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modern engineering mathematics

Questions and Answers of Modern Engineering Mathematics

Consider the following short procedure:List the first four printed values of S, and prove by induction that S = n2 the nth time the procedure reaches step 2.
If A = {3, 4, 5, 6} and B = {1, 5, 7, 9}, determine(a) A ∪ B(b) A ∩ B
Using the laws (6.1)–(6.6), verify the statementstating clearly the law used in each step.Data from Laws
When carrying out a survey on the popularity of three different brands X, Y and Z of washing powder, 100 users were interviewed, and the results were as follows: 30 used brand X only, 22 used brand Y
A company manufactures cranes. There are three basic types of crane, labelled A, B and C. Each crane is assembled from a subassembly set {a, b, c, d, e, f } as follows:In turn, the subassemblies are
Draw up the truth table that determines the state of the switching circuit given by the Boolean function
Construct truth tables to verify the De Morgan laws for the algebra of switching circuits analogous to (6.7) for the algebra of sets.
Simplify the Boolean functionstating the law used in each step of the simplication.
A machine contains three fuses p, q and r. It is desired to arrange them so that if p blows then the machine stops, but if p does not blow then the machine only stops when both q and r have blown.
In a large hall there are three electrical switches next to the three doors to operate the central lights. The three switches operate alternatively; that is, each can switch on or switch off the
Build a logic circuit to represent the Boolean function f = p̅ · q + p
Build a logic circuit to represent the Boolean function f = ( p + q̅) · (r + s · q)
Using only NOR gates, build a logic circuit to represent the Boolean function f = p̅· q + p · q̅
Let A, B and C be the following propositions:(a) Translate the following statements into logical statements using the notation of this section.(i) It is not frosty.(ii) It is frosty and after 11.00
In a piece of software, we have the following three propositions:Translate the following into symbols:(a) If the flag is set then I = 0.(b) Subroutine S is completed if the flag is set.(c) The flag
Construct the truth table determining the truth values of the compound proposition
Construct a truth table to verify the De Morgan laws for the algebra of statements analogous to (6.1) for the algebra of sets.
Use truth tables to show that the following are tautologies:(a) A → A,(b) A Λ (A → B) → B
If a, b, c, d ∈ R, prove that the inverse of the 2 × 2 matrix
Prove that if a + b ≥ 15 then either a ≥ 8 or b ≥ 8, where a and b are integers.
Prove that √2 is irrational.
Use mathematical induction to show thatfor any natural number N.
Verify that cos2θ − sin2θ = 2cos2θ – 1
Solve the equation 2cos2θ + 3sinθ = 3 for θ in the interval 0 ≤ θ ≤ 2π
Verify identity (6.4a) when A = π/3 and B = π/6
Using your knowledge of the trigonometric functions for π/6 and π/4 show that
Solve the trigonometric equationfor values of θ in the interval 0 ≤ θ ≤ 2π.
Solve the following trig equations for values of θ in the interval 0 ≤ θ ≤ 2π.
If sin A = 4/5 sin B = and 5/13 and if both A and B are acute angles, evaluate the following:
1. Express in terms of cosθ only the expression2. Express in terms of sinθ only the expression3. If x = tan π/8, use formula (6.5c) to show that 2x = 1 − x2. Thus, determine tan π/8.4. Solve
Given the matricesevaluate, where possible,(a) a + b,(b) bT + a,(c) b + CT,(d) C + D,(e) DT + C.
Given the matricesevaluate C in the three cases.(a) C = A + B(b) 2A + 3C = 4B(c) A – C = B + C
Solve for the matrix X
If(a) Show that trace(A + B) = trace A + trace B(b) Find D so that A + D = C(c) Verify the associative law (A + B) + C = A + (B + C)
Find the values of x, y, z and t from the equation
Find the values of α, β, γ that satisfy
(a) Show that the vectorsare linearly independent.(b) Show that the vectorsare linearly dependent.
Show that, for any vectorconstants α, β, γ can always be found so that
Given the matrix(a) Find the value of λ, μ, ν so that(b) Show that no solution is possible if
Market researchers are testing customers’ preferences for five products. There are four researchers who are allocated to different groups: researcher R1 deals with men under 40, R2 deals with men
A builder’s yard organizes its stock in the form of a vectorThe current stock, S, and the minimum stock, M, required to avoid running out of materials, are given asThe firm has five lorries which
Given the matricesevaluate AB, AC, BC, CA and BAT. Which if any of these are diagonal, unit or symmetric?
The matricesare given.(a) Which of the following make sense: AB, AC, BC, ABT, ACT and BCT?(b) Evaluate those products that do exist.(c) Evaluate (ATB)C and AT(BC) and show that they are equal.
(a) Represent each of the linear transformationsin matrix form and find the composite transformation that expresses z1, z2 in terms of x1, x2.(b) Represent each of the linear transformationsin
Givenevaluate AB and BA and hence show that these two matrices commute. Solve the equationfor the vector X by multiplying both sides by B.
Show that for any x the matrixsatisfies the relation A2 = I.
Ifshow that the product AB has exactly the same form.
Givenevaluate XTX and XTAX and write out the equations given by AX = b.
Given the matricesevaluate where possible AB, BA, BC, CB, CA, AC
Show that for a square matrix (A2)T = (AT)2.
Show that AAT is a symmetric matrix.
Find all the 2 × 2 matrices that commute (that is n, AB = BA) with
A matrix with m rows and n columns is said to be of type m × n. Give simple examples of matrices A and B to illustrate the following situations:(a) AB is defined but BA is not;(b) AB and BA are both
Givendetermine a symmetric matrix C and a skew-symmetric matrix D such that A = C + D
Given the matricesdetermine the elements of G where (ab)l + C2 = CT + G and I is the unit matrix.
A firm allocates staff into four categories: welders, fitters, designers and administrators. It is estimated that for three main products the time spent, in hours, on each item is given in the
Givenevaluate A2 and A3. Verify that A3 – A2 – 3A + I = 0
GivenUnder the transformation X = BY show that (5.6) becomes YT(BTAB)Y = 27 Ifevaluate BTAB, and hence show that
A well-known problem concerns a mythical country that has three cities, A, B and C, with a total population of 2400. At the end of each year it is decreed that all people must move to another city,
Find values of h, k, l and m so that A ≠ 0, B ≠ 0, A2 = A, B2 = B and AB = 0, where
A computer screen has dimensions 20 cm × 30cm. Axes are set up at the centre of the screen, as illustrated in Figure 5.5. A box containing an arrow has dimensions 2 cm × 2 cm and is situated with
Given the matrixit is known that An = I, the unit matrix, for some integer n; find this value.
Find all the minors and cofactors of the determinantHence evaluate the determinant.
Evaluate the determinants of the following matrices:
Given the matrixdetermine |A|, |AAT|, |A2| and |A + A|.
Find a series of row manipulations that takesand hence evaluate the determinant.
Determine adj A when
Determine adj A whenCheck that A(adjA) = (adj A)A = |A|I.
For the matrixevaluate |A|, adj(A),
Show that the matrixis non-singular and verify Cauchy’s theorem, namely |adjB| = |B|2.
If |A| = 0 deduce that |An| = 0 for any integer n.
Givenverify that adj(AB) = (adjB)(adj A).
Find the values of λ that make the following determinants zero:
Evaluate the determinants of the square matrices
Show that the area of a triangle with vertices (x1, y1), (x2, y2) and (x3, y3) is given by the absolute value of
Show that x + x2 – 2x3 is a factor of the determinant D whereand hence express D as a product of linear factors.
Show thatSuch an exercise can be solved in two lines of code of a symbolic manipulation package such as MAPLE or MATLAB’s Symbolic Math Toolbox.
Verify that if A is a symmetric matrix then so is adj A.
If A is a skew-symmetric n × n matrix, verify that adj A is symmetric or skew-symmetric according to whether n is odd or even.
Determine whether the following matrices are singular or non-singular and find the inverse of the non-singular matrices.
Verify thathas an inverseand hence solve the equation
(a) If a square matrix A satisfies A2 = A and has an inverse, show that A is the unit matrix.(b) Show thatsatisfies A2 = A.
Ifshow that AB = C. Find the inverse of A and B and hence of C.
Given the matrixand the elementary matricesevaluate E1A, E2E1A, E3E2E1A and E4E3E2E1A and hence find the inverse of A.
For the matrixshow that A2 – 4A – 5I = 0 and hence that A–1 = 1/5 (A – 4I). Calculate A–1 from this result. Further show that the inverse of A2 is given by 1/25 (21I – 4A) and evaluate.
Givenfind A–1 and B–1. Verify that (AB)–1 = B–1A–1.
Given the matricesshow that A2 = I and B3 = I, and hence find A–1, B–1 and (AB)–1.The matrices A and B in this exercise are examples of permutation matrices. For instance, A givesand the
Solve the matrix equation AX = b for the vector X in the following:
Ifshow thatand hence solve for the vector X in the equation
Solve the complex matrix equation
Find the inverse of the matrixand hence solve the equations
Show that there are two values of α for which the equationshave non-trivial solutions. Find the solutions corresponding to these two values of α.
Iffind the values of λ for which the equation AX = λX has non-trivial solutions.
Given the matrix(a) Solve |A| = 0 for real a,(b) If a = 2, find A–1 and hence solve(c) if a = 0, find the general solution of(d) if a = 1, show thatcan be solved for non-zero x, y and z.
Use MATLAB or a similar package to find the inverse of the matrixand hence solve the matrix equation AX = c where cT = [1 0 0 0 0 1].
In finite-element calculations the bilinear functionis commonly used for interpolation over a quadrilateral and data is always stored in matrix form. If the function fits the data u(0, 0) = u1, u(p,
In an industrial process, water flows through three tanks in succession, as illustrated in Figure 5.6. The tanks have unit cross-section and have heads of water x, y and z respectively. The rate of
A function is known to fit closely to the approximate functionIt is fitted to the three points (z = 0, f = 1), (z = 0.5, f = 1.128) and (z = 1.3, f = 1.971). Show that the parameters satisfyFind a, b
A cantilever beam bends under a uniform load w per unit length and is subject to an axial force P at its free end. For small deflections a numerical approximation to the shape of the beam is given by
Use elimination with or without partial pivoting, to solve the equations

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