Get premium membership and access revision papers, questions with answers as well as video lessons.

Math 123: Vectors And Mechanics Question Paper

Math 123: Vectors And Mechanics

Course:Bachelor Of Science

Institution: Chuka University question papers

Exam Year:2012

1
CHUKA UNIVERSITY
And
COLLEGE
UNIVERSITY EXAMINATIONS
FIRST YEAR EXAMINATION FOR THE AWARD OF DEGREE OF
BACHELOR OF EDUCATION (ARTS)
MATH 123: VECTORS AND MECHANICS
STREAM: B.ED(ARTS) Y1S2 TIME: 2 HOURS
DAY/DATE: TUESDAY 6/4/2010 2.30P.M.-4.30P.M.
INSTRUCTIONS:
ANSWER QUESTION ONE AND ANY OTHER TWO QUESTIONS.
QUESTION ONE – 30 MARKS
(a) Find the angle between the vectors
A
~
= 2
i
ˆ
-j
ˆ
3
+
k
ˆ
and
B
~
= -i
ˆ
+ 4
j
ˆ
+ 2
k
ˆ
[3 marks]
(b) Find the projection of a vector A =
k j i
ˆ
3
ˆ
3
ˆ
2 ? ?
on an assigned direction U
which make an angle of 60°. [3 marks]
(c) A necessary and sufficient condition for two non-null vectors
a
~
and
b
~
to be
orthogonal is that their scalar product should vanish. Prove. [4 marks]
(d) A necessary and sufficient condition for two non-null vectors
a
and
b
to be
parallel is that their vector product should vanish. Prove. [4 marks]
2
(e) Let F1 and F
2
be two forces acting at a point O, and let
?
be the angle
between them. If
?
is the angle that the resultant force make with
Horizontal, prove that
?
?
?
Cos F F
Sin F
Tan
2 1
2
?
?
(f) Find the position vector of the point where the line
?
=
)
ˆ
2
ˆ 3
ˆ 5 (
ˆ
3
ˆ ˆ 2 k j i k j i ? ? ? ? ? ?
meets the plane
?
~
.
. 15 )
ˆ ˆ
2
ˆ
( ? ? ? k j i
[5 marks]
(g) Resolve
a
~
j i
ˆ
3
ˆ
4 ? ?
into components
1
~
a
and
2
~
a
where
1
~
a
is parallel to
b
~
and
2
~
a
is perpendicular to
.
~
b
[5 marks]
(h) Find the work done in moving an object along a vector
j i a
ˆ
4
ˆ
3
~
? ?
if the
Force applied if
j i F
ˆ ˆ
2
~
? ?
. [3 marks]
QUESTION TWO – 20 MARKS
(a) The acceleration of a particle at any time
t
is given by
2 2
(
ˆ ˆ
) 1 (
~
t j t i t a ? ? ? ? k
ˆ
) 2 ?
. If at
, 0 ? t
the displacement
0 ? S
, and
Velocity is
,
ˆ ˆ
k i V ? ?
Find
V
and
S
at any time
. t
[6 marks]
(b) Given
? ,
3 2 Z
e xy ?
find grad
?
. [2 marks]
(c) Given a vector
,
ˆ ˆ ˆ
2 2 2
k Z j x y i Z x A ? ? ?
find
(i) div A [2 marks]
(ii) Curl A [3 marks]
(d) A particle moves along the curve
. 6 int, 4 , 4 t Z S Y Cost X ? ? ?
Find
the magnitude of the velocity and acceleration at times
0 ? t
and
.
1
?
? t
[7 marks]
3
QUESTION THREE – 20 MARKS
(a) Find the area of a triangle whose vertices are
) 1 , 0 , 4 ( ), 5 , 3 , 2 ( ? ? B A
and
). 1 , 2 , 0 ( ? C
[5 marks]
(b) Given vectors
k c j b i a A
ˆ ˆ ˆ
? ? ?
and
,
ˆ ˆ ˆ
k f j e i d B ? ? ?
show that
. . cf be ad B A ? ? ?
[3 marks]
(c) Prove the Sine rule of triangle:
c
SinC
b
SinB
a
SinA
? ?
[7 marks]
(d) Given vectors
k c j b i a A
ˆ ˆ ˆ
~
1 1 1
? ? ?
and
k c j b i a B
ˆ ˆ ˆ
~
2 2 2
? ? ?
, prove that
. ˆ ) ( ˆ ) ( ˆ ) (
~ ~
2 1 2 1 2 1 2 1 2 1 2 1 k a b b a j c a a c i b c c b B A ? ? ? ? ? ? ?
[5 marks]
QUESTION FOUR – 20 MARKS
(a) Prove the Lamis theorem. [4 marks]
(b) A weight W hangs from a fixed point O by an inextensible string. It is
pushed aside by a horizontal force P and rests at equilibrium with a string
inclined at an angle of 30° to the vertical. Find in term of W.
(i) The horizontal force P.
(ii) The tension in the string [5 marks]
(c) A force whose point of application is
) 1 , 4 , 2 ( ?
is given by
.
ˆ ˆ
7
ˆ
6
~
k j i F ? ? ?
Find the magnitude of the moment of Force at the point
). 3 , 1 , 0 (
[4 marks]
(d) Let vectors a1
and a
2
be the components of vector a. If a
1
is parallel to
Vector b and a2
is perpendicular to b, use a suitable sketch to prove that
(i)
b
b
b a
a
~
~
~
.
~
~
2 1
?
?
?
?
?
?
?
?
?
4
(ii)
b
b
b a
a a
~
~
~
.
~
~
2 2
?
?
?
?
?
?
?
?
? ?
[7 marks]
QUESTION FIVE -20 MARKS
(a) A particle moves in such a way that its displacement at any time t is S.
If S=0 at t=0, prove that
(i) V = V0
+at
(ii) S = Vo
t +
2
1
at
2
(iii) V
2
= Vo
2
+ 2aS. [8 marks]
(b) Find the acute angle between the line
4
2
1
1
2
? ?
?
?
?
?
Z
y x
and the plane
10 3 2 ? ? ? Z y x
[5 marks]
(c) Find the point where the line
)
ˆ 3
ˆ ˆ 2 (
ˆ 4
ˆ 3
ˆ k j i k j i ? ? ? ? ? ? ? ? ?
intersects the plane
17 3 2 ? ? ? Z y x
[4 marks]
(d) Find the vector perpendicular to the surface
Z y x F
2 2
? ?
at the point
(-1, 1,2). [3 marks]
---------------------------------------------------------------------------------

Math 123: Vectors And Mechanics Question Paper

More Question Papers

Popular Exams