This page contains all questions and answers asked by students from this class. The most recent questions will be at the top.
Differentiate
![Rendered by QuickLaTeX.com x=e^t](http://theculture.sg/wp-content/ql-cache/quicklatex.com-9eaffa7601f72a6beac6772c5dbbb85b_l3.png)
![Rendered by QuickLaTeX.com \frac{dx}{dt} = e^t](http://theculture.sg/wp-content/ql-cache/quicklatex.com-40dec61af7ad7cdf0a422ea52ca90f65_l3.png)
Substitute
![Rendered by QuickLaTeX.com x = e^t](http://theculture.sg/wp-content/ql-cache/quicklatex.com-0a7f304b9fb9d2ce66ce9cf5c92c28c4_l3.png)
![Rendered by QuickLaTeX.com \Rightarrow e^t \frac{dy}{dt} = \mathrm{cos}(\mathrm{ln}e^t)\mathrm{sin}(\mathrm{ln}(e^t)^3)](http://theculture.sg/wp-content/ql-cache/quicklatex.com-c1a79a88bfc7d1656fbb684e042d3f03_l3.png)
![Rendered by QuickLaTeX.com e^t \frac{dy}{dt} = \mathrm{cos}t \mathrm{sin} (3t)](http://theculture.sg/wp-content/ql-cache/quicklatex.com-e7dcc3a295b80d86db4eea1c869d67af_l3.png)
Since
![Rendered by QuickLaTeX.com \frac{dy}{dt} = \frac{dy}{dx} \times \frac{dx}{dt}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-e096885bba790aba8d58cdcb2940571e_l3.png)
![Rendered by QuickLaTeX.com \frac{dy}{dt} = \frac{1}{e^t} \mathrm{cos}t \mathrm{sin} (3t) \times e^t](http://theculture.sg/wp-content/ql-cache/quicklatex.com-f805a773e472fbdab381c6b8c174d861_l3.png)
![Rendered by QuickLaTeX.com \frac{dy}{dt} = \mathrm{cos}t \mathrm{sin} (3t)](http://theculture.sg/wp-content/ql-cache/quicklatex.com-b07e6f926702eedb095872ec6b2c637d_l3.png)
![Rendered by QuickLaTeX.com \int \frac{dy}{dt} dt = \int \mathrm{cos}t \mathrm{sin} (3t) dy](http://theculture.sg/wp-content/ql-cache/quicklatex.com-c26d5506bbfa044d3f2293eef9ef08e6_l3.png)
Using the product to sum formula as shown here, we have
![Rendered by QuickLaTeX.com \int \frac{dy}{dt} dt = \frac{1}{2} \int \mathrm{sin} 4t + \mathrm{sin} 2t dt](http://theculture.sg/wp-content/ql-cache/quicklatex.com-fc5e8259cc7ec1ba8c4baf4d9b338ba2_l3.png)
![Rendered by QuickLaTeX.com y = \frac{1}{2}(-\frac{\mathrm{cos}4t}{4} - \frac{\mathrm{cos}2t}{2}) + C](http://theculture.sg/wp-content/ql-cache/quicklatex.com-09662df299bfb268a7c2d9986a3b173d_l3.png)
![Rendered by QuickLaTeX.com y = \frac{1}{2}(-\frac{\mathrm{cos}(4 \mathrm{ln}x)}{4} - \frac{\mathrm{cos}(2\mathrm{ln}x)}{2}) + C](http://theculture.sg/wp-content/ql-cache/quicklatex.com-a407a2952ed3067b6a939c3da3e17b3c_l3.png)
Note:
![Rendered by QuickLaTeX.com i = \begin{bmatrix}1\\ 0\\ 0\end{bmatrix}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-bea5b61c61bffdde5e50dcacf4833514_l3.png)
![Rendered by QuickLaTeX.com j = \begin{bmatrix}\\ 1\\ 0\end{bmatrix}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-5bb14e7e2e7d29686d6d142556adc9f9_l3.png)
![Rendered by QuickLaTeX.com \pi: 2x + 3y = -6](http://theculture.sg/wp-content/ql-cache/quicklatex.com-560b84e3e1d46df2bdb9545e6d8b079d_l3.png)
![Rendered by QuickLaTeX.com \pi \prime: x + y + z= 5](http://theculture.sg/wp-content/ql-cache/quicklatex.com-c3b11a78a895f6ba69a0ed8606c42cf8_l3.png)
![Rendered by QuickLaTeX.com \pi_1: x = 0](http://theculture.sg/wp-content/ql-cache/quicklatex.com-87b88f8ee4ec9cc04a02b7228ed1c27b_l3.png)
Using GC,
![Rendered by QuickLaTeX.com \vec{OA} = \begin{bmatrix}0\\ -2\\ 7\end{bmatrix}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-9b59b75aae5b3456b0d1387fcf6ecf5a_l3.png)
— (1)
— (2)
— (3)
Using GC,
Qn11
(i)
![Rendered by QuickLaTeX.com l: r = \begin{bmatrix}-1\\ -1\\ 0\end{bmatrix} + \alpha \begin{bmatrix}1\\ 1\\ 1\end{bmatrix}, \alpha \in \mathbb{R}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-0df5e3d23041d53a465c9cfcc39a0800_l3.png)
(ii) Since l is the common line of intersection on
![Rendered by QuickLaTeX.com \pi_1](http://theculture.sg/wp-content/ql-cache/quicklatex.com-28740f166c71e0b9ac90721a20713c27_l3.png)
![Rendered by QuickLaTeX.com \pi_2](http://theculture.sg/wp-content/ql-cache/quicklatex.com-2fc30a62ef805c9ef7b59ffc7a8dd6c2_l3.png)
![Rendered by QuickLaTeX.com \pi_3](http://theculture.sg/wp-content/ql-cache/quicklatex.com-09ce0d7b4ca11e00d270bd8d9cf549f3_l3.png)
1. l must be parallel to
![Rendered by QuickLaTeX.com \pi_3](http://theculture.sg/wp-content/ql-cache/quicklatex.com-09ce0d7b4ca11e00d270bd8d9cf549f3_l3.png)
![Rendered by QuickLaTeX.com \pi_3](http://theculture.sg/wp-content/ql-cache/quicklatex.com-09ce0d7b4ca11e00d270bd8d9cf549f3_l3.png)
![Rendered by QuickLaTeX.com \Rightarrow \begin{bmatrix}1\\ 1\\ 1\end{bmatrix} \bullet \begin{bmatrix}5\\ {\lambda}\\ 17\end{bmatrix} = 0](http://theculture.sg/wp-content/ql-cache/quicklatex.com-0884f71705524a1e1ae68dd75c14ea7c_l3.png)
![Rendered by QuickLaTeX.com \lambda = -22](http://theculture.sg/wp-content/ql-cache/quicklatex.com-d2504a7b3d4f9a2bf46768aded2aa11d_l3.png)
2. Given that l is parallel to
![Rendered by QuickLaTeX.com \pi_3](http://theculture.sg/wp-content/ql-cache/quicklatex.com-09ce0d7b4ca11e00d270bd8d9cf549f3_l3.png)
![Rendered by QuickLaTeX.com \lambda = -22](http://theculture.sg/wp-content/ql-cache/quicklatex.com-d2504a7b3d4f9a2bf46768aded2aa11d_l3.png)
![Rendered by QuickLaTeX.com \pi_3](http://theculture.sg/wp-content/ql-cache/quicklatex.com-09ce0d7b4ca11e00d270bd8d9cf549f3_l3.png)
![Rendered by QuickLaTeX.com \begin{bmatrix}-1\\ -1\\ 0\end{bmatrix}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-f853998c25001ccb05fb24901611f860_l3.png)
![Rendered by QuickLaTeX.com \pi_3](http://theculture.sg/wp-content/ql-cache/quicklatex.com-09ce0d7b4ca11e00d270bd8d9cf549f3_l3.png)
![Rendered by QuickLaTeX.com \Rightarrow \begin{bmatrix}-1\\ -1\\ 0\end{bmatrix} \bullet \begin{bmatrix}5\\ -22\\ 17\end{bmatrix} = \mu](http://theculture.sg/wp-content/ql-cache/quicklatex.com-b064796801fdc73b3bb47efe7b3d70ef_l3.png)
![Rendered by QuickLaTeX.com \mu = 17](http://theculture.sg/wp-content/ql-cache/quicklatex.com-224992d7d4ce74631266d68f53a3ee09_l3.png)
(iii)
For 3 planes to have nothing in common, then l must be parallel to
![Rendered by QuickLaTeX.com \pi_3](http://theculture.sg/wp-content/ql-cache/quicklatex.com-09ce0d7b4ca11e00d270bd8d9cf549f3_l3.png)
![Rendered by QuickLaTeX.com \pi_3](http://theculture.sg/wp-content/ql-cache/quicklatex.com-09ce0d7b4ca11e00d270bd8d9cf549f3_l3.png)
![Rendered by QuickLaTeX.com \Rightarrow \lambda = -22](http://theculture.sg/wp-content/ql-cache/quicklatex.com-c7185c275d5b1fcc82f33429230f129b_l3.png)
But
![Rendered by QuickLaTeX.com \mu \neq 17, \mu \in \mathbb{R}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-27abf11733874c34f6d5378d2a04529f_l3.png)
12.
(i)
![Rendered by QuickLaTeX.com \pi_1 : r = \begin{bmatrix}4\\ 2\\ 12\end{bmatrix} + \lambda \begin{bmatrix}2\\ 2\\ 1\end{bmatrix} + \mu \begin{bmatrix}5\\ -1\\ -5\end{bmatrix}, \mu , \lambda \in \mathbb{R}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-64f0a0b3668b6a11d606c1267ecd0f38_l3.png)
![Rendered by QuickLaTeX.com \vec{OP} = \begin{bmatrix}4\\ 2\\ 12\end{bmatrix}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-d83423bf05b11c22e3345f9b18109939_l3.png)
![Rendered by QuickLaTeX.com \pi_1](http://theculture.sg/wp-content/ql-cache/quicklatex.com-28740f166c71e0b9ac90721a20713c27_l3.png)
![Rendered by QuickLaTeX.com \lambda = \mu = 0](http://theculture.sg/wp-content/ql-cache/quicklatex.com-4b670263ade2d9393ed57608ceda2e2c_l3.png)
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