Higher Engineering Mathematics, Sixth Edition
222 Higher Engineering Mathematics Similarly, for theR−Ccircuit shown in Fig. 20.8(b), VC lags I by 90◦ (i.e. I leads VC by 90◦) ...
Complex numbers 223 Current I= V Z . Impedance Z for the three-branch parallel circuit is given by: 1 Z = 1 Z 1 + 1 Z 2 + 1 Z 3 ...
224 Higher Engineering Mathematics Two impedances, Z 1 =( 3 +j 6 ) and Z 2 =( 4 −j 3 )are connected in series to a supply vol ...
Chapter 21 De Moivre’s theorem 21.1 Introduction From multiplicationof complex numbers in polar form, (r∠θ)×(r∠θ)=r^2 ∠ 2 θ Simi ...
226 Higher Engineering Mathematics Now try the following exercise Exercise 90 Further problems on powers of complex numbers Det ...
De Moivre’s theorem 227 arguments,θ, are different. It is shown in Problem 3 that arguments are symmetrically spaced on an Argan ...
228 Higher Engineering Mathematics (− 2 +j) 1 (^4) ⎡ ⎣ Modulus 1. 223 ,arguments 38. 36 ◦, 128. 36 ◦, 218. 36 ◦and 308. 36 ◦ ⎤ ...
De Moivre’s theorem 229 Problem 6. Change( 3 −j 4 )into (a) polar form, (b) exponential form. (a) ( 3 −j 4 )= 5 ∠− 53. 13 ◦or 5 ...
230 Higher Engineering Mathematics Ifz=7ej^2.^1 , determine lnz(a) in Cartesian form, and (b) in polar form.⎡ ⎣ (a)ln7+j 2. 1 ...
Chapter 22 The theory of matrices and determinants 22.1 Matrix notation Matrices and determinants are mainly used for the solu- ...
232 Higher Engineering Mathematics (a) Adding the corresponding elements gives: ( 2 − 1 − 74 ) + ( − 30 7 − 4 ) = ( 2 +(− 3 ) − ...
The theory of matrices and determinants 233 For scalar multiplication,each element is multiplied by the scalar quantity, hence 2 ...
234 Higher Engineering Mathematics = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ [( 3 × 2 ) [( 3 ×(− 5 )) +( 4 × 5 ) +( 4 ×(− 6 )) +( 0 ×(− 1 )) ...
The theory of matrices and determinants 235 D+E ⎡ ⎢ ⎣ ⎛ ⎜ ⎝ 7 − 18 317 47 − 2 ⎞ ⎟ ⎠ ⎤ ⎥ ⎦ A−B [( − 2 − 3 − 31 )] A+B−C [( ...
236 Higher Engineering Mathematics ∣ ∣ ∣ ∣ 5 ∠ 30 ◦ 2 ∠− 60 ◦ 3 ∠ 60 ◦ 4 ∠− 90 ◦ ∣ ∣ ∣ ∣=(^5 ∠^30 ◦)( 4 ∠− 90 ◦) −( 2 ∠− 60 ◦)( ...
The theory of matrices and determinants 237 The inverse of matrix ( pq rs ) is obtained by inter- changing the positions ofpands ...
238 Higher Engineering Mathematics the products of the elements and their cofactors is 2 × 11 + 7 ×13, i.e., ∣ ∣ ∣ ∣ ∣ ∣ 34 − 1 ...
The theory of matrices and determinants 239 Evaluate ∣ ∣ ∣ ∣ ∣ ∣ 3 ∠ 60 ◦ j 21 0 ( 1 +j) 2 ∠ 30 ◦ 02 j 5 ∣ ∣ ∣ ∣ ∣ ∣ [ 26. 94 ...
240 Higher Engineering Mathematics The determinant of ⎛ ⎝ 15 − 2 3 − 14 − 36 − 7 ⎞ ⎠ = 1 ( 7 − 24 )− 5 (− 21 + 12 )− 2 ( 18 − 3 ...
Chapter 23 The solution of simultaneous equationsbymatricesand determinants 23.1 Solution of simultaneous equations by matrices ...
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