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modern engineering mathematics

Questions and Answers of Modern Engineering Mathematics

Express in the forms r cos(θ – α) and r sin(θ – β) (a) 3 sin 0 - cos (c) sin cos (b) sin cos (d) 2 cose + 3 sin 0 -
Sketch the graphs of the functions with formula y = f(x), where f(x) is (a) H(x - 1) - H(x - 2) (b) [x] - 2x]
Show that –3/2 ≤ 2 cos x + cos 2x ≤ 3 for all x, and determine those values of x for which the equality holds. Plot the graph of y = 2 cos x + cos 2x for 0 ≤ x ≤ 2π.
Use linear interpolation and the data of Figure 2.101 to estimate the value of(a) e–x where x = 0.235 (b) x where e–x = 0.7107Figure 2.101 X 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45
Evaluate(a) sin–1(0.5) (b) sin–1(–0.5)(c) cos–1(0.5)(d) cos–1(–0.5)(e) tan–1(√3) (f) tan–1(–√3)
Sketch the graph of the functions(a) y = sin–1(cos x)(b) y = cos–1(sin x)(c) y = cos–1(cos x)(d) y = cos–1(cos x) – sin–1(sin x)
If tan–1x = α and tan–1y = β, show thatDeduce thatwhere k = –1, 0, 1 depending on the values of x and y. tan(a +)= = x + y 1- xy
Sketch the curve with polar form r = 1 + 2 cos θ
Sketch the curve whose polar form is r = 1/(1 + 2 cos θ) Show that its cartesian form is 3x2 – 4x – y2 + 1 = 0
Simplify(a) (e2)3 + e2 · e3 + (e3)2 (b) e7x/e3x(c) (e3)2 (d) exp(32) (e) √(ex)
Find the following logarithms without using a calculator: (a) log8 (c) log/2 (e) log,3 (b) log (d) log381 (f) log40.5
Sketch the graphs of y = e–2x and y = e–x2 on the same axes. Note that (e–x)2 ≠ e–x2.
Express in terms of ln x and ln y(a) ln(x2y) (b) ln √(xy) (c) ln(x5/y2)
Simplify(a)(b) e2 ln x x+I X- expIn
Express as a single logarithm(a) ln 14 – ln 21 + ln 6(b) 4 ln 2 – 1/2 ln 25(c) 1.5 ln 9 – 2 ln 6(d) 2 ln(2/3) – ln(8/9)
Sketch carefully the graphs of the functions(a) y = 2x, y = log2x (on the same axes)(b) y = ex, y = ln x (on the same axes)(c) y = 10x, y = log x (on the same axes)
Express ln y as simply as possible when y = (x + 1)/2 4 (x + 1)/3(x4 + 4)/5
Sketch the graph of y = e–x – e–2x. Prove that the maximum of y is 1/4 and find the corresponding value of x. Find the two values of x corresponding to y = 1/40 .
In each of the following exercises a value of one of the six hyperbolic functions of x is given. Find the remaining five. (a) cosh x = 5 7 (c) tanh x = -25 (e) cosech x = - (b) sinh x = (d) sech x =
Use Osborn’s rule to write down formulae corresponding to (3 - tanx)tan x 1 - 3 tanx (b) cos(x + y) = cos x cos y sin x sin y (c) cosh 2x = 1 + 2 sinhx (d) sin x - sin y = 2 sin(x - y) cos(x + y)
Prove that(a) cosh–1x = ln[x + √(x2 – 1)] (x ≥ 1)(b) = + n(1+x) 1- tanh-x=;In| (x| < 1)
The speed V of waves in shallow water is given bywhere d is the depth and L the wavelength. If d = 30 and L = 270, calculate the value of V. 6.3d L V2 = 1.8L tanh-
Find to 4dp(a) sinh–10.8(b) cosh–12(c) tanh–1(–0.5)
The formulagives the increase in resistance of strip conductors due to eddy currents at power frequencies. Calculate λ when α = 1.075 and t = 1. 2 = ot sinh at + sin at 2 cosh at cos at
The functionsare two different forms of activating functions representing the output of a neuron in a typical neural network. Sketch the graphs of f1(x) and f2(x) and show that f1(x) – f2(x) = 1/2
The potential difference E (in V) between a telegraph line and earth is given bywhere A and B are constants, x is the distance in km from the transmitting end, r is the resistance per km of the
The potential difference E (in V) between a telegraph line and earth is given bywhere A and B are constants, x is the distance in km from the transmitting end, r is the resistance per km of the
Sketch the curves represented by the following equations, locating their turning points and asymptotes:(a) x3 + y3 = 6x2(b) I- x z.-t X || y
Sketch the curves represented by(a) y2 = x(x2 – 1)(b) y2 = (x – 1)(x – 3)/x2
Sketch the graph of the functions f(x) with formulae (a) f(x) = (x) ax 1 ax (b) f(x)=[H [H(x) - H(x - 1)] ax a (c) f(x) = (x) - (x - x - 1)H(x - 1) 1 ax - (d) f(x) = - -H(x) - 1 2a 1 -(x - 1)H(x - 1)
Show that the function g(x) = [H(x – a) – H(x – b)] f(x), a In other words, g(x) is a function that is identical to the function f(x) in the interval [a, b] and zero elsewhere. Hence express as
Sketch the graphs of(a) y = |x| (b) y = 1/2(x + |x|)(c) y = |x + 1| (d) y = |x| + |x + 1| – 2|x + 2|+ 3(e) |x + y| = 1
Sketch the graph of the functionExpress the formula for y in terms of Heaviside functions. X y = {0 (x 0) (0 < x1) (1-x (1
It is a familiar observation that spoked wheels do not always appear to be rotating at the correct speed when seen on films. Show that if a wheel has s spokes and is rotating at n revolutions per
The function INT(x) is defined as the ‘nearest integer to x, with rounding up in the ambiguous case’. Sketch the graph of this function and express it in terms of [x].
Sketch the graphs of the functions(a) y = [x] – [x – 12](b) y = |FRACPT(x) – 1/2|
Tabulate the function f(x) = sin x for x = 0.0(0.2)1.6. From this table estimate, by linear interpolation, the value of sin 1.23. Construct a table equivalent to Figure 2.102, and so estimate the
The function f(x) is tabulated at unequal intervals as follows:Use linear interpolation to estimate f(17), f(16.34) and f–1(0.3). X f(x) 15 0.2316 18 0.3464 20 0.4864
Tabulate the function f(x) = x3 for x = 4.8(0.1)5.6. Construct a table equivalent to Figure 2.102, and hence estimate the largest error that would be incurred in using linear interpolation in your
Assess the accuracy of the answers obtained in Question 96 using quadratic interpolation (Lagrange’s formula, (2.11)).Data from Question 96The function INT(x) is defined as the ‘nearest integer
Show that Lagrange’s interpolation formula for cubic interpolation isUse this formula to find a cubic polynomial that fits the function f given in the following table:Draw the graph of the cubic
Construct a critical table for y = 3√x for y = 14.50(0.01)14.55.
Starting at the point (x0, y0) = (1, 0), a sequence of right-angled triangles is constructed as shown in Figure 2.111. Show that the coordinates of the vertices satisfy the recurrence relationsAny
Calculate the rate of change of the linear functions given by(a) f(x) = 3x – 2(b) f(x) = 2 – 3x(c) f(–1) = 2 and f(3) = 4
(a) Show that a root x0 of the equation x4 – px3 + q = 0 is a repeated root if and only if 4x0 – 3p = 0(b) The stiffness of a rectangular beam varies with the cube of its height h and directly
Obtain the formula for the linear functions f(x) such that(a) f(0) = 3 and f(2) = –1(b) f(–1) = 2 and f(3) = 4(c) f(1.231) = 2.791 and f(2.492) = 3.112
By setting t = tan 1/2 x, find the maximum value of (sin x)/(2 – cos x).
Draw up a table of values of the function f(x) = x2e–x for x = –0.1(0.1)1.1. Determine the maximum error incurred in linearly interpolating for the function f(x) in this table, and hence estimate
Three different functions, f(x), g(x) and h(x), have the same graph on [0, 2] as shown in Figure 2.28. On separate diagrams, sketch their graphs for [–4, 4] given that(a) f(x) is periodic with
Which of the functions y = f(x) whose graphs are shown in Figure 2.27 are odd, even or neither odd nor even? (a) VA X YA * * O (c) (e) Figure 2.27 X (b) (d) (f) y 44. X YA y O O x X
A beam is used to support a building as shown in Figure 2.17. The beam has to pass over a 3 m brick wall which is 2 m from the building. Show that the minimum length of the beam is associated with
Simplify the following algebraic expressions(a) 5x + 2x −x(b) (2a + 3b) − (a − b)(c) 2m − [3m + 2n − (4m + n)]
Find the radius and the coordinates of the centre of the circle whose equation is 2x2 + 2y2 – 3x + 5y + 2 = 0
The functions f and g are defined byLet h(x) and k(x) be the compositions f ° g(x) and g ° f(x) respectively. Determine h(x) and k(x). Is the composite function k(x) defined for all x in the domain
Simplify the following algebraic expressions:(a) 6a − 7a + 2a (b) 10x −3x + 7x −13x (c) (7a + 3b) − (3a − 5b)(d) (m − n) + (2m − 2n) − (5m − 7n) (e) 5a + b − (2a + 3b)(f) 3x +
Determine the largest valid domains for the functions whose formulae are given below. Identify the corresponding codomains and ranges and evaluate f(5), f (–4), f (–x). (a) f(x) = (25-x) (b) fx
Using the general binomial expansion expand the following expressions:(a) (x – 3)4 (b) (x + 1/2)3(c) (2x + 3)5 (d) (3x + 2y)4
(a) A formula in the theory of ventilation isExpress A in terms of the other symbols.(b) Solve the equation Q = H_AD VH KVA + D
Express as fractions in their simplest form: (Do not use a calculator.)1.2. The line segment AC as a fraction of the line segment AB in Figure 1.13. The shaded area as a fraction of the rectangle
Write (a) The binary number 10111012 as a decimal number(b) The decimal number 11510 as a binary number.
Find the decimal equivalent of 110110.1012.
Determine 3/4 + 1/5
Calculate the following arithmetic expressions. (Do not use a calculator.)(a) 7 + 5 − 3 (b) 5 + 6 × 2 (c) 10 − 12 ÷ 3(d) (−5) × (−7 + 8) (e) (−8) ÷ 4 + 2 (f) 1 − (−2) ÷ (−3
Factorize the following:(a) ax – 2x – a + 2(b) a2 – b2 + 2bc – c2(c) 4k2 + 4kl + l2 – 9m2(d) p2 – 3pq + 2q2(e) l2 + lm + ln + mn
Represent the numbers (a) Two hundred and one, (b) Two hundred and seventy-five,(c) Five and three-quarters(d) One-third in(i) Decimal form using the figures 0, 1, 2, 3, 4, 5, 6, 7, 8, 9;(ii)
Find the binary and octal (base eight) equivalents of the decimal number 16 321. Obtain a simple rule that relates these two representations of the number, and hence write down the octal equivalent
Determine 2/3 + 1/4 – 2/5.
(a) Two small pegs are 8 cm apart on the same horizontal line. An inextensible string of length 16 cm has equal masses fastened at either end and is placed symmetrically over the pegs. The middle
Find the value of (100 + 20 + 3) × 456.
Find the binary and octal equivalents of the decimal number 30.6. Does the rule obtained in Question 2 still apply?Data from Question 2Find the binary and octal (base eight) equivalents of the
1. Simplify the following:2. Evaluate the following:3. Find the values of: (a) 2x2+ (d) (5) (b) (e) 32 (c) 3 x 3 3x35x36 3 X34
Calculate 2/3 × 5/6 × 7/15.
1. Express 5/8 as a decimal.2. Express 0.15 as a proper fraction in its simplest form.3. Evaluate the following:(a) 0.375 + 0.625 (b) 10.24 + 2.341 + 0.027 (c) 0.156 − 0.045(d) 18.231 −
The impedance Z ohms of a circuit containing a resistance R ohms, inductance L henries and capacity C farads, when the frequency of the oscillation is n per second, is given by(a) Make L the subject
Rewrite (a + b) × (c + d) as the sum of products.
Use binary arithmetic to evaluate(a) 100011.0112 + 1011.0012(b) 111.100112 × 10.1112
Evaluate 3.125 + 0.32 + 0.056.
1. Round off the number 0.05651 to 3 decimal places.2. Round off the number 0.05649 to 3 decimal places.3. Round off the number −0.0035 to 2 decimal places.4. Express 2/23 as a decimal number
Find integers m and n such that √(11 + 2√30) = √m + √n. Expand out (a) and (b) and rationalize (c) to (e).(a) (3√2 – 2√3)2(b) (√5 + 7√3)(2√5 – 3√3) (c) (d) (e) 4 + 32 5 + 2
Find the values of(a) 271/3 (b) (–8)2/3 (c) 16–3/2(d) (–2)–2(e) (–1/8)–2/3 (f) (9)–1/2
Simplify the following expressions, giving the answers with positive indices and without brackets:(a) 23 × 2–4 (b) 23 ÷ 2–4 (c) (23)–4(d) 31/3 × 35/3 (e) (36)–1/2 (f) 163/4
Evaluate 6.32 × 0.6.
How many decimal places (dp) and how many significant figures (sf) do the following correctly rounded numbers have?(a) 13.0567 (b) 0.345 (c) −0.0034 (d) 251
The expression 7 – 2 × 32 + 8 may be evaluated using the usual implicit rules of precedence. It could be rewritten as ((7 – (2 × (32))) + 8) using brackets to make the precedence explicit.
Find integers m and n such that √(11 + 2√30) = √m + √n.
Express (a) in terms of √2 and simplify (b) to (f ). (a) 18+ 32 50 2 1-3 (d) (b) 6/2 (e) (1 + 6)(1-6) (c) (1-3)(1+3) (f) 1-2 1 + 6
Evaluate 19.11 ÷ 1.5.
Express in scientific notation:(a) 0.345 (b) −0.0035 (c) 251 (d) 532.21(e) 1.5 × 105 + 2 × 102 (f) (1.5 × 105) × (2 × 102) (g) (1.5 × 105) ÷ (2 × 102)
Show thatand deduce thatfor any integer n ≥ 1. Deduce that the sumlies between 198 and 200. (n + 1) = n= 1 (n + 1) + n
1. Express the following fractions as percentages:2. Express the following decimals as percentages:(a) 0.52 (b) 0.03 (c) 0.4553. Express the following percentages as decimals and as proper
Express the following in the form x + y√2 with x and y rational numbers:(a) (7 + 5√2)3 (b) (2 + √2)4(c) 3√(7 + 5√2) (d) √(11/2 – 3√2)
Show thatHence express the following numbers in the form x + y√n where x and y are rational numbers and n is an integer: 1 a+bc a - bc a-bc
What is 25% of £50?
Evaluate 7 – 5 × 3 ÷ 22.
Express each of the following subsets of R in terms of intervals:(a) {x: 4x2 – 3 < 4x, x in R}(b) {x:1/(x + ) > 2/(x – 1), x in R}(c) {x: |x + 1| < 2, x in R}(d) {x: |x + 1| < 1 + 1/2x, x in
23% of a consignment of bananas is bad. There is 34.5 kg of bad bananas. How many kilograms of bananas are there in the consignment?
Show, without using a calculator, that √2 + √3 > 2 (4√6).
1. In Figure 1.6, what is the ratio AB:AC?2. Mark on Figure 1.6 the point D which is such that AD:DC = 2:53. A mortar mixture contains cement and sand in the ratio 2:3. The total weight is 10 kg.
Find the values of x so that |x – 4.3| = 5.8
Find the difference between 2 and the squares of(a) Verify that successive terms of the sequence stand in relation to each other as m does to (m + 2n)/(m + n).(b) Verify that if m is a good

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