Understanding Engineering Mathematics

Thereciprocalof the equivalent resistance of two resistors inparallelis equal to the

sum of their reciprocals:

1

R

=

1

R 1

+

1

R 2

(see Section 1.2.5)

(a) If, in the circuit of Figure 1.1,R 1 = 1 ,R 2 = 2 ,R 3 = 3 determine the overall

equivalent resistance of the circuit, working entirely with fractions.

(b) Repeat the calculation using three decimal places accuracy and compare your result

with (a).

(c) Suppose now thatR 2 is not fixed, but can vary. Obtain the equivalent resistance in

terms ofR 2 using exact fractions.

(d) Using the result of (c), plot a graph of the equivalent resistanceRfor integer values

ofR 2 from 1 to 10.

(e) Discuss the calculations and results of (c), (d) in the light of the need to plot a

useful graph.

(f) If we wish to increase slightly the equivalent resistance above that whenR 2 = 2 

what should we do withR 2 , increase or decrease it?

2.Using the expressions fornCrtry a selection of values ofnandrto investigate whether
the following relations might be true.

(i) nCr+nCn−r=rCn (ii) n−^1 Cr+n−^1 Cn−r=nCr

(iii) nCr=nCn−r

Prove any result that you suspect is true.

Answers to reinforcement exercises

1.3.1 Types of numbers

A. (i) 2 is a positive prime and even integer.

(ii) −3 is a negative integer.

(iii) 11 – integer, positive, prime, odd.

(iv) 21 – integer, positive, odd, composite (3×7).

(v) −0 – zero, both positive and negative.

(vi)^23 – proper fraction, rational number, positive.

(vii)^52 – improper fraction, positive, rational.

(viii) 1^25 is a positive mixed fraction expressible as the improper fraction^72.

(ix) −^37 – negative proper fraction.