5. (a) Suppose a Hamming code over GF(2) has codewords length n = 7 and dimen-sion k. What is k?(b) On the other hand, suppose a Hamming code over GF(2) has a parity checkmatrix of size l × 15. What is l?
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5. (a) Suppose a Hamming code over GF(2) has codewords length n = 7 and dimen-
sion k. What is k?
(b) On the other hand, suppose a Hamming code over GF(2) has a parity check
matrix of size l × 15. What is l?
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- Suppose a special type of small data processing firm is so specialized that some have difficulty making a profit in their first year of operation. what is the value of k?4. Find the interpolating polynomial y = ax² + bx+c that passes through all three of the following data points: (1, 13), (2, 15), and (3, 19). You may use a computer or calculator to perform elimination in this problem only.6. Let p(x) = 3x* + 7x² + 4x +1. (a) Write the polynomial p(x) in the telescoping form, showing each step of the factorisation. (b) Evaluate the polynomial p(r) when x = 8 by completing the following memory table for Horner's algorithm (add more rows if needed). Hint: first identify N, x and the coefficients A(0), A(1), ..., A(N). Instruction KN-K A(N-K) S 1. S+ A(N) 2. For K = 1 to N a. S+ x*S+ A(N-K)|1 (c) With no calculation, convert 307418 to base 10, and explain why your answer is correct.
- 6. Let p(a) = 3r* + 7a² + 4x + 1. %3D (a) Write the polynomial p(x) in the telescoping form, showing each step of the factorisation. (b) Evaluate the polynomial p(z) when r = 8 by completing the following memory table for Horner's algorithm (add more rows if needed). Hint: first identify N, x and the coefficients A(0), A(1),..., A(N). Instruction KN-K A(N-K) 1. S+ A(N) 2. For K = 1 to N a. S+r*S+ A(N-K) 1 (c) With no calculation, convert 30741s to base 10, and explain why your answer is correct.Q1-Using the nodes (x,2, x=2.5 and x; =4), to find the second interpolation polynomial for flx) =-If the sum of the co-efficients in the expansion of (a + b)" is 1024, then the greatest co-efficient in the expansion is (a) 252 (b) 352 (c) 452 (d) 552
- 9.TOblems for Friday T0/1 5/6 Set background Clear frame Jack has a large baseball card collection of 2640 cards. He has them organized alphabetically from A-> Z in groups of 30 cards. How many groups does he have?10. Research has shown that losing even one night's sleep can have a significant effect on performance of complex tasks, such as problem solving (Linde & Bergstroem, 1992). To demonstrate this phenom- enon, a sample of n 25 college students was given a problem-solving task at noon on one day and again at noon on the following day. The students were not permitted any sleep between the two tests. For cach student, the difference between the first and second score was recorded. For this sample, the students averaged M,4.7 points better on the first test with a variance of s 64 for the difference scores a. Do the data indicate a significant change in problem-solving ability? Use a rwo-cailed test with a 05. b. Compute an estimated Cohen's d to measure the size of the effect
- 10 In each part, use the Lagrange interpolation formula to construct the polynomial of smallest degree whose graph contains the following points. (a) (-2, –6), (-1,5), (1, 3) (Ъ) (-4, 24), (1, 9), (3, 3) (c) (-2,3), (–1, –6), (1,0), (3, –2) (d) (-3,–30), (-2, 7), (0, 15), (1, 10)= (1) As usual, let (a, b) stand for the Greatest Common Divisor of a and b. Is it true that k(a, b) = (ka, kb) for all nonzero integers k, and pairs of integers a and b, not both zero? Either show that it is always true, or describe all the cases in which it is true, and all the cases in which it is false. (2) Show that if a is odd, then a4 - 1 is divisible by 8.2. Consider the following R code. Some of the output is missing. For (a)-(d), (i) describe what the code is doing, (ii) determine the exact or approximate value and (iii) indicate if the output would be exactly equal or approximately equal to the value you compute. Clearly show your steps and justify your answer. > x sqrt (var (x)) [1] output a (b) > mean (x qnbinom (0.04,2,0.2) +2 [1] output d (d) > mean (x) [1] output c
