All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions.
This is answers for H2 Mathematics (9740). H2 Mathematics (9758), click here.
Numerical Answers (click the questions for workings/explanation)
Question 1:
; ![Rendered by QuickLaTeX.com a = 4](http://theculture.sg/wp-content/ql-cache/quicklatex.com-47613e882fe0912b06790c8c6664ec1a_l3.png)
Question 2:
or ![Rendered by QuickLaTeX.com x \textless a](http://theculture.sg/wp-content/ql-cache/quicklatex.com-7ee7b45f524b4671239ae5976bec5e1a_l3.png)
Question 3: 2
Question 4:
; translate the graph 4 units in negative y-direction and translate the graph 2 units in positive x-direction.
Question 5:
;
;
or ![Rendered by QuickLaTeX.com x \approx 1.15](http://theculture.sg/wp-content/ql-cache/quicklatex.com-064e43125b49b91a543d16733584d7d6_l3.png)
Question 6: ![Rendered by QuickLaTeX.com r = a + (\frac{d - a \cdot n}{b \cdot n}) b](http://theculture.sg/wp-content/ql-cache/quicklatex.com-3de42190f2086623b4ea7e24692a578e_l3.png)
Question 7: ![Rendered by QuickLaTeX.com (\frac{1}{a}, \frac{1}{ae}); \frac{1}{a^2}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-b48448fb68f4e30f1dc56c1a6453a43b_l3.png)
Question 8:
or
;
; ![Rendered by QuickLaTeX.com (w^2 - 2w+2)(w^2-4w+29)](http://theculture.sg/wp-content/ql-cache/quicklatex.com-d4e82e71d0378ee1f33bc9f60c82e618_l3.png)
Question 9:
;
;
;
; ![Rendered by QuickLaTeX.com e^x](http://theculture.sg/wp-content/ql-cache/quicklatex.com-e939fbad1f1705af3cbc3b2d3ef1c062_l3.png)
Question 10:
;
; ![Rendered by QuickLaTeX.com \frac{1}{2}\sqrt{10}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-a867e4fa6f026b53ee95975a7a79673d_l3.png)
Question 11:
;
; ![Rendered by QuickLaTeX.com v = \frac{1}{k}(10- 10 e^{-kt})](http://theculture.sg/wp-content/ql-cache/quicklatex.com-ba4fa393ab95377cc4181f0d538fff07_l3.png)
![Rendered by QuickLaTeX.com e^{2x} \text{ln}(1+ax)](http://theculture.sg/wp-content/ql-cache/quicklatex.com-c2dfb68b00b1743f45ac6d962208f6f7_l3.png)
![Rendered by QuickLaTeX.com = (1 + 2x + 2x^2 + ... )(ax - \frac{a^2 x^2}{2} + \frac{a^3 x^3}{3} + ... )](http://theculture.sg/wp-content/ql-cache/quicklatex.com-070247d75f71140da8ff15cb654f8f8a_l3.png)
![Rendered by QuickLaTeX.com \approx ax + (2a - \frac{a^2}{2})x^2 + (\frac{a^3}{3} + 2a - a^2) x^3](http://theculture.sg/wp-content/ql-cache/quicklatex.com-791cce5770e459a19a32f33c8a92e31e_l3.png)
Let ![Rendered by QuickLaTeX.com 2a - \frac{a^2}{2} = 0](http://theculture.sg/wp-content/ql-cache/quicklatex.com-06929a653dac4e88d56bb1d9de4860a9_l3.png)
or ![Rendered by QuickLaTeX.com 4](http://theculture.sg/wp-content/ql-cache/quicklatex.com-e625bc76ce6e840b4341d38022d598b3_l3.png)
Since
is non-zero,
.
Let
be the preposition “
” for all ![Rendered by QuickLaTeX.com n \in \mathbb{Z}^+](http://theculture.sg/wp-content/ql-cache/quicklatex.com-1bbdf2a6a78e0da22a99b31c47860e69_l3.png)
Let ![Rendered by QuickLaTeX.com n = 1](http://theculture.sg/wp-content/ql-cache/quicklatex.com-9f2587028f08346e04e23bbccacfcb02_l3.png)
LHS ![Rendered by QuickLaTeX.com = \frac{1}{2}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-3a2d99f47aea6c47037c966bc9efa000_l3.png)
RHS ![Rendered by QuickLaTeX.com = 2 - \frac{1}{2}(1+2) = \frac{1}{2}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-3a0fb308bc8ca7bf51b37a025d5d9bbe_l3.png)
Since LHS = RHS,
is true.
Assume
is true for some
, ![Rendered by QuickLaTeX.com k \in \mathbb{Z}^+](http://theculture.sg/wp-content/ql-cache/quicklatex.com-26d9e9819921b522240b8f5378fcf554_l3.png)
![Rendered by QuickLaTeX.com \sum_{r=1}^k \frac{r}{2^r} = 2 - (\frac{1}{2})^k(k+2)](http://theculture.sg/wp-content/ql-cache/quicklatex.com-90258e6e91701475c3efe240e01bb713_l3.png)
Want to show
is true
![Rendered by QuickLaTeX.com \sum_{r=1}^{k+1} \frac{r}{2^r} = 2 - (\frac{1}{2})^{k+1}(k+3)](http://theculture.sg/wp-content/ql-cache/quicklatex.com-d442d14e7f300a50170c74cce0e9b469_l3.png)
LHS
![Rendered by QuickLaTeX.com = \sum_{r=1}^{k+1} \frac{r}{2^r}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-017b4329986cf7e5a62d88b09b9e012a_l3.png)
![Rendered by QuickLaTeX.com = \sum_{r=1}^k (\frac{r}{2^r}) + \frac{k+1}{2^{k+1}}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-25f6074237bf0522ca37cf265a6e1c77_l3.png)
![Rendered by QuickLaTeX.com = 2 - (\frac{1}{2})^k(k+2) + \frac{k+1}{2^{k+1}}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-1952239e6eb50f6afccc5e02d6deb1ce_l3.png)
![Rendered by QuickLaTeX.com = 2 - \frac{1}{2^{k+1}} [ 2(k+2) - (k+1)]](http://theculture.sg/wp-content/ql-cache/quicklatex.com-ffff795cc9c4351fef080b8676f03ad2_l3.png)
![Rendered by QuickLaTeX.com = 2 - \frac{1}{2^{k+1}} ( k+3)](http://theculture.sg/wp-content/ql-cache/quicklatex.com-1252dc83ea1eb4542d1667df64221aa1_l3.png)
![Rendered by QuickLaTeX.com = 2 - (\frac{1}{2})^{k+1}(k+3)](http://theculture.sg/wp-content/ql-cache/quicklatex.com-eff3785bfdacab27fc6da987b1276ce7_l3.png)
RHS
Hence, by Mathematical Induction, since
is true,
is true
is true,
is true for all ![Rendered by QuickLaTeX.com n \in \mathbb{z}^+](http://theculture.sg/wp-content/ql-cache/quicklatex.com-f2779b134b43c71478ec9702664cc7cf_l3.png)
(ii)
As ![Rendered by QuickLaTeX.com n \to \infty, (\frac{1}{2})^n \to 0](http://theculture.sg/wp-content/ql-cache/quicklatex.com-bfdb83dcef1dd2118f1432ee28d2b280_l3.png)
![Rendered by QuickLaTeX.com \Rightarrow \sum_{r=1}^{\infty} \frac{r}{2^r} = 2](http://theculture.sg/wp-content/ql-cache/quicklatex.com-6877805e13f4ea4983e5bb6316355a39_l3.png)
(i)
![Rendered by QuickLaTeX.com y = \frac{4x+9}{x+2}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-897ca83b881ec79525a2fac0794662b2_l3.png)
![Rendered by QuickLaTeX.com y = 4 + \frac{1}{x+2}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-07d2660fc12762af2c6b7a1c6a8e5536_l3.png)
for all ![Rendered by QuickLaTeX.com x \in \mathbb{R}, x \neq -2](http://theculture.sg/wp-content/ql-cache/quicklatex.com-775b1d3c6a5a73c46604724ec66aa860_l3.png)
(ii)
![Rendered by QuickLaTeX.com a = 4, b = 1](http://theculture.sg/wp-content/ql-cache/quicklatex.com-f48fc5a947adc8c2b46cea4fd26a0c1b_l3.png)
Asymptotes are
and ![Rendered by QuickLaTeX.com x = -2](http://theculture.sg/wp-content/ql-cache/quicklatex.com-375caac7c7302d29a3546525d4baae54_l3.png)
(iii)
First, translate the graph 4 units in negative y-direction.
Then, translate the graph 2 units in positive x-direction.
(i)
Let
![Rendered by QuickLaTeX.com f(x) = x^3 + ax^2 + bx + c](http://theculture.sg/wp-content/ql-cache/quicklatex.com-f476296f4f19d06924d818e1c66499dd_l3.png)
—(1)
—(2)
—(3)
![Rendered by QuickLaTeX.com a = -1.5, b = 1.5, c = 7](http://theculture.sg/wp-content/ql-cache/quicklatex.com-dd11ce49c61447b2bb90db72eea14b0e_l3.png)
(ii)
![Rendered by QuickLaTeX.com f(x) = x^3 - 1.5x^2 + 1.5x + 7](http://theculture.sg/wp-content/ql-cache/quicklatex.com-2b8fabd4816de5a7437d93592b45450d_l3.png)
![Rendered by QuickLaTeX.com f'(x) = 3x^2 - 3x + 1.5](http://theculture.sg/wp-content/ql-cache/quicklatex.com-76df9285185752d04bbcd95a1bb6d489_l3.png)
![Rendered by QuickLaTeX.com f'(x) = 3 ( x - 0.5)^2 + 0.75 \textgreater 0](http://theculture.sg/wp-content/ql-cache/quicklatex.com-7d49623851335b6e4e9e7d560e33df5c_l3.png)
Since
is a strictly increasing function with no stationary point. Thus, it can only have one real root.
![Rendered by QuickLaTeX.com x \approx - 1.33](http://theculture.sg/wp-content/ql-cache/quicklatex.com-df79d3fa02f3f8fa55a3df4d3ba0f992_l3.png)
(iii)
![Rendered by QuickLaTeX.com f'(x) = 3x^2 - 3x + 1.5 = 2](http://theculture.sg/wp-content/ql-cache/quicklatex.com-c8bb92e07a52e2260ec035552062ecae_l3.png)
![Rendered by QuickLaTeX.com 3x^2 - 3x - 0.5 = 0](http://theculture.sg/wp-content/ql-cache/quicklatex.com-89e329fa97c51381a8f265fb86cb86cf_l3.png)
or ![Rendered by QuickLaTeX.com x \approx 1.15](http://theculture.sg/wp-content/ql-cache/quicklatex.com-064e43125b49b91a543d16733584d7d6_l3.png)
(i)
Set of points lying on the line which passes through a and is parallel to b.
(ii)
Set of points on the plane which has normal vector, n.
is the displacement of the plane from Origin.
(iii)
![Rendered by QuickLaTeX.com \textbf{r} = \textbf{a} + t \textbf{b}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-17f5c6c8ad94e438bb84ba1dbb728d7c_l3.png)
![Rendered by QuickLaTeX.com \textbf{r} \cdot \textbf{n} = d](http://theculture.sg/wp-content/ql-cache/quicklatex.com-7e176d18b50cfb130ddc80393eaa0548_l3.png)
![Rendered by QuickLaTeX.com ( \textbf{a} + t \textbf{b} )\cdot \textbf{n} = d](http://theculture.sg/wp-content/ql-cache/quicklatex.com-2779fef0e999a4adddb4c409568e560b_l3.png)
![Rendered by QuickLaTeX.com \textbf{a} \cdot \textbf{n} + t \textbf{b} \cdot \textbf{n} = d](http://theculture.sg/wp-content/ql-cache/quicklatex.com-4967bae790c86f68f006ef1920dce9da_l3.png)
![Rendered by QuickLaTeX.com t = \frac{d - \textbf{a} \cdot \textbf{n}}{\textbf{b} \cdot \textbf{n}}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-fd221ad2350418f1ce5c640c285d8305_l3.png)
![Rendered by QuickLaTeX.com \textbf{r} = \textbf{a} + (\frac{d - \textbf{a} \cdot \textbf{n}}{\textbf{b} \cdot \textbf{n}} )\textbf{b}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-3b7f559c46e0eacf3a674910d71306d6_l3.png)
Since
, line is not parallel to plane, the solution represents the point of intersection between the line and plane.
(a)
![Rendered by QuickLaTeX.com z^2 (1-i) - 2z + (5+5i) = 0](http://theculture.sg/wp-content/ql-cache/quicklatex.com-8bde0b61dc42060df8b5091ec4b6dbc9_l3.png)
![Rendered by QuickLaTeX.com z = \frac{2 \pm \sqrt{4 - 4(1-i)(5+5i)}}{2(1-i)}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-60fceafaea5e191c28c3136c2bef5c6e_l3.png)
![Rendered by QuickLaTeX.com z = \frac{2 \pm \sqrt{4-40}}{2(1-i)}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-3ab93cf4c9e15616271775fe5760e986_l3.png)
![Rendered by QuickLaTeX.com z = \frac{2 \pm \sqrt{-36}}{2(1-i)}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-b1dd25acf25bf882329e66dfac31faeb_l3.png)
![Rendered by QuickLaTeX.com z = \frac{2 \pm 6i}{2(1-i)}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-bcd24f261630efa933b3dac8766019f0_l3.png)
![Rendered by QuickLaTeX.com z = \frac{1+3i}{1-i} \times \frac{1+i}{1+i}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-ab3c04be63534213442bd5b042aa4609_l3.png)
![Rendered by QuickLaTeX.com z = - 1 + 2i](http://theculture.sg/wp-content/ql-cache/quicklatex.com-1afea68af23a6500215aedc5ae72a68b_l3.png)
![Rendered by QuickLaTeX.com z = \frac{1-3i}{1-i} \times \frac{1+i}{1+i}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-8a471e6199d821e857e111690cc498cd_l3.png)
![Rendered by QuickLaTeX.com z = 2 - i](http://theculture.sg/wp-content/ql-cache/quicklatex.com-6d058e8a428dcf371cc4dc1fae125a54_l3.png)
or ![Rendered by QuickLaTeX.com z = 2 - i](http://theculture.sg/wp-content/ql-cache/quicklatex.com-6d058e8a428dcf371cc4dc1fae125a54_l3.png)
(b)
(i)
![Rendered by QuickLaTeX.com w^2 = (1-i)^2 = 1 - 2i -1 = -2i](http://theculture.sg/wp-content/ql-cache/quicklatex.com-7f94f572b135e42e9204c961f93d0501_l3.png)
![Rendered by QuickLaTeX.com w^3 = w^2 \times w = -2i \times (1-i) = -2 - 2i](http://theculture.sg/wp-content/ql-cache/quicklatex.com-352d9fe99f231cc036e265d554cee169_l3.png)
![Rendered by QuickLaTeX.com w^4 = (w^2)^2 = (-2i)^2 = -4](http://theculture.sg/wp-content/ql-cache/quicklatex.com-6cd845978c76cde06461f00caa1ea91f_l3.png)
![Rendered by QuickLaTeX.com w^4 + pw^3 +39w^2 + qw + 58 =0](http://theculture.sg/wp-content/ql-cache/quicklatex.com-60076f0558d68143f3209edb66b7e50e_l3.png)
![Rendered by QuickLaTeX.com \Rightarrow -4 + p(-2-2i) + 39(-2i) + q(1-i) + 58 = 0](http://theculture.sg/wp-content/ql-cache/quicklatex.com-b9cfede90c59b9388744a58e5d758355_l3.png)
Comparing real and imaginary parts,
![Rendered by QuickLaTeX.com - 4 - 2p + q + 58 = 0](http://theculture.sg/wp-content/ql-cache/quicklatex.com-f48f542ca799f81e30d579ce7b44cd3b_l3.png)
—(1)
![Rendered by QuickLaTeX.com -2p - 78 - q = 0](http://theculture.sg/wp-content/ql-cache/quicklatex.com-69ebe7dea72fd2d9cd2a46db91ccb5da_l3.png)
—(2)
(1) + (2): ![Rendered by QuickLaTeX.com p = -6](http://theculture.sg/wp-content/ql-cache/quicklatex.com-d961ee28d3df5ce8b6998ddc44208045_l3.png)
![Rendered by QuickLaTeX.com \Rightarrow q = -66](http://theculture.sg/wp-content/ql-cache/quicklatex.com-e5579ede00b8b5a4ba2e1a3ebbd480cd_l3.png)
(ii)
![Rendered by QuickLaTeX.com w^4 - 6w^3 +39w^2 - 66w + 58 =0](http://theculture.sg/wp-content/ql-cache/quicklatex.com-9994ea805b017c0037bbd4cbe2620426_l3.png)
Since all coefficients of
are real,
is a root
is also a root.
![Rendered by QuickLaTeX.com [w - (1+i)][w-(1-i)] = w^2 - 2w + 2](http://theculture.sg/wp-content/ql-cache/quicklatex.com-eb3f4e70d0ce4bc0e03391bb23444f32_l3.png)
![Rendered by QuickLaTeX.com (w^2 -2w +2)(w^2 + aw+ b) = w^4 - 6w^3 +39w^2 - 66w + 58 =0](http://theculture.sg/wp-content/ql-cache/quicklatex.com-127bc9d6768c18d8da5c5dcdbed146c2_l3.png)
![Rendered by QuickLaTeX.com \Rightarrow 2b = 58](http://theculture.sg/wp-content/ql-cache/quicklatex.com-5421dacb4fd9331ddae7e9e0e4daee74_l3.png)
![Rendered by QuickLaTeX.com \Rightarrow b = 29](http://theculture.sg/wp-content/ql-cache/quicklatex.com-cdc22a10d1e6f672076a33b44479a43b_l3.png)
![Rendered by QuickLaTeX.com \Rightarrow -2(29)+2a = -66](http://theculture.sg/wp-content/ql-cache/quicklatex.com-04ffd30036e59e3026f1b819f744d596_l3.png)
![Rendered by QuickLaTeX.com \Rightarrow a = -4](http://theculture.sg/wp-content/ql-cache/quicklatex.com-41822276fa28a4e9f1a2b72381857590_l3.png)
![Rendered by QuickLaTeX.com \therefore (w^2 - 2w + 2)(w^2 - 4w + 29)](http://theculture.sg/wp-content/ql-cache/quicklatex.com-4ec617e4647c676b7839d2313414761d_l3.png)
(a)
(i)
![Rendered by QuickLaTeX.com U_n = S_n - S_{n-1}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-e071fd37e33a6ecff52653e2da4d3181_l3.png)
![Rendered by QuickLaTeX.com U_n = An^2 + Bn -[A(n-1)^2 + B(n-1)]](http://theculture.sg/wp-content/ql-cache/quicklatex.com-753663c18d8c7637cccd57794dead32e_l3.png)
![Rendered by QuickLaTeX.com U_n = 2An - A +B](http://theculture.sg/wp-content/ql-cache/quicklatex.com-b40bda1c46d53b98406fd43e8b5266ab_l3.png)
(ii)
![Rendered by QuickLaTeX.com U_{10} = 48](http://theculture.sg/wp-content/ql-cache/quicklatex.com-d4ef03d70ae7fb647fc1bc089646df47_l3.png)
—(1)
![Rendered by QuickLaTeX.com U_{17} = 90](http://theculture.sg/wp-content/ql-cache/quicklatex.com-85d5eaa57e5b438207b937446e9f2cc0_l3.png)
—(2)
(2) – (1): ![Rendered by QuickLaTeX.com A = 3](http://theculture.sg/wp-content/ql-cache/quicklatex.com-ec13f175fa9bbc20978602c914870195_l3.png)
![Rendered by QuickLaTeX.com \Rightarrow B = -9](http://theculture.sg/wp-content/ql-cache/quicklatex.com-eadb1ecee85ea5626aac2f0281981fb6_l3.png)
(b)
LHS
![Rendered by QuickLaTeX.com = r^2(r+1)^2 - (r-1)^2 r^2](http://theculture.sg/wp-content/ql-cache/quicklatex.com-e5b14583d42cad46a5787a66ad735285_l3.png)
![Rendered by QuickLaTeX.com = r^2 ( r^2 + 2r + 1) - r^2 ( r^2 - 2r + 1)](http://theculture.sg/wp-content/ql-cache/quicklatex.com-de21147d551a9ee59213a31ec9894664_l3.png)
![Rendered by QuickLaTeX.com = 4r^3](http://theculture.sg/wp-content/ql-cache/quicklatex.com-63bd455b6b815ac82bfe2a075b3fbf1d_l3.png)
![Rendered by QuickLaTeX.com \therefore k = 4](http://theculture.sg/wp-content/ql-cache/quicklatex.com-01d954eed78b079388217ef0c91b30bb_l3.png)
![Rendered by QuickLaTeX.com \sum_{r=1}^n r^3 = \frac{1}{4} \sum_{r=1}^n [r^2(r+1)^2 - (r-1)^2 r^2]](http://theculture.sg/wp-content/ql-cache/quicklatex.com-0bfe15349c894ae89df2cf064acb11a6_l3.png)
Using method of difference,
![Rendered by QuickLaTeX.com \sum_{r=1}^n r^3 = \frac{1}{4} (n+1)^2 n^2](http://theculture.sg/wp-content/ql-cache/quicklatex.com-73400aff724c1b000d8ed9ed38cc6feb_l3.png)
![Rendered by QuickLaTeX.com \sum_{r=1}^n r^3 = \frac{1}{4} (n^4 + 2n^3 + n^2)](http://theculture.sg/wp-content/ql-cache/quicklatex.com-23195c16636362f86d742b88cf10962d_l3.png)
(c)
Let ![Rendered by QuickLaTeX.com a_r = \frac{x^r}{r!}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-4a4725a3cd3f9217ebe142edb6f489c6_l3.png)
![Rendered by QuickLaTeX.com \frac{a_{n+1}}{a_n} = \frac{\frac{x^{n+1}}{(n+1)!}}{\frac{x^n}{n!}}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-2e0a64441375f3651e30f1adb55a9bb6_l3.png)
![Rendered by QuickLaTeX.com \frac{a_{n+1}}{a_n} = \frac{x}{n+1}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-5dfc46ac70190cbc40d94c0f9840afab_l3.png)
![Rendered by QuickLaTeX.com \lim_{n \to \infty} \frac{1}{n+1} = 0](http://theculture.sg/wp-content/ql-cache/quicklatex.com-8ba11b73373de7832b210e989d3083fd_l3.png)
![Rendered by QuickLaTeX.com \Rightarrow \lim_{n \to \infty} \frac{x}{n+1} = 0 \textless 1](http://theculture.sg/wp-content/ql-cache/quicklatex.com-5a28e6cc2bd7a54fb44b6e56cc6ca15c_l3.png)
Thus, it converges.
From MF26,
![Rendered by QuickLaTeX.com \sum_{r=0}^{\infty} \frac{x^r}{r!} = e^x](http://theculture.sg/wp-content/ql-cache/quicklatex.com-fa86c60e83a6b900f0703c322aea3441_l3.png)
![Rendered by QuickLaTeX.com l_L: \textbf{r} = \lambda \begin{pmatrix}{3}\\{1}\\{-2}\end{pmatrix}, \lambda \in \mathbb{R}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-efeb3491801b74948fb6c43ee73fdd2c_l3.png)
![Rendered by QuickLaTeX.com \vec{PQ} = \begin{pmatrix}{4}\\{5}\\{a+1}\end{pmatrix}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-86f443637ef3de8d7995e0bddd50b0a2_l3.png)
![Rendered by QuickLaTeX.com l_{PQ}: \textbf{r} = \begin{pmatrix}{1}\\{2}\\{-1}\end{pmatrix} + \mu \begin{pmatrix}{4}\\{5}\\{a+1}\end{pmatrix}, \mu \in \mathbb{R}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-f56f33cbec6f4ef1b3db06316e3b0cb3_l3.png)
For them to intersect,
for some
and ![Rendered by QuickLaTeX.com \mu](http://theculture.sg/wp-content/ql-cache/quicklatex.com-243abb230e11149a610dd2033f7db411_l3.png)
—(1)
—(2)
—(3)
Solving, ![Rendered by QuickLaTeX.com \lambda = - \frac{3}{11}, \mu = - \frac{5}{11}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-180a5efe199fd1a1a501312ef806b25c_l3.png)
![Rendered by QuickLaTeX.com \Rightarrow a = -4.4](http://theculture.sg/wp-content/ql-cache/quicklatex.com-a25c876e72a24761c4f567b2281dc48f_l3.png)
(ii)
Let
for some ![Rendered by QuickLaTeX.com \lambda](http://theculture.sg/wp-content/ql-cache/quicklatex.com-ab48baf331239642a00255b86324280a_l3.png)
![Rendered by QuickLaTeX.com \vec{PR} = \lambda \begin{pmatrix}{3}\\{1}\\{-2}\end{pmatrix} - \begin{pmatrix}{1}\\{2}\\{-1}\end{pmatrix}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-f7bdd3806f1fe5b3db7c82cd838c836d_l3.png)
![Rendered by QuickLaTeX.com \vec{QR} = \lambda \begin{pmatrix}{3}\\{1}\\{-2}\end{pmatrix} - \begin{pmatrix}{5}\\{7}\\{-3}\end{pmatrix}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-f53e46fb0c56f6ef8c8845a9d149e052_l3.png)
![Rendered by QuickLaTeX.com \vec{PR} \cdot \vec{QR}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-87a4683271b4f9073e636626124b0d5e_l3.png)
![Rendered by QuickLaTeX.com = \begin{pmatrix}{3 \lambda -1}\\{\lambda -2}\\{22 \lambda +1}\end{pmatrix} \cdot \begin{pmatrix}{3 \lambda - 5}\\{\lambda - 7}\\{22 \lambda +3}\end{pmatrix}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-fed9bf00655f462de8418cacf172932a_l3.png)
![Rendered by QuickLaTeX.com = 14 \lambda ^2 - 35 \lambda + 22](http://theculture.sg/wp-content/ql-cache/quicklatex.com-d392d1f0d340e48cf8b156da1969a8c1_l3.png)
Since discriminant ![Rendered by QuickLaTeX.com = -7 \textless 0](http://theculture.sg/wp-content/ql-cache/quicklatex.com-049f5fe7f6dd8f3bd07bd0103e858895_l3.png)
There are no solutions for ![Rendered by QuickLaTeX.com \vec{PR} \cdot \vec{QR} = 0](http://theculture.sg/wp-content/ql-cache/quicklatex.com-38968eaae4b2e124511cfbac1e6eb7b0_l3.png)
![Rendered by QuickLaTeX.com \Rightarrow \angle PRQ \neq 90^{\circ}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-c945b12dfaadb9b117c8d300d10ae3b9_l3.png)
(iii)
![Rendered by QuickLaTeX.com |\vec{PR}| = \sqrt{(3 \lambda -1)^2 + (\lambda -2)^2 + (-2 \lambda +1)^2}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-050e45fc63caf17b6fa5ecb5b8c39b34_l3.png)
![Rendered by QuickLaTeX.com |\vec{PR}|^2 = 14 \lambda ^2 - 14 \lambda + 6](http://theculture.sg/wp-content/ql-cache/quicklatex.com-d33c315764f5f3cf2bc15bd6a3938053_l3.png)
Let ![Rendered by QuickLaTeX.com |\vec{PR}|^2 = L](http://theculture.sg/wp-content/ql-cache/quicklatex.com-cac4930617a69cde83d253fad14a057b_l3.png)
![Rendered by QuickLaTeX.com \frac{dM}{d \lambda} = 28 \lambda - 14](http://theculture.sg/wp-content/ql-cache/quicklatex.com-8b621f2f10f2b49ded83506854a6ba39_l3.png)
Let ![Rendered by QuickLaTeX.com \frac{dM}{d \lambda} = 0](http://theculture.sg/wp-content/ql-cache/quicklatex.com-8dd24fa4f6270e7f814015e99ec7e243_l3.png)
![Rendered by QuickLaTeX.com \lambda = \frac{1}{2}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-e6c49baf2be241678cc83c17e3a84bc9_l3.png)
![Rendered by QuickLaTeX.com \frac{d^2M}{d \lambda ^2} = 28 \textgreater 0](http://theculture.sg/wp-content/ql-cache/quicklatex.com-e707edcaee2d1d6b4fc22c317895cc2a_l3.png)
gives minimum
.
![Rendered by QuickLaTeX.com \therefore R(1.5, 0.5, -1)](http://theculture.sg/wp-content/ql-cache/quicklatex.com-b2a32800bb81fd95c5ede635b2216bdf_l3.png)
m.
(i)
![Rendered by QuickLaTeX.com \frac{dv}{dt} = c](http://theculture.sg/wp-content/ql-cache/quicklatex.com-6e3535e470b66a1dd66eb60c2211321c_l3.png)
(ii)
![Rendered by QuickLaTeX.com \int 1 ~dv = \int c ~dt](http://theculture.sg/wp-content/ql-cache/quicklatex.com-32ed7309080552ec7919fb104bb84ef8_l3.png)
, where
is an arbitrary constant.
When ![Rendered by QuickLaTeX.com t = 0, v= 4, D=4](http://theculture.sg/wp-content/ql-cache/quicklatex.com-92120fc5e0296008d8d02f0d6d892cee_l3.png)
When
,
![Rendered by QuickLaTeX.com 29 = 2.5c + 4](http://theculture.sg/wp-content/ql-cache/quicklatex.com-e7e943d274cc37136d3d3fbb503d83b4_l3.png)
![Rendered by QuickLaTeX.com c =10](http://theculture.sg/wp-content/ql-cache/quicklatex.com-a66d5c4b951e4899942acdaa1274efff_l3.png)
![Rendered by QuickLaTeX.com \therefore v = 10t + 4](http://theculture.sg/wp-content/ql-cache/quicklatex.com-83878066bfaf2bd84b2db6e8907ea8e2_l3.png)
(iii)
![Rendered by QuickLaTeX.com \frac{dv}{dt} = 10 - kv](http://theculture.sg/wp-content/ql-cache/quicklatex.com-ad644273f60523c228872a4020e52f11_l3.png)
![Rendered by QuickLaTeX.com \int \frac{1}{10-kv} ~ dv = \int 1 ~dt](http://theculture.sg/wp-content/ql-cache/quicklatex.com-d58831554c8ed9c9adf9500621e75c63_l3.png)
, where
is an arbitrary constant.
![Rendered by QuickLaTeX.com 10 - kv = \pm e^{-kE}e^{-kt}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-b075973c735839f6822912b846937c1a_l3.png)
Let ![Rendered by QuickLaTeX.com \pm e^{-kE} = A](http://theculture.sg/wp-content/ql-cache/quicklatex.com-3a52e0f5ae817cc69dc6a99dd155daf2_l3.png)
![Rendered by QuickLaTeX.com \Rightarrow 10 - kv = A e^{-kt}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-5712a2c6eceb96b1262f1c947a657365_l3.png)
When
.
![Rendered by QuickLaTeX.com \Rightarrow v = \frac{1}{k} (10 - 10 e^{-kt})](http://theculture.sg/wp-content/ql-cache/quicklatex.com-9b383beb2a68eca5f1569504b3a5447d_l3.png)
(iv)
As
,
![](https://theculture.sg/wp-content/uploads/2017/11/Screen-Shot-2017-11-09-at-2.46.23-PM.png)
Graph of 11(iv)
Relevant materials
MF26
KS Comments
To be honest, this paper is really the same as the H2 Mathematics (9758). They just rephrased everything. You can see for yourself here.
hello there are no answers 🙁
hi for Q10(ii) would it be sufficient to explain that for both P and Q do not lie on line L hence they are both not points of intersection, PRQ will never be 90degrees?
vectors need not intersect to be perpendicular actually.
Hello, got a quick question! For q6(ii) i remembered my teacher specifically saying r.n = d the ‘d’ is simply a constant with no meaning… so shouldnt displacement of plane from origin be d/(n^)?
Thanks!
yes. but n is given to be a unit vector.. it gives a lot of meaning then
Hi, according to the 9740 papers, may I know what are ur thoughts on the expected range of grade to score A grade? Is 70 marks overall for Ppr 1 & 2 guarantee an A?