Week 2 Quiz: Equations and Graphs, Functions, and Systemsof Equations

SGPE Summer School

Lines: Slopes and Intercepts

Question 1: Find the slope, y-intercept, and x-intercept of the following line:

15x− 3y = 60

(A) slope= 5, y-intercept= −20, and x-intercept= 4

(B) slope= 6, y-intercept= −22, and x-intercept= 5

(C) slope= 15, y-intercept= 20, and x-intercept= −4

(D) slope= 5.5, y-intercept= −20, and x-intercept= 8

(E) None of the above

Question 2: Find the equation of a straight line with slope of 12, and y-intercept of -33.

(A) y = 112x + 33

(B) y = −12x + 33

(C)y = 12x− 33

(D) y = 12x− 133

(E) None of the above

Question 3: Find the equation of a straight line passing through the point (-2, 7) and perpendicularto a line with equation 24x + 6y = 30.

(A) y = 4x + 132

(B) y = −14x + 15

2

(C)y = −4x + 15

(D) y = 112x + 15

2

(E) None of the above

Question 4: Find the x and y intercepts (crossing points) of the following line in terms of a,b and c:

ax− by = c

Question 5: With $100,000 to invest, how much should a broker invest in a Portuguese bond thatpays 11% and how much in an Italian bond that pays 15% in order to earn an expected return of 14%on the total investment?

1

(A) 23,000 in Portuguese bond and 77,000 in Italian bond

(B) 50,000 in Portuguese bond and 50,000 in Italian bond

(C) 75,000 in Portuguese bond and 25,000 in Italian bond

(D) 35,000 in Portuguese bond and 65,000 in Italian bond

(E) None of the above

Question 6: Write down the equation of the line given below:

x axis

y axis

(2,3)

(0,1)

Solving Quadratic Equations

Question 7: Solve the following quadratic equation:

x2 + 6x + 9 = 0

(A) x = 3 and x = −3

(B) x = 3

(C) x = −3

(D) x = 2 and x = 3

(E) None of the above

Question 8: Solve the following quadratic equation:

5x2 + 47x + 18 = 0

(A) x = −25

and x = −9

(B) x = −25

and x = 9

(C) x = 25

and x = 9

(D) x = 25

and x = −9

(E) None of the above

Question 9: Consider the equation ax2 + bx + c = 0 with a 6= 0. Follow the steps given below to getthe quadratic formula:

2

(i) Multiply both sides of equation 1/a.

(ii) Add − ca

to both sides of the equation.

(iii) Add(

b2a

)2

to both sides of the equation.

(iv) Try to obtain a full square such as (x + d)2 and solve for x.

Non-linear Functions

Question 10: Items such as automobiles are subject to accelerated depreciation whereby they losevalue faster than they do under linear depreciation. Suppose that a car with initial value of $100, 000value depreciates at 10% per year (continuously compounded) (i.e., the cars value after t years isV (t) = 100000e−0.10t). Further, suppose that there always exists the option to sell the car for scrap toa “chop-shop” for $2000. After how many years will in be optimal to sell the car for scrap?

(A) 39

(B) 41

(C) Never!

(D) 40

(E) None of the above

Question 11: A factory’s cost function C(Q) is a function of the number (quantity) of units produced;suppose that the quantity of units produced is itself a function of time, Q(t). Specifically, assume thatC(Q) = 1900 + 50Q and that Q(t) = 16t− 1

4t2. Find the function that expresses the factory’s cost as

a function of time, and then find out the factory’s costs are after t = 1 and t = 10 periods.

(A) 1900 + 300t− 12t2; C(1) = 2188; C(10) = 1000

(B) 1900 + 800t− 252t2; C(1) = 2687.5; C(10) = 40650

(C) 1900 + 800t− 10t2; C(1) = 2690; C(10) = 31900

(D) 1900 + 300t− 252t2; C(1) = 2187.5; C(10) = 3650

(E) None of the above

Systems of Equations

Question 12: Suppose that supply and demand are described by the following set of equations:

Supply: Qs = 2P − 2

Demand: Qd = −8

5P + 16

By equating supply and demand find the market clearing price Pe and quantity Qe.

(A) Pe = 6; Qe = 9

(B) Pe = 4; Qe = 7

(C) Pe = 5; Qe = 8

(D) Pe = 8; Qe = 5

3

(E) None of the above

Question 13: In a race a turtle can run 0.01 meters every minute and a rabbit can run 1.5 metersevery minute. But it takes 3 minutes for rabbit to stop. How far can the turtle get before the rabbitcatches him? You can round your answer to two decimal places.

Question 14: Solve the following system of equations for the equilibrium levels of income, Y andinterest rate, i (round your answers to the nearest hundredth):

0.1Y + 80i− 75 = 0

0.4Y − 124i− 260 = 0

(A) Y = 677.93; i = 0.09

(B) Y = 505.84; i = 0.22

(C) Y = 604.83; i = 0.11

(D) Y = 605.99; i = 0.24

(E) None of the above

Question 15: Solve the following system of equations for the the equilibrium level of income, Y , interms of given levels of government spending, G = G0, and investment, I = I0.

Y = C + I + G

C = C0 + bY where 0 < b < 1

(A) Ye = 1b−1

(C0 + I0 + G0)

(B) Ye = 11−b

(C0 + I0 + G0)

(C) Ye = 1−b(C0+I0+G0)

(D) Ye = (1− b)(C0 + I0 + G0)

(E) None of the above

Question 16: Solve the following system of equations:

x− 2y = 14

x + 3y = 9

4