equation of the plane passing through the points D(3,1, 4), E (6, 0,1) and F (1, 5, 2).
4. Find the vector and symmetric equation for the line parallel to the vector
3i  4 j  k , passing through the
point (2,3, 2).
5. The lines and are contained within the plane . Find the equation of the plane.
6. The coordinates of the points
L is
x 3 y 2 z 2


.
2
2
1
A and B are (0, 2, 5) and (-1, 3, 1) respectively. The equation of the line
a) Find the equation of the plane  which contains A and is perpendicular to L .
b) Verify that B lies in  .
c) Show that the point C in which L meets  is (1, 4, 3), and find the angle between
CA and CB .
Unit 3 Outcome 2
 3
1 4 
 10 2 
, B 
 & C 
 find, where possible;
1
 0 3 
 8 1
T
a) A  B
b) BC
c) BA
d) AB
e) CB
f) B
1
g) det B
h) B
i) the 2x2 matrix D such that BD  C .
 x2 3 
3 6
2. Find the possible values x can take given that A  
, B
 and AB  BA .
2
x
1
3
x




 3 2
2
3. If A  
 find the values of m and n if A  mA  nI .
 4 1 
1. If A  
4. Find the 2x2 matrix which will transform the point (1, 2) to (3, 3) and the point (-1, 1) to (-3, 3).
5. Find the matrices corresponding to the following linear transformations.
a) 180o rotation about the origin;
b) Enlargement with a scale factor of 3, centre (0, 0);
c) Reflection in the line y  x ;
d) 45o rotation about the origin.
6. Give a geometrical description of the effect of the following transformation matrices;
a)
0 1


1 0
b)
5 0


0 5
c)
3 0


0 1
d)
0 3


3 0
e)
1 0


2 1
f)
1 0

.
0 0
1 0 0


2
7. Given that A  0 k 1 , calculate A and find the values of k for which the determinant of the matrix


0 0 k 


2
A  2I is zero. ( I is the 3x3 identity matrix.)
1 1 0


1
8. Calculate A where A  2 3 1


 2 2 1


Hence solve the system of equations
x

y
 1
2x  3y  z  2
2x  2 y  z  1
9. For the matrix
2 

A
 , find the values of  such that the matrix is singular.
 2   3
Write down the matrix
A1 when   3.
Unit 3 Outcome 3
56. Use Maclaurin’s theorem to find series expansions for each of the following, giving all terms up to and
including that in
x5 ;
a)
cos( x)
2. Find the Maclaurin expansion of
3. Prove that the equation
b)
ex
c)
e2 x
d)
(1  x)n .
(2  x)2 e x as far as the term in x3
x3  2 x  1  0 has only one root in the interval 1  x  3. Verify that the equation
1
1
can be rewritten as
x  (2 x  1) 3 . By using the iterative scheme xn1  (2 xn  1) 3 with xo  2 , obtain an
approximation to the root which is correct to two decimal places.
1
5
xn 1   xn   with x0  2 to calculate x1 , x2 and x3 .
2
xn 
4. Apply the recurrence relation
Find the fixed points of this recurrence relation.
Suggest an initial value to generate a sequence converging to the negative fixed point.
Unit 3 Outcome 4
1. An industrial scientist finds that the differential equation
t
dx
 2 x  3t 2 models a production process.
dt
Find the general solution of the differential equation.
Hence find the particular solution given x  1 when t
 1.
.
2
dy
d y
 6 x  2 , given that when x  1,  2 and y  3.
2
dx
dx
2. Solve the differential equation
3. Find the general solution to each of the following differential equations.
d2y
dy
 3  10 y  0
a)
2
dx
dx
d2y
dy
 8  16 y  0
b)
2
dx
dx
c)
d2y
dy
 2  15 y  e4 x
2
dx
dx
d)
2
e)
d2y
dy
 2  5 y  10 x  1
2
dx
dx
f)
2
d2y
dy
 5  3 y  4e5 x
2
dx
dx
d2y
dy
 11  12 y  2 x 2  5 x  7 .
2
dx
dx
Unit 3 Outcome 5
1. Change 523 into a) binary
b) octal c) Hexadecimal.
2. Find the gcd of 286 and 142
3. Express the gcd of 132 and 424 in the form
132s  424t where s and t 
and are to be calculated.