MaxWong

\(V = \dfrac{1}{3}Ah\)

\(A = 32 \cdot 66 = 2112\)

\(h = \sqrt{65^2-\left(\dfrac{32}{2}\right)^2-\left(\dfrac{66}{2}\right)^2} = 24\sqrt5\)

\(V = \dfrac{1}{3}(2112)(24\sqrt5) = 16896\sqrt5\)

\(f(x) = q(x)d(x)+r(x)\)

deg f = 9, deg r = 3, deg d > deg r.

For maximum deg q. deg d is minimum. => deg d = 4.

\(O(x^9) = q(x) O(x^4) + O(x^3)\)

q(x) = O(x⁵)

Therefore deg q = 5.

\(P(x) = 3x^3+a_2x^2+a_1x-6\)

Product of roots = \(-\dfrac{-6}{3}\)= 2.

Possible roots = {1,-1,2,-2}.

\(\quad (\sqrt[3]{7}+\sqrt[3]{49})^3\\ = 7 + 49 + 3\sqrt[3]{7\cdot49}(\sqrt[3]{7}+\sqrt[3]{49})\\ = 56 + 21(\sqrt[3]{7}+\sqrt[3]{49})\)

Therefore P(x) is x³ - 21x - 56.

Product of roots = \(-\dfrac{-56}{1} = 56\)

\(\text{Circumference} = \pi (\text{Diameter})\\ \text{Circumference} = 13\pi \text{ cm}\)

\(9x^2-18x-16=0\\ (3x-8)(3x+2) = 0\\ x = \dfrac{8}{3}\text{ OR }x = \dfrac{-2}{3}\)

\(15x^2+28x+12 = 0\\ (3x+2)(5x+6) = 0\\ x = \dfrac{-2}{3} \text{ OR } x = \dfrac{-6}{5}\)

Only x = -2/3 satisfies both equations. Therefore the answer is -2/3.

\((x-p)(x-q) = x^2+4x+6\\ (x-p)(x-q) = x^2+bx+c\\ \text{Comparing coefficients,} \\ (b,c) = (4,6)\\ \therefore b + c = 10\)

5)

\(\begin{cases} x+y=3\\ x^2+y^2=6\\ x^4=y^4+18\sqrt3 \end{cases}\\ (x + y)^2 = 9\\ x^2 + y^2 + 2xy = 9\\ 2xy = 3\\ x = \dfrac{3}{2y}\\ \dfrac{3}{2y} + y = 3\\ 2y^2 - 6y + 3 = 0\\ y = \dfrac{6\pm2\sqrt{3}}{4} = \dfrac{3\pm\sqrt{3}}{2}\\ \text{When } y = \dfrac{3+\sqrt3}{2}\text{,}\\ x = \dfrac{3-\sqrt3}{2}\\ \text{When } y = \dfrac{3-\sqrt3}{2}\text{,}\\ x = \dfrac{3+\sqrt3}{2}\\ \therefore (x,y) = \left(\dfrac{3+\sqrt3}{2},\dfrac{3-\sqrt3}{2}\right) \text{ OR }\left(\dfrac{3-\sqrt3}{2},\dfrac{3+\sqrt3}{2}\right)\)

4)

\(m+n = \dfrac{-5}{4}\\ mn = \dfrac{3}{4}\\ (m+7)(n+7) = mn + 7(m+n) + 49 = \dfrac{3}{4}+7\left(\dfrac{-5}{4}\right) + 49 = 41\)

3)

\(s+t = -\dfrac{9}{4}\\ st = \dfrac{-6}{4} = -\dfrac{3}{2}\\ \dfrac{s}{t} + \dfrac{t}{s} = \dfrac{(s+t)^2 - 2st}{st} = \dfrac{\left(\frac{-9}{4}\right)^2 -2\left(\frac{-3}{2}\right)}{\frac{-3}{2}} =\dfrac{-43}{8}\)