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modern engineering mathematics

Modern Engineering Mathematics 6th Edition Glyn James - Solutions

Two skew lines L1, L2 have respective equationsObtain the equation of a plane through L1 parallel to L2 and show that the shortest distance between the lines is 6.
The three vectors a = (1, 0, 0), b = (1, 1, 0) and c = (1, 1, 1) are given. Evaluate(a) a × b, b × c, c × a(b) a · (b × c)For the vector d = (2, –1, 2) calculate(c) The parameters a, b, g in the expression(d) The parameters p, q, r in the expressionand show that
Given the line with parametric equation r = a + λd show that the perpendicular distance p from the origin to this line can take either of the formsFind the parametric equation of the straight line through the points A(1, 0, 2) and B(2, 3, 0) and determine(a) The length of the perpendicular from
Given the three non-coplanar vectors a, b, c, and defining v = a · b × c, three further vectors are defined asShow thatIf a vector is written in terms of a, b, c asevaluate α, β, γ in terms of a´, b´ and c´.
An unbalanced machine can be approximated by two masses, 2 kg and 1.5 kg, placed at the ends A and B respectively of light rods OA and OB of lengths 0.7 m and 1.1 m. The point O lies on the axis of rotation and OAB forms a plane perpendicular to this axis; OA and OB are at right angles. The machine
Represent on an Argand diagram the complex numbers(a) 3 + j2 (b) –5 + j3 (c) 8 – j5 (d) –2 – j3
If the two complex numbersare equal(a) Find the values of the real numbers a and b;(b) Write down the real and imaginary parts of z1 and z2. Z = (3a + 2) + j(3b-1) and 2 = (b + 1) j(a + 2 b)
If z1 = 3 + j2 and z2 = 5 – j3 determine(a) z1 + z2 (b) z1 – z2
If z1 = 3 + j2 and z2 = 5 + j3 determine z1z2.
If z1 = 3 + j2 and z2 = 5 + j3 determine z1/z2.
Find the real and imaginary parts of the complex number z + 1/z for z = (2 + j)/(1 – j).
Express the zeros of f(x) = x2 – 6x + 13 as complex numbers.
Find all the roots of the quartic equation x4 + 4x2 + 16 = 0
For the complex numbers z1 = 5 + j3 and z2 = 3 – j2 verify the identity (Zzz)* = zz*
Determine the modulus and argument of(a) 3 + j2 (b) 1 – j (c) –1 + j (d) –√6 – j√2
Express the following complex numbers in polar form.(a) 12 + j5 (b) –3 + j4 (c) –4 – j3
If z1 = –12 + j5 and z2 = –4 + j3, determine, using (3.5a) and (3.5b), |z1z2| and arg(z1z2).Data from 3.5a and b | 2| = rr = |2||Z| (3.5a)
For the following pairs of complex numbers obtain z1/z2 and z2/z1. (a) Z = 4(cos/2 + jsin/2), (b) z = cos 3/4 + jsin 3/4, Z = 9(cos/3 + jsin 7/3) Z Z = 2(cos /8 + jsin/8)
Find the modulus and argument of Z = (1 + j2) (4- j3) (3 + j4)4(2-j)
Express the following complex numbers in exponential form:(a) 2 + j3 (b) –2 + j
Express in cartesian form the complex number e2+jπ/3.
Find the values of(a) sin[1/4π(1 + j)] (b) sinh(3 + j4)(c) tan(π/4 – j3) (d) z such that cos z = 2(e) z such that tanh z = 2
Evaluate ln(–3 + j4) in the form x + jy.
Express 1 – j in the form r(cos θ + j sin θ) and hence evaluate (1 – j)12.
Given evaluate(a) z1/2 (b) z1/3and display the roots on an Argand diagram. +-=2
Evaluate and display the roots on an Argand diagram. +) -2/3
Solve the quadratic equation z2 + (2j – 3)z + (5 – j) = 0
Expand in terms of sines and cosines of multiple angles(a) cos5θ (b) sin6θ
Expand cos 4θ as a polynomial in cos θ.
Describe the locus of z given by(a) Re(z) = 4 (b) arg(z – 1 – j) = π/4(c)(d) Im((1 – j2)z) = 3 - j2 z- z-1 = 1
Find the cartesian equation of the circle|z – (2 + j3)| = 2
Find the cartesian equation of the curve whose equation on the Argand diagram is z-j z-1-j2 = 2
Find the locus of z in the Argand diagram such that Re[(z – j)/(z + 1)] = 0
Find the cartesian equation of the locus of z given by |z + 1| + |z – 1|= 4
Express u and v in terms of x and y where w = u + jv, z = x + jy, w = f(z) and (a) f(z) = z (b) f(z) = z-j z+1 Z = -1
Find the image on the w plane of the strip between x = 1 and x = 2 on the z plane under the mapping defined by w= z + 2 N
Calculate the complex impedance of the element shown in Figure 3.20 when an alternating current of frequency 100 Hz flows. 15 Figure 3.20 41.3 mH
Using the formula find the solutions of the following quadratic equations, if they exist:(a) 2x2 − 5x + 3 = 0 (b) 4x2 + 3x − 1 = 0(c) 5x2 −7x + 2 = 0(d) 2x2 + 3x − 1 = 0 (e) 4x2 +12x + 9 = 0
Solve the following quadratic equations by completing the square:(a) x2 + 2x − 8 = 0(b) p2 − p − 2 = 0 (c) x2 − 4x − 2 = 0(d) 4m2 + 9m + 2 = 0 (e) 3x2 − 14x + 8 = 0 (f) x2 + 4x +1 = 0
Solve the following quadratic equations by factorisation:(a) x2 + 12x + 35 = 0 (b) 4m2 − 4m − 3 = 0 (c) 3p2 − p − 2 = 0(d) 2x2 + 9x − 5 = 0 (e) 3y2 − 7y + 2 = 0 (f) 6x2 − x −1 = 0
1. Solve the simultaneous equations2. Solve the simultaneous equations3. Solve the simultaneous equations4. Solve the simultaneous equations 3x-2y = 11 5x+2y=29
1. Find the value of x which satisfies the equation x + 4 = 5 − 2x2. Find the value of m which satisfies the equation 5m − 2 = 4m + 13. Find the value of p such that4. Find the value of y which satisfies the equation5. Find the value of x which satisfies the equation6. Find the value of q such
Find the solutions of the following quadratics, if they exist:(a) 2x2 + 3x + 7 = 0 (b) 2x2 + 7x + 3 = 0 (c) x2 − 4x + 4 = 0
Solve the quadratic equation 2m2 + 3m − 2 = 0
Solve the quadratic equation x2 + 6x + 7 = 0
Complete the square for the following quadratic expressions x2 + 6x + 7
Solve the quadratic equation 6x2 + 19x + 10 = 0
Solve the equation 7x + 5 = 4x − 6
Show that as θ varies, the point z = a(h + cos θ) + ja(k + sin θ) describes a circle. The Joukowski transformation u + jv = z + l2/z is applied to this circle to produce an aerofoil shape in the u–v plane. Show that the coordinates of the aerofoil can be written in the formTaking the case a =
Show that the functionwhere z = x + jy and w = u + jv, maps the circle |z| = 3 on the z plane onto a circle in the w plane. Find the centre and radius of this circle in the w plane and indicate, by means of shading on a sketch, the region in the w plane that corresponds to the interior of the
Show that the function w = (1 + j)z + 1 where z = x + jy and w = u + jv, maps the line y = 2x – 1 in the z plane onto a line in the w plane and determine its equation.
Show that the function w/z = 4 where z = x + jy and w = u + jv, maps the line 3x + 4y = 1 in the z plane onto a circle in the w plane and determine its radius and centre.
Show that as z describes the circle z = bejθ, u + jv = z + a2/z describes an ellipse (a ≠ b). What is the image locus when a = b?
If z1 = 3 + j2 and z2 = 1 + j, and O, P, Q, R represent the numbers 0, z1, z1z2, z1/z2 on the Argand diagram, show that RP is parallel to OQ and is half its length.
ABCD is a square, lettered anticlockwise, on an Argand diagram. If the points A, B represent 3 + j2, –1 + j4 respectively, show that C lies on the real axis, and find the number represented by D and the length of AB.
Show that if ω is a complex cube root of unity, then ω2 + ω + 1 = 0. Deduce thatHence show that the three roots ofExpress the remaining two roots in terms of u, v and ω and find the condition that all three roots are real. (x + y + z)(x + wy + wz)(x + wy + wz) = = x3 + y3 + z3 - 3
Find the real part ofand deduce that if R2 is negligible compared with (ωL)2 and (LCω2)2 is negligible compared with unity then the real part is approximately R(1 + 2LCω2). (R+ joL)/jwC jL + R + 1/jwC
Determine the six roots of the complex number –1 + j√3, in the form rejθ where –π 2Z + 1 + j3 = 0
Find, in exponential form, the four values ofDenoting any one of these by p, show that the other three are given by jnp (n = 1, 2, 3). 7+ j24 25 71/4
Given Z = (1 + j)/(3 – j4) obtain(a) Z (b) √Z (c) eZ(d) ln Z (e) sin Zin the form a + jb, a, b real, giving a and b correct to 2dp.
Show that if the propagation constant of a cable is given bywhere R, G, ω, L and C are real, then the value of X2 is given by X + JY = [(R + jwL)(G + jwC)]
Express Z = cosh(0.5 + j1/4π) in the forms(a) x + jy (b) rejθThe current in a cable is equal to the real part of the expression ej0.7/Z. Calculate the current, giving your answer correct to 3dp.
The voltage in a cable is given by the expressionCalculate its value in the form a + jb, giving a and b correct to 2dp, when cosh nx + Zo NIN Z sinh nx
In a transmission line the voltage reflection equation is given bywhere K is a real constant, Z = R + jX and Z0 = R0 + jX0. Obtain an expression for θ, the phase angle, in terms of R0, R, X0 and X. Hence show that if Z0 is purely resistive (that is, real) then Keje Z - Zo Z + Zo
Two impedances Z1 and Z0 are related by the equationwhere α, β and l are real. If al is so small that we may take sinh αl = αl, cosh αl = 1 and (αl)2 as negligible, show that Z = Z, tanh(al +jBI)
Using complex numbers, show that sin'0 (35 sin0 - 21 sin 30 + 7 sin 50 sin 70) = = -
Express in the form a + jb, with a and b expressed to 2dp(a) sin(0.2 + j0.48) (b) cosh–1(j2)(c) cosh(3.8 – j5.2) (d) ln(2 + j)(e) cos(1/4π – j)
Find the modulus and argument of (3 + j4)*(12 - j5) (3 - j4)(12 + j5)
The input impedance Z of a particular network is related to the terminating impedance z by the equationFind Z when z = 0, 1 and jΩ and sketch the variation of |Z| and arg Z as z moves along the positive real axis from the origin. Z = (1 + j) z - 2 + j4 z + 1 + j
The characteristic impedance Z0 and the propagation constant C of a transmission line are given bywhere Z is the series impedance and Y the admittance of the line, and Re(Z0) > 0 and Re(C) > 0. Find Z0 and C when Z = 0.5 + j0.3Ω and Y = (1 – j250) × 10–8Ω. Zo = (Z/Y) and C = (ZY)
For a certain network the impedance Z is given bySketch the variation of |Z| and arg Z with the frequency ω. (Take values of ω ≥ 0.) Z = 1 + jo 2 1 + ja - w
Prove that the statements(a) |z + 1| > |z – 1|(b) Re(z) > 0are equivalent.
Show that , (a) sin0 [cos 40 - 4 cos 20 + 3] (b)sin0 = [sin 50 5 sin 30 + 10 sin 0] (c) cos'0 = 3/2 [cos 60+ 6 cos 40 + 15 cos 20 + 10] (d) cos0 sin0 = [2 sin0+ sin 30 - sin 50] 16 =
Prove that if p(z) is a polynomial in z with real coefficients then [p(z)]* = p(z*). Deduce that the roots of a polynomial equation with real coefficients occur in complex-conjugate pairs.
Show that the solutions of z4 – 3z2 + 1 = 0 are given by z = 2 cos 36°, 2 cos 72°, 2 cos 216°, 2 cos252° Hence show that (a) cos 36 =(5 + 1) (b) cos 72 =(5 - 1) 4
(a) Express cos 6θ as a polynomial in cos θ.(b) Given z = cos θ + j sin θ show, by expanding (z + 1/z)5(z –1/z)5 or otherwise, that sin 0 cos0 = 1 -(sin 100 5 sin 60 2 + 10 sin 20)
A circuit consists of a resistance R1 and an inductance L in parallel connected in series with a second resistance R2. When a voltage V of frequency ω/2π is applied to the circuit the complex impedance Z is given byShow that if R1 varies from zero to infinity the locus of Z on the Argand diagram
Writing ln[(x + jy + a)/(x + jy – a)] = u + jv, show that (a) x + y - 2ax coth u + a = 0 (b) x = a sinh ul(cosh u cos v) (c) |x + jy| = a(cosh u + cos v)/(cosh u - cos v)
(a) Find the loci in the Argand diagram corresponding to the equation(b) If the point z = x + jy describes the circle |z – 1| = 1, show that the real part of 1/(z – 2) is constant. |-1| = 2|z-jl
Given z = (2 + j)/(1 – j), find the real and imaginary parts of z + z–1.
For x and y real solve the equation jy jx+1 3y + j4 = 0 4 3x + y
Let z = 4 + j3 and w = 2 – j. Calculate(a) 3z (b) w* (c) zw(d) z2 (e) |z| (f) w/z(g) z – 1/w (h) arg z (i) z3/2
The complex impedance of two circuit elements in series as shown in Figure 3.22 (a) is the sum of the complex impedances of the individual elements, and the reciprocal of the impedance of two elements in parallel is the sum of the reciprocals of the individual impendances, as shown in Figure
Calculate the complex impedance for the circuit shown in Figure 3.21 when an alternating current of frequency 50 Hz flows. 572 mH Figure 3.21 100 40 F
By writing z = x + jy and w = u + jv, show that the line y = π/4 on the z plane is transformed into the line v = u on the w plane by the function w = ez Find the image of the line x = 0 under the same function.
Show that the line x = 1 on the z plane is transformed into the circle u2 + v2 – u = 0 on the w plane by the function w = (z – 1)/(z + 1)
Show that the function w = (jz – 1)/(z – 1) maps the line y = x on the z plane onto the circle (u – 1)2 + (v – 1)2 = 1 on the w plane.
Show that the line y = 1 on the z plane is transformed into the line u = 1 on the w plane by the function w = (z + j)/(z – j).
Find the values of the complex numbers a and b such that the function w = az + b maps the point z = 1 + j to w = j and the point z = –1 to the point w = 1 + j.
Find u and v in terms of x and y where w = f(z), z = x + jy, w = u + jv and (a) f(z) = (1 - j)z 1 (c) f(z) = z + Z (b) f(z) = (z 1)
Find the cartesian equation of the locus of the point z = x + jy that moves in the Argand diagram such that |(z + 1)/(z – 2)| = 2.
Given that the argument of (z – 1)/(z + 1) is 1/4π, show that the locus of z in the Argand diagram is part of a circle of centre (0, 1) and radius √2.
Find the cartesian equation of the circle given byand give two other representations of the circle in terms of z. z+j z-1 = 2
Find the locus of the point z in the Argand diagram which satisfies the equation (a) |z 1 = 2 (c) |z2j3|= 4 (b) |2z1 = 3 (d) arg(z) = 0 (e) |z4|3z +1 (f) arg Z - z-j =/
Express as simply as possible the following loci in terms of a complex variable:(a) y = 3x – 2 (b) x2 + y2 + 4x = 0(c) x2 + y2 + 2x – 4y – 4 = 0(d) x2 – y2 = 1
Identify and sketch the loci on the complex plane given by (a) Re ()=1 z-j (c) Z +J = 3 (e) Im(z) = 2 (g) |z+j|-|z 1 (i) arg(2z - 3) = = z b) Re ( + 1) = 2 j (d) tan arg z+j Z - = (f) (h) arg(z + j2) = (j) |z-j2| = 1 3 |z+j|+|z1| = 2
The circle x2 + y2 + 4x = 0 and the straight line y = 3x + 2 are taken to lie on the Argand diagram. Describe the circle and the straight line in terms of z.
Describe the locus of z when (a) Rez = 5 (c) Z z+1 = 3 (b) |z1|= 3 (d) arg(z - 2) = /4
Describe the locus of z when (a) Rez = 5 (c) Z z+1 = 3 (b) |z1|= 3 (d) arg(z - 2) = /4
Let z = 8 + j and w = 4 + j4. Calculate the distance on the Argand diagram from z to w and from z to –w.

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