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Question 1. Ali has been hired by alpha company. The company asked him to give demonstration of RSA algorithm to the developers where they further use in the application. Using your discrete mathematics number theory demonstrate the following.
Hint (you can take two prime numbers as 19 and 23 and M/P is 20).
a. Calculate the public key (e,n) and private key (d,n)
b. Encrypt the message using (e,n) to get C
c. Apply (d,n) to obtain M/P back.
d. Write a Java program to demonstrate the keys you generated in a and b, and encrypt the message “20” and then decrypt it back to get the original message.
Can ∫ (x^6 +8)^2 dx be integrated with u = x^6+8? Explain
Project costs Rs 180,000 and is expected to generate cash inflows as:
Year Cash Inflow ( Rs )
1 20000
2 24000
3 30000
4 36000
5 40000
a. Discuss the characteristics of long term capital budgeting decisions
b. calculate the Net Present Value of the project if the cost of capital is 12% and conclude.
3. According to the report of National Economic Development Authority (NEDA) last year, a Filipino household spends an average of 1333 a day. You took a random sample of 20 households and determined the amount of their allotted budget each day revealing a mean of P420 and standard deviation of P120. Using 0.01 level of significance, can it be concluded that the average amount spent per day by a Filipino household has increased? Assume normality over the population.
(D^4 − 18D^3 + 119D^2 − 342D + 360)y = 0
1/x+ye^xy+2x)dx +(1/y+xe^xy+2y)dy=0
Consider the vector space V = C2with scalar multiplication over the real numbers R, and let T : V → V be the linear operator defined by T (z1, z2) = (z1 − iz2, z2 − z2) . Let W be the cyclic subspace of V generated by w = (1 + 2i, 1 + i).
5.1 Find the T–cyclic basis β for W generated by w.
5.2 Find the characteristic polynomial of TW .
5.3 Find [TW ]β. 5.4 Explain whether T = TW
Let T : V → V be a linear operator on a finite-dimensional vector space V over C such that T2 = T.
7.1 Show that R(T) ⊆ N(T − I) and R(T − I) ⊆ N(T).
7.2 Show that V = R(T) + R(T − I).
7.3 Show that V = N(T) ⊕ N(T − I).
7.4 Show that T is diagonalisable
Express the null hypothesis and alternative hypothesis in notation form of the scenario “It is
believed that the mean yearly salary of professors in the Philippines is Php480,000.00. A
random sample of 65 professors revealed a mean salary of Php500,000.00 Can it be
concluded that the mean salary is greater than Php480,000.00?
A sample size n = 22 is a simple random sample selected from a normally distributed
population. Find the t such that the shaded area to the left of t is 0.05.
