answer:To solve the given problem, we start by substituting 1-x with t, which gives us x=1-t. This substitution simplifies the original expression to: [ (3-2t)(-1-t)^5 = a_0 + a_1t + a_2t^2 + ldots + a_6t^6 ] **Step 1: Finding a_0** By setting t=0, we can directly find the value of a_0: [ a_0 = (3-2cdot0)(-1-0)^5 = 3 cdot (-1)^5 = -3 ] Thus, we have a_0 = -3, which means option A is correct. So, we can encapsulate this as boxed{A}. **Step 2: Calculating a_1** The expansion of (-1-t)^5 involves a general term {T}_{r+1} = {C}_{5}^{r} cdot (-1)^{5-r} cdot (-t)^{r}. When this is paired with 3 (for r=1), the coefficient is {C}_{5}^{1} cdot (-1) cdot 3 = -15. When paired with -2t (for r=0), the coefficient is {C}_{5}^{0} cdot (-1) cdot (-2) = 2. Therefore, a_1 = -15 + 2 = -13, which shows that option B is incorrect. **Step 3: Verifying Option D** When t=1, we substitute it into the equation to get the sum of all coefficients: [ a_0 + a_1 + ldots + a_6 = (3-2cdot1)(-1-1)^5 = -32 ] This confirms that option D is correct, so we encapsulate this as boxed{D}. **Step 4: Evaluating a_2 + a_4 + a_6** When t=-1, we have: [ a_0 - a_1 + a_2 - a_3 + a_4 - a_5 + a_6 = 0 quad (1) ] Adding this equation to the equation from Step 3, we get: [ 2(a_0 + a_2 + a_4 + a_6) = -32 ] Solving for a_0 + a_2 + a_4 + a_6 gives us: [ a_0 + a_2 + a_4 + a_6 = -16 ] Since we know a_0 = -3, we find: [ a_2 + a_4 + a_6 = -16 + 3 = -13 ] This shows that option C is incorrect since it states a_2 + a_4 + a_6 = -124. Therefore, the correct options are boxed{A text{ and } D}.

answer:First, we find that |overrightarrow{a}|=5, and since |overrightarrow{b}|=2 and the angle <overrightarrow{a}, overrightarrow{b}>=frac{pi}{3}; Therefore, overrightarrow{a}cdotoverrightarrow{b}=5times2timesfrac{1}{2}=5; Thus, (overrightarrow{a}+2overrightarrow{b})^2=overrightarrow{a}^2+4overrightarrow{a}cdotoverrightarrow{b}+4overrightarrow{b}^2=25+20+16=61; Therefore, |overrightarrow{a}+2overrightarrow{b}|=sqrt{61}. Hence, the correct option is boxed{A}. By calculating |overrightarrow{a}|=5, we can further find overrightarrow{a}cdotoverrightarrow{b}=5, which allows us to calculate the value of (overrightarrow{a}+2overrightarrow{b})^2, and thus find the value of |overrightarrow{a}+2overrightarrow{b}|. This problem tests the ability to calculate vector lengths and the scalar product of vectors based on their coordinates.

answer:Start with the new expression given by Jason's procedure: z = (x+3) - 2(y-5) Simplify and expand the expression: z = (x + 3) - 2(y - 5) = x + 3 - 2y + 10 = x - 2y + 13 Compare z with x - 2y, we see that: z = x - 2y + 13 Since 13 is a positive constant, z is always 13 units greater than x - 2y. Therefore, z is always greater than x - 2y. Conclusion: z = x - 2y + 13 z is always greater than x - 2y by 13. The final answer is boxed{textbf{(A)}} boxed{z = x - 2y + 13}

answer:Since the real part of the complex number z is a and the imaginary part is 1, we have: |z| = sqrt{a^2 + 1^2}. Since 0 < a < 2, we can determine the bounds for |z|: 1 < |z| = sqrt{a^2 + 1} < sqrt{4 + 1} = sqrt{5}. Therefore, the range of values for |z| is: boxed{(1, sqrt{5})}. This problem examines the properties and calculation of the modulus of complex numbers, which is a fundamental concept. When solving this problem, it is important to carefully read the question and provide a detailed solution.