Math Answers

Suppose that 𝑓 is differentiable and 𝑓′′(𝑎) exists. Prove that 𝑓′′(𝑎)=lim ℎ→𝑎[(𝑓(𝑎+ℎ)−2𝑓(𝑎)+𝑓(𝑎−ℎ))/h^2.] Give an example where the above limit exists, but 𝑓′′(𝑎) does not exist.

Solve

(D^(2)-DD'-2D'^(2)+2D+2D')z=sin(2x+y)

Solve

(D^(2)+DD'-6D'^(2))z=y cosx

Retro Company has cash of TK 100,000. its sales are Tk 600,000of which 87% is cost of goods sold and 43% is the accounts receivable. Assuming a 360 day year, the average number of days that retro takes to collect its outstanding sales is

In the manufacture of car tyres, a particular production process is know to yield 10 tyres with defective walls in every batch of 100 tyres produced. From a production batch of 100 tyres, a sample of 4 is selected for testing to destruction. Find: (i) the probability that the sample contains 1 defective tyre. (ii) the expectation of the number of defectives in samples of size 4. (iii) the variance of the number of defectives in samples of size 4.

A man expects to receive Php 125,000 in eight years. How much is that money worth now considering an interest rate of 12% compounded monthly?

Suppose that an emergency medicine unit registers on the average 3 patients with paracot poisoning per week. Calculate the probability that in a given week the unit will register, (i) one or more cases of paracot poisoning, (ii) two or more cases but less than 5 cases of paracot poisoning

A man expects to receive Php 125,000 in eight years. How much is that money worth now considering an interest rate of 12% compounded monthly?

18. Suppose R^2 has weighted inner product given as <u, v> =(3u base 1 v base 1 + 2u base 2 V base 2 for u = (u base 1, u base 2), v = (V base 1, v base 2). Let u = (1, 2), v = ( 2, - 1) and K = 3. Then the valued of <u, kv> is.....

(i) 4

(ii) 6

(iii) 18

(iv) None

19. Suppose that u, v "\\isin" V are such that ||u|| = 2, ||u +v|| = 3 and ||u - v|| = 4. Then ||v|| is?

(i) 17/2

(ii) √17

(iii) Does not exist

(iv) None

20. For a given matrix A="\\begin{bmatrix}\n3 & 1 \\\\\n 1 & 3\n\\end{bmatrix}", the matrix P that is orthogonally diagonalizes A is of the following matrices are diagonalisable

(i)P= "\\begin{bmatrix}\n1\/\u221a2 & 1\/\u221a2 \\\\\n 1\/\u221a2 & - 1\/\u221a2\n\\end{bmatrix}"

(ii)P= "\\begin{bmatrix}\n 0 & 1 \\\\\n 1 & 0 \n\\end{bmatrix}"

(iii)P="\\begin{bmatrix}\n-1\/\u221a2 & 1\/\u221a2 \\\\\n - 1\/\u221a2 & 1\/\u221a2\n\\end{bmatrix}"

15. Suppose T : R^2"\\to" R^3 is linear defined by T(x, y) =(x + 3y, x - y, x). Then

(i) 1

(ii) 2

(iii) 3

(iv) None

16. Suppose T : R^3"\\to" R^3 is linear and has an upper-triangular matrix with respect to the basis (1,0,0),(1,1,1),(1,1,2). Then, the orthonormal basis of R^3 with respect to which T has an upper-triangular matrix is...

(i) (1, 0, 0), (0, 1/(√2), 1/(√2)), (0, - 1/(√2), 1/(√2))

(ii) (1, 0, 0), (0, 1, 0), (0, 1/(√2), - 1/(√2)

(iii) (1, 0, 0), (0, - 1, 1), (0, 1, 1)

(iv) None

17. Which of the following defines an inner product

(i) <(x base 1, x base 2), y base 1, y base 2)>2x base 1 y base 1 +x base 2 y base 2 in R^2

(ii) <(x base 1, x base 2), y base 1, y base 2)>x base 1 y base 1 +2x base 2 y base 2 - 1 in R^2

(iii) <a base 1 + b base 1 x +c base 1 x^2, a base 2 +b base 2 x + c base 2 x^2 > = a base 1 b base 1 +a base 2 b base 2 +c base 1 c base 2 in P base 2