Math Answers

7.

a. A standardized STF1093 basic statistics test was given to 50 girls and 75 boys. The girls got an average grade of 76 with a standard deviation of 6. The boys made an average grade of 82 with a standard deviation of 8. Find the 96% confidence interval for the difference in the mean score of all boys and the mean score of all girls.

(7 marks)

b. From the above scores, a further sampling of 50 girls and 75 boys revealed an average score of boys is 79, and for girls is 78. Test the hypothesis the mean differences of scores between boys and girls are not significant at 0.04 level of significance.

(8 marks)

5) A fruit crate has square ends and is twice as long as it is wide.

(a) Find the volume of the crate if its width is 20 inches.

(b) Find a formula for the volume V of the crate in terms of its width x.

Let A be a given finite set and P(A) its power set. Let ⊆ be the inclusion relation on the elements of P(A). Draw Hasse diagrams of (P(A), ⊆) for A={a}; A={a,b}; A={a,b,c} and A={a,b.c.d}.

A $100,000 mortgage is to be amortized by making monthly payments for 20 years. Interest is 5.1% compounded semi-annually for a six-year term.

(a)

(b)

(c)

Compute the size of the monthly payment.

Determine the balance at the end of the six-year term.

If the mortgage is renewed for a six-year term at 5% compounded semi-annually, what is the size of the monthly payment for the renewal term?

(a) The size of the monthly payment is $

(Round the final answer to the nearest cent as needed. Round all intermediate values to six decimal places as needed.)

(b) The balance at the end of the six-year term is $

(Round the final answer to the nearest cent as needed. Round all intermediate values to six decimal places as needed.)

(c) The size of the monthly payment for the renewal term is $

(Round the final answer to the nearest cent as needed. Round all intermediate values to six decimal places as needed.)

How much will deposits of $35 made at the end of each month amount to after 10 years if interest is 6% compounded semi-annually?

Jim remembers to re-torque his wheels 95% of the time. When he does remember there is only a 1% chance he will lose a wheel. When he doesn't remember there is a 15% chance he will lose a wheel. Jim lost a wheel. Determine the probability that he forgot to re-torque his wheels.

Showing all working, find a formula for the general term of the sequence (sn) = s3, s4, s5 . . . defined by

s3 = π and sn = sn-1 - 2 for n >= 4.

Differentiate of the following functions with respect to x:

i) 𝑡𝑎𝑛𝑥∙ 𝑙𝑛(𝑠𝑖𝑛𝑥)

ii) √𝑐𝑜𝑡√𝑥

iii) 𝑒 ln (𝑡𝑎𝑛5𝑥)

iv) 𝑆𝑖𝑛2

{ln(𝑠𝑒𝑐𝑥)}

v) ln (𝑐𝑜𝑠𝑒𝑐𝑥)/𝑥

If B(u)

 is a differentiable vector function of u

 and ||B(u)||=1

, prove that du

 is perpendicular to B

Let E:=E

x

i

^

+E

y

j

^

+E

z

k

^

 and H:=H

x

i

^

+H

y

j

^

+H

z

k

^

 be two vectors assumed to have continuous partial derivatives (of second order at least) with

respect to position and time. Suppose further that E

 and H

 satisfy the equations:

∇⋅E=0,∇⋅H=0,∇×E=−1

c

∂H

∂t

,∇×H=1

c

∂E

∂t

prove that E

 and H

 satisfy the equation

∇

2

E

i

=1

c

2

∂

2

E

i

∂t

2

 and ∇

2

H

i

=1

c

2

∂

2

H

i

∂t

2

Here, i=x,y

 or z

.

Hint: Use the fact that∇×(∇×V)=∇(∇⋅V)−∇

2

V.