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If the beta cells of the pancreas were destroyed, which of the following would be the result?

The lubber grasshopper is a very large grasshopper, and is black with red and yellow stripes. Red stripes are expressed from the homozygous RR genotype, yellow stripes from the homozygous YY genotype, and both red and yellow stripes from the heterozygous genotype. What will be the phenotypic ratio of the offspring resulting from a cross of two grasshoppers – one has only yellow stripes and the other has red and yellow stripes?

A student goes ice fishing on Lake Mendota. After several hours in the cold the student experiences bouts of tingling and numbness in their toes. To relieve the numbness the student stomps their feet and wiggles their toes.  In this example the student has impairment in their ___________ but not their ______________ .

For arbitrary integers a, b, and c, with a ≠ 0, if GCD(a, b) = 1 and a | (b c), then a | c.

The function f : ℝ ⟶ ℝ defined by f(x) = 2×4 – 5×3 + x2 + 10x – 30 is O(x3).

The octal expansion of the decimal number 107 is ___________eight. Only type the digits; do not include the base.

For arbitrary integers a, b, and c, with a ≠ 0, if GCD(a, b) = 1 and a | (b c), then a | c.

To complete the division algorithm equation, a = mq + r, using a = – 56 and m = 5, which of the following gives appropriate values for integers q and r, with r expressed as a non-negative integer between 0 and (m-1), inclusive.

For arbitrary integers a, b, s, t, and d, with d ≠ 0, such that d | (as + bt), then d | a and d | b.

Let the function f : ℕ → ℝ be defined recursively as follows:      Initial Condition:  f (0) = 1 Recursive Part:    f (n + 1) = 3 * f (n), for n ≥ 0 Consider how to prove the following statement about this given function f using induction. f (n) = 3n, for all nonnegative integers n. Select the best response for each question below about how this proof by induction should be done.  Q1. Which is a correct way to prove the Basis Step for this proof?  [Basis] A. For n = 1, f(n) = f(1) = 3*f(0) = 3; also 3n= 31 = 3, so f(n) = 3n for n = 1.B. For n = 0, f(n) = f(0) = 1; also 3n = 30 = 1, so f(n) = 3n for n = 0.C. For n = k+1, f(k+1) = 3(k+1) when f(k) = 3k for some integer k ≥ 0, so f(n) = 3n for n = k+1.D. For n = k, assume f(k) = 3k for some integer k ≥ 0, so f(n) = 3n for n = k. Q2.  Which is a correct way to state the Inductive Hypothesis for this proof?  [InductiveHypothesis] A. Prove f(k) = 3k for some integer k ≥ 0. B. Prove f(k) = 3k for all integers k ≥ 0. C. Assume f(k) = 3k for some integer k ≥ 0. D. Assume f(k+1) = 3(k+1) when f(k) = 3k for some integer k ≥ 0. Q3.  Which is a correct way to complete the Inductive Step for this proof?  [InductiveStep] A. When the inductive hypothesis is true, f(k+1) = 3*f(k) = 3*3k  = 3(k+1). B. f(k+1) = 3*f(k), which confirms the recursive part of the definition. C. When f(k+1) = 3(k+1) = 3*3k; also f(k+1) = 3*f(k), so f(k) = 3k, confirming the induction hypothesis. D. When the inductive hypothesis is true, f(k+1) = 3(k+1) = 3*3k = 3*f(k), which confirms the recursive part of the definition. Q4.  Which is a correct way to state the conclusion for this proof?  [Conclusion] A. By the principle of mathematical induction, f(k) = 3k implies f(k+1) = 3(k+1) for all integers k ≥ 0. B. By the principle of mathematical induction, f(k) = f(k+1) for all integers k ≥ 0. C. By the principle of mathematical induction, f(n+1) = 3*f(n) for all integers n ≥ 0. D. By the principle of mathematical induction, f(n) = 3n for all integers n ≥ 0.