Question: Given that Z_1=3(cos-45°+jcos-45°)andZ_2=6(cos102°+jsin102°), calculate the following, leaving your answer in polar form

Solution:

Question :

In the following series circuit, the e.m.f. source E(t)=20sinwt at w=2pif and f=1.65KHZ

Calculate:

(a)the total associated phasor impedance

(b)the current in the circuit

(c)the fall in p.d across the Resistor

(d)the fall in p.d. across the Capacitor

(e)the fall in p.d. across the Inductor.

Solution:

Question.

(a) If you throw two dice, what is the probability that their product is less than 15?

There is the following combination which has a product less than 15

(1,1) , (1,2) , (1, 3), (1,4) , (1,5), (1,6) , (2,1 ) , (2,2) , (2,3) ,(2,4) ,(2,5) ,(2,6) ,(3,1) ,(3,2) (3,3) , (3,4) , (4,1) , (4,2) , (4,3) , (5,1) , (5,2) , (6,1) and (6,2) .

So, 23 out of 36 possible combinations which have product less than the 15.

So, probability is 23/36.

P= = 0.6388

So, the probability is 63.89%.

 (b) If a card is selected at random from a shuffled deck of cards, what is the probability that the value of the card is greater than 4 and less than 10?

The standard deck, there are total card is 52, in which four types of card in a deck is, Diamond, Club, Viz.spade, Heart. Where each type contains 13 cards, the number of card with a value greater than 4 and less than 10 is 5. Thus total number of card is with a value greater than 4 and less than 20

Question.

A combination lock for a bicycle has 4 dials, each with the numbers 1 to 6. If you have forgotten the combination that opens the lock, how long will it take you to try all combinations, at a rate of one every 5 seconds?

There are 9999 different combinations in the 4-digit lock

Where each number is 1-6

There are 1 million of them (999999+1). If repletion is not allowed. So we picked the any of ten digits for the first, and any of nine remaining for eth seconds at the rate of every 5 seconds.

So, approximately 10 time and 9 times so on, 151200 

Question.

Consider the following three vectors:

p = 3i + 7j – 2k

 q = 4i – j + k

r = 5i –2 j + 6k

Calculate each of the following:

(a) The angle between the vectors p and q;

Calculate the position vector 

(d) The volume of the tetrahedron formed using the vectors p, q and r;

p = 3i + 7j – 2k

 q = 4i – j + k

r = 5i –2 j + 6k

Solution

Solution

Planes have normal vectors

a = (1, 6,-5) and b (7,-1,-11)

L denote the line of intersection

V= a*b   

Question

Write down the equation of any two planes that are parallel to each other.

The equation of planes are given below after solving the points for equation 1 is

A = (12, -4, 2),

B = (-6, 3,-3)

C = (10, 15, 9).

And the points of the equations 2 is

A = (-1, 2, 1),

B = 0, -3, 2)

C = (1, 1, -4).

68x+128y-328z= -352                        (1)

26x + 7y +9z = -3                               (2)

Question

Using the Bisection method, determine, correct to 1 decimal place, the root of the equation

Question

(a) Plot a graph of the function f(t) = Cost-t over the range  0 £ t £ 1  radians.

f(t) = Cost-t

f `(t) = Sint-1

here graph of y = Sint

Height of curve at any point is the line value of the Sine function

y=Sint  is an odd function

Sin(-t) = -Sint

y= Cost is an even function

Independent variable t is measured in radian

(b) By selecting a suitable starting point from your graph, use the Newton Raphson method to determine, correct to 5 decimal places, the root of the equation:

The life of a drill bit has a mean of 23 hours and a standard deviation of 2.8 hours. Assuming a Normal distribution, calculate the probability that a bit selected at random will have a life of:-

(a) Greater than 24 hours (3 marks)

Question

If 15% of the bolts produced by a machine are defective, find the probability that out of 5 bolts chosen at random

No bolts will be defective.