1 Million+ Step-by-step solutions

Without using a calculator, evaluate

(a) 10 × (−2)

(b) (−1) × (−3)

(c) (−8) ÷ 2

(d) (−5) ÷ (−5)

(e) 24 ÷ (−2)

(f) (−10) × (−5)

(g) 20/-4

(h) -27/-9

(i) (−6) × 5 × (−1)

( j) 2 x (-6) x 3 (-9)

Without using a calculator, evaluate

(a) 5 − 6

(b) −1 − 2

(c) 6 − 17

(d) −7 + 23

(e) −7 − (−6)

(f) −4 − 9

(g) 7 − (−4)

(h) −9 − (−9)

(i) 12 − 43

( j) 2 + 6 − 10

Without using a calculator, evaluate

(a) 5 × 2 − 13

(b) -30-6/-18

(c) (-3) x (-6) x (-1)/2 - 3

(d) 5 × (1 − 4)

(e) 1 − 6 × 7

(f) −5 + 6 ÷ 3

(g) 2 × (−3)^{2} (h) −10 + 2^{2}

(i) (−2)^{2} − 5 × 6 + 1

(j)(-4)^{2} x (-3) x (-1)/(-2)^{3}

Simplify each of the following algebraic expressions:

(a) 2 × P × Q

(b) I × 8

(c) 3 × x × y

(d) 4 × q × w × z

(e) b × b

(f) k × 3 × k

Simplify the following algebraic expressions by collecting like terms:

(a) 6w − 3w + 12w + 4w

(b) 6x + 5y − 2x − 12y

(c) 3a − 2b + 6a − c + 4b − c

(d) 2x^{2} + 4x − x^{2} − 2x

(e) 2cd + 4c − 5dc

(f) 5st + s^{2} − 3ts + t^{2} + 9

Without using a calculator, find the value of the following:

(a) 2x − y when x = 7 and y = 4.

(b) x^{2} − 5x + 12 when x = 6.

(c) 2m^{3} when m = 10.

(d) 5fg^{2} + 2g when f = 2 and g = 3.

(e) 2v + 4w − (4v − 7w) when v = 20 and w = 10.

If x = 2 and y = −3, evaluate

(a) 2x + y

(b) x − y

(c) 3x + 4y

(d) xy

(e) 5xy

(f) 4x − 6xy

(a) Without using a calculator, work out the value of (−4)^{2}.

(b) Press the following key sequence on your calculator:

(-) 4 x^{2}

Explain carefully why this does not give the same result as part (a) and give an alternative key sequence that does give the correct answer.

Without using a calculator, work out

(a) (5 − 2)^{2}

(b) 5^{2} − 2^{2}

Is it true in general that (a − b)^{2} = a^{2} − b^{2}?

Use your calculator to work out the following. Round your answer, if necessary, to two decimal places.

(a) 5.31 × 8.47 − 1.012

(b) (8.34 + 2.27)/9.41

(c) 9.53 − 3.21 + 4.02

(d) 2.41 × 0.09 − 1.67 × 0.03

(e) 45.76 − (2.55 + 15.83)

(f) (3.45 − 5.38)2

(g) 4.56(9.02 + 4.73)

(h) 6.85/(2.59 + 0.28)

Multiply out the brackets:

(a) 7(x − y)

(b) 3(5x − 2y)

(c) 4(x + 3)

(d) 7(3x − 1)

(e) 3(x + y + z)

(f) x(3x − 4)

(g) y + 2z − 2(x + 3y − z)

Factorise

(a) 25c + 30

(b) 9x − 18

(c) x^{2} + 2x

(d) 16x − 12y

(e) 4x^{2} − 6xy

(f) 10d − 15e + 50

Multiply out the brackets:

(a) (x + 2)(x + 5)

(b) (a + 4)(a − 1)

(c) (d + 3)(d − 8)

(d) (2s + 3)(3s + 7)

(e) (2y + 3)(y + 1)

(f) (5t + 2)(2t − 7)

(g) (3n + 2)(3n − 2)

(h) (a − b)(a − b)

Simplify the following expressions by collecting together like terms:

(a) 2x + 3y + 4x − y

(b) 2x^{2} − 5x + 9x^{2} + 2x − 3

(c) 5xy + 2x + 9yx

(d) 7xyz + 3yx − 2zyx + yzx − xy

(e) 2(5a + b) − 4b

(f) 5(x − 4y) + 6(2x + 7y)

(g) 5 − 3(p − 2)

(h) x(x − y + 7) + xy + 3x

Use the formula for the difference of two squares to factorise

(a) x^{2} − 4

(b) Q^{2} − 49

(c) x^{2} − y^{2}

(d) 9x^{2} − 100y^{2}

Simplify the following algebraic expressions:

(a) 3x − 4x^{2} − 2 + 5x + 8x^{2}

(b) x(3x + 2) − 3x(x + 5)

A law firm seeks to recruit top-quality experienced lawyers. The total package offered is the sum of three separate components: a basic salary which is 1.2 times the candidate’s current salary together with an additional $3000 for each year worked as a qualified lawyer and an extra $1000 for every year that they are over the age of 21. Work out a formula that could be used to calculate the total salary, S, offered to someone who is A years of age, has E years of relevant experience and who currently earns $N. Hence work out the salary offered to someone who is 30 years old with five years’ experience and who currently earns $150 000.

Write down a formula for each situation:

(a) A plumber has a fixed call-out charge of $80 and has an hourly rate of $60. Work out the total charge, C, for a job that takes L hours in which the cost of materials and parts is $K.

(b) An airport currency exchange booth charges a fixed fee of $10 on all transactions and offers an exchange rate of 1 dollar to 0.8 euros. Work out the total charge, C, (in $) for buying x euros.

(c) A firm provides 5 hours of in-house training for each of its semi-skilled workers and 10 hours of training for each of its skilled workers. Work out the total number of hours, H, if the firm employs a semi-skilled and b skilled workers.

(d) A car hire company charges $C a day together with an additional $c per mile. Work out the total charge, $X, for hiring a car for d days and travelling m miles during

that time.

Reduce each of the following numerical fractions to their lowest terms:

(a) 13/26

(b) 9/12

(c) 18/30

(d) 24/72

(e) 36/27

In 2011 in the United States, 35 out of every 100 adults owned a smartphone. By 2013 this figure increased to 56 out of every 100.

(a) Express both of these figures as fractions reduced to their lowest terms.

(b) By what factor did smartphone ownership increase during this period? Give your answer as a mixed fraction in its lowest terms.

Reduce each of the following algebraic fractions to their lowest terms:

(a) 6x/9

(b) x/2x^{2}

(c) b/abc

(d) 4x/6x^{2}y

(e) 15a^{2}b/20ab^{2}

By factorising the numerators and/or denominators of each of the following fractions, reduce each to its lowest terms:

(a) 2p/4q + 6r

(b) x/x^{2} - 4x

(c) 3ab/6a^{2} + 3a

(d) 14d/21d - 7de

(e) x + 2/x^{2} - 4

Which one of the following algebraic fractions can be simplified? Explain why the other two fractions cannot be simplified.

x - 1/2x - 2, x - 2/x + 2, 5t/10t - s

(1) Without using a calculator, work out the following, giving your answer in its lowest terms:

(2) Use your calculator to check your answers to part (1).

It takes 11/4 hours to complete an annual service of a car. If a garage has 471/2 hours available, how many cars can it service?

Work out each of the following, simplifying your answer as far as possible:

Solve each of the following equations. If necessary give your answer as a mixed fraction reduced to its lowest terms.

Which of the following inequalities are true?

(a) −2 < 1

(b) −6 > −4

(c) 3 < 3

(d) 3 ≤ 3

(e) −21 ≥ −22

(f) 4 < 25

Simplify the following inequalities:

(a) 2x > x + 1

(b) 7x + 3 ≤ 9 + 5x

(c) x − 5 > 4x + 4

(d) x − 1 < 2x − 3

Simplify the following algebraic expression:

4/x^{2}y / 2x/y

(a) Solve the equation

6(2 + x) = 5(1 − 4x)

(b) Solve the inequality

3x + 6 ≥ 5x − 14

On graph paper draw axes with values of x and y between −3 and 10, and plot the following points:

P(4, 0), Q(−2, 9), R(5, 8), S(−1, −2) Hence find the coordinates of the point of intersection of the line passing through P and Q, and the line passing through R and S.

An airline charges $300 for a flight of 2000 km and $700 for a flight of 4000 km.

(a) Plot these points on graph paper with distance on the horizontal axis and cost on the vertical axis.

(b) Assuming a linear model, estimate

(i) The cost of a flight of 3200 km;

(ii) The distance travelled on a flight costing $400.

By substituting values into the equation, decide which of the following points lie on the line, x + 4y = 12:

A(12, 0), B(2, 2), C(4, 2), D(−8, 5), E(0, 3)

For the line 3x − 5y = 8,

(a) find the value of x when y = 2;

(b) find the value of y when x = 1.

Hence write down the coordinates of two points which lie on this line.

If 4x + 3y = 24, complete the following table and hence sketch this line.

Solve the following pairs of simultaneous linear equations graphically:

(a) −2x + y = 2

2x + y = −6

(b) 3x + 4y = 12

x + 4y = 8

(c) 2x + y = 4

4x − 3y = 3

(d) x + y = 1

6x + 5y = 15

State the value of the slope and y intercept for each of the following lines:

(a) y = 5x + 9

(b) y = 3x − 1

(c) y = 13 − x

(d) −x + y = 4

(e) 4x + 2y = 5

(f) 5x − y = 6

Use the slope–intercept approach to produce a rough sketch of the following lines:

(a) y = −x

(b) x − 2y = 6

A taxi firm charges a fixed cost of $4 plus a charge of $2.50 a mile.

(a) Write down a formula for the cost, C, of a journey of x miles.

(b) Plot a graph of C against x for 0 ≤ x ≤ 20.

(c) Hence, or otherwise, work out the distance of a journey which costs $24.

The number of people, N, employed in a chain of cafes is related to the number of cafes, n, by the equation:

N = 10n + 120

(a) Illustrate this relation by plotting a graph of N against n for 0 ≤ n ≤ 20.

(b) Hence, or otherwise, calculate the number of

(i) employees when the company has 14 cafes;

(ii) cafes when the company employs 190 people.

(c) State the values of the slope and intercept of the graph and give an interpretation.

Monthly sales revenue, S (in $), and monthly advertising expenditure, A (in $), are modelled by the linear relation, S = 9000 + 12A.

(a) If the firm does not spend any money on advertising, what is the expected sales revenue that month?

(b) If the firm spends $800 on advertising one month, what is the expected sales revenue?

(c) How much does the firm need to spend on advertising to achieve monthly sales revenue of $15 000?

(d) If the firm increases monthly expenditure on advertising by $1, what is the corresponding increase in sales revenue?

Use the elimination method to solve the following pairs of simultaneous linear equations:

The total annual sales of a book in either paper or electronic form are 3500. Each paper copy of the book costs $30 and each e-book costs $25. The total cost is $97 500.

(a) If x and y denote the number of copies in paper and electronic form, write down a pair of simultaneous equations.

(b) Solve the equations to find the number of e-books sold.

Sketch the following lines on the same diagram:

2x − 3y = 6, 4x − 6y = 18, x -3/2y = 3

Hence comment on the nature of the solutions of the following systems of equations:

Use the elimination method to attempt to solve the following systems of equations.

Comment on the nature of the solution in each case.

(a) −3x + 5y = 4

9x − 15y = −12

(b) 6x − 2y = 3

15x − 5y = 4

If f (x) = 3x + 15 and g(x) = 1/3x − 5, evaluate

(a) f(2)

(b) f(10)

(c) f(0)

(d) g(21)

(e) g(45)

(f) g(15)

What word describes the relationship between f and g?

Sketch a graph of the supply function P = 1/3Q + 7 Hence, or otherwise, determine the value of

(a) P when Q = 12

(b) Q when P = 10

(c) Q when P = 4

The demand function of a good is

Q = 100 − P + 2Y + 1/2A

where Q, P, Y and A denote quantity demanded, price, income and advertising expenditure, respectively.

(a) Calculate the demand when P = 10, Y = 40 and A = 6. Assuming that price and income are fixed, calculate the additional advertising expenditure needed to raise demand to 179 units.

(b) Is this good inferior or normal?

The demand, Q, for a certain good depends on its own price, P, and the price of an alternative good, PA, according to Q = 30 − 3P + PA

(a) Find Q if P = 4 and PA = 5.

(b) Is the alternative good substitutable or complementary? Give a reason for your answer.

(c) Determine the value of P if Q = 23 and P_{A} = 11.

The demand for a good priced at $50 is 420 units, and when the price is $80, demand is 240 units. Assuming that the demand function takes the form Q = aP + b, find the values of a and b.

(a) Copy and complete the following table of values for the supply function

P = 1/2Q + 20

Hence, or otherwise, draw an accurate sketch of this function using axes with values of Q and P between 0 and 50.

(b) On the same axes draw the graph of the demand function P = 50 − Q and hence find the equilibrium quantity and price.

(c) The good under consideration is normal. Describe the effect on the equilibrium quantity and price when income rises.

The demand and supply functions of a good are given by

P = −3Q_{D} + 48

P = 1/2Q_{S} + 23

Find the equilibrium quantity if the government imposes a fixed tax of $4 on each good.

The demand and supply functions for two interdependent commodities are given bywhere Q_{Di}, Q_{Si} and P_{i} denote the quantity demanded, quantity supplied and price of good i, respectively. Determine the equilibrium price and quantity for this two-commodity model.

Make Q the subject of

P = 2Q + 8

Hence find the value of Q when P = 52.

Write down the formula representing each of the following flow charts

Draw flow charts for each of the following formulae:

Transpose the formulae:

(a) Q = aP + b to express P in terms of Q

(b) Y = aY + b + I to express Y in terms of I

(c) Q = 1/aP + b to express P in terms of Q

Make x the subject of each of the following formulae:

Make x the subject of the formula

y = 3/x - 2

If the consumption function is given by C = 4200 + 0.75Y, state the marginal propensity to consume and deduce the marginal propensity to save.

If the national income, Y, is 1000 units, then consumption, C, is 800 units. Also whenever income rises by 100, consumption increases by 70. Assuming that the consumption function is linear:

(a) state the marginal propensity to consume and deduce the marginal propensity to save;

(b) find an expression for C in terms of Y.

If the consumption function is given by C = 0.7 Y + 40 state the values of

(a) autonomous consumption;

(b) marginal propensity to consume.

Transpose this formula to express Y in terms of C and hence find the value of Y when C = 110.

Write down expressions for the savings function given that the consumption function is

(a) C = 0.9Y + 72

(b) C = 0.8Y + 100

For a closed economy with no government intervention, the consumption function is C = 0.6Y + 30 and planned investment is I = 100 Calculate the equilibrium level of

(a) national income;

(b) consumption;

(c) savings.

A consumption function is given by C = aY + b.

It is known that when Y = 10, the value of C is 28, and that when Y = 30, the value of C is 44.

By solving a pair of simultaneous equations, find the values of a and b, and deduce that the corresponding savings function is given by S = 0.2Y − 20

Determine the equilibrium level of income when planned investment I = 13.

Given that

G = 50

I = 40

C = 0.75Y_{d} + 45

T = 0.2Y + 80

calculate the equilibrium level of national income.

Solve the following quadratic equations:

(a) x^{2} = 81

(b) x^{2} = 36

(c) 2x^{2} = 8

(d) (x − 1)^{2} = 9

(e) (x + 5)^{2} = 16

Write down the solutions of the following equations:

(a) (x − 1)(x + 3) = 0

(b) (2x − 1)(x + 10) = 0

(c) x(x + 5) = 0

(d) (3x + 5)(4x − 9) = 0

(e) (5 − 4x)(x − 5) = 0

Use ‘the formula’ to solve the following quadratic equations. (Round your answers to

two decimal places.)

(a) x^{2} − 5x + 2 = 0

(b) 2x^{2} + 5x + 1 = 0

(c) −3x^{2} + 7x + 2 = 0

(d) x^{2} − 3x − 1 = 0

(e) 2x^{2} + 8x + 8 = 0

(f) x^{2} − 6x + 10 = 0

Solve the equation f (x) = 0 for each of the following quadratic functions:

(a) f (x) = x^{2} − 16

(b) f (x) = x(100 − x)

(c) f (x) = −x^{2} + 22x − 85

(d) f (x) = x^{2} − 18x + 81

(e) f (x) = 2x^{2} + 4x + 3

Sketch the graphs of the quadratic functions given in Question 4.

__Data from Question 4__

Use the results of Question 5 to solve each of the following inequalities:

(a) x^{2} − 16 ≥ 0

(b) x (100 − x) > 0

(c) −x^{2} + 22x − 85 ≥ 0

(d) x^{2} − 18x + 81 ≤ 0

(e) 2x^{2} + 4x + 3 > 0

The production levels of coffee in Mexico, Q (in suitable units) depends on the average summer temperature, T (in °C). A statistical model of recent data shows that Q = −0.046T2 + 2.3T + 27.6.

(a) Complete the table of values and draw a graph of Q against T in the range, 23 ≤ T ≤ 30.(b) Average summer temperatures over the last few decades have been about 25°C. However, some climate change models predict that this could rise by several degrees over the next 50 years. Use your graph to comment on the likely impact that this may have on coffee growers in Mexico.

Use a sign diagram to solve the following inequalities:

Given the quadratic supply and demand functionsdetermine the equilibrium price and quantity.

Given the supply and demand functions

determine the equilibrium price and quantity.

A clothing supplier sells T-shirts to retailers for $7 each. If a store agrees to buy more than 30, the supplier is willing to reduce the unit price by 3 cents for each shirt bought above 30, with a maximum single order of 100 shirts.

(a) How much does an order of 40 shirts cost?

(b) If the total cost of an order is $504.25, how many T-shirts did the store buy altogether?

(a) If the demand function of a good is given by

P = 80 − 3Q

find the price when Q = 10, and deduce the total revenue.

(b) If fixed costs are 100 and variable costs are 5 per unit, find the total cost when

Q = 10.

(c) Use your answers to parts (a) and (b) to work out the corresponding profit.

Given the following demand functions, express TR as a function of Q and hence sketch the graphs of TR against Q:

(a) P = 4

(b) P = 7/Q

(c) P = 10 − 4Q

Given the following total revenue functions, find the corresponding demand functions:

(a) TR = 50Q − 4Q^{2}

(b) TR = 10

Given that fixed costs are 500 and that variable costs are 10 per unit, express TC and AC as functions of Q. Hence sketch their graphs.

Given that fixed costs are 1 and that variable costs are Q + 1 per unit, express TC and AC as functions of Q. Hence sketch their graphs.

The total cost, TC, of producing 100 units of a good is 600 and the total cost of producing 150 units is 850. Assuming that the total cost function is linear, find an expression for TC in terms of Q, the number of units produced.

The total cost of producing 500 items a day in a factory is $40 000, which includes a fixed cost of $2000.

(a) Work out the variable cost per item.

(b) Work out the total cost of producing 600 items a day.

Find an expression for the profit function given the demand function

2Q + P = 25

and the average cost function

AC = 32/Q + 5

Find the values of Q for which the firm

(a) breaks even;

(b) makes a loss of 432 units;

(c) maximises profit.

A taxi firm charges a fixed cost of $10 together with a variable cost of $3 per mile.

(a) Work out the average cost per mile for a journey of 4 miles.

(b) Work out the minimum distance travelled if the average cost per mile is to be less than $3.25.

Sketch, on the same diagram, graphs of the total revenue and total cost functions,

TR = −2Q^{2} + 14Q

TC = 2Q + 10

(1) Use your graphs to estimate the values of Q for which the firm

(a) Breaks even;

(b) Maximises profit.

(2) Confirm your answers to part (1) using algebra.

The demand function for a firm’s product is given by P = 60 − Q.

Fixed costs are 100, and the variable costs per good are Q + 6.

(a) Write down an expression for total revenue, TR, in terms of Q and sketch a graph

of TR against Q, indicating clearly the intercepts with the coordinate axes.

(b) Write down an expression for total costs, TC, in terms of Q and deduce that the

average cost function is given by

AC = Q + 6 + 100/Q

Copy and complete the following table of function values:

Draw an accurate graph of AC against Q and state the value of Q that minimises average cost.

(c) Show that the profit function is given by

π = 2(2 − Q)(Q − 25) State the values of Q for which the firm breaks even, and determine the maximum profit.

(1) Without using your calculator, evaluate

(a) 8^{2}

(b) 2^{1 }

(c) 3^{−1}

(d) 17^{0 }

(e) 1^{1/5 }

(f) 36^{1/2 }

(g) 8^{2/3 }

(h) 49^{−3/2}

Use the rules of indices to simplify

Write the following expressions using index notation:

For the production function, Q = 200K1/4L2/3, find the output when

(a) K = 16, L = 27

(b) K = 10 000, L = 1000

Which of the following production functions are homogeneous? For those functions which are homogeneous, write down their degrees of homogeneity and comment on their returns to scale.

(a) Q = 500K^{1/3}L^{1/4}

(b) Q = 3LK + L^{2}

(c) Q = L + 5L^{2}K^{3}

Write down the values of x which satisfy each of the following equations:

(a) 5^{x} = 25

(b) 3^{x} = 1/3

(c) 2^{x} = 1/8

(d) 2^{x }= 64

(e) 100^{x} = 10

(f) 8^{x} = 1

Write down the value of

(a) log_{b}b^{2}

(b) log_{b}b

(c) log_{b}1

(d) log_{b} √b

(e) log_{b}(1/b)

Use the rules of logs to express each of the following as a single log:

(a) log_{b}x + log_{b}z

(b) 3log_{b}x − 2log_{b}y

(c) log_{b}y − 3log_{b}z

Express the following in terms of logbx and logby:

(a) log_{b}x^{2}y

(b) log_{b}(x/y^{2})

(c) log_{b}x^{2}y^{7}

Solve the following equations for x. Give your answers to two decimal places.

(a) 5x = 8

(b) 10^{x} = 50

(c) 1.2^{x} = 3

(d) 1000 × 1.05^{x} = 1500

(1) State the values of

(a) log_{2}32 (b) log_{9 }(1/3)

(2) Use the rules of logs to express 2log_{b}x = 4log_{b}y as a single logarithm.

(3) Use logs to solve the equation 10(1.05)^{x} = 300 Give your answer correct to one decimal place.

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