Assign-02-H8152.tex 23/6/2006 15: 6 Page 75
A
Number and Algebra
Assignment 2
This assignment covers the material contained
in Chapters 6 to 8.The marks for each question are shown in
brackets at the end of each question.- Determine the 20th term of the series 15.6, 15,
14 .4, 13.8,... (3) - The sum of 13 terms of an arithmetic progres-
sion is 286 and the common difference is 3.
Determine the first term of the series. (4) - An engineer earns £21000 per annum and receives
annual increments of £600. Determine the salary
in the 9th year and calculate the total earnings in
the first 11 years. (5) - Determine the 11th term of the series 1.5, 3, 6,
12,... (2) - Find the sum of the first eight terms of the series
1, 2.5, 6.25,..., correct to 1 decimal place. (4) - Determine the sum to infinity of the series
5, 1,^15 ,... (3) - A machine is to have seven speeds ranging from
25 rev/min to 500 rev/min. If the speeds form
a geometric progression, determine their value,
each correct to the nearest whole number. (8) - Use the binomial series to expand (2a− 3 b)^6
(7) - Determine the middle term of
(
3 x−1
3 y) 18(6)- Expand the following in ascending powers oft
as far as the term int^3
(a)1
1 +t(b)1
√
(1− 3 t)For each case, state the limits for which the
expansion is valid. (12)- Whenxis very small show that:
1
(1+x)^2√
(1−x)≈ 1 −3
2x (5)- The modulus of rigidityGis given byG=
R^4 θ
L
whereRis the radius,θthe angle of twist and
Lthe length. Find the approximate percentage
error inGwhenRis measured 1.5% too large,
θis measured 3% too small andLis measured
1% too small. (7)- Use Maclaurin’s series to determine a power
series for e^2 xcos 3xas far as the term inx^2.
(10) - Show, using Maclaurin’s series, that the first four
terms of the power series for cosh 2xis given by:
cosh 2x= 1 + 2 x^2 +2
3x^4 +4
45x^6 (11)- Expand the function x^2 ln(1+sinx) using
Maclaurin’s series and hence evaluate:
∫ 1
2
0x^2 ln(1+sinx)dxcorrect to 2 significantfigures. (13)