MATHS ADVANCED (1 Viewer)

hiiitsme · Nov 12, 2021

Hi This is a thread to post semi-hard/challenging/out of the box/difficult/you get the picture questions. Anything of the sort you have or can think of we could post and then practice before Monday?

No questions that are just straight theory application

jimmysmith560 · Nov 13, 2021

Here is a question:

Evaluate $\int _0^{\frac{\pi }{2}}sin^2x\:dx+\int _0^{\frac{\pi }{2}}cos^2x\:dx$

5uckerberg · Nov 13, 2021

jimmysmith560 said:
Here is a question:

Evaluate $\int _0^{\frac{\pi }{2}}sin^2x\:dx+\int _0^{\frac{\pi }{2}}cos^2x\:dx$

Guys, this has to be a joke right

$\frac{\pi}{2}$

icycledough · Nov 13, 2021

5uckerberg said:
Guys, this has to be a joke right

$\frac{\pi}{2}$

You'd think that people would recognize what it equates to, but you may be surprised at the fact that some people may go about this in a very long and unnecessary manner.

Trebla · Nov 13, 2021

Look at the questions here - provided you know which parts are still relevant to the current syllabus (which would be most of the questions)

BoS Trials Mathematics 2013-2015 Exams and Solutions

BoS Trials Mathematics 2013-2015 Exams and Solutions NOTE: This is a legacy resource migrated from an old version of BoredOfStudies. Original Uploader: Trebla

www.boredofstudies.org

CM_Tutor · Nov 14, 2021

8=D said:
is there one for just advanced questions

Those are just advanced questions, the BoS trials included an advanced / 2u level paper in 2013, 2014, and 2015.

CM_Tutor · Nov 14, 2021

Ok, here are some questions to try, taken from a variety of topics:

BoS Advanced questions - challenge 1.png

There is a typo in the question... it should say that

$F(x) = \int \cfrac{x + 1.5}{x^2 + 3x}\ dx$

for the rest of the question to work.

$\text{(a)} \qquad F(x) = \cfrac{1}{2}\ln{\left|x^2+3x\right|} + C,\ \text{for some constant $C$}$
(c) The integrand has discontinuities at $x=0$ and $x=3$ so the Fundamental Theorem of Calculus, which states that $\int_a^b f(x)\ dx = F(b) - F(a)$ when $F(x)$ is the primitive of $f(x)$ and $f(x)$ is continuous on $[a,\ b]$ , does not apply.

CM_Tutor · Nov 14, 2021

BoS Advanced questions - challenge 2.png

$\text{(a)} \qquad M\ \text{is at}\ \left(-\cfrac{\sqrt{3}}{2},\ \cfrac{3}{2}\right) \qquad \text{and} \qquad N\ \text{is at}\ \left(\cfrac{\sqrt{3}}{2},\ \cfrac{3}{2}\right)$
$\text{(b)} \qquad \cfrac{\pi}{3}$
$\text{(d)} \qquad \left(\cfrac{\pi}{3} - \cfrac{3\sqrt{3}}{8}\right)\ \text{square units}$

CM_Tutor · Nov 14, 2021

BoS Advanced questions - challenge 3.png

$\text{(i)} \quad f(n) = \cfrac{\sqrt{n+1} - \sqrt{n-1}}{2}\ \text{has domain}\ \left\{n \in \mathbb{R} : n \geqslant 1\right\}\ \text{or}\ n \in \big[1,\ \infty\big)$
$\text{(ii)} \qquad f(3) = 1 - \cfrac{\sqrt{2}}{2}$

CM_Tutor · Nov 14, 2021

BoS Advanced questions - challenge 4.png

$\text{(a)} \qquad \cfrac{1}{4}$
$\text{(b)} \qquad \cfrac{9}{40}$
$\text{(c)} \qquad \cfrac{61}{160}$
$\text{(e)} \qquad \cfrac{10}{19}\ \text{and 5 turns}$
$\text{(f)} \qquad 1 - A_n - O_n = \left(\cfrac{21}{40}\right)^n$
This represents the probability of no winner after both Ava and Owen have had $n$ turns.
$\text{(g)} \qquad 33\cfrac{1}{3}\%$
Owen's chance of winning increases from $\cfrac{9}{19}$ if Ava has the first throw of the dice to $\cfrac{12}{19}$ if Owen's turn comes before Ava's.

CM_Tutor · Nov 14, 2021

BoS Advanced questions - challenge 5.png

$\text{(a)} \qquad \text{351 marbles}$
$\text{(b)} \qquad \cfrac{5}{32}$
$\text{(c)(i)} \qquad \cfrac{65}{416} \times \cfrac{64}{415} = \cfrac{2}{83}$
$\text{(c)(ii)} \qquad 1 - \cfrac{351}{416} \times \cfrac{350}{415} = \cfrac{383}{1328}$
$\text{(c)(iii)} \qquad X$ is incorrect as the answer to (c)(i) is determined when both marbles could be red. X has shown that at least one of the marbles is blue, so we seek the conditional probability that both are blue given one is blue, which is:
$P(\text{both blue}\ |\ \text{at least one blue}) = \cfrac{P(\text{both blue})}{P(\text{at least one blue})} = \cfrac{\frac{2}{83}}{\frac{383}{1328}} = \cfrac{32}{383}$

CM_Tutor · Nov 14, 2021

BoS Advanced questions - challenge 6.png

$\text{a} \qquad \theta \approx 53^\circ 8'\ \text{(nearest minute)}$
$\text{b} \qquad \cfrac{5}{2}\sin\theta$

CM_Tutor · Nov 14, 2021

BoS Advanced questions - challenge 7.png

$\text{c} \qquad \text{Ratio of Sides}\ = 3 : 5 : 7$
This ratio of sides is independent of $a$ and $d$ , so all $\Delta ABC$ are similar as they have sides in the same ratio.
Alternatively, they are all equiangular with angles $\cos^{-1}\cfrac{13}{14} \approx 21^\circ 47'$ , $\cos^{-1}\cfrac{11}{14} \approx 38^\circ 13'$ , and $120^\circ$ .

CM_Tutor · Nov 14, 2021

BoS Advanced questions - challenge 8.png

$\text{(a)} \qquad \cfrac{1}{6}$
$\text{(c)} \qquad ABCG,\ ABFG,\ ADCG,\ ADHG,\ AEFG,\ \text{and}\ AEHG.$
$\text{(d)} \qquad \cfrac{11}{27}$
$\text{(e)} \qquad \cfrac{73}{81}$

CM_Tutor · Nov 14, 2021

BoS Advanced questions - challenge 9.png

$\text{(a)} \qquad \cfrac{d}{dx}\bigg(3 + \sin{2x}\bigg) = 2\cos{2x}$
$\text{(b)} \qquad k = \cfrac{\pi}{4}$

CM_Tutor · Nov 14, 2021

BoS Advanced questions - challenge 10.png

No answers provided as all parts are "Show"

CM_Tutor · Nov 14, 2021

azure_ said:
what are the topics we should be skipping/not relevant to the current syllabus?

Commenting just from reading the paper, not checking answers / methods in detail:

2013 paper: I would not worry about Q8, plane geometry is now only considered as prior knowledge. Q11(e) and Q12(a) is no longer in the syllabus. Q13(b) part (i) is possible to ask, though unlikely, and part (ii) is outside the syllabus. Q13(c) is possible but unlikely (as both 13(b) and 13(c) use concepts that are still in the syllabus, but greatly de-emphasised). Q13(d) is possible as a trigonometry exercise but the extent to which trig will be explored in plane geometry contexts is unclear. Q15(a) I wouldn't worry about (plane geometry). Some very nice questions here, but challenging.

2014 paper: Q2 is outside the syllabus, sigma notation is now MX1. Q4 is outside the new syllabus. Q11(d) us outside the syllabus as written, but inside it if the sigma notation was replaced. Q11(e) and 11(g) are outside the new syllabus. Q12(b) and 12(c) are outside the new syllabus. Q13(b) is possible but unlikely as it uses concepts that are still in the syllabus, but greatly de-emphasised. Q15(d) and (f) are outside the new syllabus, though 15(e) is doable if you assume the result from 15(d). I would not worry about Q16(c) as it is plane geometry. Again, some very nice questions here, but seriously challenging in places.

2015 paper: I would not worry about Q2. Q4 is still in the syllabus, even though limits are basically gone otherwise. Q5 uses heavily de-emphasised materials. Q9 is outside the new syllabus. Q11(d) requires the difference of two cubes, which appears rarely. Q12(c)(ii), volumes by revolution, is now only MX1 content. Q12(d)(i) could be asked but (ii) could not... and there is not much point asking (i) on its own. 13(b) is in the de-emphasised materials category. Q13(c) is plane geometry and so unlikely. The rest are ok, though some very difficult.

CM_Tutor · Nov 14, 2021

azure_ said:
thank you so much!

FYI, I've edited to add the 2015 paper as well.

5uckerberg · Nov 14, 2021

CM_Tutor said:
Ok, here are some questions to try, taken from a variety of topics:

View attachment 33695

$\text{(a)} \qquad F(x) = \cfrac{1}{2}\ln{\left|x^2+3x\right|} + C,\ \text{for some constant $C$}$
(c) The integrand has discontinuities at $x=0$ and $x=3$ so the Fundamental Theorem of Calculus, which states that $\int_a^b f(x)\ dx = F(b) - F(a)$ when $F(x)$ is the primitive of $f(x)$ and $f(x)$ is continuous on $[a,\ b]$ , does not apply.

Is the first one just 2 times the natural log of x^2+3x

d1zzyohs · Nov 14, 2021

CM_Tutor said:
View attachment 33704

$\text{c} \qquad \text{Ratio of Sides}\ = 3 : 5 : 7$
This ratio of sides is independent of $a$ and $d$ , so all $\Delta ABC$ are similar as they have sides in the same ratio.
Alternatively, they are all equiangular with angles $\cos^{-1}\cfrac{13}{14} \approx 21^\circ 47'$ , $\cos^{-1}\cfrac{11}{14} \approx 38^\circ 13'$ , and $120^\circ$ .

How does one do part a?

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