Hopefully, you guys have started on the Set B. You will find the following solutions useful. Click on the question. Please do attempt them during this December Holidays. 🙂
If you do have any questions, please WhatsApp me. 🙂
(i)
By Pythagoras’ Theorem, Length of side of isosceles ![Rendered by QuickLaTeX.com = \sqrt{ \frac{(2x)^2}{2} } = \sqrt{2} x](https://theculture.sg/wp-content/ql-cache/quicklatex.com-95cfdb3742f0b1ac15c1b5968115b97c_l3.png)
Area ![Rendered by QuickLaTeX.com = 3 \frac{1}{2} (\sqrt{2}x)(\sqrt{2}x) + 6x \times y](https://theculture.sg/wp-content/ql-cache/quicklatex.com-6b720191e5a08f13699154238f4c808b_l3.png)
![Rendered by QuickLaTeX.com 120 = 3x^2 + 6xy](https://theculture.sg/wp-content/ql-cache/quicklatex.com-8bd1e85f63b7b1763ac11547bda0b254_l3.png)
![Rendered by QuickLaTeX.com \Rightarrow y = \frac{120-3x^2}{6x}](https://theculture.sg/wp-content/ql-cache/quicklatex.com-89d31ac3fc2e8292dbd433fecb54baeb_l3.png)
![Rendered by QuickLaTeX.com P = 6x + 6(\sqrt{2}x) + 2 y](https://theculture.sg/wp-content/ql-cache/quicklatex.com-9cedc1c9195e6b6de46a0b9967a709e7_l3.png)
![Rendered by QuickLaTeX.com P = 6x + 6\sqrt{2}x + 2(\frac{120-3x^2}{6x})](https://theculture.sg/wp-content/ql-cache/quicklatex.com-7b10141023e9d6eac59c48add75944d0_l3.png)
![Rendered by QuickLaTeX.com P = 5x + 6\sqrt{2}x + \frac{40}{x}](https://theculture.sg/wp-content/ql-cache/quicklatex.com-1ee24709b148ddb362057ea487ee65c1_l3.png)
(ii)
![Rendered by QuickLaTeX.com \frac{dP}{dx} = 5 + 6 \sqrt{2} - \frac{40}{x^2}](https://theculture.sg/wp-content/ql-cache/quicklatex.com-f440aab4b8e46686d30dba6403288281_l3.png)
Let ![Rendered by QuickLaTeX.com \frac{dP}{dx} = 0](https://theculture.sg/wp-content/ql-cache/quicklatex.com-0232cd1112b698ae19c62a9fcd59c119_l3.png)
![Rendered by QuickLaTeX.com x^2 = \frac{40}{5 + 6\sqrt{2}}](https://theculture.sg/wp-content/ql-cache/quicklatex.com-9515ceaa1b66fdc155167c23646c0641_l3.png)
![Rendered by QuickLaTeX.com x = 1.722265053](https://theculture.sg/wp-content/ql-cache/quicklatex.com-d33d7a2dba2233cbdb8401053f7d3de0_l3.png)
![Rendered by QuickLaTeX.com \frac{d^2P}{dx^2} = \frac{80}{x^3} = \frac{80}{{1.722265053}^3} \textgreater 0](https://theculture.sg/wp-content/ql-cache/quicklatex.com-f412055662b5e116ffa3190e5ed8aadf_l3.png)
Thus
is minimum when ![Rendered by QuickLaTeX.com x = 1.722265053](https://theculture.sg/wp-content/ql-cache/quicklatex.com-d33d7a2dba2233cbdb8401053f7d3de0_l3.png)
.
(i)
![Rendered by QuickLaTeX.com \frac{AP}{PB} = \frac{2}{3}](https://theculture.sg/wp-content/ql-cache/quicklatex.com-0e4b40842d09d7032ae224e3a4a26e51_l3.png)
![Rendered by QuickLaTeX.com 3 \vec{AP} = 2 \vec{PB}](https://theculture.sg/wp-content/ql-cache/quicklatex.com-4f648a8cd914b0a45f477b994a585bc7_l3.png)
![Rendered by QuickLaTeX.com 3 ( \vec{OP} - \vec{OA}) = 2 ( \vec{OB} - \vec{OP} )](https://theculture.sg/wp-content/ql-cache/quicklatex.com-b3864cf02bcf34dcb5f39c5009255a57_l3.png)
![Rendered by QuickLaTeX.com \vec{OP} = \frac{2 \vec{OB} + 3 \vec{OA} }{5}](https://theculture.sg/wp-content/ql-cache/quicklatex.com-631d46551b339f592044021fe10fad36_l3.png)
![Rendered by QuickLaTeX.com \vec{OP} = \frac{1}{5} ( \begin{pmatrix}{8}\\{8}\\{20}\end{pmatrix} + \begin{pmatrix}{-3}\\{-18}\\{-15}\end{pmatrix})](https://theculture.sg/wp-content/ql-cache/quicklatex.com-fe4a21564e3e81ffabd8d8a18b31732b_l3.png)
![Rendered by QuickLaTeX.com \vec{OP} = \begin{pmatrix}{1}\\{-2}\\{1}\end{pmatrix}](https://theculture.sg/wp-content/ql-cache/quicklatex.com-757292f5c401f02580498691140a7591_l3.png)
(ii)
![Rendered by QuickLaTeX.com \vec{AB} = \begin{pmatrix}{5}\\{10}\\{15}\end{pmatrix}](https://theculture.sg/wp-content/ql-cache/quicklatex.com-645dd621cb116f8a5b75adcf56a54ceb_l3.png)
![Rendered by QuickLaTeX.com \vec{AB} \cdot \vec{OP}](https://theculture.sg/wp-content/ql-cache/quicklatex.com-a1b737feef4a8868217d422d7bc7b9e2_l3.png)
![Rendered by QuickLaTeX.com = \begin{pmatrix}{5}\\{10}\\{15}\end{pmatrix} \cdot \begin{pmatrix}{1}\\{-2}\\{1}\end{pmatrix}](https://theculture.sg/wp-content/ql-cache/quicklatex.com-d209e46b4b48e86351467209c0ad29ba_l3.png)
![Rendered by QuickLaTeX.com = 5 - 20 + 15](https://theculture.sg/wp-content/ql-cache/quicklatex.com-711cb02afb5a8ff032252ab12f9fffa4_l3.png)
![Rendered by QuickLaTeX.com = 0 \Rightarrow AB \perp OP](https://theculture.sg/wp-content/ql-cache/quicklatex.com-a9299627357e50076b9c6cb9714b3a19_l3.png)
(iii)
![Rendered by QuickLaTeX.com |a \cdot c|](https://theculture.sg/wp-content/ql-cache/quicklatex.com-9a8621b14b303fa8541770aad04f4a4a_l3.png)
![Rendered by QuickLaTeX.com = \frac{\big| \begin{pmatrix}{-1}\\{-6}\\{-5}\end{pmatrix} \cdot \begin{pmatrix}{1}\\{-2}\\{1}\end{pmatrix} \big|}{\big| \begin{pmatrix}{1}\\{-2}\\{1}\end{pmatrix} \big|}](https://theculture.sg/wp-content/ql-cache/quicklatex.com-0fb61fece10cb78a588c30dab4756e87_l3.png)
![Rendered by QuickLaTeX.com = \frac{| -1 + 12 - 5|}{\sqrt{6}}](https://theculture.sg/wp-content/ql-cache/quicklatex.com-52827d792dab5630889803393ce46a9c_l3.png)
![Rendered by QuickLaTeX.com = \sqrt{6}](https://theculture.sg/wp-content/ql-cache/quicklatex.com-8bfcbac63df24394e2b97d1c113c62b0_l3.png)
is the length of projection of
onto
.
![Rendered by QuickLaTeX.com \big| \vec{OP} \big| = \sqrt{6}](https://theculture.sg/wp-content/ql-cache/quicklatex.com-6faa3984de320c5580c6b051f8828b97_l3.png)
Area
![Rendered by QuickLaTeX.com = \frac{1}{2} \big| \begin{pmatrix}{-1}\\{-6}\\{-5}\end{pmatrix} \times \begin{pmatrix}{4}\\{4}\\{10}\end{pmatrix} \big|](https://theculture.sg/wp-content/ql-cache/quicklatex.com-35b29ef8e2e2dc736584a65a76de6043_l3.png)
![Rendered by QuickLaTeX.com = \frac{1}{2} \big| \begin{pmatrix}{-40}\\{-10}\\{20}\end{pmatrix} \big|](https://theculture.sg/wp-content/ql-cache/quicklatex.com-a5c478db9cc9d86529e791a6da01706b_l3.png)
![Rendered by QuickLaTeX.com = 5 \sqrt{21}](https://theculture.sg/wp-content/ql-cache/quicklatex.com-207e75084d0e2e7c12b830eded6e6c3a_l3.png)
(i)
Since
![Rendered by QuickLaTeX.com R_g = ( - \infty, 2) \subset ( - \infty, 3 ] = D_f](https://theculture.sg/wp-content/ql-cache/quicklatex.com-637256918529c9515d73aab706db86e0_l3.png)
, fg exists.
![Rendered by QuickLaTeX.com fg(x) = (2-e^x)^2 - 2(2-e^x) - 24](https://theculture.sg/wp-content/ql-cache/quicklatex.com-67300a32787a973b798613303d7a7ef2_l3.png)
![Rendered by QuickLaTeX.com fg(x) = e^{2x} - 2e^x - 24, x \in \mathbb{R}](https://theculture.sg/wp-content/ql-cache/quicklatex.com-ceba73b9970f1cd1100758689b7c0b9f_l3.png)
(ii)
![](https://theculture.sg/wp-content/uploads/2017/11/Screen-Shot-2017-12-04-at-1.35.38-PM.png)
Since the line
cuts
at more than one point,
is not a one-one function, thus the inverse does not exist.
(iii)
![Rendered by QuickLaTeX.com \Rightarrow a = 1](https://theculture.sg/wp-content/ql-cache/quicklatex.com-cfcf59c991831b0c9663cd44bd705322_l3.png)
Let ![Rendered by QuickLaTeX.com f(x) = y](https://theculture.sg/wp-content/ql-cache/quicklatex.com-361803e959988dba23c81a2e3b508f26_l3.png)
![Rendered by QuickLaTeX.com y = (x-1)^2 - 25](https://theculture.sg/wp-content/ql-cache/quicklatex.com-7033ba32d2af7c60e30c96f2c0f6efee_l3.png)
![Rendered by QuickLaTeX.com x = 1 \pm \sqrt{25 + y}](https://theculture.sg/wp-content/ql-cache/quicklatex.com-af339a94b533ff1ce47c87eff658abd7_l3.png)
Since ![Rendered by QuickLaTeX.com x \le 1, ~ x = 1 - \sqrt{25 + y}](https://theculture.sg/wp-content/ql-cache/quicklatex.com-4fe8994f39880261b63878fe0d9e4bcc_l3.png)
![Rendered by QuickLaTeX.com f^{-1}(x) = 1 - \sqrt{25 + x}](https://theculture.sg/wp-content/ql-cache/quicklatex.com-5240ee046bb43a78720b48dcf7ba8ca0_l3.png)
![Rendered by QuickLaTeX.com D_{f^{-1}} = R_f = [-25, \infty)](https://theculture.sg/wp-content/ql-cache/quicklatex.com-a00be1365367b4ffd4b4ed1ac23e90df_l3.png)
(a)
(i)
![Rendered by QuickLaTeX.com 15 + (50-1)(0.4) = 34.6 \text{km}](https://theculture.sg/wp-content/ql-cache/quicklatex.com-1b10be019e093674e09bb51210b2c788_l3.png)
(ii)
Let
be number of sessions he needs to run
![Rendered by QuickLaTeX.com \frac{n}{2} [2(15) + (n-1)(0.4)] = 560](https://theculture.sg/wp-content/ql-cache/quicklatex.com-a186bed9f17c0afc1ada6122b80782f2_l3.png)
Using GC, ![Rendered by QuickLaTeX.com n = 29](https://theculture.sg/wp-content/ql-cache/quicklatex.com-f0585f927d5474b162972c937b9b5d4a_l3.png)
(b)
(i)
m
(ii)
![Rendered by QuickLaTeX.com S_{\infty} = \frac{30}{1-0.95} = 600](https://theculture.sg/wp-content/ql-cache/quicklatex.com-4329753c8e2cb1b071f1843d1637b2ca_l3.png)
Thus, he can be cycle more than 600km in total.
(i)
![Rendered by QuickLaTeX.com y = \frac{2x+9}{x+2} = 2 + \frac{5}{x+2}](https://theculture.sg/wp-content/ql-cache/quicklatex.com-87b3aac373d99f05a5f22bea5fb334e0_l3.png)
![Rendered by QuickLaTeX.com \Rightarrow A = 2, B = 5](https://theculture.sg/wp-content/ql-cache/quicklatex.com-29a71976a3fa44f057d2ae9496bececc_l3.png)
First, scale the curve 5 units parallel to the y = axis.
Then, translate the curve 2 units in the positive y – direction.
(ii)
![](https://theculture.sg/wp-content/uploads/2017/11/Screen-Shot-2017-12-07-at-12.02.58-PM.png)
(iii)
Using GC,
and ![Rendered by QuickLaTeX.com -2.95, -3.26)](https://theculture.sg/wp-content/ql-cache/quicklatex.com-8961a23f1beb5b1030e5144beb85fc3a_l3.png)
(i)
![Rendered by QuickLaTeX.com \text{Required~angle~} = \text{cos}^{-1} \bigg| \frac{\begin{pmatrix}{-1}\\{2}\\{1}\end{pmatrix} \cdot \begin{pmatrix}{1}\\{0}\\{0}\end{pmatrix} }{\sqrt{6} \sqrt{1} } \bigg| = 65.9 ^{\circ}](https://theculture.sg/wp-content/ql-cache/quicklatex.com-485835402aa3b962fa14a2239b7375f7_l3.png)
(ii)
Normal to ![Rendered by QuickLaTeX.com P_2 = \begin{pmatrix}{-1}\\{2}\\{1}\end{pmatrix} \times \begin{pmatrix}{1}\\{1}\\{-1}\end{pmatrix} = \begin{pmatrix}{-3}\\{0}\\{-3}\end{pmatrix}](https://theculture.sg/wp-content/ql-cache/quicklatex.com-1484be01a0f1b534680de8f81bae4ae6_l3.png)
![Rendered by QuickLaTeX.com P_2 : r \cdot \begin{pmatrix}{-3}\\{0}\\{-3}\end{pmatrix} = \begin{pmatrix}{3}\\{5}\\{-2}\end{pmatrix} \cdot \begin{pmatrix}{-3}\\{0}\\{-3}\end{pmatrix}](https://theculture.sg/wp-content/ql-cache/quicklatex.com-87366411db1a2e3d3806ce88a76b552f_l3.png)
![Rendered by QuickLaTeX.com \Rightarrow P_2: x + z = 1](https://theculture.sg/wp-content/ql-cache/quicklatex.com-81df46aa68c0dd15a9d5338ecb11c6e2_l3.png)
(iii)
Using GC: ![Rendered by QuickLaTeX.com m: r = \begin{pmatrix}{1}\\{3}\\{0}\end{pmatrix} + \lambda \begin{pmatrix}{-1}\\{2}\\{1}\end{pmatrix}, \lambda \in \mathbb{R}](https://theculture.sg/wp-content/ql-cache/quicklatex.com-bd311367d48b696dc4b214f02838b2b8_l3.png)
(iv)
Let F be the foot of perpendicular from A to
.
for some ![Rendered by QuickLaTeX.com \lambda](https://theculture.sg/wp-content/ql-cache/quicklatex.com-ab48baf331239642a00255b86324280a_l3.png)
![Rendered by QuickLaTeX.com \bigg[ \begin{pmatrix}{3}\\{5}\\{-2}\end{pmatrix} + \lambda \begin{pmatrix}{1}\\{1}\\{-1}\end{pmatrix} \bigg] \cdot \begin{pmatrix}{1}\\{1}\\{-1}\end{pmatrix} = 4](https://theculture.sg/wp-content/ql-cache/quicklatex.com-92d451878f1fb79fb2f808fb76ca364b_l3.png)
![Rendered by QuickLaTeX.com 10 + 3 \lambda = 4](https://theculture.sg/wp-content/ql-cache/quicklatex.com-0d556df0d3a5043dbafc3fc99d38919b_l3.png)
![Rendered by QuickLaTeX.com \Rightarrow \lambda = - 2](https://theculture.sg/wp-content/ql-cache/quicklatex.com-3d76ff17afd526f383b9e4ceb77595bf_l3.png)
![Rendered by QuickLaTeX.com \Rightarrow \vec{OF} = \begin{pmatrix}{1}\\{3}\\{0}\end{pmatrix}](https://theculture.sg/wp-content/ql-cache/quicklatex.com-24d963ca2fed6a0cf1e176df7db9e7e8_l3.png)
![Rendered by QuickLaTeX.com \vec{OF} = \frac{\vec{OA} + \vec{OB} }{2}](https://theculture.sg/wp-content/ql-cache/quicklatex.com-bf62db912e3a0a879f66c4170257ebcd_l3.png)
![Rendered by QuickLaTeX.com \vec{OB} = 2 \times \begin{pmatrix}{1}\\{3}\\{0}\end{pmatrix} - \begin{pmatrix}{3}\\{5}\\{-2}\end{pmatrix}](https://theculture.sg/wp-content/ql-cache/quicklatex.com-78e6ce6b680acf989df74fddefb068b0_l3.png)
![Rendered by QuickLaTeX.com \vec{OB} = \begin{pmatrix}{-1}\\{1}\\{2}\end{pmatrix} = (-1, 1, 2)](https://theculture.sg/wp-content/ql-cache/quicklatex.com-5070c63d3b4fe88c86d4433c8034d575_l3.png)