Question 1

Consider the two- sector model
Yt = Ct + It
Ct = 0.75Yt - 1 + 400
It = 200
Find the value of C2, given that Y0 = 4,000.

Accepted Answer

A) 3,600 
B) 3,200 
C) 2,400 
D) 2,700 
E) 3,100 
A) 3,600 
B) 3,200 
C) 2,400 
D) 2,700 
E) 3,100 E

Question 2

The value of an economic variable over time satisfies the relation 
$R ( t ) = \frac { 6 } { 1 + t ^ { 2 } } + 3 e ^ { - 0.4 t }$, Find the equilibrium value of $R$.

Accepted Answer

A)  it does not exist 
B)  3 
C)  0 
D)  9 
E)  6 
A)  it does not exist 
B)  3 
C)  0 
D)  9 
E)  6 C

Question 3

A principal of $60 is invested. The value, I(t), of investment, t days later satisfies the differential equation $\frac { d I } { d t } = 0.02 I + 5$ Find the value of the investment after 27 days, correct to two decimal places.

Accepted Answer

A) $301.38 
B) $97.65 
C) $176.34 
D) $234.98 
E) $281.96 
A) $301.38 
B) $97.65 
C) $176.34 
D) $234.98 
E) $281.96 E

Question 4

Which of the following gives the solution of the difference equation 
Yt = bYt - 1 + c, With initial condition, Y0 = a?
 $Y _ { t } = b Y _ { t } - 1 + c$, with initial condition, $Y _ { 0 } = a$ ?

Accepted Answer

A)  $\frac { c ( 1 - a + c ) b ^ { t } + c } { 1 - b }$ 
B)  $\frac { c \left( 1 - b ^ { t } \right) } { 1 - b } + a b ^ { t }$ 
C)  $\frac { a ( 1 - c ) b ^ { t } + a b } { 1 - b }$ 
D)  $\frac { c b ^ { t } } { 1 - b } + a ( 1 - b )$ 
E)  $\frac { a ( 1 - b + c ) b ^ { t } + c } { 1 - b }$ 
A)  $\frac { c ( 1 - a + c ) b ^ { t } + c } { 1 - b }$ 
B)  $\frac { c \left( 1 - b ^ { t } \right) } { 1 - b } + a b ^ { t }$ 
C)  $\frac { a ( 1 - c ) b ^ { t } + a b } { 1 - b }$ 
D)  $\frac { c b ^ { t } } { 1 - b } + a ( 1 - b )$ 
E)  $\frac { a ( 1 - b + c ) b ^ { t } + c } { 1 - b }$

Question 5

Find the solution of the differential equation $\frac { d y } { d t } = 3 t ^ { 2 } - \frac { 4 } { \sqrt { t } }$
with initial condition, $y ( 0 ) = 4$.

Accepted Answer

A)  $t ^ { 3 } - 2 \sqrt { t } + 4$ 
B)  $t ^ { 3 } - 8 \sqrt { t } + 4$ 
C)  $t ^ { 3 } + 2 t ^ { \frac { 3 } { 2 } } + 4$ 
D)  $t ^ { 3 } - 2 t \sqrt { t } + 4$ 
E)  $t ^ { 3 } + 4 \sqrt { t } + 4$ 
A)  $t ^ { 3 } - 2 \sqrt { t } + 4$ 
B)  $t ^ { 3 } - 8 \sqrt { t } + 4$ 
C)  $t ^ { 3 } + 2 t ^ { \frac { 3 } { 2 } } + 4$ 
D)  $t ^ { 3 } - 2 t \sqrt { t } + 4$ 
E)  $t ^ { 3 } + 4 \sqrt { t } + 4$

Question 6

Which of the following statements describes the behaviour of the sequence of numbers which satisfy $Y _ { 1 } = - \frac { 1 } { 2 } Y _ { t - 1 } ^ { 2 }$ with initial condition, Y0 = - 1?

Accepted Answer

A) Uniform convergence 
B) Oscillatory convergence 
C) None of these 
D) Oscillatory divergence 
E) Uniform divergence 
A) Uniform convergence 
B) Oscillatory convergence 
C) None of these 
D) Oscillatory divergence 
E) Uniform divergence

Question 7

Consider the market model 
$\begin{array} { l } 
Q _ { S } = 4 P - 3 \
Q _ { D } = - 2 P + 13 \
\frac { d P } { d t } = 0.4 \left( Q _ { D } - Q _ { S } \right)
\end{array}$
Find an expression for $Q _ { D } ( t )$ when $P ( 0 ) = 2$.

Accepted Answer

A)  $\frac { 2 } { 3 } \left( 4 - e ^ { - 2.4 t } \right)$ 
B)  $\frac { 1 } { 3 } \left( 23 + 4 e ^ { - 2.4 t } \right)$ 
C)  $\frac { 1 } { 3 } \left( 4 + e ^ { - 2.4 t } \right)$ 
D)  $\frac { 2 } { 3 } \left( 4 + e ^ { - 2.4 t } \right)$ 
E)  $\frac { 1 } { 3 } \left( 23 - 8 e ^ { - 2.4 t } \right)$ 
A)  $\frac { 2 } { 3 } \left( 4 - e ^ { - 2.4 t } \right)$ 
B)  $\frac { 1 } { 3 } \left( 23 + 4 e ^ { - 2.4 t } \right)$ 
C)  $\frac { 1 } { 3 } \left( 4 + e ^ { - 2.4 t } \right)$ 
D)  $\frac { 2 } { 3 } \left( 4 + e ^ { - 2.4 t } \right)$ 
E)  $\frac { 1 } { 3 } \left( 23 - 8 e ^ { - 2.4 t } \right)$

Question 8

Which of the following is a particular solution of the differential equation $\frac { d y } { d t } = - 5 y + 3 t ^ { 2 } + 2 ?$

Accepted Answer

A) 0.6t2 - 0.448 
B) 0.6t2 + 0.24t - 0.448 
C) 0.6t2 + 0.448 
D) 0.6t2 - 0.24t + 0.448 
E) 0.6t2 + 0.24t + 0.448 
A) 0.6t2 - 0.448 
B) 0.6t2 + 0.24t - 0.448 
C) 0.6t2 + 0.448 
D) 0.6t2 - 0.24t + 0.448 
E) 0.6t2 + 0.24t + 0.448

Question 9

Which of the following difference equations represents the reduced form of the market model 
$\begin{array} { l } 
Q _ { S t } = a P _ { t } - b \
Q _ { D t } = - c P _ { t } + d \
P _ { t } = P _ { t } - 1 - e ( Q S t - 1 - Q D t - 1 ) ?
\end{array}$

Accepted Answer

A)  $P _ { t } = ( 1 - a e - c e ) P _ { t } - 1 - e ( b + d )$ 
B)  $P _ { t } = ( 1 - a e - c e ) P _ { t } - 1 - e ( b - d )$ 
C)  $P _ { t } = ( 1 + a e - c e ) P _ { t } - 1 + e ( b + d )$ 
D)  $P _ { t } = ( 1 - a e + c e ) P _ { t } - 1 + e ( b - d )$ 
E)  $P _ { t } = ( 1 - a e - c e ) P _ { t } - 1 + e ( b + d )$ 
A)  $P _ { t } = ( 1 - a e - c e ) P _ { t } - 1 - e ( b + d )$ 
B)  $P _ { t } = ( 1 - a e - c e ) P _ { t } - 1 - e ( b - d )$ 
C)  $P _ { t } = ( 1 + a e - c e ) P _ { t } - 1 + e ( b + d )$ 
D)  $P _ { t } = ( 1 - a e + c e ) P _ { t } - 1 + e ( b - d )$ 
E)  $P _ { t } = ( 1 - a e - c e ) P _ { t } - 1 + e ( b + d )$

Question 10

Consider the two- sector model $\frac { d Y } { d t } = 0.4 ( C + I - Y )$ 
C = 0.6Y + 400
I = 0.8Y + 500
Given that Y(0)= 100, find the value, correct to the nearest whole number, of Y(2.4).

Accepted Answer

A) 1,000 
B) 1,200 
C) 850 
D) 950 
E) none of the above 
A) 1,000 
B) 1,200 
C) 850 
D) 950 
E) none of the above

Consider the two- sector model Y_t = Ct + It C_t = 0.75Yt - 1 + 400 I_t = 200 Find the value of C2, given that Y0 = 4,000.

The value of an economic variable over time satisfies the relation $R ( t ) = \frac { 6 } { 1 + t ^ { 2 } } + 3 e ^ { - 0.4 t }$ , Find the equilibrium value of $R$ .

A principal of $60 is invested. The value, I(t), of investment, t days later satisfies the differential equation $\frac { d I } { d t } = 0.02 I + 5$ Find the value of the investment after 27 days, correct to two decimal places.

Which of the following gives the solution of the difference equation Yt = bYt - 1 + c, With initial condition, Y0 = a? $Y _ { t } = b Y _ { t } - 1 + c$ , with initial condition, $Y _ { 0 } = a$ ?

Find the solution of the differential equation $\frac { d y } { d t } = 3 t ^ { 2 } - \frac { 4 } { \sqrt { t } }$ with initial condition, $y ( 0 ) = 4$ .

Which of the following statements describes the behaviour of the sequence of numbers which satisfy $Y _ { 1 } = - \frac { 1 } { 2 } Y _ { t - 1 } ^ { 2 }$ with initial condition, Y₀ = - 1?

Consider the market model =4P-3 =-2P+13 =0.4 - Find an expression for $Q _ { D } ( t )$ when $P ( 0 ) = 2$ .

Which of the following is a particular solution of the differential equation $\frac { d y } { d t } = - 5 y + 3 t ^ { 2 } + 2 ?$

Which of the following difference equations represents the reduced form of the market model =a-b =-c+d =-1-e(QSt-1-QDt-1)?

Consider the two- sector model $\frac { d Y } { d t } = 0.4 ( C + I - Y )$ C = 0.6Y + 400 I = 0.8Y + 500 Given that Y(0)= 100, find the value, correct to the nearest whole number, of Y(2.4).

Linear Equations

Non-Linear Equations

Mathematics of Finance

Differentiation

Partial Differentiation

Integration

Matrices

Linear Prgamming

Filters

Exam 9: Dynamics

Consider the two- sector model Yt = Ct + It Ct = 0.75Yt - 1 + 400 It = 200 Find the value of C2, given that Y0 = 4,000.

The value of an economic variable over time satisfies the relation R(t)=61+t2+3e−0.4tR ( t ) = \frac { 6 } { 1 + t ^ { 2 } } + 3 e ^ { - 0.4 t }R(t)=1+t26​+3e−0.4t , Find the equilibrium value of RRR .

A principal of $60 is invested. The value, I(t), of investment, t days later satisfies the differential equation dIdt=0.02I+5\frac { d I } { d t } = 0.02 I + 5dtdI​=0.02I+5 Find the value of the investment after 27 days, correct to two decimal places.

Which of the following gives the solution of the difference equation Yt = bYt - 1 + c, With initial condition, Y0 = a? Yt=bYt−1+cY _ { t } = b Y _ { t } - 1 + cYt​=bYt​−1+c , with initial condition, Y0=aY _ { 0 } = aY0​=a ?

Find the solution of the differential equation dydt=3t2−4t\frac { d y } { d t } = 3 t ^ { 2 } - \frac { 4 } { \sqrt { t } }dtdy​=3t2−t​4​ with initial condition, y(0)=4y ( 0 ) = 4y(0)=4 .

Which of the following statements describes the behaviour of the sequence of numbers which satisfy Y1=−12Yt−12Y _ { 1 } = - \frac { 1 } { 2 } Y _ { t - 1 } ^ { 2 }Y1​=−21​Yt−12​ with initial condition, Y0 = - 1?

Consider the market model =4P-3 =-2P+13 =0.4 - Find an expression for QD(t)Q _ { D } ( t )QD​(t) when P(0)=2P ( 0 ) = 2P(0)=2 .

Which of the following is a particular solution of the differential equation dydt=−5y+3t2+2?\frac { d y } { d t } = - 5 y + 3 t ^ { 2 } + 2 ?dtdy​=−5y+3t2+2?

Which of the following difference equations represents the reduced form of the market model =a-b =-c+d =-1-e(QSt-1-QDt-1)?

Consider the two- sector model dYdt=0.4(C+I−Y)\frac { d Y } { d t } = 0.4 ( C + I - Y )dtdY​=0.4(C+I−Y) C = 0.6Y + 400 I = 0.8Y + 500 Given that Y(0)= 100, find the value, correct to the nearest whole number, of Y(2.4).

Linear Equations

Non-Linear Equations

Mathematics of Finance

Differentiation

Partial Differentiation

Integration

Matrices

Linear Prgamming

Filters

Consider the two- sector model Y_t = Ct + It C_t = 0.75Yt - 1 + 400 I_t = 200 Find the value of C2, given that Y0 = 4,000.

The value of an economic variable over time satisfies the relation $R ( t ) = \frac { 6 } { 1 + t ^ { 2 } } + 3 e ^ { - 0.4 t }$ , Find the equilibrium value of $R$ .

A principal of $60 is invested. The value, I(t), of investment, t days later satisfies the differential equation $\frac { d I } { d t } = 0.02 I + 5$ Find the value of the investment after 27 days, correct to two decimal places.

Which of the following gives the solution of the difference equation Yt = bYt - 1 + c, With initial condition, Y0 = a? $Y _ { t } = b Y _ { t } - 1 + c$ , with initial condition, $Y _ { 0 } = a$ ?

Find the solution of the differential equation $\frac { d y } { d t } = 3 t ^ { 2 } - \frac { 4 } { \sqrt { t } }$ with initial condition, $y ( 0 ) = 4$ .

Which of the following statements describes the behaviour of the sequence of numbers which satisfy $Y _ { 1 } = - \frac { 1 } { 2 } Y _ { t - 1 } ^ { 2 }$ with initial condition, Y₀ = - 1?

Consider the market model =4P-3 =-2P+13 =0.4 - Find an expression for $Q _ { D } ( t )$ when $P ( 0 ) = 2$ .

Which of the following is a particular solution of the differential equation $\frac { d y } { d t } = - 5 y + 3 t ^ { 2 } + 2 ?$

Consider the two- sector model $\frac { d Y } { d t } = 0.4 ( C + I - Y )$ C = 0.6Y + 400 I = 0.8Y + 500 Given that Y(0)= 100, find the value, correct to the nearest whole number, of Y(2.4).