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modern engineering mathematics

Questions and Answers of Modern Engineering Mathematics

1) Find the angles X° and Y° in the following diagram in which AP and BP are tangents to the circle and O is its centre.2) In the following diagram, PC is a tangent to the circle and is of length 8
Obtain the modulus and argument of z whereand write z in the form x + jy. Z= (2 + j)(-3 + j4) (12 - j5)*(1-j)*
Express the following complex numbers in exponential form:(a) 3 + j4 (b) –1 + j√3
Express the following complex numbers in cartesian form:(a) e3+jπ/4 (b) e–1+jπ/3
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 z = (2 – j)(3 + j2)/(3 – j4) in the form x + jy and also in polar form.
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
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
Express in the form x + jy (a) sin(+j) (c) sinh[;(1+ j] (b) cos(j) (d) cosh(j4)
Solve z = x + jy when(a) sin z = 2 (b) cos z = j3/4(c) sin z = 3 (d) cosh z = –2
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,
Show that(a) ln(5 + j12) = ln 13 + j1.176(b) In(-/- j3)=-jt
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.
Use de Moivre’s theorem to calculate the third and fourth powers of the complex numbersData from De Moivre’s theorem
Expand in terms of multiple angles(a) cos4θ (b) sin3θ
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 cos 0 + 160 cos0 - 32 cos0 + 1 (c) tan 50 = 5 tan 0 10 tan0+ tan50 1-10 tan0
Find the three values of (8 + j8)1/3 and show them on an Argand diagram.
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
Solve the differential equation (In sint - 3x) d.x dt +xcott + 4t = 0
Determine the values of the appropriate parameters needed to give the systems governed by the following second-order linear constant-coefficient differential equations the damping parameters and
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)6 - z = 0
Find the damping parameters and natural frequencies of the systems governed by the following second-order linear constant-coefficient differential equations: dx (a) + dt dx + 2x=0 dt dx (b) + p- dt
Solve the differential equation dx 2xt + x - 2t = 0 dt
Solve the differential equation 1x - 3dx= x+cost dt
Obtain the four solutions of the equation z4 = 3 – j4 giving your answers to three decimal places.
Let z = 8 + j and w = 4 + j4. Calculate the distance on the Argand diagram from z to w and from z to –w.
Solve the quadratic equation z2 – (3 + j5)z + j8 – 5 = 0
For the following pairs of numbers obtain z1z2, z1/z2, and z2/z1: 2 = = = (a) Z - Z 2 cos | + jsin ? ?] ) T + jsin
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)
Error-correcting codes are widely used for data transmission. A message consisting of N binary bits is partitioned into blocks of k bits, and each block is transmitted with some additional parity
A large number N of people are subjected to a blood investigation to test for the presence of an illegal drug. This investigation is carried out by mixing the blood of k persons at a time and testing
In the game of craps, two dice are tossed. A total of 7 or 11 wins immediately, a total of 2, 3 or 12 loses. For remaining outcomes, both dice are tossed repeatedly until either a total of 7 appears,
Find the expected value of the maximum of four independent exponential random variables, each with parameter λ. In particular, if the time taken for a routine test and service of a jet aircraft
If X1, . . ., Xn are independent exponentially distributed random variables, each with parameter λ, prove that the random variable whose value is given by the minimum of {X1, . . ., Xn} also has an
The functionis known as the gamma function, and the probability density functiondefines the gamma distribution. Prove that 00. J = 6 I (a) = yledy (a: (0
Suppose that X is a continuous random variable with mean μX and variance σ2X. By separating the integral in the definition of σ2X into three parts and substituting the respective bounds for (x –
Ten thousand numbers are to be added, each rounded to the sixth decimal place. Assuming that the errors arising from rounding the numbers are mutually independent and uniformly distributed on (–0.5
A manufacturer has agreed to dispatch small servomechanisms in cartons of 100 to a distributor. The distributor requires that 90% of cartons contain at most one defective servomechanism. Assuming the
The binomial is a special case of the more general multinomial distribution:where p1 + . . . + pk = 1 and n1 + . . . . + nk = n. Each observation of a random variable has k possible outcomes, with
The City Engineer’s department installs 10 000 fluorescent lamp bulbs in street lamp standards. The bulbs have an average life of 7000 operating hours with a standard deviation of 400 hours.
If there are 720 personal computers in an office building and they each break down independently with probability 0.002 per working day, use the Poisson approximation to the binomial distribution to
A continuous random variable X has probability density function given bywhere c is constant. Find(a) The value of the constant c;(b) The cumulative distribution function of X;(c) P(X > 2);(d) The
Find the UCL and apply it to the data in Example 13.33.Example 13.33Regular samples of fifty are taken from a process making electronic components, for which an acceptable proportion of defectives is
Regular samples of fifty are taken from a process making electronic components, for which an acceptable proportion of defectives is 5%. Successive counts of defectives in each sample are as
If 70% of airline passengers using a particular route are members of a frequent-flyer club, find the probability that out of a sample of fifty chosen independently, more than forty will be members of
It was noted that the maximum error that would occur in the sum of 100 numbers, each of which was rounded to three decimal places, is 0.05. Find the probability of the error in the sum exceeding
In a quality control scheme at a factory, batches of components are accepted or rejected depending on the number of defective items counted in a sample. Rejected batches are inspected and all
The burning time X of an experimental rocket is a random variable having (approximately) a normal distribution with mean 600 s and standard deviation 25 s. Find the probability that such a rocket
If X∼ N(4, 4), find(a) P(X ≤ 6.7)(b) the constant c such that P(X > c) = 0.1.
A machine produces components that have defect A with probability 0.015 and defect B with probability 0.020, the two defects being independent. If fifty-four components are packed into a batch, what
If 0.04% of cars break down while driving through a certain tunnel, find the probability that at most two break down out of 2000 cars entering the tunnel on a given day.
A component supplier claims that 95% of its catalogue items are in stock at any time. A particular order for twenty different components is returned with three items missing as being out of stock. Is
Find the sample median and range for the data in Examples 13.22 and 13.23.Data from Example 13.23Measured values of resistance (inΩ) for twelve nominally 100Ω resistors were as follows:106, 98, 95,
Measured values of resistance (inΩ) for twelve nominally 100Ω resistors were as follows:106, 98, 95, 109, 99, 102, 101, 108, 94, 99, 96, 102Find the sample average and both forms of sample variance
A die was tossed twenty-four times, producing the following results: 4, 6, 2, 4, 2, 1, 5, 1, 3, 1, 3, 4, 5, 4, 3, 1, 6, 5, 6, 3, 1, 2, 4, 6Find the sample average and standard deviation.
If a mean temperature is 58°F, what is the mean temperature in degrees Celsius?
Find the distribution of total time for the situation described in Example 13.19, and the expected value of this time.Data from Example 13.19A new plant at a manufacturing site has to be first
A new plant at a manufacturing site has to be first installed and then commissioned. The times required for these two stages depend upon different random factors, and can therefore be regarded as
Find the mean and standard deviation of the waiting time in Example 13.15.Data from Example 13.15Two people have agreed to meet in a definite place between six and seven o’clock. Their actual times
Find the variance and standard deviation for each of the random variables in Example 13.16, and the interquartile range for the lifetime distribution.Data from Example 13.6Find the mean, median and
Find the mean, median and mode for(a) A simple die toss,(b) The number of ship arrivals(c) The lifetime distribution
Two people have agreed to meet in a definite place between six and seven o’clock. Their actual times of arrival are independent and entirely random (no arrival time more likely than any other)
For the distribution of component lifetimes in Example 13.13 find the proportion of components that last longer than 6000 hours.Data from Example 13.13 The lifetime of an electronic component (in
The number of ships arriving at a container terminal during any one day can be any integer from zero to four, with respective probabilities 0.1, 0.3, 0.35, 0.2, 0.05. Plot the probability and
If n people are independently selected, how large does n have to be before there is a better than even chance that at least two of them have the same birthday (not necessarily in the same year and
If two fair dice are tossed, find the probability of at least one six occurring.
A card is selected at random from an ordinary pack of fifty-two playing cards. Find the probabilities that the card drawn is(a) An ace and a club, (b) An ace or a club,(c) An ace and a king, (d) An
Items from a production line can have defects A or B. Some items have both, some just one, but most have neither. Tables (a) and (b) show two alternative sets of joint probabilities: (a) A Total B B
Suppose that on a small tropical island there are only two kinds of day: sunny days and rainy days. The probability that a sunny day is followed by a rainy day is 0.6, and the probability that a
The probability that a regularly scheduled flight departs on time is P(D) = 0.83, the probability that it arrives on time is P(A) = 0.92, and the probability that it both departs and arrives on time
Someone tosses a die, covers it up and tells you that the number shown is less than four. How does this change the probability that the number is even?
During the assessment of a class of students, 80% passed the examination in mathematics, 85% passed in laboratory work, and 75% passed both. For a student chosen at random from the class, find the
A fair six-sided die is tossed. Find the probability of the event ‘even number or number less than four’.
Two adjacent areas recorded two incidents in 250 flights and one incident in 85 flights respectively. Test the combination of the two areas.
A third area in the near-miss survey recorded five incidents in 800 flights. Should this area also be regarded as unusually risky?
Thirty-two successive samples of 100 castings each, taken from a production line, contained numbers of defectives as follows:If the proportion defective is to be maintained at 0.02, use the Shewhart
It is intended that 90% of electronic devices emerging from a machine should pass a simple on-the-spot quality test. The numbers of defectives among samples of 50 taken by successive shifts are as
A major airline operates 350 flights a day throughout the world. The probability that a flight will be delayed for more than one hour, for any reason, is 0.7%. If more than four flights suffer such
The diameter of ball-bearings produced by a machine is a random variable having a normal distribution with mean 6.00mm and standard deviation 0.02 mm. If the diameter tolerance is ;1%, find the
In firing at a target, a marksman scores at each shot either 10, 9, 8, 7 or 6, with respective probabilities 0.5, 0.3, 0.1, 0.05, 0.05. If he fires 100 shots, what is the approximate probability that
A fleet car operator has n cars, each of which has probability 8% of being broken down on any particular day. Find the smallest value of n that gives probability 90% that at least forty cars will be
Prove by making the substitution u = (x – μX)/σX in the integrals concerned that the mean and variance of the normal distribution are μX and σ2X respectively. (For the variance, integrate u[u
Suppose that the actual amount of cement that a filling machine puts into ‘six-kilogram’ bags is a normal random variable with σ = 0.05 kg. If only 3% of bags are to contain less than 6 kg, what
Assume that in the composition of a book there exists a constant probability 0.0001 that an arbitrary letter will be set incorrectly. After composition, the proofs are read by a proofreader, who
A Geiger counter and a source of radioactive particles are so situated that the probability that a particle emanating from the radioactive source will be registered by the counter is 1/10 000.Assume
If on average 7% of airline passengers order special meals, find the approximate probability that on a particular flight carrying eighty-five passengers, eight or more will order special meals.
A machine makes components, and the probability that a component is defective is p. If components are packed in cartons of 20, what value of p will ensure that 90% of cartons contain at most one

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