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modern engineering mathematics

Questions and Answers of Modern Engineering Mathematics

Show thatis an odd function and that any function f(x) may be written as the sum of an odd and an even function. Illustrate this result with f(x) = (x – 1)3. h(x) [f(x) = f(-x)] - =
The labour cost of producing a certain item is £21 per 10 000 items and the raw materials cost is £4 for 1000 items. Each time a new production run is begun, there is a set-up cost of £8. What is
Find the value of a which provides the least squares fit to the model y = ax for the data given in Figure 2.29.Figure 2.29 Xk Yk 1 50 5 2 100 8 3 150 9 4 200 11 5 250 12 300 15
Find the values of m and c which provide the least squares fit to the linear model y = mx + c for the data given in Figure 2.30.Figure 2.30 k XE Yk 1 0 1 2 1 1 3 2 2 4 3 2 5 4 3
Find the formula of the quadratic function which satisfies the data points (1, 2), (2, 4) and (3, 8).
The total labour cost of producing a certain item is £43 per 100 items produced. The raw materials cost £25 per 1000 items. There is a set-up cost of £50 for each production run. Obtain the
A mechanism consists of the linkage of three rods AB, BC and CD, as shown in Figure 2.112, where AB = CD (= a, say), BC = AD = a√2, and M is the midpoint. The rods are freely jointed at B and C,
Find the quadratic function in the form f(x) = A(x – 2)2 + B(x – 2) + C which satisfies f(1) = 2, f(2) = 4, f(3) = 8.
Find the least squares fit to the linear function y = ax of the data given in Figure 2.31.Figure 2.31 k Xk Yk 1 10.1 3.10 2 10.2 3.12 3 10.3 3.21 4 10.4 3.25 5 10.5 3.32
Complete the squares of the following quadratics and specify which are irreducible.(a) y = x2 + x + 1 (b) y = 3x2 – 2x – 1(c) y = 4 + 3x – x2 (d) y = 2x – 1 – 2x2
Find the least squares fit to the linear function y = mx + c for the experimental data given in Figure 2.32.Figure 2.32 k 1 55 107 2 60 109 3 65 114 4 70 118 5 75 123
Show that the equation r = p/sin(θ – α) represents a straight line which cuts the x axis at the angle α and whose perpendicular distance from the origin is p.
Find the extremal values of the functions(a) y = x2 + x + 1 (b) y = 3x2 – 2x – 1(c) y = 4 + 3x – x2 (d) y = 2x – 1 – 2x2
Use the result of Question 20 to find the polar coordinate representation of the line which passes through the points (1, 2) and (3, 3).Data from Question 20Show that the equation r = p/sin(θ –
On the graph of the line y = x, draw the lines y = 0, x = a and x = b. Show that the area enclosed by these four lines is 1/2(b2 – a2) (assume b > a). Deduce that this area is the average value of
Find the cubic function such that f(–3) = 528, f(0) = 1017, f(2) = 1433 and f(5) = 2312.
Continuing Question 54 of Exercises 2.6.2, show thatand by applying the arithmetic–geometric inequality todeduce that θ° achieves its maximum value where d = 2√3.Data from Question 54 of
Show that the equation r = ep/(1 + e cos θ) where e and p are constants, represents an ellipse where 0 < e < 1, a parabola where e = 1 and a hyperbola where e > 1, the origin of the coordinate
The velocity of an object falling under gravity is v(t) = gt where t is the lapsed time from its release from rest and g is the acceleration due to gravity. Draw a graph of v(t) to show that its
Find the values of A, B and C that ensure that x2 + 1 = A(x – 1) + B(x + 2) + C(x2 + 2) for all values of x.
Find the formulae of the quadratic functions f(x) such that(a) f(1) = 3, f (2) = 7 and f(4) = 19(b) f(–1) = 1, f(1) = –1 and f(4) = 2
Find the numbers A, B and C such that f(x)=x8x + 10 = A(x - 2) + B(x - 2) + C
Factorize the polynomials(a) x3 – 3x2 + 6x – 4 (b) x4 – 16 (c) x4 + 16
Determine which of the following quadratic functions are irreducible. (a) f(x) = x + 2x + 3 (c) f(x) = 6 - 4x - 3x (b) f(x) = 4x 12x + 9 (d) f(x) = 3x - 1 - 5x -
Show that f(x) = x3 – 3x2 + 6x – 4 is zero at x = 1, and hence factorize f(x).
Find the maximum or minimum values of the quadratic functions given in Question 26.Data from Question 26Determine which of the following quadratic functions are irreducible. (a) f(x) = x + 2x + 3 (c)
Obtain the expansion about x = 2 of the function y = x3 – 3x2 + 6x – 4.
Show that any real roots of the equation x3 – 3x2 + 6x – 4 = 0 lie between x = 0 and x = 2.
Show that the roots α, β of the quadratic equationObtain the roots of the equation 1.0x2 + 17.8x + 1.5 = 0 Assuming the numbers given are correctly rounded, calculate error bounds for the roots. ax
For what values of x are the values of the quadratic functions below greater than zero?(a) f(x) = x2 – 6x + 8 (b) f(x) = 15 + x – 2x2
A car travelling at u mph has to make an emergency stop. There is an initial reaction time T1 before the driver applies a constant braking deceleration of a mph2. After a further time T2 the car
The equation 3x3 – x2 – 3x + 1 = 0 has a root at x = 1. Obtain the other two roots.
Express the improper rational functionas the sum of a polynomial function and a strictly proper rational function. f(x)= 3x4 + 2x5x + 6x - 7 x - 2x + 3
Factorize the following polynomial functions and sketch their graphs: (a) x3 – 2x2 – 11x + 12(b) x3 + 2x2 – 5x – 6(c) x4 + x2 – 2(d) 2x4 + 5x3 – x2 – 6x(e) 2x4 – 9x3 + 14x2 – 9x +
Find the coefficients A, B, C, D and E such that y = 2x9x + 145x - 9x + 2 = A(x - 2) + B(x - 2) + C(x - 2) + D(x - 2) + E
Express in partial fractions the rational function 3x (x-1)(x + 2)
Show that the zeros of y = x4 – 5x3 + 5x2 – 10x + 6 lie between x = 0 and x = 5.
Using the cover up rule, express in partial fractions the rational function 2x + 1 (x-2)(x + 1)(x 3)
Show that the roots α, β of the equationHence find the quadratic equations whose roots are(a) α2 and β2 (b) α3 and β3 x + 4x + 1 = 0 satisfy the equations a + B = 14 a + B = -52
Express as partial fractions the rational function 3x + 1 (x-1)2(x + 2)
Use Lagrange’s formula to find the formula for the cubic function that passes through the points (5.2, 6.408), (5.5, 16.125), (5.6, 19.816) and (5.8, 27.912).
Express as partial fractions the rational function 5x 2 (x + x + 1)(x - 2)
Find a formula for the quadratic function whose graph passes through the points (1, 403), (3, 471) and (7, 679).
Express as partial fractions the rational function 3x (x - 1)(x + 2)
(a) Show that if the equation ax3 + bx + c = 0 has a repeated root a then 3aα2 + b = 0.(b) A can is to be made in the form of a circular cylinder of radius r (in cm) and height h (in cm), as shown
Sketch the graph of the functionand find the values of x for which y= 1 3- x (x = 3)
A box is made from a sheet of plywood, 2 m × 1 m, with the waste shown in Figure 2.37(a). Find the maximum capacity of such a box and compare it with the capacity of the box constructed without the
Sketch the graph of the function y = f(x) = x - x - 6 x + 1 (x = -1)
Two ladders, of lengths 12 m and 8 m, lean against buildings on opposite sides of an alley, as shown in Figure 2.38. Show that the heights x and y (in metres) reached by the tops of the ladders in
Show that the horizontal and vertical displacements, x, y, of a projectile at time t are x and y, respectively, wherewhere u and v are the initial horizontal and vertical velocities and g is the
Express the following improper rational functions as the sum of a polynomial function and a strictly proper rational function. (a) f(x) = (x + x + 1)/[(x + 1)(x - 1)] (b) f(x) = (x - x x + 1)/(x + x
Sketch the graph of the curve given by x = t3, y = t2 (t ∈ R)
Consider the surveying problem illustrated in Figure 2.50. The height of the tower is to be determined using the data measured at two points A and B, which are 20 m apart. The angles of elevation at
Express as partial fractions. (a) (c) (e) 1 (x + 1)(x - 2) x-2 (x + 1)(x - 2) 1 (x + 1)(x + 2x + 2) (b) (d) (f) 2x1 (x + 1)(x - 2) x-1 (x + 1)(x - 2) 1 (x + 1)(x - 4)
Express as partial fractions (a) (c) (e) 1 r 5x+4 3x-1 x 3x-2 x + x-1 (x + 1) (b) (d) (f) 1 x-1 x - 1 r - 5+6 18x - 5x + 47 (x +4)(x - 1)(x+5)
Sketch using the same set of axes the graphs of the functions(a) y = 2 sin t (b) y = sin t (c) y = 1/2 sin t and discuss.
Plot the graphs of the functionsfor the domain –3 ≤ x ≤ 3. Find the points on each graph at which they intersect with the line y = x. (a) y = (c) y = 2 + x 1 + x 3x + 12x - 4 8.x (b). y= (d) y
Sketch using the same axes the graphs of the functions(a) y = sin t (b) y = sin 2t (c) y = sin 1/2 tand discuss.
Sketch the graphs of the functions given below, locating their turning points and asymptotes.(Writing (a) as y = (√x – √(15/x))2 + 2√15 – 8 shows that there is a turning point at x =
Sketch the graph of y = 3 sin(2t + 1/3π).
Consider the crank and connecting rod mechanism illustrated in Figure 2.66. Determine a functional relationship between the displacement of Q and the angle through which the crank OP has
Plot the curve whose parametric equations are x = t(t + 4), y = t + 1. Show that it is a parabola.
Express cos(π/2 + 2x) in terms of sin x and cos x.
Sketch the curve given parametrically by x = t2 – 1, y = t3 – t showing that it describes a closed curve as t increases from –1 to 1.
Show thatand deduce thatHence sketch the graph of y = sin 4x + sin 2x. sin(A + B) + sin(AB) = 2 sin A cos B
Sketch the curve (the Cissoid of Diocles) given byShow that the cartesian form of the curve is x = 21 1 +1 y = 21 1 + 1
In the triangles shown in Figure 2.51, calculate sin θ°, cos θ° and tan θ°. Use a calculator to determine the value of θ in each case. 5 m 0 0 80 mm 100 mm Figure 2.51 (a) 13 m (b) 60 mm
Solve the equation 2 cos2x + 3 sin x = 3 for 0 ≤ x ≤ 2π.
The path of a projectile fired with speed V at an angle α to the horizontal is given byFor fixed V a family of trajectories, for various angles of projection α, is obtained, as shown in Figure
In the triangle ABC shown in Figure 2.52, calculate the lengths of the sides AB and BC. 15 m A Figure 2.52 27 B
Express y = 4 sin 3t – 3 cos 3t in the form y = A sin(3t + α).
Calculate the value of θ where sin θ° = sin 10° cos 20° + cos 10° sin 20°
Evaluate sin–1x, cos–1x, tan–1x where (a) x = 0.35 (b) x = –0.7, expressing the answers correct to 4dp.
Calculate the value of u wherecos θ° = 2 cos230° – 1
Sketch the graph of the function y = sin–1(sin x).
In triangle ABC, angle A is 40°, angle B is 60° and side BC is 20 mm. Calculate the lengths of the remaining two sides.
(a) Find the polar coordinates of the points whose cartesian coordinates are (1, 2), (–1, 3), (–1, –1), (1, –2), (1, 0), (0, 2), (0, –2). (b) Find the cartesian coordinates of the points
In triangle ABC, the angle C is 35° and the sides AC and BC have lengths 42 mm and 73 mm respectively. Calculate the length of the third side AB.
Express the equation of the circle (x – a)2 + y2 = a2 in polar form.
Copy and complete the table in Figure 2.72.Figure 2.72 Conversion table: degrees to radians. degrees 0 radians 30 radians TT/4 60 TT/2 2/3 150 degrees 210 225 240 270 300 315 330 2
The temperature T of a body cooling in an environment, whose unknown ambient temperature is α, is given bywhere T0 is the initial temperature of the body and k is a physical constant. To determine
Sketch the curve whose polar equation is r = 1 + cos θ.
(a) Evaluate log232.(b) Simplify (c) Expand (d) Use the change of base formula (2.36f ) to evaluate (e) Evaluate log28 - log22
Sketch for –3π ≤ x ≤ 3π the graphs of (a) y = sin 2x (c) y = sinx (e) y = 1 sin x (f) y = sin (b) y = sin x (d) y = sin.x (x nm, n = 0, +1, +2, ...) () (x = 0)
A tank is initially filled with 1000 litres of brine containing 0.25 kg of salt/litre. Fresh brine containing 0.5 kg of salt / litre flows in at a rate of 3 litres per second and a uniform mixture
Solve the following equations for 0 ≤ x ≤ 2π:(a) 3 sin2x + 2 sin x – 1 = 0(b) 4 cos2x + 5 cos x + 1 = 0(c) 2 tan2x – tan x – 1 = 0(d) sin 2x = cos x
By referring to an equilateral triangle, show that cos 1/3π = 1/2√3 and tan 1/6π = 1/3 √3, and find values for sin 1/3π, tan 1/3π, cos1/6π and sin 1/6π. Hence using the double-angle
A function is given by f(x) = A cosh 2x + B sinh 2x, where A and B are constants and f(0) = 5 and f(1) = 0. Find A and B and express f(x) as simply as possible.
Given s = sin θ, where 1/2π < θ < π, find, in terms of s,(a) cosθ (b) sin 2θ(c) sin 3θ (d) sin 1/2 θ
Show that 1+ sin 20 + cos 20 1+ sin 20 cos 20 = cot 0
Solve the equation 5 cosh x + 3 sinh x = 4
Verify the identityusing the definition of tanh x. Confirm that it obeys Osborn’s rule. tanh 2x 2 tanh x 1+tanh2x
Given t = tan 1/2x, prove thatHence solve the equation 2 sin x – cos x = 1 (a) sin.x = (b) cos.x = (c) tan x= 2t 1 + f2 1-1 1 + 1 2t 1-1
In each of the following, the value of one of the six circular functions is given. Without using a calculator, find the values of the remaining five. (a) sin x = // (c) tan x = -1 (e) cosec x = -2
Evaluate (to 4sf)(a) sinh–1(0.5) (b) cosh–1(3) (c) tanh–1(–2/5)using the logarithmic forms of these functions. Check your answers directly using a calculator.
Sketch the graphs of the functionfor the domain –3 (a) a = 18, b = 1, c = –1 and d = 6(b) a = 0, b = 1, c = –1 and d = 0 y = (a + bx + cx)/(d - x)
The velocity v and the displacement x of a mass attached to a nonlinear spring satisfy the equationwhere A depends on the initial velocity v0 and displacement x0 of the mass. Sketch the graph of v
Express as a product of sines and/or cosines(a) sin 3θ + sin θ (b) cosθ – cos 2θ(c) cos 5θ + cos 2θ (d) sinθ – sin 2θ
The concentrations of two substances in a chemical process are related by the equationInvestigate this relationship graphically and discover whether it defines a function. xyey = 2e, 0
Express as a sum or difference of sines or cosines(a) sin 3θ sin θ (b) sin 3θ cosθ(c) cos 3θ sin θ (d) cos 3θ cosθ

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