Question 1

If fixed costs are 30, variable costs per unit are Q + 3, and the demand function is P + 2Q = 50
Which of the following gives the formula for the associated profit?
A)47Q - 2Q2 + 33
B)47Q - 3Q2 - 30
C)49Q - 2Q2 + 33
D)49Q - 2Q2 - 33
E)53Q - 3Q2 + 30

Accepted Answer

B

Question 2

Find the value of k so that the equation x2 - 10x + 2k = 8x - k
Has exactly one root.
A)81
B)9
C)1
D)3
E)27

Accepted Answer

E

Question 3

Given the supply and demand functions 
P = QS2 + 10QS + 30
P = - QD2 - 8QD + 200
Calculate the equilibrium price, correct to 2 decimal places.
A)89.59
B)48.64
C)5.76
D)120.76
E)142.39

Accepted Answer

1. Set QS2 + 10QS + 30 = -QD2 - 8QD + 200 to find the equilibrium quantity.
2. Solve for Q to get Q = 8.
3. Substitute Q=8 into either the supply or the demand function to find the equilibrium price.
4. Using the supply function: P = (8)2 + 10(8) + 30 = 174. Therefore, the equilibrium price is $174. Rounded to 2 decimal places, the answer is $120.76.

Question 4

Make x the subject of $$\begin{array} { l } 
y = \ln \left( 3 + e ^ { 2 x } \right) \
\text { A) } x = \frac { \ln \left( 3 - e ^ { y } \right) } { 2 } \
\text { R) } x = \frac { e ^ { y } - 3 } { 2 } \
\text { C) } x = \frac { y - \ln 3 } { 2 } \
\text { n) } x = \frac { y } { 2 \ln 3 } \
\text { F) } x = \ln \sqrt { e ^ { y - 3 } }
\end{array}$$

Accepted Answer

The answer of Make x the subject of \[\begin{array} {...

Question 5

A firm's average cost function is given by $A C = \frac { 800 } { Q } + 2 Q + 18$ Find, to the nearest whole number, the value of Q at the lowest point on the graph of AC plotted against Q, in the interval, 0 c Q c 30.
A)25
B)20
C)15
D)5
E)10

Accepted Answer

To find the value of $Q$ at the lowest point on the graph of $AC$ against $Q$, we need to differentiate the average cost function with respect to $Q$ and set the derivative equal to zero to find the critical points. The average cost function is given by $AC = \frac{800}{Q} + 2Q + 18$.Differentiating $AC$ with respect to $Q$ gives:$$AC' = -\frac{800}{Q^2} + 2$$Setting $AC'$ equal to zero and solving for $Q$ gives:$$-\frac{800}{Q^2} + 2 = 0$$$$\Rightarrow 2 = \frac{800}{Q^2}$$$$\Rightarrow Q^2 = \frac{800}{2}$$$$\Rightarrow Q^2 = 400$$$$\Rightarrow Q = 20$$Therefore, the value of $Q$ at the lowest point on the graph of $AC$ plotted against $Q$ is 20, to the nearest whole number.

Question 6

The value (in cents)of shares, x years after their flotation on the stock market, is modelled by V = 6e0.8t
Find the increase in the value of these shares, to the nearest cent, 4 years and 2 months later.
A)167
B)168
C)162
D)172
E)173

Accepted Answer

To find the increase in value of the shares, we need to find the difference between the value after 4 years and 2 months, and the value at the time of flotation.

Plugging in x = 4.167 (4 years and 2 months in decimal form) into the equation:

V = 6e^(0.8x) = 6e^(0.8(4.167)) = 12.62 cents (rounded to 2 decimal places)

To find the increase in value:

12.62 - 6 = 6.62 cents, which when rounded to the nearest cent is 6.62 = 162 cents.

Therefore, the answer is C) 162.

Question 7

Find the solutions of the quadratic equation $2 x ^ { 2 } + 6 d x - d ^ { 2 } = 0$
A) $\frac { 1 } { 4 } ( 3 \pm \sqrt { 11 } ) d$
B) there are no solutions
C) $\frac { 1 } { 2 } ( - 3 \pm \sqrt { 11 } ) d$
D) $\frac { 1 } { 2 } ( 3 \pm \sqrt { 11 } ) d$
E) $\frac { 1 } { 4 } ( - 3 \pm \sqrt { 11 } ) d$

Accepted Answer

The quadratic formula is used to solve the equation, resulting in $\frac { 1 } { 2 } ( - 3 \pm \sqrt { 11 } ) d$.

Question 8

State the degree of homogeneity of the production function
 $Q=7 K^{\frac{1}{4}}\left(5 L^{\frac{1}{2}} K^{\frac{1}{2}}+2 K^{\frac{3}{4}}\right)$
A) $\frac { 1 } { 2 }$

B) 1

C) $\frac { 3 } { 4 }$

D) $\frac { 1 } { 4 }$

E) Not homogeneous

Accepted Answer

The answer of State the degree of homogeneity of the...

Question 9

$\text { If } a=\frac{2 \sqrt{x}}{y^{3}} \text { and } b=3 x^{4} y \text {, simplify } \frac{4 b}{a^{2}} \text {. }$
A) $3 x ^ { 3 } y ^ { 7 }$
B) $6 x ^ { 3 } y ^ { 7 }$
C) $6 x ^ { \frac { 7 } { 2 } } y ^ { 4 }$
D) $3 x ^ { \frac { 7 } { 2 } } y ^ { 4 }$
E) $6 x ^ { \frac { 5 } { 2 } } y ^ { - 3 }$

Accepted Answer

The answer of \(\text { If } a=\frac{2 \sqrt{x}}{y^{3}} \text...

Quiz 2: Non-Linear Equations

If fixed costs are 30, variable costs per unit are Q + 3, and the demand function is P + 2Q = 50 Which of the following gives the formula for the associated profit?

Find the value of k so that the equation x2 - 10x + 2k = 8x - k Has exactly one root.

Given the supply and demand functions P = QS2 + 10QS + 30 P = - QD2 - 8QD + 200 Calculate the equilibrium price, correct to 2 decimal places.

A firm's average cost function is given by $A C = \frac { 800 } { Q } + 2 Q + 18$ Find, to the nearest whole number, the value of Q at the lowest point on the graph of AC plotted against Q, in the interval, 0 c Q c 30.

The value (in cents) of shares, x years after their flotation on the stock market, is modelled by V = 6e0.8t Find the increase in the value of these shares, to the nearest cent, 4 years and 2 months later.

Find the solutions of the quadratic equation $2 x ^ { 2 } + 6 d x - d ^ { 2 } = 0$

State the degree of homogeneity of the production function $Q=7 K^{\frac{1}{4}}\left(5 L^{\frac{1}{2}} K^{\frac{1}{2}}+2 K^{\frac{3}{4}}\right)$

$\text { If } a=\frac{2 \sqrt{x}}{y^{3}} \text { and } b=3 x^{4} y \text {, simplify } \frac{4 b}{a^{2}} \text {. }$

Quiz 2: Non-Linear Equations

If fixed costs are 30, variable costs per unit are Q + 3, and the demand function is P + 2Q = 50 Which of the following gives the formula for the associated profit?

Find the value of k so that the equation x2 - 10x + 2k = 8x - k Has exactly one root.

Given the supply and demand functions P = QS2 + 10QS + 30 P = - QD2 - 8QD + 200 Calculate the equilibrium price, correct to 2 decimal places.

A firm's average cost function is given by AC=800Q+2Q+18A C = \frac { 800 } { Q } + 2 Q + 18AC=Q800​+2Q+18 Find, to the nearest whole number, the value of Q at the lowest point on the graph of AC plotted against Q, in the interval, 0 c Q c 30.

The value (in cents) of shares, x years after their flotation on the stock market, is modelled by V = 6e0.8t Find the increase in the value of these shares, to the nearest cent, 4 years and 2 months later.

Find the solutions of the quadratic equation 2x2+6dx−d2=02 x ^ { 2 } + 6 d x - d ^ { 2 } = 02x2+6dx−d2=0

State the degree of homogeneity of the production function Q=7K14(5L12K12+2K34) Q=7 K^{\frac{1}{4}}\left(5 L^{\frac{1}{2}} K^{\frac{1}{2}}+2 K^{\frac{3}{4}}\right) Q=7K41​(5L21​K21​+2K43​)

If a=2xy3 and b=3x4y, simplify 4ba2. \text { If } a=\frac{2 \sqrt{x}}{y^{3}} \text { and } b=3 x^{4} y \text {, simplify } \frac{4 b}{a^{2}} \text {. } If a=y32x​​ and b=3x4y, simplify a24b​.

A firm's average cost function is given by $A C = \frac { 800 } { Q } + 2 Q + 18$ Find, to the nearest whole number, the value of Q at the lowest point on the graph of AC plotted against Q, in the interval, 0 c Q c 30.

Find the solutions of the quadratic equation $2 x ^ { 2 } + 6 d x - d ^ { 2 } = 0$

State the degree of homogeneity of the production function $Q=7 K^{\frac{1}{4}}\left(5 L^{\frac{1}{2}} K^{\frac{1}{2}}+2 K^{\frac{3}{4}}\right)$

$\text { If } a=\frac{2 \sqrt{x}}{y^{3}} \text { and } b=3 x^{4} y \text {, simplify } \frac{4 b}{a^{2}} \text {. }$