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Questions and Answers of Mathematics Economics Business

(a) Find the determinant of(b) Find the inverse of each matrix in part (a). 5 6 –10 (ii) (iv) (iii) 2 7 -4 -7 -8 -7 -6 -2 -10 (i) 4 3 4
(a) Evaluate the matrix product Ax, whereHence show that the system of linear equations7x + 5y = 3x + 3y = 2can be written as Ax = b where(b) The system of equations2x + 3y − 2z = 6x − y + 2z =
Iffind AB and BA. A = [1 2 -4 3] and B = 3
Verify the equations(a) A(B + C) = AB + AC (b) (AB)C = A(BC) in the case when c-[ ] -3 1 1 and C = %3D 4 0 A B:
(1) LetFind(a) AT (b) BT (c) A + B (d) (A + B)TDo you notice any connection between (A + B)T, AT and BT?(2) LetFind(a) CT (b) DT (c) CD (d) (CD)TDo you notice any
A firm orders 12, 30 and 25 items of goods G1, G2 and G3. The cost of each item of G1, G2 and G3 is $8, $30 and $15, respectively.(a) Write down suitable price and quantity vectors, and use matrix
A firm manufactures three products, P1, P2 and P3, which it sells to two customers, C1 and C2. The number of items of each product that are sold to these customers is given byThe firm charges both
If A, B and C are matrices with orders, 3 × 3, 2 × 3 and 4 × 2, respectively, which of the following matrix calculations are possible? If the calculation is possible, state the order of the
Ifwork out(a) 2A(b) 2B (c) 2A + 2B(d) 2(A + B)Do you notice any connection between your answers to parts (c) and (d)? 1 7 9 6 B B = 2 1 0 5 2 3 2 3 1 A =10 5 0 6 7 8 6 4 5 3
The monthly sales (in thousands) of burgers (B1) and bites (B2) in three fast-food restaurants (R1, R2, R3) are as follows:(a) Write down two 2 × 3 matrices J and F, representing sales in January
The demand function of a good is given by P = 80/3√Q(a) Show that the price elasticity of demand is a constant.(b) Sketch a graph of the demand curve.(c) Shade the area which gives the consumer’s
A hospital trust decides to install a new rooftop solar power system which will save money on its energy bills. It estimates that t years after installation the savings flow (in $1000s per year) will
(a) Sketch a graph of the supply curve P = 4 + Q2 on the interval 0 ≤ Q ≤ 3.(b) On the same diagram, sketch the line, P = 13.(c) Work out the exact area bounded between the curve P = 4 + Q2, the
The price elasticity of demand is given by E = -P/2Q2, where P and Q denote price and quantity demanded, respectively. It is known that quantity demanded is 5 when the price is 10. Find an expression
(a) Find the total cost function given that the marginal cost function MC = 2Q + 3√Q and that total costs are 40 when Q = 4.(b) The inverse demand function for a good is Q = 10 - P/2P. Find the
(a) Find the following indefinite integrals:(b) The supply of a good can be modelled by a linear function, aP + bQ = 1 where a and b are constants. It is known that when the price, P is 12, the
A fund is established to provide a continuous revenue stream at a constant rate for a prescribed number of years when the discount rate is 4%.(a) Calculate the present value if the fund provides
The supply and demand functions of a good are given bywhere P, QS and QD denote the price, quantity supplied and quantity demanded, respectively.(a) Draw sketch graphs of these functions on the
A firm’s marginal revenue and marginal cost functions are given byMR = 140 − 6Q and MC = Q2 + Q + 20Fixed costs are 10.(a) Write down an expression for total revenue and deduce the corresponding
An investment flow is I(t) = 900√t where t is measured in years.(a) Calculate the total capital formation during the first four years.(b) Calculate the total capital formation from the end of the
(a) Find the area under the curve y = x2 + 2x + 3 between x = 1 and x = 2.(b) Integrate each of the following functions with respect to x:(c) Ф е сф и* 3 (ii) e += () х5 — 2х? + 9 (it)
A monopolist has a marginal cost function, 6Q2 + 5Q, and the fixed costs are 25. The demand function is 2P + Q = 510.(a) Find an expression for the profit function in terms of Q.(b) Hence find the
A company begins its extraction of oil from a newly discovered oil field at t = 0. The rate of extraction, measured in thousands of barrels per year, is given byCalculate the number of barrels
Calculate the present value of a revenue stream for eight years at a constant rate of $12 000 per year if the discount rate is 7.5%.
Given the investment flowI(t) = 2400√t(a) Calculate the total capital formation during the first four years.(b) Find an expression for the annual capital formation during the Nth year and hence
If the investment flow isI(t) = 5000t1/4calculate the capital formation from the end of the second year to halfway through the fifth year. Give your answer to the nearest whole number.
Given the demand functionand the supply functionFind(a) the consumer’s surplus(b) the producer’s surplus assuming pure competition. P = -Q – 4Qp + 68 P = Q + 2Qs + 12
Find the consumer’s surplus for the demand functionP = 50 − 2Q − 0.01Q2 when(a) Q = 10 (b) Q = 11
Find the producer’s surplus at Q = 9 for the following supply functions:(a) P = 12 + 2Q (b) P = 20√Q + 15
Find the consumer’s surplus at P = 5 for the following demand functions:(a) P = 25 − 2Q (b) P = 10/√Q
Evaluate each of the following definite integrals:By sketching a rough graph of the cube function between and x = −2 and 2, suggest a reason for your answer to part (b). What is the actual area
Find the exact areas under each of the following curves:(a) y = 2x2 + x + 3 between x = 1 and x = 5(b) y = (x − 2)2 between x = 2 and x = 3(c) y = 3√x between x = 4 and x = 25(d) y = ex between x
Evaluate each of the following integrals: 6. (c) (d) | 4x3 – 3x² + 4x + 2dx (a) | 4x*dr (b) -dp- dpr
Find the short-run production functions corresponding to each of the following marginal product of labour functions:(a) 1000 − 3L2 (b) 6/√L - 0.01
Find the consumption function if the marginal propensity to consume is 0.6 and consumption is 10 when income is 5. Deduce the corresponding savings function.
Find the total revenue and demand functions corresponding to each of the following marginal revenue functions:(a) MR = 20 − 2Q (b) MR = 6/√Q
The marginal cost function is given byMC = 2Q + 6If the total cost is 212 when Q = 8, find the total cost when Q = 14.
(a) Find the total cost if the marginal cost isMC = Q + 5and fixed costs are 20.(b) Find the total cost if the marginal cost isMC = 3e0.5Qand fixed costs are 10.
Find (b) (c) (10el dr (a) 6r'dr (d) х -dp- 3 dx (1) || 7x' + 4e * &x + 3)dx (h) |(ax + b)dr x?
(a) A function is defined implicitly byx3 + 2xy + y4 = 61Verify that the point (−2, 3) lies on this curve and find the gradient at this point.(b) Find and classify the stationary point on the
An individual needs to allocate the 168 hours available each week to work, W, hours and free time, F, hours. He earns $20 an hour (after deductions), and his utility function is given by U = 6E1/3
A firm’s production function is given by Q = K2L/K + 4L where K and L denote capital and labour, respectively.(a) Show that this production function is homogeneous of degree 2.(b) Calculate the
A firm’s production function is given by Q = 16√K + 6√L where Q, K and L denote output, capital and labour, respectively. The cost of providing each unit of capital and labour is $80 and $27,
Find and classify the stationary points of the function C(x, y) = (2xy − x2)e−y + 100.
A firm’s unit capital and labour costs are $4 and $1, respectively. If the production function is given by Q √LK and total input costs are $120, use the method of Lagrange multipliers to find the
A firm is the sole producer of two goods labelled 1 and 2. The demand functions aregiven bywhere Qi and Pi denote the quantity demanded and price of good i, respectively. The totalcost function is(a)
A firm’s production function is given by (a) Write down expressions for the marginal product of labour and marginal product of capital.(b) The firm currently uses 18 units of capital and 50
Sophie has $50 to spend on goods 1 and 2. The unit prices of these goods are $1 and $2, respectively. Her utility function is given by U = x1x2 + 2x1 + 2x2(a) Write down Sophie’s budgetary
The demand function of a good is given byQ = 1000 − 2P − 0.5P2A + 0.01Ywhere Q, P, PA and Y denote the quantity demanded, the price of the good, the price of an alternative good and income,
function of two variables is given by f(x, y) = 2x + 3x2y − y2.(a) Find expressions for the first-order and second-order partial derivatives of f.(b) Evaluate fx(1, 2) and fy(1, 2), and use the
A monopolistic producer of two goods, G1 and G2, has a total cost functionTC = 5Q1 + 10Q2where Q1 and Q2 denote the quantities of G1 and G2, respectively. If P1 and P2 denote the corresponding
A firm’s production function is given by Q = 80KLwhich is subject to a budgetary constraint, 3K + 5L = 1500.(a) Work out the first-order partial derivatives of the Lagrangian functiong(K, L,
A firm’s production function is given by Q = 80KL. Unit capital and labour costs are $2 and $1, respectively. The firm is contracted to provide 4000 units of output and wants to fulfil this
A firm’s production function is given byQ = KLUnit capital and labour costs are $2 and $1, respectively. Find the maximum level of output if the total cost of capital and labour is $6.
(a) Use Lagrange multipliers to find the maximum value ofz = 4xysubject to the constraintx + 2y = 40State the associated values of x, y and l.(b) Repeat part (a) when the constraint is changed tox +
Use Lagrange multipliers to find the maximum value ofz = x + 2xysubject to the constraintx + 2y = 5
A firm’s production function is given by Q = AKL where A is a positive constant. Unit costs of capital and labour are $2 and $1, respectively. The firm spends a total of $1000 on these inputs.(a)
A firm produces two goods A and B. The weekly cost of producing x items of A and y items of B isTC = 0.2x2 + 0.05y2 + 0.1xy + 2x + 5y + 1000(a) State the minimum value of TC in the case when there
Find the maximum value of the utility function, U = x1x2, subject to the budgetary constraint, x1 + 4x2 = 360.
The total cost of producing x items of product A and y items of product B isTC = 22x2 + 8y2 − 5xyIf the firm is committed to producing 20 items in total, write down the constraint connecting x and
A firm’s production function is given by Q = 50KL Unit capital and labour costs are $2 and $3, respectively. Find the values of K and L which minimise total input costs if the production quota is
Find the maximum value ofz = 80x − 0.1x2 + 100y − 0.2y2subject to the constraintx + y = 500
Find the maximum value ofz = 6x − 3x2 + 2ysubject to the constrainty − x2 = 2
(a) Make y the subject of the formula 9x + 3y = 2.(b) The function,z = 3xyis subject to the constraint9x + 3y = 2Use your answer to part (a) to show thatz = 2x − 9x2Hence find the maximum value of
(a) If the monopolist in Question 7 is no longer allowed to discriminate between the two markets and must charge the same price, P, show that the total demand, Q = Q1 + Q2, is given byQ = 52 - 3/2
A monopolist sells its product in two isolated markets with demand functionsP1 = 32 − Q1 and P2 = 40 − 2Q2The total cost function is TC = 4(Q1 + Q2).(a) Show that the profit function is given
A monopolist produces the same product at two factories. The cost functions for eachfactory are as follows:The demand function for the good isP = 100 − 2Qwhere Q = Q1 + Q2. Find the values of Q1
An individual’s utility function is given bywhere x1 is the amount of leisure measured in hours per week and x2 is earned income measured in dollars per week. Find the values of x1 and x2 which
A perfectly competitive producer sells two goods, G1 and G2, at $70 and $50, respectively. The total cost of producing these goods is given bywhere Q1 and Q2 denote the output levels of G1 and G2.
A firm’s profit function for the production of two goods is given byFind the output levels needed to maximise profit. Use second-order derivatives to confirm that the stationary point is a maximum.
Find and classify the stationary points of the following functions:(a) f (x, y) = x3 + y3 − 3x − 3y (b) f (x, y) = x3 + 3xy2 − 3x2 − 3y2 + 10
(a) Find the first-order partial derivatives,of the function z = 2x2 + y2 − 12x − 8y + 50 and hence find the stationary point.(b) Find the second-order partial derivatives,and hence show that the
A firm’s production function is given by Q = 18K1/6L5/6.(a) Show that this function displays constant returns to scale.(b) Find expressions for the marginal products of capital and labour.(c) State
The demand functions for two commodities, A and B, are given by QA = AP−0.5Y0.5 and QB = BP−1.5Y1.5 where A and B are positive constants(a) Find the price elasticity of demand for each
If Q = 2K3 + 3L2K, show that K(MPK) + L(MPL) = 3Q.
Evaluate MPK and MPL for the production functionQ = 2LK +√Lgiven that the current levels of K and L are 7 and 4, respectively. Hence(a) write down the value of MRTS;(b) estimate the increase in
Given the utility function
Given the demand function Q = PAY/P2 find the income elasticity of demand.
The satisfaction gained by consuming x units of good 1 and y units of good 2 is measured by the utility function U = 2x2 + 5y3 Currently an individual consumes 20 units of good 1 and 8 units of good
A utility function is given by U = 2x2 + y2.(a) State the equation of the indifference curve which passes through (4, 2). (b) Calculate the marginal utilities at (4, 2) and hence work out the
Given the demand function Q = 200 − 2P − PA + 0.1Y2 where P = 10, PA = 15 and Y = 100, find(a) the price elasticity of demand;(b) the cross-price elasticity of demand;(c) the income
Given the demand functionQ = 1000 − 5P − P2A + 0.005Y3 where P = 15, PA = 20 and Y = 100, find the income elasticity of demand and explain why this is a superior good. Give your answer
(a) Use the small increments formula to estimate the change in z = x3 − 2xy when x increases from 5 to 5.5 and y increases from 8 to 8.8.(b) By evaluating z at (5, 8) and (5.5, 8.8), work out the
Find the first-order partial derivatives,For each of the following functions:(a) z = u + v2 − 5w3 + 2uv (b) z = 6u1/2v1/3w1/6 ди' ду дw zę zę że
(a) If f (x, y) = y − x3 + 2x write down expressions for fx and fy. Hence use implicit differentiation to find dy/dx given that y − x3 + 2x = 1(b) Confirm your answer to part (a) by rearranging
Use the small increments formula to estimate the change in z = x2y4 − x6 + 4y when(a) x increases from 1 to 1.1 and y remains fixed at 0; (b) x remains fixed at 1 and y decreases from 0 to
If f (x, y) = x4y5 − x2 + y2 write down expressions for the first-order partial derivatives, fx and fy. Hence evaluate fx(1, 0) and fy(1, 1).
Write down expressions for the first-order partial derivatives, −z −x −z −y and , for(a) z = x2 + 4y5(b) z = 3x3 − 2ey (c) z = xy + 6y(d) z = x6y2 + 5y3
If f (x, y) = xy2 + 4x3 show that f (2x, 2y) = 8f (x, y)
If f (x, y) = 2x2 + xy write down an expression for (a) f (a, a) (b) f (b, −b)
If f (x, y) = 3x2y3 evaluate f (2, 3), f (5, 1) and f (0, 7).
The demand function of a good is given byP = 100e−0.1QShow that demand is unit elastic when Q = 10.
If a firm’s production function is given byQ = 700Le−0.02Lfind the value of L that maximises output.
Find the output needed to maximise profit given that the total cost and total revenue functions areTC = 2Q and TR = 100 ln(Q + 1) respectively.
Since the beginning of the year, weekly sales of a luxury good are found to have decreased exponentially. After t weeks, sales can be modelled by 3000e−0.02t. (a) Work out the weekly sales
Find and classify the stationary points of(a) y = xe−x (b) y = ln x − xHence sketch their graphs.
Use the quotient rule to differentiate(a) y = e4x/x2 + 2(b) y = ex/In x
Use the product rule to differentiate(a) y = x4e2x (b) y = x ln x
Use the chain rule to differentiate(a) y = ex3 (b) y = ln(x4 + 3x2)
Write down the derivative of(a) y = ln(3x) (x > 0) (b) y = ln(−13x) (x < 0)

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