1. Explain the importance of error analysis in numerical methods with suitable example.
2. Out of Bisection method and secant method which one is better and why? Solve one application based problem using that method.
3. Use Newton-Raphson Method to find the root of trigonometric function correct up to seven decimal places. (Trigonometric function should be complex)
4. Solve one problem which is based on the application of Interpolation.
5. Using numerical differentiation solve one application based problem. (Use central difference approximation and problem must include first order as well as second order derivatives)
6. Out of Trapezoidal rule and Simpson’s 1/3^rd rule which one is better explain in detail. Also solve one application based problem using that rule. Compare the exact and approximate result to compute the relative error.

(I saw the same question already uploaded in the chegg but i need other numbers )  

If p,p+2 are twin primes, prove 4((p−1)!+1)+p≡0 modp(p+2)

Convert the base-10 number to Two’s Complement. Show your work.

(a) −12 (b) −32 (c) −57 (d) −112 (e) −24 (f) −85

3. Suppose A and B are non-empty sets of real numbers that are both bounded above.

(a) Prove that, if A ⊆ B, then supA ≤ supB.

(b) Prove that supA∪B = max{supA,supB}.

(c) Prove that, if A∩B 6= ∅, then supA∩B ≤ min{supA,supB}. Give an example to show that equality need not hold.

Use the Intermediate value theorem to find an interval of length one that contains a root of the equation. a) x^3=9 b) 3x^3+x^2=x+5 2. Still consider equations 1a and 1b A)How many iterations are required to find the roots by the Bisection Method within six decimal places B) Use the interval you find in problem 1 that contains a root to compute the first 4 iterations and table the results

We will write Δ(m,n) for the set of order preserving functions with source m and target n.

Let f ∈ ∆(k, n), g ∈ ∆(m, n), and h ∈ ∆(n, p). Suppose im f = im g. Prove that im(h ◦ f) = im(h ◦ g).

Consider all positive integers less than 100. Find the number of integers divisible by 3 or 5?

Consider strings formed from the 26 English letters. How many strings are there of length 5?

How many ways are there to arrange the letters `a',`b', `c', `d', and `e' such that `a' is not immediately followed by`e' (no repeats since it is an arrangement)?

3. Let a and b be positive integers. Is gcd(5a + b, 11a + 2b) = gcd(2a + b, 3a + 2b)?

If yes provide a proof. If not, provide a counterexample.

How to solve it by wolfram mathematica:

The fraction 49/98 is a curious fraction, as an inexperienced mathematician in attempting to simplify it may incorrectly believe that 49/98 = 4/8, which is correct, is obtained by cancelling the 9s.
We shall consider fractions like, 30/50 = 3/5, to be trivial examples.
There are exactly four non-trivial examples of this type of fraction, less than one in value, and containing two digits in the numerator and denominator.
You have to identify these four fractions.

(This is "almost" Problem 33 from Euler project, with only the requirement modified).

Indicate whether each statement is True or False. Briefly justify your answers ( Please answer C and E both )

(c) The set of all 3 × 3 matrices A with zero trace (T r(A) = 0) is a vector space under the usual matrix operations of addition and scalar multiplication. (

e) If x⃗1 and x⃗2 are in the null space of a square matrix A, then any linear combination of x⃗1 and x⃗2 is also in the null space of A.

Let N be a normal subgroup of the group G.

(a) Show that every inner automorphism of G defines an automorphism of N.
(b) Give an example of a group G with a normal subgroup N and an automorphism of N that is not defined by an inner automorphism of G

Let v1 be an eigenvector of an n×n matrix A corresponding to λ1, and let v2, v3 be two linearly independent eigenvectors of A corresponding to λ2, where λ1 is not equal to λ2. Show that v1, v2, v3 are linearly independent.

As the new owner of a supermarket, you have inherited a large inventory of unsold imported Limburger cheese, and you would like to set the price so that your revenue from selling it is as large as possible. Previous sales figures of the cheese are shown in the following table.

Price per Pound, p $3.00 $4.00 $5.00

Monthly Sales, q (pounds) 402 284 222

(a) Use the sales figures for the prices $4 and $5 per pound to construct a demand function of the form q = Ae−bp, where A and b are constants you must determine. (Round A and b to two significant digits.) q =

(b) Use your demand function to find the price elasticity of demand at each of the prices listed. (Round your answers to two decimal places.) p = $3, E = p = $4, E = p = $5, E =

(c) At what price should you sell the cheese to maximize monthly revenue? (Round your answer to the nearest cent.) $

(d) If your total inventory of cheese amounts to only 200 pounds, and it will spoil one month from now, how should you price it to receive the greatest revenue? (Round your answer to the nearest cent.)

Fake news reporter Chocolate Medaughter has had another interesting scoop to tell to the world. Suppose that the rate of exposure of this new story to new people is proportional to the number of people who have not seen the story out of L possible viewers (limited growth). Suppose that initially, at time t = 0 days, no one has heard of the story, and it takes 10 days to expose the story to 10% of L. How long will it take to expose the story to 50% of L?