EBK BASIC TECHNICAL MATHEMATICS
11th Edition
ISBN: 9780134508290
Author: Evans
Publisher: PEARSON CUSTOM PUB.(CONSIGNMENT)
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Chapter 18, Problem 44RE
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The number of revolutions
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1.2.19. Let and s be natural numbers. Let G be the simple graph with vertex set
Vo... V„−1 such that v; ↔ v; if and only if |ji| Є (r,s). Prove that S has exactly k
components, where k is the greatest common divisor of {n, r,s}.
Question 3
over a field K.
In this question, MË(K) denotes the set of n × n matrices
(a) Suppose that A Є Mn(K) is an invertible matrix. Is it always true that A is
equivalent to A-¹? Justify your answer.
(b) Let B be given by
8
B = 0 7 7
0 -7 7
Working over the field F2 with 2 elements, compute the rank of B as an element
of M2(F2).
(c) Let
1
C
-1 1
[4]
[6]
and consider C as an element of M3(Q). Determine the minimal polynomial
mc(x) and hence, or otherwise, show that C can not be diagonalised.
[7]
(d) Show that C in (c) considered as an element of M3(R) can be diagonalised. Write
down all the eigenvalues. Show your working.
[8]
16. Solve the given differential equation:
y" + 4y sin (t)u(t 2π),
-
y(0) = 1, y'(0) = 0
Given,
1
(x² + 1)(x²+4)
1/3
-1/3
=
+
x²+1 x² +4
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Chapter 18 Solutions
EBK BASIC TECHNICAL MATHEMATICS
Ch. 18.1 - Practice Exercise
In a certain electric field a...Ch. 18.1 - Prob. 2PECh. 18.1 - Prob. 1ECh. 18.1 - Prob. 2ECh. 18.1 - Prob. 3ECh. 18.1 - Prob. 4ECh. 18.1 - Prob. 5ECh. 18.1 - Prob. 6ECh. 18.1 - Prob. 7ECh. 18.1 - Prob. 8E
Ch. 18.1 - Prob. 9ECh. 18.1 - Prob. 10ECh. 18.1 - Prob. 11ECh. 18.1 - Prob. 12ECh. 18.1 - Prob. 13ECh. 18.1 - Prob. 14ECh. 18.1 - Prob. 15ECh. 18.1 - Prob. 16ECh. 18.1 - In Exercises 11–26, find the required...Ch. 18.1 - Prob. 18ECh. 18.1 - In Exercises 11–26, find the required...Ch. 18.1 - Prob. 20ECh. 18.1 - Prob. 21ECh. 18.1 - Prob. 22ECh. 18.1 - Prob. 23ECh. 18.1 - Prob. 24ECh. 18.1 - Prob. 25ECh. 18.1 - Prob. 26ECh. 18.1 - Prob. 27ECh. 18.1 - Prob. 28ECh. 18.1 - Prob. 29ECh. 18.1 - Prob. 30ECh. 18.1 - Prob. 31ECh. 18.1 - Prob. 32ECh. 18.1 - Prob. 33ECh. 18.1 - Prob. 34ECh. 18.1 - Prob. 35ECh. 18.1 - Prob. 36ECh. 18.1 - Prob. 37ECh. 18.1 - Prob. 38ECh. 18.1 - Prob. 39ECh. 18.1 - Prob. 40ECh. 18.1 - Prob. 41ECh. 18.1 - Prob. 42ECh. 18.1 - Prob. 43ECh. 18.1 - Prob. 44ECh. 18.1 - Prob. 45ECh. 18.1 - Prob. 46ECh. 18.1 - Prob. 47ECh. 18.1 - Prob. 48ECh. 18.1 - Prob. 49ECh. 18.1 - Prob. 50ECh. 18.1 - Prob. 51ECh. 18.1 - Prob. 52ECh. 18.2 - Express the relationship that y varies directly as...Ch. 18.2 - Prob. 2PECh. 18.2 - Prob. 3PECh. 18.2 - Prob. 4PECh. 18.2 - Prob. 1ECh. 18.2 - Prob. 2ECh. 18.2 - Prob. 3ECh. 18.2 - Prob. 4ECh. 18.2 - Prob. 5ECh. 18.2 - Prob. 6ECh. 18.2 - Prob. 7ECh. 18.2 - Prob. 8ECh. 18.2 - Prob. 9ECh. 18.2 - Prob. 10ECh. 18.2 - Prob. 11ECh. 18.2 - Prob. 12ECh. 18.2 - Prob. 13ECh. 18.2 - Prob. 14ECh. 18.2 - Prob. 15ECh. 18.2 - Prob. 16ECh. 18.2 - Prob. 17ECh. 18.2 - In Exercises 17–20, give the specific equation...Ch. 18.2 - Prob. 19ECh. 18.2 - Prob. 20ECh. 18.2 - Prob. 21ECh. 18.2 - Prob. 22ECh. 18.2 - Prob. 23ECh. 18.2 - Prob. 24ECh. 18.2 - Prob. 25ECh. 18.2 - Prob. 26ECh. 18.2 - Prob. 27ECh. 18.2 - Prob. 28ECh. 18.2 - Prob. 29ECh. 18.2 - Prob. 30ECh. 18.2 - Prob. 31ECh. 18.2 - Prob. 32ECh. 18.2 - Prob. 33ECh. 18.2 - Prob. 34ECh. 18.2 - Prob. 35ECh. 18.2 - Prob. 36ECh. 18.2 - Prob. 37ECh. 18.2 - Prob. 38ECh. 18.2 - Prob. 39ECh. 18.2 - Prob. 40ECh. 18.2 - Prob. 41ECh. 18.2 - Prob. 42ECh. 18.2 - Prob. 43ECh. 18.2 - Prob. 44ECh. 18.2 - In Exercises 31–64, solve the given applied...Ch. 18.2 - Prob. 46ECh. 18.2 - Prob. 47ECh. 18.2 - Prob. 48ECh. 18.2 - Prob. 49ECh. 18.2 - Prob. 50ECh. 18.2 - Prob. 51ECh. 18.2 - Prob. 52ECh. 18.2 - Prob. 53ECh. 18.2 - Prob. 54ECh. 18.2 - Prob. 55ECh. 18.2 - Prob. 56ECh. 18.2 - Prob. 57ECh. 18.2 - Prob. 58ECh. 18.2 - Prob. 59ECh. 18.2 - Prob. 60ECh. 18.2 - Prob. 61ECh. 18.2 - Prob. 62ECh. 18.2 - Prob. 63ECh. 18.2 - Prob. 64ECh. 18 - Prob. 1RECh. 18 - Prob. 2RECh. 18 - Prob. 3RECh. 18 - Prob. 4RECh. 18 - Prob. 5RECh. 18 - Prob. 6RECh. 18 - Prob. 7RECh. 18 - Prob. 8RECh. 18 - Prob. 9RECh. 18 - Prob. 10RECh. 18 - Prob. 11RECh. 18 - Prob. 12RECh. 18 - Prob. 13RECh. 18 - Prob. 14RECh. 18 - Prob. 15RECh. 18 - Prob. 16RECh. 18 - Prob. 17RECh. 18 - Prob. 18RECh. 18 - Prob. 19RECh. 18 - Prob. 20RECh. 18 - Prob. 21RECh. 18 - In Exercises 21–36, answer the given questions by...Ch. 18 - Prob. 23RECh. 18 - Prob. 24RECh. 18 - Prob. 25RECh. 18 - Prob. 26RECh. 18 - Prob. 27RECh. 18 - Prob. 28RECh. 18 - Prob. 29RECh. 18 - Prob. 30RECh. 18 - Prob. 31RECh. 18 - Prob. 32RECh. 18 - Prob. 33RECh. 18 - Prob. 34RECh. 18 - Prob. 35RECh. 18 - Prob. 36RECh. 18 - Prob. 37RECh. 18 - Prob. 38RECh. 18 - Prob. 39RECh. 18 - Prob. 40RECh. 18 - Prob. 41RECh. 18 - Prob. 42RECh. 18 - Prob. 43RECh. 18 - Prob. 44RECh. 18 - Prob. 45RECh. 18 - Prob. 46RECh. 18 - Prob. 47RECh. 18 - Prob. 48RECh. 18 - Prob. 49RECh. 18 - Prob. 50RECh. 18 - Prob. 51RECh. 18 - Prob. 52RECh. 18 - Prob. 53RECh. 18 - Prob. 54RECh. 18 - Prob. 55RECh. 18 - In Exercises 41–82, solve the given applied...Ch. 18 - Prob. 57RECh. 18 - In Exercises 41–82, solve the given applied...Ch. 18 - Prob. 59RECh. 18 - Prob. 60RECh. 18 - Prob. 61RECh. 18 - Prob. 62RECh. 18 - Prob. 63RECh. 18 - Prob. 64RECh. 18 - Prob. 65RECh. 18 - Prob. 66RECh. 18 - Prob. 67RECh. 18 - Prob. 68RECh. 18 - Prob. 69RECh. 18 - Prob. 70RECh. 18 - Prob. 71RECh. 18 - Prob. 72RECh. 18 - Prob. 73RECh. 18 - Prob. 74RECh. 18 - Prob. 75RECh. 18 - Prob. 76RECh. 18 - Prob. 77RECh. 18 - Prob. 78RECh. 18 - Prob. 79RECh. 18 - Prob. 80RECh. 18 - Prob. 81RECh. 18 - Prob. 82RECh. 18 - Prob. 83RECh. 18 - Prob. 1PTCh. 18 - Prob. 2PTCh. 18 - Prob. 3PTCh. 18 - Prob. 4PTCh. 18 - Prob. 5PTCh. 18 - Prob. 6PTCh. 18 - Prob. 7PT
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- R denotes the field of real numbers, Q denotes the field of rationals, and Fp denotes the field of p elements given by integers modulo p. You may refer to general results from lectures. Question 1 For each non-negative integer m, let R[x]m denote the vector space consisting of the polynomials in x with coefficients in R and of degree ≤ m. x²+2, V3 = 5. Prove that (V1, V2, V3) is a linearly independent (a) Let vi = x, V2 = list in R[x] 3. (b) Let V1, V2, V3 be as defined in (a). Find a vector v € R[×]3 such that (V1, V2, V3, V4) is a basis of R[x] 3. [8] [6] (c) Prove that the map ƒ from R[x] 2 to R[x]3 given by f(p(x)) = xp(x) — xp(0) is a linear map. [6] (d) Write down the matrix for the map ƒ defined in (c) with respect to the basis (2,2x + 1, x²) of R[x] 2 and the basis (1, x, x², x³) of R[x] 3. [5]arrow_forwardQuestion 4 (a) The following matrices represent linear maps on R² with respect to an orthonormal basis: = [1/√5 2/√5 [2/√5 -1/√5] " [1/√5 2/√5] A = B = [2/√5 1/√5] 1 C = D = = = [ 1/3/5 2/35] 1/√5 2/√5 -2/√5 1/√5' For each of the matrices A, B, C, D, state whether it represents a self-adjoint linear map, an orthogonal linear map, both, or neither. (b) For the quadratic form q(x, y, z) = y² + 2xy +2yz over R, write down a linear change of variables to u, v, w such that q in these terms is in canonical form for Sylvester's Law of Inertia. [6] [4]arrow_forwardpart b pleasearrow_forward
- Question 5 (a) Let a, b, c, d, e, ƒ Є K where K is a field. Suppose that the determinant of the matrix a cl |df equals 3 and the determinant of determinant of the matrix a+3b cl d+3e f ГЪ e [ c ] equals 2. Compute the [5] (b) Calculate the adjugate Adj (A) of the 2 × 2 matrix [1 2 A = over R. (c) Working over the field F3 with 3 elements, use row and column operations to put the matrix [6] 0123] A = 3210 into canonical form for equivalence and write down the canonical form. What is the rank of A as a matrix over F3? 4arrow_forwardQuestion 2 In this question, V = Q4 and - U = {(x, y, z, w) EV | x+y2w+ z = 0}, W = {(x, y, z, w) € V | x − 2y + w − z = 0}, Z = {(x, y, z, w) € V | xyzw = 0}. (a) Determine which of U, W, Z are subspaces of V. Justify your answers. (b) Show that UW is a subspace of V and determine its dimension. (c) Is VU+W? Is V = UW? Justify your answers. [10] [7] '00'arrow_forwardGood explanation it sure experts solve itarrow_forward
- Best explains it not need guidelines okkarrow_forwardTask number: A1.1, A1.7 Topic: Celestial Navigation, Compass - Magnetic and Gyro Activ Determine compass error (magnetic and gyro) using azimuth choosing a suitable celestial body (Sun/ Stars/ Planets/ Moon). Apply variation to find the deviation of the magnetic compass. Minimum number of times that activity should be recorded: 6 (2 each phase) Sample calculation (Azimuth- Planets): On 06th May 2006 at 22h20m 10s UTC, a vessel in position 48°00'N 050°00'E observed Mars bearing 327° by compass. Find the compass error. If variation was 4.0° East, calculate the deviation. GHA Mars (06d 22h): Increment (20m 10s): 089° 55.7' 005° 02.5' v (0.9): (+) 00.3' GHA Mars: 094° 58.5' Longitude (E): (+) 050° 00.0' (plus- since longitude is easterly) LHA Mars: 144° 58.5' Declination (06d 22h): d (0.2): N 024° 18.6' (-) 00.1' Declination Mars: N 024° 18.5' P=144° 58.5' (If LHA<180°, P=LHA) A Tan Latitude/ Tan P A Tan 48° 00' Tan 144° 58.5' A = 1.584646985 N (A is named opposite to latitude, except when…arrow_forwardTask number: A1.1, A1.7 Topic: Celestial Navigation, Compass - Magnetic and Gyro Activ Determine compass error (magnetic and gyro) using azimuth choosing a suitable celestial body (Sun/ Stars/ Planets/ Moon). Apply variation to find the deviation of the magnetic compass. Minimum number of times that activity should be recorded: 6 (2 each phase) Sample calculation (Azimuth- Planets): On 06th May 2006 at 22h20m 10s UTC, a vessel in position 48°00'N 050°00'E observed Mars bearing 327° by compass. Find the compass error. If variation was 4.0° East, calculate the deviation. GHA Mars (06d 22h): Increment (20m 10s): 089° 55.7' 005° 02.5' v (0.9): (+) 00.3' GHA Mars: 094° 58.5' Longitude (E): (+) 050° 00.0' (plus- since longitude is easterly) LHA Mars: 144° 58.5' Declination (06d 22h): d (0.2): N 024° 18.6' (-) 00.1' Declination Mars: N 024° 18.5' P=144° 58.5' (If LHA<180°, P=LHA) A Tan Latitude/ Tan P A Tan 48° 00' Tan 144° 58.5' A = 1.584646985 N (A is named opposite to latitude, except when…arrow_forward
- Activ Determine compass error using amplitude (Sun). Minimum number of times that activity should be performed: 3 (1 each phase) Sample calculation (Amplitude- Sun): On 07th May 2006 at Sunset, a vessel in position 10°00'N 010°00'W observed the Sun bearing 288° by compass. Find the compass error. LMT Sunset: LIT: (+) 00d 07d 18h 00h 13m 40m UTC Sunset: 07d 18h 53m (added- since longitude is westerly) Declination (07d 18h): N 016° 55.5' d (0.7): (+) 00.6' Declination Sun: N 016° 56.1' Sin Amplitude = Sin Declination/Cos Latitude = Sin 016°56.1'/ Cos 10°00' = 0.295780189 Amplitude=W17.2N (The prefix of amplitude is named easterly if body is rising, and westerly if body is setting. The suffix is named same as declination) True Bearing=287.2° Compass Bearing= 288.0° Compass Error = 0.8° Westarrow_forwardOnly sure experts solve it correct complete solutions okkarrow_forward4c Consider the function f(x) = 10x + 4x5 - 4x³- 1. Enter the general antiderivative of f(x)arrow_forward
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