Math Answers

13. Suppose T, S : R^2"\\to" R^2 are linear defined by T(u, v) =(3u + v, u + 2v) and S(x, y) =(2x - y, x + y). Also the matrices of T and S with respect to the standard bases of R^2 and R^2 are given as

M(T) ="\\begin{bmatrix}\n 3 & 1 \\\\\n 1 & 2\n\\end{bmatrix}" and M(S) ="\\begin{bmatrix}\n 2 & -1 \\\\\n 1 & 1\n\\end{bmatrix}"Then M(TS) =

(i)"\\begin{bmatrix}\n 1 & 0 \\\\\n 0 & 1\n\\end{bmatrix}"

(ii)"\\begin{bmatrix}\n 3 & 1 \\\\\n 1 & 5\n\\end{bmatrix}"

(iii)"\\begin{bmatrix}\n 3 & 1 \\\\\n 4 & 1\n\\end{bmatrix}"

(iv) None

14. Suppose T : R^2"\\to" R^2 is linear defined by T(x, y) = (y, x). Then the eigenvalues of T is...

(i) 1 and - 1

(ii) 0 and 2

(iii) Does not exist

(iv) None

8. Let W be the subspace of R^5 defined by W={x base 1,x base 2,x base 3,

x base 4 ,x base 5) "\\isin" R^5: x base 1 = 3x base 2 and x base 3 =7x base 4}. Then the basis of W is

(i) (3,1,0,0,1), (3,1,3,0,0), (3,1,0,0,1)

(ii) (3,1,0,1,1), (0,0,3,0,1), (0,0,1,3,1)

(iii) (3,1,1,0,1), (0,1,1,0,3), (0,0,1,0,1)

9. The basis of a solution space of given homogeneous linear system

X base 1 + x base 2 - x base 3 =0 X base 1 + x base 2 - x base 3 =0 X base 1 - x base 3 =0

-X base 1 + x base 3 =0 "\\implies"X base 2 =0 =0 "\\implies" X base 2 =0

-2X base 1 - x base 2 + 2x base 3 =0 -2X base 1 - x base 2 + 2x base 3 =0 - 2X base 1 + 2x base 3 =0

is

(i) {(1, 0, 1)}

(ii) {(1, 0, 1), (0, 1, 0)}

(iii) {(1, 1, -1), (-1, 0, 1), (-2, -1, 2)}

(iv) None

10. For a given matrix A "\\begin{bmatrix}\n 1 & 0 & 2 & - 3 \\\\\n 2 & 0 & 4 & - 6 \\\\\n - 3 & 0 & - 6 & 9\n\\end{bmatrix}". Which of the following is true

(i) rank (A) =3, nullity (A) =1

(ii) rank (A) =2, nullity (A) =2

(iii) rank (A) =1, nullity (A) =3

(iv) None

A stock with a current price of $300, can go either up by 20%, or down by 15%. You have the opportunity to buy an American call option on this stock for $65, which matures at the end of one period, with an exercise price of 275.

a-      Please draw the trees of the stock and option values

b-      What is the price of the call option today if the risk free rate is 8%.

c-      What is your decision for this investment opportunity? Explain your answer

You have bought a stock for a $250. You are expecting to get a return of 12/% while the volatility of your stock is 22%. if you intend to sell the stock after one year.

a-      What is the stock price range (upper and lower bound) assuming that the stock returns are normally distributed, and a two-tailed confidence interval of 90%? explain your answer.

b-      What is the Value at Risk of this investment at a confidence interval of 90%? explain your answer.

c-      What are the similarities and differences in calculating the above questions (a and b)? Please support your answer by drawing the needed figures showing each case.

found two call options A and B, both options sell in the market for $4, for an underlying stock which currently sells at $15. Both options mature in one year, and the risk free rate is 6% and a variance of 4%. Upon maturity, you can exercise option A for $10, and option B for $20.

Which option you choose to buy? Explain your answer.

A one litre box of juice is three quarters full.you pour out one fifth of the litre of juice.how much of the litre of juice is now left in the box

A shopping mall has setup a promotion as follows .with any mall purchase of 50 or more the customer gets to spin the wheel shown here.if a number 1 comes up the customer wins 10 .if the number 2 comes up the customer wins 5 and If the number 3 or 4 comes up the customer wins a discount coupon .find the customer wins 10

An investment of Ksh1,000,000 in an account accumulated to Ksh 2,500,000 after 5 years.

(i) State the accumulation factor A(0,5) . (2 marks)

(ii) (a) Find the simple annual interest rate which would give the accumulation

factor in part (i). (3 marks)

(b) Find the annual compound interest rate which would give the

accumulation factor in part (i). (3 marks)

Determine the value of the following:

1. P36

2. D5

3. Q3

The efficacy of rating of 155 faculty members of a certain college was taken and shown below.

Classes f

73-75 2

76-78 6

79-81 12

82-84 16

85-87 18

88-90 39

91-93 36

94-96 21

97-99 5

Determine the value of the following:

1. P36

2. D5

3. Q3