answer:Answer: left{2,-1, frac{1}{2}right}.

answer: Step-by-Step Solution# Part (1): Rectangular Coordinate Equation of Curve C and Parametric Equation of Line l- Rectangular Coordinate Equation of Curve C:Given the polar coordinate equation of curve C: rho ^{2}cos ^{2}theta +3rho ^{2}sin ^{2}theta =12.We know that in rectangular coordinates, rho^2 = x^2 + y^2, costheta = frac{x}{rho}, and sintheta = frac{y}{rho}. Substituting these into the given equation, we get:x^{2}+3y^{2}=12This can be rewritten as:frac{x^2}{12}+frac{y^2}{4}=1- Parametric Equation of Line l:Point P has polar coordinates (2,pi). Converting to rectangular coordinates:x = 2cospi = -2, y = 2sinpi = 0Thus, point P in rectangular coordinates is P(-2,0).The line l with inclination angle alpha passing through P can be described in parametric form as:left{begin{array}{l}x=-2+tcosalphay=tsinalphaend{array}right.where t is a parameter.# Part (2): Range of frac{1}{{|PA|}}+frac{1}{{|PB|}}- Substituting Line l into Curve C's Equation:Substitute the parametric equations of line l into the equation of curve C:(1+2sin ^{2}alpha )t^{2}-4tcos alpha -8=0- Discriminant Delta:The discriminant of this quadratic equation is:Delta = 48+48sin ^{2}alpha > 0This shows that the equation always has real solutions, meaning line l always intersects curve C at two points, A and B.- Solving for t_1 and t_2:Let t_1 and t_2 be the parameters corresponding to points A and B respectively. From the quadratic formula, we have:{t_1}+{t_2}=frac{4cosalpha}{1+2sin^2alpha}{t_1}{t_2}=frac{-8}{1+2sin^2alpha}<0- Finding the Range of frac{1}{{|PA|}}+frac{1}{{|PB|}}:The sum of the reciprocals of the distances from P to A and B can be expressed as:frac{1}{{|PA|}}+frac{1}{{|PB|}}=frac{|t_1|+|t_2|}{|t_1t_2|}=frac{|t_1-t_2|}{|t_1t_2|}=frac{sqrt{(t_1+t_2)^2-4t_1t_2}}{|t_1t_2|}=frac{sqrt{3+3sin^2alpha}}{2}Given that sin^2alpha varies between 0 and 1, the range of this expression is:left[frac{sqrt{3}}{2},frac{sqrt{6}}{2}right]Therefore, the range of frac{1}{{|PA|}}+frac{1}{{|PB|}} is boxed{left[frac{sqrt{3}}{2},frac{sqrt{6}}{2}right]}.

answer:Given that |x| > 2 is a necessary but not sufficient condition for x 2 can be broken down into two cases: - Case 1: x > 2 - Case 2: x 2 is necessary but not sufficient for x 2, but not all x satisfying |x| > 2 necessarily satisfy x 2 or x 2 or x 2, contradicting the given condition.4. Therefore, to maximize a while ensuring that all x 2, we find that the maximum value of a is -2.Hence, the maximum value of a is boxed{-2}.

answer:Answer: 16.Solution. The goat eats hay at a rate of 1 / 6 cart per week, the sheep - at a rate of 1 / 8 cart per week, the cow - at a rate of 1 / 3 cart per week. Then 5 goats, 3 sheep, and 2 cows together will eat hay at a rate of frac{5}{6}+frac{3}{8}+frac{2}{3}=frac{20+9+16}{24}=frac{45}{24}=frac{15}{8} carts per week. Therefore, 30 carts of hay they will eat in 30: frac{15}{8}=16 weeks.

answer:14.2left(x-frac{1}{2}right)^{2}+frac{(y+1)^{2}}{2}=1.From a-2 b+c=0, we know that the line a x+b y+c=0 passes through the fixed point P(1,-2). Since point P lies on the ellipse frac{x^{2}}{2}+frac{y^{2}}{8}=1, P is one endpoint of the intercepted segment. Let the other endpoint be Qleft(x_{1}, y_{1}right), and the midpoint of segment P Q be Mleft(x_{0}, y_{0}right), thenleft{begin{array} { l } { x _ { 0 } = frac { x _ { 1 } + 1 } { 2 } , } { y _ { 0 } = frac { y _ { 1 } - 2 } { 2 } , }end{array} text { i.e., } left{begin{array}{l}x_{1}=2 x_{0}-1 y_{1}=2 y_{0}+2 .end{array}right.right.Since point Qleft(x_{1}, y_{1}right) lies on the ellipse frac{x^{2}}{2}+frac{y^{2}}{8}=1, we havefrac{left(2 x_{0}-1right)^{2}}{2}+frac{left(2 y_{0}+2right)^{2}}{8}=1 text {. }Therefore, the equation of the trajectory of the midpoint M is 2left(x-frac{1}{2}right)^{2}+frac{(y+1)^{2}}{2}=1.

answer:To evaluate which of the given operations is correct, let's analyze each option step by step:Option A: 3sqrt{2}-sqrt{2}- Start with distributing the square root: 3sqrt{2}-sqrt{2}- Combine like terms: 2sqrt{2}- Thus, 3sqrt{2}-sqrt{2} = 2sqrt{2}, not 3. So, option A is incorrect.Option B: (a-1)^{2}- Apply the square of a binomial formula: (a-1)^{2} = a^{2} - 2a + 1- Thus, (a-1)^{2} neq a^{2} - 1. So, option B is incorrect.Option C: (a^{3})^{2}- Apply the power of a power rule: (a^{3})^{2} = a^{3cdot2} = a^{6}- Thus, (a^{3})^{2} neq a^{5}. So, option C is incorrect.Option D: 4a^{2}cdot a- Apply the product of powers rule: 4a^{2}cdot a = 4a^{2+1} = 4a^{3}- Thus, 4a^{2}cdot a = 4a^{3}, which matches the original statement. So, option D is correct.Therefore, the correct operation among the given options is:boxed{D}