Download t dx dt − x = t2. ( 2xt3 − tsinx ) dx dt + 3x2t2 + cosx = 0 d2y dx2 + 2 dy

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MATH1015W1
SECTION A
A1. (2 marks) Find an integrating factor for the differential equation
t
dx
− x = t2 .
dt
(Do NOT solve the equation.)
A2. (2 marks) Determine whether the equation
¡
¢ dx
2xt3 − t sin x
+ 3x2 t2 + cos x = 0
dt
is exact. (Do NOT solve the equation).
A3. (2 marks) Find a particular integral of the differential equation
dy
d2 y
+
2
+ 5y = 20x + 23 .
dx2
dx
A4. (2 marks) Find a particular integral of the differential equation
d2 x
dx
+
2
− x = −2e−3t .
2
dt
dt
A5. (2 marks) If a = −i + 2j + 3k and b = j − 2k, and c = −2i + 3j − k , obtain
a · b × c.
A6. (2 marks) Points A and B have position vectors 2i + j − 3k and i − j + k . Find the
distance between the two points.
A7. (2 marks) A particle P has position vector 3 sin t i − 3 cos t k at time t. Calculate
the velocity and speed of the particle.
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MATH1015W1
A8. (2 marks) Find the vector equation of the plane which is perpendicular to the vector
3i − 4j − 2k and passes through the point (−1, 2, −3).
A9. (2 marks) Obtain the product AB of the matrices


−2 1
A =  4 −1 
−2 3
µ
and B =
0 −1
2 −3

0
 1
A10. (2 marks) Calculate the determinant of the matrix C = 
 1
−1
¶
.

0 0
1
0 −2 −1 
.
3 0
2 
1 −4 2
A11. (2 marks) State a condition on the matrix A, for the system of n homogeneous equations in n unknowns given by AX = 0 to have a non-zero solution vector X .
¡
¢
A12. (2 marks) If det D2 = 25, obtain det(D).
¡ 2¢
∂ 2f
A13. (2 marks) Given that f (x, y) = y x − x y + x sin y , find
.
∂x∂y
3
3
A14. (2 marks) If f is a function of u and v where both u and v depend only on t, obtain
an expression for
df
in terms of derivatives of f , u and v .
dt
∞
X
A15. (2 marks) Determine whether the series
k=1
k
is convergent.
k+1
x2
A16. (2 marks) Using l’Hopital’s rule, or otherwise, evaluate lim
.
x→0 1 − cos 2x
TURN OVER
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MATH1015W1
A17. (2 marks) Write −2j in complex exponential form and hence, or otherwise, calculate
the principal value of ln(−2j).
¡ ¡ ¢
¡ ¢¢4
A18. (2 marks) Using de Moivre’s Theorem, or otherwise, calculate cos π6 + j sin π6
.
A19. (2 marks) Find the Laplace transform of the function t2 e−t .
1
.
A20. (2 marks) Find the inverse Laplace transform of the function 2
s + 4s + 5
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MATH1015W1
SECTION B
B1. (a) (7 marks) Find the general solution of differential equation
³
´ dx
2
2
2
x2
2xte − t sin x + sec x
+ ex + 2t cos x + cos t = 0.
dt
(b) (8 marks) Obtain the general solution of the second order differential equation
d2 x
dx
+
5
+ 6x = 2e−t .
2
dt
dt
B2. A line L1 passes through the points A (1, 2, 3) and B (−2, 5, 0). A second line L2
passes through the origin and the point C (0, 6, 4).
(a) (5 marks) Obtain the vector equations of L1 and L2 . If L1 and L2 intersect, find the
point of intersection.
(b) (5 marks) Determine the angle between the lines L1 and L2 .
(c) (5 marks) Find a vector perpendicular to the lines L1 and L2 . Hence, or otherwise,
obtain the cartesian equation of the plane containing the two lines.
TURN OVER
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MATH1015W1
B3. (a) (7 marks) Find the eigenvalues and eigenvectors of the matrix
µ
1 −2
−2 4
¶
.
(b) (8 marks) Using the notation z = x + jy , find the locus of the point z in the
Argand diagram satisfying the equation
¯
¯
¯z − j ¯
¯
¯
¯ z − 1 ¯ = 2.
Sketch the result on an Argand diagram and describe the locus geometrically.
B4. (a) (4 marks) Evaluate the determinant of the matrix


1 1 1
A= 1 2 3.
α 1 2
(b) (6 marks) Find A−1 when α = 1.
(c) (5 marks) Hence, or otherwise, solve the system of linear equations
x + y + z = 2
x + 2y + 3z = 2
x + y + 2z = 1
for the unknowns x, y and z .
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MATH1015W1
B5. (a) (4 marks) Use the table of Laplace transforms to find
½
(i) L−1
(ii) L−1
(iii) L−1
¾
1
,
s−1
½
¾
1
,
(s − 1)2
¾
½
1
.
s+3
(b) (6 marks) Use partial fractions and part (a) to obtain
½
L−1
8
(s + 3)(s − 1)2
¾
.
(c) (5 marks) Use Laplace transforms and part (b) to find the solution of the differential equation
d2 x
dx
+
2
− 3 x = 8 et
2
dt
dt
dx
= 0 when t = 0 .
which satsifies the initial conditions x = 0 and
dt
1
B6. (a) (8 marks) Use Maclaurin’s theorem to obtain the expansion of f (x) = (1 − x) 2
as
1
1
(1 − x) 2 = 1 − x + ax2 + bx3 + R3 .
2
Give the values of the coefficients a and b and find the form of the Lagrange
remainder R3 , stating any associated inequality. What is the maximum
error if
√
3
we use this expansion up to the x term to estimate the value of 0.95 ?
(b) (7 marks) The constant density ρ of a sphere is related to its mass m and its
radius r by the formula
ρ=
3m
.
4πr3
If the mass is measured with a maximal error of 1% and the radius with a maximal
error of 0.5%, use partial derivatives to estimate the maximum percentage error
in the density.
END OF PAPER
MATH1015W1
Full marks may be obtained for complete answers to ALL questions in
Section A and FOUR questions in Section B.
All questions in Section A are worth 2 marks and all questions in
Section B are worth 15 marks.
Only your best FOUR answers to Section B questions will be taken into
account.
The Engineering Data Book (Calvert and Farrar) will be provided.
The Official University Calculator MAY be used
c University of Southampton
Copyright 2006 °
Number of
Pages 7