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modern engineering mathematics

Questions and Answers of Modern Engineering Mathematics

Find the values of z1/3, where z = cos 2π + j sin 2π. Generalize this to an expression for 11. Hence solve the equations (a) 2 2+2 Z = 1 (b) (z 3) z = 0 -
Solve the quadratic equation z2 – (3 + j5)z + j8 – 5 = 0
Obtain the four solutions of the equation z4 = 3 – j4 giving your answers to three decimal places.
Find the following complex numbers in their polar forms:(a) (√3 – j)1/4 (b) (j8)1/3(c) (3 – j3)–2/3 (d) (–1)1/4(e) (2 + j2)4/3(f) (5 – j3)–1/2
Find the three values of (8 + j8)1/3 and show them on an Argand diagram.
Use the method of Section 3.3.2 to prove the following results: (a) sin 30 = 3 cos0 sin 0 sin0 (b) cos 80 = 128 cos0 - 256 cose + 160 cos0 - 32 cos0 + 1 (c) tan 50 = 5 tan 0 10 tan 0+ tan50 1-10 tan0
Expand in terms of multiple angles(a) cos4θ (b) sin3θ
Use de Moivre’s theorem to calculate the third and fourth powers of the complex numbersData from De Moivre’s theorem (a) 1 + j (d) 1-j3 (b) 3 - j (e) 1+ j3 (c) 3+j4 (f) -1 - j3
In a certain cable of length l the current I0 at the sending end when it is raised to a potential V0 and the other end is earthed is given byCalculate the value of I0 when V0 = 100, Z0 = 500 + j400,
Writing tanh(u + jv) = x + jy, with x, y, u and v real, determine x and y in terms of u and v. Hence evaluate tanh(2 + j 1/4π) in the form x + jy.
Show that(a) ln(5 + j12) = ln 13 + j1.176(b) In(-/- j3)=-jt
Solve z = x + jy when(a) sin z = 2 (b) cos z = j3/4(c) sin z = 3 (d) cosh z = –2
Express in the form x + jy (a) sin(+j) (c) sinh[;(1+ j] (b) cos(j) (d) cosh(j)
Using the exponential forms of cos θ and sin θ given in (3.11a, b), prove the following trigonometric identities:Data from 3.11a and b (a) sin(a + B) = sin a cos + cos a sin (b) sin0 = sin 0 -sin
Given z1 = 2ejπ/3 and z2 = 4e–2jπ/3, find the modulus and argument of (a) (b) z (c) z/z
Given z1 = ejπ/4 and z2 = e–jπ/3, find(a) The arguments of z1z22 and z31/z2(b) The real and imaginary parts of z21 + jz2
Express z = (2 – j)(3 + j2)/(3 – j4) in the form x + jy and also in polar form.
Express in polar form the complex numbers(a) j (b) 1(c) –1 (d) 1 – j(e) √3 – j√3 (f) –2 + j(g) –3 – j2 (h) 7 – j5(i) (2 – j)(2 + j) (j) (–2 + j7)2
Express the following complex numbers in cartesian form:(a) e3+jπ/4 (b) e–1+jπ/3
Express the following complex numbers in exponential form:(a) 3 + j4 (b) –1 + j√3
Obtain the modulus and argument of z whereand write z in the form x + jy. Z= (2 + j)(-3 + j4) (12 - j5)*(1-j)*
For the following pairs of numbers obtain z1z2, z1/z2, and z2/z1: 2 = = = (a) Z - Z 2 cos | + jsin ? ?] ) T + jsin
If z1 = 1 + j and z2 = √3 + j, determine |z1z2|, |z1/z2|, arg(z1z2) and arg(z1/z2).
Find z3 in the form x + jy, where x and y are real numbers, given thatwhere z1 = 3 – j4 and z2 = 5 + j2. zz + N I I || Z3 1
Find the values of the real numbers x and y which satisfy the equation 2 + x - Jy = 1 + j2 3x + jy
Find z = z1 + z2z3/(z2 + z3) when z1 = 2 + j3, z2 = 3 + j4 and z3 = –5 + j12.
Find the real and imaginary parts of z when 1 2 || Z 2+j3 + 1 3-j2
Given z = 2 – j2 is a root offind the remaining roots of the equation. 2z 9z + 20z - 8 = 0
For z = x + jy (x and y real) satisfyingfind x and y. 22 1 + j 2z j || 5 2 + j
Find the complex numbers w, z which satisfy the simultaneous equations 4z + 3w = 23 z + jw = 6 + j8
Find the modulus and argument of each of the complex numbers given in Question 1.Data from question 1Show in an Argand diagram the points representing the following complex numbers: (a) 1 + j (c) 3+
With z = 2 – j3, find(a) jz (b) z* (c) 1/z (d) (z*)*
Find z such that zz* + 3(z – z*) = 13 + j12
Find the roots of the equations(a) x2 + 2x + 2 = 0 (b) x3 + 8 = 0
Determine the complex conjugate of(a) 2 + j7(b) –3 – j (c) –j6 (d) 1/00 v/m
Express in the form x + jy where x and y are real numbers: (a) (5+j3)(2-j)-(3+j) 5-j8 3-j4 (e) / (1+j) 1 5-j3 1 5+j3 (b) (1 - j2) 1-j 1 + j (f) (3-j2) (d) (h) 1 2 3-j4 5-j8
Express in the form x + jy:(a) (4 – j6)/(1 + j) (b) (5 + j3)/(3 – j2)(c) (1 – j)/(4 + j3) (d) (–4 – j3)/(2 – j)
Express in the form x + jy:(a) (6 – j3)(2 + j4) (b) (7 + j)(2 – j3)(c) (–1 + j)(–2 + j3) (d) (–3 + j2)(4 + j7)
Obtain the roots of the equations below using complex numbers where necessary: (a) x + 6x + 13 = 0 (b) x-x+ 2 = 0 (c) 4x + 4x + 5 = 0 (d) x + 2x - 3=0 (e) xx - 6 = 0
Find where z1 and z2 are the complex numbers z1 = 1 + j2, z2 = 3 – j. Z + Z2, 21-22, 22, -322, 52 - 222, 22 +2
Show in an Argand diagram the points representing the following complex numbers: (a) 1 + j (c) 3+ j4 (e) 1+ j3 (b) 3 - j (d) 1 - j3 (f) -1- j3
Obtain the graph of f–1(x) when (a) f(x) = 9/5 x + 32, (b)(c) f(x) = x2. x + 2 x+1 -x-1,
Multiply out the following:(a) (2x + 1)(y − 3) (b) (−3a + 4)(2a − 7)(c) (m − 3)(2m + 1) − (2 − m)(m + 5) (d) (a + 2b − c)(a − 2b)(e) (x + y)(y + z) − (x + y)(x + z) + (z + y)(y +
Factorise 6x2 + 19x + 10
Without using a calculator calculate 1192
Express as a single term 9x2 − 30x + 25
The function f (x) has formula y = x2 for 0 ≤ x < 1. Sketch the graphs of f (x) for –4 < x < 4 when(a) f(x) is periodic with period 1;(b) f(x) is even and periodic with period 2;(c) f(x) is odd
Sketch the graphs of the functions(a) y = √(x2)(b) y = √(x2 + x3), x ≥ –1(c) y = x√(1 + x), x ≥ –1(d) y = √(1 + x) + √(1 – x), –1 ≤ x ≤ 1
The lower edge of a mural, which is 4 m high, is 2 m above an observer’s eye level, as shown in Figure 2.53. Show that the optical angle θ° is given bywhere d m is the distance of the observer
The initial cost of buying a car is £6000. Over the years, its value depreciates and its running costs increase, as shown in the table below.Draw up a table showing (a) The cumulative running cost
If the number whose decimal representation is 14732 has the representation 152112b to base b, what is b?
For the functions with formulae below, identify their domains, codomains and ranges and calculate values of f(2), f(–3) and f(–x).(a) f(x) = 3x2 + 1(b) f:x → √[(x + 4)(3 – x)]
A straight horizontal road is to be constructed through rough terrain. The width of the road is to be 10 m, with the sides of the embankment sloping at 1 (vertical) in 2 (horizontal), as shown in
Simplify the following expressions:(a) 4 (c − d) − 3 (c − 2d) (b) 3x(y + 2) + 2y (x − 1)(c) m (m + 2n) − n (3n + 2m − 4) (d) 3x(x − y) − 2y (2x − 3y)(e) 7 (2a − 4b − 5) − 3
The function y = f(x) is given by the minimum diameter y of a circular pipe that can contain x circular pipes of unit diameter, where x = 1, 2, 3, 4, 5, 6, 7. Find the domain, codomain and range of
The perimeter of an ellipse depends on the lengths of its major and minor axes, and is given byand E is the function whose graph is given in Figure 2.109.(a) Calculate the perimeter of the ellipse
Simplify the following expressions(a) 2(2a + 3) + 3(3a + 4)(b) x(3 + 2x) + 2(3 + 5x2)
A hot-water tank has the form of a circular cylinder of internal radius r, topped by a hemisphere as shown in Figure 2.8. Show that the internal surface area A is given byFind the formula relating
The relationship between the temperature T1 measured in degrees Celsius (°C) and the corresponding temperature T2 measured in degrees Fahrenheit (°F) isInterpreting this as a function with T1 as
The sales volume of a product depends on its price as follows:The cost of production is £1 per unit. Draw up a table showing the sales revenue, the cost and the profits for each selling price, and
Multiply out the following(a) (2x + 1)(y − 3)(b) (m + n)(n − m) − (n − 2m)(m − 3n)
An oil storage tank has the form of a circular cylinder with its axis horizontal, as shown in Figure 2.9. The volume of oil in the tank when the depth is h is given in the table below.Draw a careful
1. Expand (x + 2y)22. Expand (2a − 1)23. Expand and simplify (2m + 3n)2 − (5m − 7n)24. Evaluate (√3 + √5)(√3 − √5) without using a calculator5. Simplify (√a − √b)(√a + √b)6.
Draw graphs of the functions below, locating their zeros, intervals in which they are increasing, intervals in which they are decreasing and their optimal values. (a) y = 2x + 3x - 12x + 32 (b) y =
A function f is defined byDraw the graphs of f(x), f(x – 2) and f(2x). The function g(x) is defined as f(x + 2) – f(2x – 1). Draw a graph of g(x). [x+1 (x < -1) f = {0 (-1x1) x-1 (x>1)
Simplify the expression (x + 3)2 − 2x − 5.
Factorise the following:(a) 36x2 − 9x (b) 16m2 − 9(c) bx − cy + by − cx (d) ac − a2 + ad − cd(e) 6cd + 2cn − 9md − 3mn (f) x2 + 12x + 35(g) 2x2 + 9x − 5 (h) p2 − 2p − 3(i)
A gas company charges its industrial users according to their gas usage. Their tariff is as follows:What is the quarterly charge paid by a user? Quarterly usage/10 units 0-19.999 20-49.999 50-99.999
Plot graphs of the functions below, locating their zeros, intervals in which they are increasing, intervals in which they are decreasing and their optimal values. (a) y = x(x - 2) (c) y = x(x - 2)
1. Simplify the following expressions:2. Simplify the following expressions:3. Divide 4x2 − 3x + 2 by x − 24. Divide x3 + 5x2 + 7x + 3 by x + 35. Divide 2x3 + 3x2 − 5x +15 by x + 3 (a) (c)
Obtain the inverse function of the real function y = f(x) =1/5 (4x – 3).
Without using a calculator calculate 132 − 112
A function f(x) is defined by f(x) = 1/2(10x + 10–x), for x in R. Show that (a) 2(f(x)) = f(2x) + 1 (b) 2f(x)f(y) = f(x + y) + f(x - y)
Express as single fractions in their lowest terms: (a) 1 + x+1x-1 (e) 2 R- x+5 (c) -+ x + 3 2-x 2x+1 2 p-3 12 p(p-3) 5 p (b) (d) m 3 m-3 m-7m +12 + 2x 1 x+x-6 x-2 +
Obtain the inverse function of y = f(x) = x + 2 x + 1 x = -1.
Sketch the functions(a) x2 – 4x + 7(b) x3 – 2x2 + 4x – 3(c)(d) x + 4 x - 1
Factorise the following expressions(a) 5x + 10y (b) 6a2 − 12ab + 3a (c) 6y2 − 2y
1. The simple interest I on a loan is given by the formulawhere P is the size of the loan, R is the rate of interest and T is the duration of the loan. If P = £3000, R = 6 % and T = 3 years,
Draw separate graphs of the functions f and g where f(x) = (x + 1)2 and g(x) = x – 2 The functions F and G are defined by F(x) = f(g(x)) and G(x) = g(f (x)) Find formulae for F(x) and G(x) and
Find the Taylor expansion of x4 + 3x3 – x2 + 2x – 1 about x = 1.
Factorise the following expressions(a) ax − ay + bx − by (b) a2 + bc + ab + ac
A function f is defined bySketch on separate diagrams the graphs of f(x), f(x + 1/2), f(x + 1), f(x + 2), f(x – 1/2), f(x – 1) and f(x – 2). f(x) 0 (x < -1) x+1 (-1 1)
Find the partial fractions of (a) (c) x + 2 (x - 1)(x-4) x - 2x + 3 (x + 2)(x - 1) (b) (d) x + 4 (x + 1)(x 3) - x(2x - 1) (x - x + 1)(x + 3)
If y = f(x) = x2 + 2x and y = g(x) = x – 1, obtain the composite functions f(g(x)) and g(f(x)).
Find the inverse function (if it is defined) of the following functions:If f (x) does not have an inverse function, suggest a suitable restriction of the domain of f(x) that will allow the definition
An open conical container is made from a sector of a circle of radius 10 cm as illustrated in Figure 2.16, with sectional angle θ (radians). The capacity Ccm3 of the cone depends on θ. Find the
Express as products of sines and/or cosines(a) sin 2θ – sin θ (b) cos 2θ + cos 3θ(c) sin 4θ – sin 7θ
Simplify the following (a) 8abc 2ac (b) 2x+1 6x-x-2
Express in the form r sin(u – a)(a) 4 sin θ – 2 cos θ (b) sin θ + 8 cos θ(c) √3 sin θ + cos θ
Show thatmay be expressed in the formInterpret this result graphically. f(x)= = 2x-3 x +4
Which of the functions y = f(x) whose graphs are shown in Figure 2.20 are odd, even or neither odd nor even?Figure 2.20 (a) (c) y 6 4 4 2 -1 1 -4 (b) -2 -2 y O 2
(a) From the definition of the hyperbolic sine function prove(b) Sketch the graph of y = x3 + x carefully, and show that for each value of y there is exactly one value of x. Setting z = 1/2x√3,
(a) Divide 2x2 − 3x + 4 by x + 1(b) Divide x3 − 3x2 + 6x − 4 by x − 1
The stiffness of a rectangular beam varies directly with the cube of its height and directly with its breadth. A beam of rectangular section is to be cut from a circular log of diameter d. Show that
A function f(x) has the graph on [0, 1] shown in Figure 2.23. Sketch its graph on [–3, 3] given that(a) f(x) is periodic with period 1;(b) f(x) is periodic with period 2 and is even;(c) f(x) is
Express as a single fraction in its lowest terms: (a) 2-x + x-2 x+3 (b) 1 2m m-1 1-m +
The parts produced by three machines along a factory aisle (shown in Figure 2.110 as the x axis) are to be taken to a nearby bench for assembly before they undergo further processing. Each assembly
Sketch the graphs of the functions(a)(b) xH(x) – (x – 1)H(x – 1) + (x – 2)H(x – 2) ] 2 -x
A manufacturer produces 5000 items at a total cost of £10 000 and sells them at £2.75 each. What is the manufacturer’s profit as a function of the number x of items sold?
Use Lagrange’s formula to find the linear function f(x) where f(10) = 1241 and f(15) = 1556.

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