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modern engineering mathematics

Modern Engineering Mathematics 6th Edition Glyn James - Solutions

Obtain an approximation to the function e–x in the formand find an estimate for the error. e-x A a + bx + cx A + Bx + Cx
Plot the points with coordinates: A(2, 3), B(−1, 4), C(−3, −1), D(3, −2)
Plot the points A(2, 2), B(−2, 2), C(−2, −2) and D(2, −2) on the following diagrams, joining the points with straight lines.
Find the distance between the points:(a) A(2, 3) and B(−1, 4) (b) B(−1, 4) and C(−3, −1)(c) C(−3, −1) and D(3, −2)
Find the equation of the line through the points:(a) (−1, 1/2) and (2, 2) (b) (−3, 4) and (3, 0)Show the lines on a diagram.
Find the intercepts of the lines with equation:(a) y = 1/2x +1 (b) y = − 2/3x + 2
Find the equation of the line with intercepts:(a) (−2, 0), (0, 1) (b) (3, 0), (0, 2)
Find the slope of the line with equation:(a) y = 1/2x +1 (b) 3y = 6 − 2x
Find the slope of the line through the points:(a) (1, −1), (3, 3) (b) (−1, 10), (2, 1)
Find the equation of the line orthogonal (perpendicular) to the lineand passing through the point (1, 2). || y=
Find the equation of the circle(a) With centre (1, 2) and radius 5(b) With centre (−1, 2) and radius 3
Find the centre and radius of the circle with equation x + y + 3x-2y+1=0 (7.13)
Plot the points P(3, 2), Q(−4, 1), R(−1, −3) and S(2, −3) on the coordinate systems given below. Join up the points PQ, QR, RS and SP with straight lines. (a) y 1 1 X (b) y 1 1 X
Find the distance between the two points:(a) P(3, 2) and Q(−4, 1) (b) R(−1, −3) and S(2, −3)(c) S(2, −3) and P(3, 2)
Find the equation of the line through the points:(a) (1, −1), (3, 3) (b) (−1, 10), (2, 1) (c) (0, 1), (5, 3)Show the lines on a diagram.
Find the intercepts of the lines with equation:(a) y = 2x − 3 (b) y = 7 − 3x (c) y = 2/5 x +1
Find the equation of the line with intercepts(a) (5, 0), (0, −3) (b) (−4, 0), (0, −2)
1. Find the slope of the line2. Write down the equations of the straight lines with the propertiesFind where these lines cut the axes and their point of intersection. Draw the lines on the same diagram.3. Find the equation of the line that passes through the point (0, 1) and makes an angle of 45°
Find the equation of the line orthogonal to the line y = 7 − 3x and passing through the point (2, 1).
Find the equation of the circle(a) With centre (3, 1) and radius 4(b) With centre (2, −1) and radius 2
Find the centre and radius of the circle with the following equations:(a) x2 + y2 − 6x − 8y −144 = 0(b) x2 + y2 + 2x − 3y + 3 = 0(c) x2 + y2 − 3x + 5y − 50 = 0(d) x2 + y2 + 2y − 3 = 0
There are two methods of assessing the value of a wasting asset. The first assumes that it decreases each year by a fixed amount; the second assumes that it depreciates by a fixed percentage. A piece of equipment costs £1000 and has a ‘lifespan’ of six years after which its scrap value is
A machine that costs £1000 has a working life of three years, after which it is valueless and has to be replaced. It saves the owner £500 per year while it is in use. Show that the true total saving £S to the owner over the three years iswhere r% is the current rate of interest. Estimate S for r
An economic model for the supply S(P) and demand D(P) of a product at a market price of P is given by(so that supply lags behind demand by 1 time unit). Show thatFind the particular solution of the recurrence relation corresponding to P0 = 0.8 and sketch it in a cobweb diagram. What is the steady
Show thatwhere Tk is the kth triangular number. (See Question 4 in Exercises 7.2.3.)Data from Question 4 in Exercises 7.2.3.Triangular numbers (Tn) are defined by the number of dots that occur when arranged in equilateral triangles, as shown in Figure 7.5. Show that Tn = 1/2n(n + 1) for every
Find the general solutions of the following linear recurrence relations: (a) fn+2 - 5fn+1+6f=0 (c) fn+25fn+1+6f=4" (b) fn+24fm+1+4f=0 (d) fn+25fn+1+6fn=3" -
Suppose that consumer spending in period t, Ct, is related to personal income two periods earlier, It – 2, byDeduce that if personal income increases by a factor 1.05 each period, that isDescribe the behaviour of Ct in the long run. C = 0.87512 0.2C_ (t2) -
An economist believes that the price Pt of a seasonal commodity in period t satisfies the recurrence relationwhere C is a positive constant. Show thatwhere A and B are complex-conjugate constants. Noting thatexplain why the economist is mistaken. P+2=2(P+1 P) + C (t> 0)
The cobweb model applied to agricultural commodities assumes that current supply depends on prices in the previous season. If Pt denotes market price in any period and QSt, QDt supply and demand in that period, thenFind the market price and comment on its form. QD1800.75P, Qst=-30+0.3P-1 where
Solve for National Income, Yt, the set of recurrence relationsComment on your solution. Y = 1 + C, +1, C = = Y, -1 I = 2(C C-1)
A sequence is defined byrepeated linear extrapolation. (γ is known as Euler’s constant.) ak = 1 + 2 + 1 3 + 1 k - Ink (k = 1, 2, ...) Given a0.626 383, a20= 0.602 009, estimate y = lim a,, using M-* a160.608 140 and
Discuss the convergence of (a) 2 (b) kal + 3 2 3 kP 4 5 4 + + (all p) (c)+ -3 + 3-+ + + 15 (d) 1-+-+... +. ...
Express the following recurring decimal numbers in the form p/q where p and q are integers:(a) 1.231 231 23 . . . (b) 0.429 429 429 . . .(c) 0.101 101 101 . . . (d) 0.517 251 72 . . .
Determine which of the following series are convergent: (a) 1 n=on + 1 in I-u 2n5 - 1 (b) n=1 () M=1 n + 2 n? n-1 n? + n 3
A rational function f (x) has the following power series representation for –1 Find a closed-form expression for f(x). f(x) = 1x + 2x + 3x + 4x + ...
The function f(x) = sinh–1x has the power series expansionObtain polynomial approximations for sinh1x for –0.5 (a) 0.005 (b) 0.000 05. sinh-x = x 1 x 23 + 1.3 x5 2.4 5 1.3.5 x 2.4.6 7 +
A chord of a circle is half a mile long and supports an arc whose length is 1 foot longer (1 mile = 5280 feet). Show that the angle θ subtended by the arc at the centre of the circle satisfiesUse the series expansion for sine to obtain an approximate solution of this equation, and estimate the
A machine is purchased for £3600. The annual running cost of the machine is initially £1800, but rises annually by 10%. After x years its secondhand value is £3600e–0.35x. Show that the average annual cost £C (including depreciation) after x years is given byShow graphically that the machine
Consider the sequence Φn defined byEvaluate Φ64 and Φ128 (without using the yx key of your calculator), and use extrapolation to estimate the value of e. on x = [(+9 + (-9] n Show that , e as no. Using the power series expansions of ln(1 +x) and e, show that (1 + )" = exp[n in (1 + )] n 1 11 +
A beam of weight W per unit length is simply supported at the same level at (N + 1) equidistant points, the extreme supports being at the ends of the beam. The bending moment Mk at the kth support satisfies the recurrence relationwhere a is the distance between the supports and M0 = 0 and MN = 0.
A complex voltage E is applied to the ladder network of Figure 7.31. Show that the (complex) mesh currents Ik satisfy the equationsShow thatsatisfies (7.20) provided that cosh θ = 1 – 1/2 LCω2. Note that this equation yields two values for u, so that in general Ik may be written aswhere A and B
A lightweight beam of length l is clamped horizontally at both ends. It carries a concentrated load W at a distance a from one end (x = 0). The shear force F and bending moment M at the point x on the beam are given byDraw the graphs of these functions. Use Heaviside functions to obtain single
(a) Show thatand explain why there is a restriction on the domain of θ.(b) Use the binomial expansion to show thatgiving the value of A.(c) Show that sin 4θ can be expressed in the form sin 40 = 4 sin 0 (1 - 2 sin0)(1-sin0), -TT/2
The seriessums to the value π2/8. Lagrange’s formula for linear interpolation iswhere p and q are integers. Choosing p = 5 and q = 10, estimate the value of π2/8. k=0 1 (2k + 1) = 1+ 1 1 3 5 + + 7 +
The expression x/1 + ax2 is to be used as an approximation to 1/2ln[(1 + x)/(1 – x)] on –1 Show thatfor |x|5. Draw on the same diagram the graphs of - + n ( 1 + x) In - =-(-+ a)x-(-a)x - ( + a)x + ... X 1 + ax
List A is a list of propositions, while list B is a list of sentences that are not propositions.(a) Determine the truth values of the propositions in list A and state their negation statements.(b) Explain why the sentences in list B are not propositions.List A:(a) Everyone can say where they were
For the matrices(a) Evaluate (A + B)2 and A2 + 2AB + B2(b) Evaluate (A + B)(A – B) and A2 – B2Repeat the calculations with the matricesand explain the differences between the results for the two sets. A = and B- [J 1 0
1) Find the value of the angle X° in each of the following diagrams:2) Find the length x cm in each of the following diagrams: (a) (c) 70 60 80 70 X to (b) (d) 30 1x 60 21
1) Calculate the area of the shaded region in each of the following diagrams (not drawn to scale).2) In the following diagram, ABCD is a parallelogram. Determine the angles X° and Y°3) Calculate the area of the parallelogram ABCD in the diagram below4) A rhombus has diagonals of length 8 cm and 6
1) Find the value of the angle X° , and when required Y° , in each of the following diagrams, in which • indicates the centre of the circle:2) In the following diagram, AC = BC and O is the centre of the circle. Determine angle ABC.3) In the following diagram, O is the centre of a circle of
1) Convert the following angles from degrees to radians, giving the answers as multiples of π.(a) 30°(b) 45°(c) 120°(d) 200°2) Convert the following angles from radians to degrees.(a) π/2(b) 3π/4(c) 5π/6(d) π/3 3) Calculate the length of the arc subtended by an angle of 20° in a
1. Given that sin θ° = 3/5, 0 < θ° < 90° determine from a suitably drawn triangle the values of cos θ° and tan θ°.2. Given that cos θ = 3/8, 0 < θ < π/2 , determine without using a calculator the values of sin θ and tan θ.3. If tan θ = 8/15 and θ is an acute angle, calculate the
Given z1 = 2ejπ/3 and z2 = 4e–2jπ/3, find the modulus and argument of (a) (b) z (c) z/z
If z1 = 1 + j and z2 = √3 + j, determine |z1z2|, |z1/z2|, arg(z1z2) and arg(z1/z2).
Express the following sets in listed form:
For the sets A, B, C and D of Question 1 list the sets A ∪ B, A ∩ B, A ∪ C, A ∩ C, B ∪ D, B ∩ D and B ∩ C.Data from Question 1Express the following sets in listed form:
If A = {1, 3, 5, 7, 9}, B = {2, 4, 6, 8, 10} and C = {1, 4, 5, 8, 9}, list the sets A ∪ B, A ∩ C, A ∩ B, B ∪ C and B ∩ C.
Illustrate the following sets using Venn diagrams:
Given
For the sets defined in Question 5, check whether the following statements are true or false:Data from Question 5Given
If the universal set is the set of all integers less than or equal to 32, and A and B are as in Question 5, interpretData from Question 5Given
(a) If A ⊂ B and A ⊂ B̅ , show that A = ∅.(b) If A ⊂ B and C ⊂ D, show that (A ∪ C) ⊂ (B ∪ D) and illustrate the result using a Venn diagram.
If A, B and C are the sets {2, 5, 6, 7, 10}, {1, 3, 4, 7, 9} and {2, 3, 5, 8, 9} respectively, verify that
Using the rules of set algebra, verify the absorption rules
Using the laws of set algebra, simplify the following:
Defining the difference A – B between two sets A and B belonging to the same universal set U to be the set of elements of A that are not elements of B, that is A – B = A ∩ B̅, verify the following properties:Illustrate the identities using Venn diagrams.
If n(X) denotes the number of elements of a set X, verify the following results, which are used for checking the results of opinion polls:Here the sets A, B and C belong to the same universal set U.
In carrying out a survey of the efficiency of lights, brakes and steering of motor vehicles, 100 vehicles were found to be defective, and the reports on them were as follows:Use a Venn diagram to determine(a) How many vehicles had defective lights, brakes and steering;(b) How many vehicles had
On carrying out a later survey on the efficiency of the lights, brakes and steering on the 100 vehicles of Question 14, the report was as follows:Use a Venn diagram to determine(a) How many vehicles were non-defective;(b) How many vehicles had defective lights only.
An analysis of 100 personal injury claims made upon a motor insurance company revealed that loss or injury in respect of an eye, an arm or a leg occurred in 30, 50 and 70 cases respectively. Claims involving the loss or injury to two of these members numbered 42. How many claims involved loss or
Bright Homes plc has warehouses in three different locations, L1, L2 and L3, for making replacement windows. There are three different styles, called ‘standard’, ‘executive’ and ‘superior’:The parts A, B, C, D, E and F are made from components a, b, c, d, e, f, g, h and i as
By setting up truth tables, find the possible values of the following Boolean functions:
Figure 6.19 shows six circuits. Write down a Boolean function that represents each by using truth tables.
Use the De Morgan laws to negate the function f = (p + q) · (r̅ · s) · (q + t̅)
Give a truth table for the expression
Simplify the following Boolean functions, stating the law used in each step of the simplification:
Write down the Boolean functions for the switching circuits of Figure 6.20.
Draw the switching circuit corresponding to the following Boolean functions:
Four engineers J, F, H and D are checking a rocket. Each engineer has a switch that he or she presses in the event of discovering a fault. Show how these must be wired to a warning lamp, in the countdown control room, if the lamp is to light only under the following circumstances:(a) D discovers a
In a public discussion a chairman asks questions of a panel of three. If to a particular question a majority of the panel answer ‘yes’ then a light will come on, while if to a particular question a majority of the panel answer ‘no’ then a buzzer will sound. The members of the panel record
Design a switching circuit that can turn a lamp ‘on’ or ‘off ’ at three different locations independently.
Design a switching circuit containing three independent contacts for a machine so that the machine is turned on when any two, but not three, of the contacts are closed.
The operation of a machine is monitored on a set of three lamps A, B and C, each of which at any given instant is either ‘on’ or ‘off’. Faulty operation is indicated by each of the following conditions:(a) When both A and B are off;(b) When all lamps are on;(c) When B is on and either A is
Write down the Boolean function for the logic blocks of Figure 6.35. Simplify the functions as far as possible and draw the equivalent logic block
Simplify the following Boolean functions and sketch the logic block corresponding to both the given and simplified functions:
Negate the following propositions:(a) Fred is my brother.(b) 12 is an even number.(c) There will be gales next winter.(d) Bridges collapse when design loads are exceeded.
Determine the truth values of the following propositions:(a) The world is flat.(b) 2n + n is a prime number for some integer n.(c) a2 = 0 implies a = 0 for all a ∈ N.(d) a + bc = (a + b)(a + c) for real numbers a, b and c.
Determine which of the following are propositions and which are not. For those that are, determine their truth values.(a) x + y = y + x for all x, y ∈ R.(b) AB = BA, where A and B are square matrices.(c) Academics are absent-minded.(d) I think that the world is flat.(e) Go fetch a
Let A, B and C be the following propositions:Translate the following into logical notation:(a) It is raining and the Sun is shining.(b) If it is raining then there are clouds in the sky.(c) If it is not raining then the Sun is not shining and there are clouds in the sky.(d) If there are no clouds
Let A, B and C be as in Question 35. Translate the following logical expressions into English sentences:Data from Question 35Let A, B and C be the following propositions:
Consider the ambiguous sentence x2 = y2 implies x = y for all x and y(a) Make the sentence into a proposition that is true.(b) Make the sentence into a proposition that is false.
The counterexample is a good way of disproving assertions. (Examples can never be used as proof.) Find counterexamples for the following assertions:(a) 2n – 1 is a prime for every n ≥ 2(b) 2n + 3n is a prime for all n ∈ N(c) 2n + n is prime for every positive odd integer n
Give the converse and contrapositive for each of the following propositions:
Construct the truth tables for the following:
Prove or disprove the following:Note that to disprove a tautology, only one line of a truth table is required.
Use contradiction to show that √3 is irrational.
Prove or disprove the following:(a) The sum of two even integers is an even integer.(b) The sum of two odd integers is an odd integer.(c) The sum of two primes is never a prime.(d) The sum of three consecutive integers is divisible by 3. Indicate the methods of proof where appropriate.
Prove that the number of primes is infinite by contradiction.
Use induction to establish the following results:
Prove that 11n – 4n is divisible by 7 for all natural numbers n.

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