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HOLIDAY ASSIGNMENT
1.
Find in dot product form, the equation of the plane
r  1  3  2 i  1    4  j     k
2a)
c)
A and B are points with position vectors a and b respectively. D is a point on the
line joining A to B such that AD : DB  3 : 4 . Find the position vector of D in terms of
a and b .
Find the symmetric equation of the straight line passing through the point (1,2,1) and
is normal to the plane 2 x  3 y  z  2 .
Find the point of intersection of the line in (b) above with the plane x  y  2 z  9 .
3.
Find the shortest distance between two skew lines:
b)
x  2 y  2 z 1


1
2
1
x 1 y  2
z


and
2
3
1
4.
A plane passes through the point (1,2,3) and is perpendicular to the vector i  5 j  4k .
The plane meets the z  plane in P and the y  plane in Q . Find:
i)
the equation of the plane.
ii)
the distance PQ .
5a)
Determine the equation of the plane through the points A(1, 1, 2) , B(2,  1, 3) and
C (1, 2,  2) .
A line through the point D(13, 1, 2) and parallel to the vector 12i  6 j  3k meets the
plane in (a) at point E . Find:
i)
the coordinates of E .
ii)
The angle between the line and the plane.
b)
6.
If
i)
ii)
iii)
iv)
v)
cos ec  sin   m and sec   cos   n, prove that m 2 n 2 (m 2  n 2  3)  1.
x  cos  cos 2 , and y  sin   sin 2 , show that
x 2  y 2  cos 2  2 cos 3  cos 4 and that 2 xy  sin 2  2 sin 3  sin 4
tan 2   3 tan 2   2 , show that cos 2  3 cos 2  2
x  a cos  b cos 3 , y  a sin   b sin 3 , show that x 2  y 2  (a  b) 2  4ab sin 2 
x  tan   sin  , y  tan   sin  , prove that ( x 2  y 2 ) 2  16 xy
k 1
sin A. . Find all the angles for
k 1
0 o  x  360 o which satisfy the equation 2 tan x  tan( 30 o  x)  0.
7.
Prove that if tan x  k tan( A  x) , then sin( 2 x  A) 
8.
Prove that if tan  tan(   )  k , then (k  1) cos( 2   )  (1  k ) cos  . Find all the
angles for 0 o  x  360 o which satisfy the equation tan  tan(   3) .
© MATHEMATICS DEPARTMENT 2011
Email: [email protected]
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HOLIDAY ASSIGNMENT
9.
If cos   cos  a , sin   sin   b , prove that cos(   )  12 (a 2  b 2  2) and
tan 12 (   )  b a . Hence solve the simultaneous equations
cos   cos  1 , sin   sin   1.5
10.
Given that sin   sin   a , cos   cos   b , prove that tan 12 (   )  a b and
sin(    )  2ab (a 2  b 2 ) .
11.
Show that the perpendicular distance from a point P( x1 , y1 ) to a line ax  by  c  0 is
ax1  by1  c
12.
13.
14.
15.
.
a2  b2
Find the equation of the line through the origin and concurrent with 2 x  5 y  3 and
3x  4 y   2 .
Prove that the quadrilateral with vertices (2, 1), (2, 3), (5, 6), (5, 4) is a parallelogram.
Find the equations of the straight lines drawn through the point (1, -2), making angles
45o with the x-axis.
ABCD is a quadrilateral with A2,  2 , B5,  1 , C 6, 2 and D3, 1 . Show that the
quadrilateral is a rhombus.
16.
One side of the rhombus is the line y  2 x , and two opposite vertices are the points
0, 0 and 4 12 , 4 12  . Find the equations of the diagonals, the coordinates of the other
two vertices and the length of the side.
17.
The curve C is given by y  ax 2  b x where a and b are constants. Given that the
gradient of C at the point 1, 1 is 5 , find a and b .
18.
A tangent to the parabola x 2  16 y is perpendicular to the line x  2 y  3  0 . Find the
equation of this tangent and the coordinates of its point of contact.
1
1
Show that the tangent to the curve y  4  2 x  2 x 2 at points  1, 4 and  , 2  ,
2
2
1
 1
respectively, pass through the point   , 5  . Calculate the area of the curve
2
 4
enclosed between the curve and the x  axis.
19.
20.
Show that the gradient of the curve y  x( x  3) 2 is zero at the point P1, 4 , and
sketch the curve. The tangent at P cuts the curve again at Q. Calculate the area
contained between the chord PQ and the curve.
© MATHEMATICS DEPARTMENT 2011
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HOLIDAY ASSIGNMENT
21.
If :i)
ii)
iii)
iv)
dy
d2y
and
in terms of  .
dx
dx 2
2
dy
3 d y
2
2
and show that (4 x  3 y)
3 0
3x  8 xy  3 y  3 find
dx
dx 2
2
d2y
 dy 


2
a
x  ay 2  by  c where a, b, c are constants, prove that


dx 2
 dx 
1
1
y  A tan 2 x  B(2  x tan 2 x), where A and B are constants, prove that
x  a cos 3  ,
(1  cos x)
y  b sin 3  find
d2y
y
dx 2
2
v)
vi)
vii)
d 2 y  dy 
y  (5 x  3) , show that y 2     5
dx
 dx 
3
d2y
dy
2
show that 4 1  x 2
 4x
 9y  0
y  x  1  x2
2
dx
dx
2
d 2 y  dy 
y  sec x , prove that y 2     y 4
dx
 dx 
2





22.
Express 12 x  8 x 2  5 in the form  a( x  b) 2  c and deduce the maximum value of the
curve.
23.
Express 2  3x  4 x 2 in the form a  b( x  c) 2 and find the turning point stating whether
its a maximum or a minimum.
24.
The sum of two numbers is 24. Find the two numbers if the sum of their squares is to
be minimum.
25.
A right circular is circumscribed about a sphere of radius a . If h is the distance form
the centre of the sphere to the vertex of the cone, show that the volume of the cone is
a 2 (a  h) 2
. Find the vertical angle of the cone when the volume is minimum.
3(h  a)
26.
A right circular cylinder is inscribed in a sphere of given radius a . Show that the
volume of the cylinder is h(a 2  14 h 2 ) , where h is the height of the cylinder. Find the
ratio of the height to the radius of the cylinder when its volume is greatest.
27.
A right circular cone of vertical angle 2 is inscribed in a sphere of radius a . Show
that the area of the curved surface of the cone is a 2 (sin 3  sin  ) and prove that its
greatest area is 8a 2 3 3 .
© MATHEMATICS DEPARTMENT 2011
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HOLIDAY ASSIGNMENT
28.
A right circular cylinder is inscribed in a sphere of given radius a . Prove that the total
area of its surface (including its ends) is 2a 2 (sin 2  cos 2  ) , where a cos is the
radius of an end. Hence prove that the maximum value of the total area is
a 2 ( 5  1) .
TOPICS YOU MUST GO THROUGH BEFORE COMING BACK TO
SCHOOL
Introducing Pure Mathematics
Advanced Level Pure Maths
TOPIC
SERIES
Book 1 and 2
Understanding Pure Mathematics
Advanced Level Pure Mathematics By Tranter
SUBTOPIC
- ARITHMETIC
PROGRESSION
- GEOMETRIC PROGRESSION
- PROOF BY INDUCTION
REFERENCE
Intrd P M Ch. 9 Under P M Ch. 8
BackHse 1 Ch. 13 Tranter Ch. 3
- PERMUTATIONS
- COMBINATIONS
BHSE 1 Ch. 12
Tranter Ch. 3
Undersg Pure Mths Ch. 7
BINOMIAL
THEOREM
- PASCAL’S TRIANGLE
- BINOMIAL EXPANSION
EXPONENTIAL
&
LOGARITHMIC
FUNCTIONS
- EXPONENTIAL FUNCTIONS
BHSE 1 Ch. 14 Tranter Ch. 3
BHSE II Ch. 4 Intrdg PM Ch. 10
Understanding PM Ch. 8
BACKHOUSE 2 Ch. 2
Understanding PM Ch .19
Tranter Ch. 13
- LOGARITHMIC FUNCTIONS
MACLAURINS’
THEOREM
INTEGRATION(II)
BACKHOUSE 2 Ch .16
Tranter Ch.13
BACK HOUSE 2 Ch. 1 , 2 & 13
DIFFERENTIAL
EQUATIONS
Intrdg Pure Mths Ch. 20
BHSE 2 Ch. 19
Understadg Pure Mths Ch. 20
I
Please engage the girls into serious research and discussions during this holiday.
NOTE: An Assessment will be done when you get back to school.
© MATHEMATICS DEPARTMENT 2011
Email: [email protected]
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