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Get HSC Ready in 28 Days | Free HSC Study Planning Kit

Need to get HSC ready in 28 days? In this Guide, we're going to show you how to plan, figure out your priorities, and allocate time.

You’re a month out from your HSC and the reality is starting to sink in – I have to get HSC ready in 28 days! 😱😱😱 Should you panic? No! Instead, you have to get organised and plan for success. To help you, in this guide, we’re going to tell you how to identify your weaknesses, plot a course to success, and action it with a 28-day plan!

Are you ready? Well, buckle up! We’re going to get you HSC ready in 28 days!

How do I get HSC ready in 28 days?

In this article, we’re going to explain to you how to assess strengths and weaknesses to formulate your plan in 5 steps,

With 28 days to go, we don’t have time to muck about. Let’s get planning.

How do you assess your strengths and weaknesses to formulate your plan?

The first thing you’ve got to do is assess where you’re at. This is the hard bit.

You need to honestly reflect upon where you’re at what and what you’re nailing and what you’re not!

However, this is easier said than done. How does one accurately and effectively explain where they’re at?

What you need to do is follow a process! And we’ve broken it down into 5 accessible steps.

How to self-assess, plan and timetable for HSC success in 5 steps!

Step 1: Figure out your aims. University. Degree. ATAR.

By now you know what you want to study and you’ve put in for places at several degrees. But what do you need to achieve to get there?

Now’s the time to double-check the entrance requirements for your degree. Check out the UAC website, and search up your course.

Then, pick the course you’re aiming for and check the ATAR entry requirement for the previous year:

For example, the Bachelor of Advanced Computing/ Bachelor Science (Health) at USYD requires a minimum ATAR of 90.00

Next, you need to use an ATAR Calculator to tell you what marks you need to achieve the ATAR for that course. The Matrix ATAR calculator (powered by Learnable) will tell you what marks you need to achieve the ATAR you for admission to your course.

You need to use this report to focus your efforts. You need to assess what marks you need and where you are at.

But how?

The AIR report from the Matrix ATAR calculator breaks down your subjects to show you the marks you need and the potential scaling (based on the previous years’ ATARs) that those subject might accrue.

Image: AIR Report scaling opportunities

This report will give you an indication of where your efforts are going to be most effective. You can target the subjects that are going to see the most significant scaling for your HSC mark.

The ATAR boost report is especially useful for this as it will show you which of your subjects will receive the most and least amounts of scaling.

For example, English is compulsory and English Advanced receives a scaling bump for higher achievers:

Image: AIR report ATAR Boosts

The ATAR Boost report can help you focus your efforts on subjects that will be rewarded with the higher proportion of scaling:

Image: AIR Report shows which subject will accrue the most scaling. This is an important tool for focusing your efforts

The final part of the AIR Report breaks each subject down by mark and provides actionable advice for how you can improve your marks.

Use the Subject Report to see how to improve your study practices and your marks.

Take the time to study all of this information. You especially want to note down what your current mark is and what the mark you need for the HSC is. The discrepancy between these will help you plan your next steps. Let’s look at how to do that.

Is the HSC creeping up on you?

Revise your years content in 6 Days to get yourself HSC ready! We will go through the content, share exam taking tips and provide you with a wide variety of practice questions for you to do. Learn more about our HSC Exam Prep Course now.

To progress and consolidate, or save, your subjects for the HSC, you need to appraise your current position.

This means rating your subjects for how you’ve succeeded.

First, grab your final school report and see where you sit within your class. What is your rank for each subject?

You want to rank your subjects according to where you sit in the cohort. For the subject where you have the highest internal rank, give it a 1. For the subject with the lowest, give it a 5 or 6 depending on how many subjects you’re taking.

Next, go through the syllabus (you can find all the HSC syllabuses here on the NESA website) for each subject and rate each dot point syllabus depending on your confidence level

• A = Super confident
• B = Barely confident
• C = Terrible at it.

Tally these up and figure out your average confidence for each subject. For example, if for Maths Ext 1 you mostly rated yourself as B confidence and it is your 2nd best subject with a rank of 5/60 you’d rank that as 2B.

Once you’ve ranked yourself for each subject, you need to see what subjects are going to get the biggest return investment on effort. To figure this out, we’re going to use the AIR report’s Boost and subject reports to see which subjects get the largest scaling hit.

What does this look like?

Okay. Let’s see how we can do this:

List your subjects from strongest to weakest – 1-5 or 1-6, etc.

What is their scaling like – good, neutral, poor? Note this, it will help you determine what to focus on. Use your AIR report to determine which subjects have the best cost (hours invested)/benefit (return in marks) outcomes.

Label them in terms of urgency – most urgent (need to make up ground), Least urgent (need to maintain marks).

Let’s look at a case study to see how this works.

Selma wants to study a Bachelor of Engineering (Hons) (Civil) / Bachelor of Surveying at the University of New South Wales (UNSW).

Her current trajectory after HSC Trials sees her at:

 Table: Selma’s current and required marks for the HSC to get an ATAR of 93 Subject Current (pre-HSC) mark Required Mark (according to ATAR Calculator) English Extension 40 44 English Adv 80 88 Maths Ext 1 39 41 Maths Adv 91 89 Chemistry 88 85 Physics 86 84 ATAR Estimate 90.55 93

Selma needs to add an additional 2.5 points on her ATAR. Selma can use the ATAR calculator to reverse engineer what she should be scoring on her weakest subjects by toggling the scores:

The ATAR Calculator is an excellent tool for study tactics and planning!

Selma then reads her AIR report to see what subjects are going to help her find the biggest gains if she focuses her final efforts on them.

The AIR Report shows which subjects will help Selma boost her ATAR!

Selma realises that she needs to improve by 4 marks for English Advanced, 2 for Maths Advanced, 2 marks for Physics, and 1 for Chemistry. As Maths Ext 1 accrues the lowest scaling, she just needs to maintain her score and similarly, she just needs to hold steady with her Ext 1 English mark as the scaling for the other subjects is better.

Selma plots all of this data into a table and also ranks all of her subjects:

 Table: Selma’s current and required marks for the HSC to get an ATAR of 93 Subject Subject Rating Current (pre-HSC) mark Required Mark (after calculating best efficiency) English Ext 1 5 B 40 40 English Adv 4 C 80 84 Maths Ext 1 6 B 39 39 Maths Adv 1 A 91 93 Chemistry 2 B 88 89 Physics 3 C 86 88 ATAR Estimate 90.55 93.25

So Selma has ranked her subjects as:

1. Maths Adv 1A – Needs 2 more marks
2. Chemistry 2B – Requires 1 mark
3. Physics 3C – Requires 2 marks
4. English Adv 4C – Needs 4 more marks
5. English Ext 1 5B – Hold steady
6. Maths Ext 1 6B – Hold steady

Using this process, Selma can now hone in on what subjects are her priorities, in terms of urgency and effort:

1. English Adv 4C – Needs 4 more marks
2. Physics 3C – Requires 2 marks
3. Maths Adv 1A – Needs 2 more marks
4. Chemistry 2B – Requires 1 mark

Selma now has her targets and is ready to timetable her next 30 days to make sure she gets into her degree. It’ll be a bit of hard work, but it’s totes going to be worth it!

Now you want to do what Selma has done, analyse your subjects and marks and identify the importance of your subjects!

Have you done that, now? Good. Let’s allocate your time!

Before you can timetable your life away, you really need to know how much time you have and how much time you can allocate to your efforts. In the 4 weeks leading up the HSC, you have to figure out how many hours a day you can study and then decide how much of that you need to allocate to each subject.

Remember, just because you need to improve some marks doesn’t mean you can neglect the subjects you only need to maintain.

The first thing you want to do is jot down your HSC exam dates in your calendar. Ideally, you want to have a big wall calendar that you can pin above your study space you’ve got a constant reminder of when your exams are.

Next, you need to figure out the available time that you have. This means deciding:

• How many hours can you allocate per day to study?
• How many days a week can you study?

How many hours a day should I study?

To make these decisions, you need to be as realistic as possible. You might think that studying 12-hours a day for a month is an effective plan to start with, but the truth is that’s a terrible idea.

Effective study means working to maximum efficiency and retention. 12 hours a day revising, writing, and doing practice exams isn’t going to help you retain or learn a huge amount of new information.

If you think about the structure of school, you spend 6 hours a day on-campus and probably do 2 – 3 hours of study off-campus. That equates to about 7-8 hours of learning each day. This works for the vast majority of students. Yes, there are outlier who score high ATARs who study less or more, but this is a good benchmark.

After all, 7 – 8 hours of effective study is going to be much more efficient than 12 hours littered with distractions or brain fades where your mind stops retaining information.

How many days a week should I study?

You don’t go to school 7-days a week, so, similarly, you should be cautious about studying every day for 4 weeks. Instead, you should strive to have at least one day off each week. You don’t want to hit the HSC exhausted!

A good aim is to study for 5 full days and maybe do some light revision over the remaining day. If you do have the irrepressible urge to study 7 days a week, make sure that at least two of those days are shorter days with plenty of breaks and R&R.

Considering this, the average Matrix student in STUVAC studies for 5 X 7-8 hour days and does one shorter day of 3-4 hours with one day off.

$$(5 \times 8) + 4 = 44 \ \text{hours a week}$$

Your individual goal might change, but 44 hours a week isn’t an unreasonable target.

Now what you need to do is figure out how much of that 44 hours you can allocate to each subject. But, how do you do this?

Well, you need to take into account:

• How many subjects you are taking
• How much you need to improve in the subject
• Whether the subject is going to accrue scaling, and if so, how much
• How many hours you have each week.

Fortunately, you’ve ranked your subjects already. Now you’ve just got to assign the time to it for each week.

It might seem straightforward to say, “I’ve 10 units, so I’ll just divide my 44 hours by 10 and spend 4.4 hours a week on each unit!” And, this isn’t a terrible idea, but it is not as effective as it could be.

Instead, you want to target your units according to their need and urgency. We’ve already considered need in Step 3, above, but what’s meant by urgency?

Well, while the HSC might be a month away, not every subject is on the same day. Instead, they’re spread over 4 weeks. For example (from the 2020 HSC timetable),

• English Adv kicks the HSC off on Day 1 and then Day 2
• Maths exams are in Week 2
• Physics is in Week 3
• Chemistry is in Week 4
• The last exam is Drama on day 17.

What does this mean? Well, it means you can plan your 28-day study timetable in the lead up to the HSC around the fact that some exams are sooner and others are later.

How do I use this information to weight my study plan? Let’s use Selma as an example again to explain.

Remember, Selma’s situation was:

 Table: Selma’s current and required marks for the HSC to get an ATAR of 93 Subject Subject Rating Current (pre-HSC) mark Required Mark (after calculating best efficiency) English Ext 1 5 B 40 40 English Adv 4 C 80 84 Maths Ext 1 6 B 39 39 Maths Adv 1 A 91 93 Chemistry 2 B 88 89 Physics 3 C 86 88 ATAR Estimate 90.55 93.25

And she’d identified her key subjects as:

1. English Adv 4C – Needs 4 more marks
2. Physics 3C – Requires 2 marks
3. Maths Adv 1A – Needs 2 more marks
4. Chemistry 2B – Requires 1 mark

Notice how she’s not included her extension 1 subjects, this isn’t because she doesn’t or isn’t going to study for them but because she isn’t focusing on them – they’ll get the minimum amount of study time.

If we use the 2020 exam calendar to see when her exams are, we can see that she has the following timetable:

 Table for allocating priority and urgency Subject (in ranked order) Exam date 1. English Adv 4C 20/10/2020 Day 1 of HSC (29 days away) 2. Physics 3C 06/11/2020 Day 14 of HSC (45 days away) 3. Maths Adv 1A 26/10/2020 Day 5 of HSC (34 days away) 4. Chemistry 2B 09/12/2020 Day 15 of HSC (49 days away) 5. English Ext 1 09/10/2020 Day 38 of HSC (49 days away) 6. Maths Ext 1 30/10/2020 Day 15 of HSC (38 days away)

So, what does this tell us?

1. English is the most pressing and important subjects – Selma needs 4 marks and only has a month!
2. While Physics is her 2nd priority, it is at the end of Week 3 – this is her 3rd priority
3. Maths Adv starts Week 2 and is important – this should receive the 2nd largest allocation of time
4. Chemistry is the least pressing of the important subjects – this is her 4th priority
5. Maths Ext 1 is at the end of the 2nd week – this is the 5th priority
6. English Ext 1 is on the same day as Chemistry, it is her last exam!

From this, we can tell that Selma needs to focus on English Advanced and Maths Advanced first and foremost. So, rather than allocating 4.4 hours a week for each unit of these subjects, she should allocate more.

A good trick is to set up a simple spreadsheet to calculate how many hours you can allocate to each subject.

Use spreadsheets to calculate time per subject

If you’re wondering how to weigh how much time you should place to each subject, try adding 1.5 hours for each unit for the most urgent subjects and taking that away from the weakest. Then, for the other units work down 1 hour and then 30 minutes per unit. (It won’t be perfect, but it is a great guide!)

This is the hardest part of setting up your timetable. The last thing to do before you put it all together is to set some study goals.

Let’s see how to do that, now.

Step 4: Set your subject study goals

Study is overwhelming unless you break it down into manageable chunks.

You’re not going to achieve much by sitting down and attempting to do an English Paper 1 and Paper 2 and read a novel in one sitting. On their own, sitting practice papers and rereading English texts are admirable study goals – you all should be aiming to to do that! But trying to CRAM it together, you’ll end up tired an irritable, won’t learn, and probably recall less. Instead, you need to break these kind of goals up over a series of days and weeks.

So, how do I do this?  ¯\_(ツ)_/¯

First, list all the study goals you have each subject in a table. For example, consider the following to do list:

 English Adv Maths Adv Physics Paper 1 practice each week Sketch practice graphs for MA-T3 Motion of projectiles/ Gravitational fields Paper 2 practice each week 2019 HSC paper Past Baulko Trial papers each week Reread CM text MA-C4 practice questions (anti-derivatives) Answer inquiry Questions for Mod 8 4 scaffolding drills for Mod B each week Past Ruse Trial Papers each week 2019 HSC paper

Next, you want to break down each subject into a set of weekly goals.

 Physics weekly goals Week 1 Week 2 Week 3 Week 4 Motion of projectiles Gravitational fields Electromagnetic spectrum Light: Wave model Baulko 2019 Trial Paper Baulko 2020 Trial Paper HSC 2019 Trial Paper Ruse 2019 Trial Paper Mod 8 inquiry q: origin of elements Mod 8 inquiry q: nature of Atom Mod 8 inquiry q:Properties of nucleus Revise Mod 8 Inquiry Qs

Then, when you’re populating your study timetable you can segment those further within your daily goals!

Remember, when planning study goals aim for realistic outcomes, not dreams. You want to achieve your aims, not be disheartened and increasingly stressed because you failed to meet your targets.

Now you’ve got your time allocations and study goals, you’re able to put it all together!

Step 5: Write your 28 day study timetable

Now, we’re ready to write up a study timetable for the next month. First, we have to set some rules for how we allocate things:

1. One day off each week
2. One half-day each week
3. Spread subjects across the week, not just one day!
4. Regular breaks
5. Regular meals
6. Consistent sleep
8. Create a routine, stick to it!

That might seem like a lot of rules, but they’re important. The most important ones are the first and last. At least one day off and,

Create a routine and stick to it!

First you should set up a broad planner that has key daily goals on it for the month. It might look like this:

 Tuesday Wednesday Thursday Friday Saturday Sunday Monday Week 1 Paper 1 Eng Adv Maths Ext 1 2019 Chem 2019 Week 2 Maths Adv 2019 Physics 2019 Paper 2 Eng Adv Week 3 Ruse Maths Ext 1 Trial Baulko Eng Ext 1 Trial Week 4

You want to plan things around an 8 hour day. 7 hours work and 40 minutes for lunch and morning tea. It might help to break your day into 7 50-minute blacks with a 10 minute breather between each subject, or maybe you’ll have shorter breathers and a longer mid-afternoon break. So, you may want to start from a timetable that looks like this:

 Table: An example of how to structure a week’s timetable Week 1: Mon Tues Wed Thurs Fri Sat Sun 9-9:55 Subject: Goal: Subject: Goal: Subject: Goal: Subject: Goal: Subject: Goal: Subject: Goal: R&R 10-10:55 Subject: Goal: Subject: Goal: Subject: Goal: Subject: Goal: Subject: Goal: Subject: Goal: R&R Morning tea 11:20- 12:15 Subject: Goal: Subject: Goal: Subject: Goal: Subject: Goal: Subject: Goal: Subject: Goal: R&R Lunch 1-1:55 Subject: Goal: Subject: Goal: Subject: Goal: Subject: Goal: Subject: Goal: Subject: Goal: R&R 2-2:55 Subject: Goal: Subject: Goal: Subject: Goal: Subject: Goal: Subject: Goal: R&R Afternoon break 3:15-4:10 Subject: Goal: Subject: Goal: Subject: Goal: Subject: Goal: Subject: Goal: R&R 4:15-5:10 Subject: Goal: Subject: Goal: Subject: Goal: Subject: Goal: Subject: Goal: R&R

What next?

Now it’s your turn. Now you know how to plan out the next 4 weeks of your life, you need to do it! If you need some subject specific advice and help, on the next pages we’ve put together some specific advice for Maths, Science, and English and included some sample timetables.

Don’t forget to to download our study pack with planning templates that you can print out and stick to!

Now, we’ll show you how to plan the next 28 days for English so you can ace your HSC.

© Matrix Education and www.matrix.edu.au, 2021. Unauthorised use and/or duplication of this material without express and written permission from this site’s author and/or owner is strictly prohibited. Excerpts and links may be used, provided that full and clear credit is given to Matrix Education and www.matrix.edu.au with appropriate and specific direction to the original content.


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# 2D bifurcation problem

I come across this problem which is about bifurcation. I am trying to take all the cases. I am expecting Hopf bifurcation to occur here but the last case I could not find the fixed point. Could you please help me?

Consider the vector field on plane \begin{align} \dot{x} &= x - xy +1\\ \label{sys1} \dot{y} &= \alpha y + \beta x^2,\nonumber \end{align}

where $$\alpha$$ and $$\beta$$ are parameters. Study the bifurcation of the dynamical system in as much details as possible. You have to study the situation in the neighbourhood of the Bogdonov-Takens bifurcation points.

[Case 1:] When $$\alpha=0$$ and $$\beta =0$$ we have not equilibrium point. [Case 2:] Assume $$\beta=0$$.

\begin{align} \dot{z} &= z - zy +1\\ \label{sys2} \dot{y} &= \alpha y\nonumber \end{align}

When $$\alpha=0$$ we go back to case 1, however when $$\alpha \not = 0$$ we have a equilibrium point $$(-1,0)$$. Shifting the fixed point to the origin by $$z= x-1$$ we have the system

\begin{aligned} \dot{x} &= x+y - x y \\ \dot{y} &= \alpha y \end{aligned}

which has the fixed point $$(0,0)$$. Finding the Jacobian for (\ref{sys3})

$$A_{(0,0)} = \begin{pmatrix} 1&&1\\ 0&& \alpha \end{pmatrix},$$

we can see that the eigenvalues are $$\lambda_1=1$$, and $$\lambda_2=\alpha$$ with the eigenvectors

$$v_1 = \begin{pmatrix} 1\\ 0 \end{pmatrix}, \text{ and } v_2 = \begin{pmatrix} \frac{1}{\alpha -1} \\1 \end{pmatrix}$$ When $$\alpha \geq 0$$ the equilibrium point (0,0) unstable (source) with two dimension unstable manifold, and when $$\alpha <0$$ the equilibrium point $$(0,0)$$ is a hyperbolic unstable (saddle) equilibrium point; and the stable and unstable eigenspaces are;

$$\mathcal{E}^s = \{ (x,y) \in \mathbb{R}^2 | y= \frac{1}{\alpha - 1} x \}, \text{ and } \mathcal{E}^u = \{ (x,y) \in \mathbb{R}^2 | y= 0 \}$$

with one dimension stable manifold and one dimension unstable manifold.

Case 3: Assume $$\alpha=0$$. When $$\beta =0$$ we go to case 1. When $$\beta \not =0$$ we have the system has no fixed points.

Case 4: When $$\alpha \not = 0$$, and $$\beta \not = 0$$

Hint.

Analyzing the intersections of

$$\cases{ x-x y+1 =0\\ \lambda y +x^2=0 }$$

we can foresight the equilibrium points qualification.

For $$\lambda \approx -1$$

For $$\lambda\approx -5$$

Here the tangency point is solved easily as follows

$$\lambda x+x^3+\lambda = (x-x_1)^2(x-x_2)$$

so equating to zero the $$x$$'s powers coefficients

$$\cases{ \lambda+x_1^2x_2 = 0\\ \lambda -x_1^2-2x_1x_2 = 0\\ 2x_1+x_2=0 }$$

we get $$x_1 = -\frac 32, x_2=3, \lambda = -\frac{27}{4}$$

For $$\lambda\approx -35$$

For $$\lambda \approx 1$$

NOTE

For $$\lambda > 0$$ we have one equilibrium point.

For $$-\frac{27}{4}<\lambda< 0$$ we have one equilibrium point.

For $$\lambda < -\frac{27}{4}$$ we have three equilibrium points.

For a given equilibrium point $$(x_0,y_0)$$ the Jacobian is

$$J=\left( \begin{array}{cc} 1-y_0 & -x_0 \\ 2 x_0 & \lambda \\ \end{array} \right)$$

with eigenvalues

$$\frac 12\left(\lambda+1-y_0\pm\sqrt{(y_0-1+\lambda)^2-8x_0^2}\right)$$

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# Understand the problem

Let ABC be a triangle with circumcircle $\Omega$ and let G be the centroid of the triangle ABC. Extend AG, BG, and CG to meet $\Omega$ again at $A_1, B_1$ and $C_1$ respectively. Suppose $\angle BAC = \angle A_1B_1C_1 , \angle ABC = \angle A_1 C_1 B_1$ and $\angle ACB = \angle B_1 A_1 C_1$. Prove that ABC and $A_1B_1C_1$ are equilateral triangles.

##### Source of the problem
Regional Math Olympiad, 2019 Problem 2
Geometry

5/10

##### Suggested Book
Challenges and Thrills in Pre College Mathematics

Do you really need a hint? Try it first!

Draw a diagram. Construction: Complete the hexagon. Can you find some parallelograms in the picture? Particularly, can you prove $BGCA_1$ is a parallelogram?

(We will prove that the marked blue angles are equal) Notice that $\angle ABC = A_1 C_1 B_1$ (given) Also $\angle A_1 C_1 B_1 = \angle A_1 B B_1$ (subtended by the chord $A_1 B_1$ at the circumference. Hence $\angle ABC = \angle A_1 B B_1$ Now substracting $\angle CBB_1$ from both side we have IMPORTANT: $\angle A_1 B C = \angle ABB_1$ Similarly notice that $\angle BAC =\ C_1 B_1 A_1$ (given) Also $\angle C_1 B_1 A_1 = \angle C_1 A A_1$ (subtended by the chord $C_1 A_1$ at the circumference. Hence $\angle BAC = \angle C_1 A A_1$ Now substracting $\angle BAA_1$ from both side we have IMPORTANT: $\angle C_1 A B = \angle A_1AC$ Now $\angle A_1 B C = \angle A_1 A C$  (subtended by the same segment $A_1 C$ at the circumference.) And $\angle C_1 A B = \angle C_1 C B$ ( subtended by the same segment $C_1 B$ ) Thus $\angle C_1 C B = \angle A_1 B C$  Thus alternate angles are equal making $BA_1$ parallel to GC.  Thus in $\Delta A_1MB$ and $\Delta GMC$ we have BM = CM, $\angle BMA_1 = \angle GMC$ and $\angle C_1 C B = \angle A_1 B C$   Hence the triangles are congruent, making $BA_1 = GC$

Since one pair of opposite sides are equal and parallel hence the quadrilateral $BGCA_1$ is a parallelogram.

Similarly, we have $CGAB_1, AGBC_1$ to be parallelogram.  Therefore $GM = MA_1$ (since diagonals of a parallelogram bisect each other).  Since G is the centroid, hence $GM = \frac{1}{2} AG$ implying $MA_1 = \frac{1}{2} AG$ implying G is the midpoint of $AA_1$.
Similarly G is the midpoint of $BB_1 , CC_1$  Can you conclude that G is the center?
If G is not center then let O be the center. Hence OG is perpendicular to both $AA_1, BB_1$ at G as line from center hits the midpoint of a chord of a circle perpendicularly. Thus contradiction.  Now notice that $\angle BGC = 2 \angle A$ (angle at center is twice at the circumference). But $\angle BGC = \angle BA_1 C$ (opposite angles of parallelogram) and $\angle BA_1C = 180 – \angle A$ since $BACA_1$ is cyclic Hence $180 – \angle A = 2 \angle A$ implying $\angle A = 60^o$  Similarly $\angle B, \angle C$ are also 60 degrees.  Hence proved!

# Connected Program at Cheenta

Math Olympiad is the greatest and most challenging academic contest for school students. Brilliant school students from over 100 countries participate in it every year. Cheenta works with small groups of gifted students through an intense training program. It is a deeply personalized journey toward intellectual prowess and technical sophistication.

# Similar Problems

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Protein sequence analysis
3
0
Entering edit mode
7.7 years ago

Hi,

I have a multiple protein sequence which looks like -

>gnl|TC-DB|198412035 1.A.70.1.3 PREDICTED: hypothetical protein [Ciona intestinalis]
>gnl|TC-DB|198415263 1.A.70.1.2 PREDICTED: hypothetical protein [Ciona intestinalis]
MDRKIAFAFLFVILQVTTVSAGWGSLLRVETKAILGTLALRRRTWNENKASQQITPEMEEKLDAEMEKLMQQLAEDQQ
>gnl|TC-DB|210060745 2.A.39.3.6 Chain A, Structure Of Mhp1, A Nucleobase-Cation-Symport-1 Family Transporter
MNSTPIEEARSLLNPSNAPTRYAERSVGPFSLAAIWFAMAIQVAIFIAAGQMTSSFQVWQVIVAIAAGCTIAVILLFFTQ
SAAIRWGINFTVAARMPFGIRGSLIPITLKALLSLFWFGFQTWLGALALDEITRLLTGFTNLPLWIVIFGAIQVVTTFYG
SPAYTLCSTFPRVFTFKTGVIVSAVVGLLMMPWQFAGVLNTFLNLLASALGPLAGIMISDYFLVRRRRISLHDLYRTKGI
>gnl|TC-DB|58177374 1.A.2.2.2 Chain B, Intermediate Gating Structure 2 Of The Inwardly Rectifying K+ Channel Kirbac3.1
MTGGMKPPARKPRILNSDGSSNITRLGLEKRGWLDDHYHDLLTVSWPVFITLITGLYLVTNALFALAYLACGDVIENARP
GSFTDAFFFSVQTMATIGYGKLIPIGPLANTLVTLEALCGMLGLAVAASLIYARFTRPTAGVLFSSRMVISDFEGKPTLM
VLFTGHHEAFAQNVHARHAYSCDEIIWGGHFVDVFTTLPDGRRALDLGKFHEIAQHHHHHH
>gnl|TC-DB|A0CIB0 1.A.17.3.1 Chromosome undetermined scaffold_19, whole genome shotgun sequence OS=Paramecium tetraurelia GN=GSPATT00007662001 PE=4 SV=1
MDDQNQPILQEQPKPKQKKPLLNTKMVKKQKMQNKKEENLREILNFYTNQVDARKFLQKMKAVVDSNQQEKKYQDD
..
..
..

and so on... (almost around 6000 sequences)

I need to do functional annotation of it, how can I do that?

Best!

Shashank

alignment blast • 2.0k views
3
Entering edit mode
7.7 years ago
cdsouthan ★ 1.9k

Start with http://www.ebi.ac.uk/Tools/picr/ FASTA to map to UniProt IDs. Then use the http://www.uniprot.org/ interface to slice and dice (e.g. by GO or anything you like). For GO you might also consider http://www.pantherdb.org/ for additional options (including working between UniProt and PANTHER)

Note this answer is for analysis of function, not annotation (sensu adding this into the records). It also assumes these are "knowns" at least somewhere in UniProt. If these include a proportion of novel sequences you have different problem for which others may suggest solutions

2
Entering edit mode
7.7 years ago
Juke34 7.5k

There is also the interproscan possibility. Look at my previous answer here: Get Associated GO terms to a transcript

You will get a tab delimited output file.

1
Entering edit mode
7.7 years ago

You can also use either the free or the pro version of Blast2Go for doing this task. Having proteins as query, use BlastP into Blat2Go, and you will be ending with a very convenient table containing the Blast hits, the E and cover numbers, the GO, EC Number, InterPro domains and even the KEGG pathways associated to those proteins having similarity to your proteins. Then you can get statistics, doing Fisher enrichment tests, and even getting graphics of your searches in a very easy and convenient way

The Pro version will allow you to work faster because it is using cloud and privileged facilities, and also will allow you to work with different algorithms of your BlastP, not just the default one

0
Entering edit mode

Thank you.

I think Blast2Go works for me.


text:
Refer to our Texas Go Math Grade 8 Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Grade 8 Module 4 Quiz Answer Key.

4.1 Representing Linear Nonproportional Relationships

Question 1.
Complete the table using the equation y = 3x + 2.

In the given equation y = 3x + 2 include the given values of x.
x1 = -1 → y = 3 • (-1) + 2 = -3 + 2 → y = -1
x2 = 0 → y = 3 • 0 + 2 = 0 + 2 → y1 = 2
x3 = 1 → y = 3 • 1 + 2 = 3 + 2 → y1 = 5
x4 = 2 → y = 3 • 2 + 2 = 6 + 2 → y1 = 8
x5 = 3 → y = 3 • 3 + 2 = 9 + 2 → y1 = 11
Obtained values of y put in the tabla

4.2 Determining Slope and y-intercept

Question 2.
Find the slope and y-intercept of the line in the graph.

The following equation for finding the slope and y-intercept is y = mx + b
The slope presents the ‘m’ in the equation, and the y-intercept represents b
The slope is calculated by:
m = $$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$$
From the given graph, find two points that will help you calculate the slope.
I’ll take (0, 1) for (x1, y1) and (1, 4) for (x2, y2).
Thus: m = $$\frac{4-1}{1-0}$$ = $$\frac{43}{1}$$ = 3
To find the y-intercept, we need to include values of x and y from one of these two points, and include the obtained slope. So:
y = mx + b
1 = 3 . 0 + b
1 = 0 + b
b = 1
m = 3, b = 1

4.3 Graphing Linear Nonproportional Relationships

Question 3.
Graph the equation y = 2x – 3 using slope and y-intercept.

y = 2x – 3
Slope = 2
y – intercept = -3

Plot the point that contains the y-intercept:
(0, -3)
The slope is m = $$\frac{2}{1}$$
Use the slope to find a second point.
From (0, -3) count $2$ unit up and $1$ unit right. The new point is (1, -1)
Draw a line through the points

4.4 Proportional and Nonproportional Situations

Question 4.
Does the table represent a proportional or a nonproportional linear relationship?

To see if some linear relationship is proportional or nonproportional, we have to use ratio $$\frac{y}{x}$$, which needs to be constant for proportionality.

From the given table, we have values for x and y, so just put them into the calculation.
$$\frac{y}{x}$$ = $$\frac{4}{1}$$ = 4
= $$\frac{8}{2}$$ = 4
= $$\frac{12}{3}$$ = 4
= $$\frac{16}{4}$$ = 4
= $$\frac{20}{5}$$ = 4
Thus, the obtained values are constant so this table represents a proportional relationship.

4.5 Solving Systems of Linear Equations by Graphing

Question 5.
A school band ordered hats for $3 and large T-shirts for$5. They bought 150 items in all for \$590. Graph a system of equations to find how many hats and T-shirts the band ordered.
____________
The graph represents the nonproportional relationship because the obtained b, ie y-intercept, is bigger than 0 (b /
= 0) and the line does not pass the origin. When the b = 0 and the line pass the origin, the linear relationship is proportional.

Essential Question

Question 6.
How can you identify a linear nonproportional relationship from a table, a graph, and an equation?
The graph represents the nonproportional reLationship because the b, i.e. y-intercept from the given equation, is
bigger than 0 (b ≠ 0) and the line does not pass the origin. When the b = 0 and the line pass the origin, the linear relationship is proportional.

Texas Go Math Grade 8 Module 4 Mixed Review Texas Test Prep Answer Key

Selected Response

Question 1.
The table below represents which equation?

(A) y = -x – 10
(B) y = -6x
(C) y = 4x – 6
(D) y = -4x + 2
(C) y = 4x – 6

Explanation:
The table is represented by Option C From the table, you can see that the y-intercept (when x = 0) is b = -6. Comparable to y = mx + b
y = 4x – 6

Question 2.
The graph of which equation is shown below?

(A) y = -2x + 3
(B) y = -2x + 1.5
(C) y = 2x + 3
(D) y = 2x + 1.5
(A) y = -2x + 3

Explanation:
Option C is rejected
Since the graph is slanting downwards, the slope is negative.
The graph represents
y = -2x + 3

Question 3.
The table below represents a linear relationship.

What is the y-intercept?
(A) -4
(B) -2
(C) 2
(D) 3

The y-intercept is -2.
Option B is the correct answer.

Question 4.
Which equation represents a nonproportional relationship?
(A) y = 3x + 0
(B) y = -3x
(C) y = 3x + 5
(D) y = $$\frac{1}{3}$$x
(C) y = 3x + 5

Explanation:
Option C represents a non-proportional relationship
y = 3x + 5

For a non-proportional relationship, the equation is y = mx + b and b ≠ 0.

Question 5.
Which statement describes the solution of a system of linear equations for two lines with the same slope and the same y-intercept?
(A) one nonzero solution
(B) infinitely many solutions
(C) no solution
(D) solution of 0


text:
# Volume of a Solid of Revolution

## Homework Statement

Find the volume generated by rotating the area bounded by $y^2 = 8x, x = 2$ and the x-axis about the y-axis.

## Homework Equations

Volume of cylinder, volume of disk.

## The Attempt at a Solution

I think this can be solved by subtracting the 'empty' volume from the volume of a cylinder of radius 2:

$$\pi r^2h - \int_0^4 \frac{\pi y^2}{8}\,dy$$

I get an answer of $\pi 40/3$ but the solution provided is $128/5\pi$.

I'm sure my arithmetic is not that far off :-)

What did I do wrong?

Thanks,
Sheldon

## Homework Statement

Find the volume generated by rotating the area bounded by $y^2 = 8x, x = 2$ and the x-axis about the y-axis.

## Homework Equations

Volume of cylinder, volume of disk.

## The Attempt at a Solution

I think this can be solved by subtracting the 'empty' volume from the volume of a cylinder of radius 2:

$$\pi r^2h - \int_0^4 \frac{\pi y^2}{8}\,dy$$

I get an answer of $\pi 40/3$ but the solution provided is $128/5\pi$.

I'm sure my arithmetic is not that far off :-)

What did I do wrong?

Thanks,
Sheldon

I see an error, but I still don't get the provided answer. I believe the following is correct:

$$\pi r^2h - \int_0^4 \frac{\pi y^4}{64}\,dy$$

However I get the answer: $64\pi/5$

Thanks!
Sheldon

HallsofIvy
Homework Helper
I see an error, but I still don't get the provided answer. I believe the following is correct:

$$\pi r^2h - \int_0^4 \frac{\pi y^4}{64}\,dy$$

However I get the answer: $64\pi/5$

Thanks!
Sheldon
That integral is the correct one. I don't see how you could get $64\pi/5$ out of it! Show exactly how you integrated.

That integral is the correct one. I don't see how you could get $64\pi/5$ out of it! Show exactly how you integrated.

When x = 2, y = 4, so that's the height of the cylinder:

$$\pi(2^2)4 - \int_0^4 \frac{\pi y^4}{64}\,dy = 16\pi -\pi\left[\frac{y^5}{(5)(64)}\right]_0^4$$

$$16\pi - \frac{16\pi}{5} = \frac{80\pi}{5} - \frac{16\pi}{5} = \frac{64\pi}{5}$$

Am I making some dumb mistake here?

Thanks for your help!
Sheldon

Your integration is actually perfect.

Just notice that your answer is 1/2 the book's answer. That should lead you in the right direction.

Your integration is actually perfect.

Just notice that your answer is 1/2 the book's answer. That should lead you in the right direction.

Hi Wolf,

I noticed that, but I don't understand it. I am starting to wonder if the book answer is wrong, but this is a new topic, and I am not feeling very confident.

If I multiply by two (or integrate from -4 to 4), then I get the book answer. But the problem states that the area is bounded by the X-axis. Multiplying by 2 would give the volume for the object both above and below the X-axis. Isn't that right?

Thanks for your help,
Sheldon


text:
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## If x â‰  0, what is the value of (x^w/x^y)^4?

tagged by: VJesus12

This topic has 1 expert reply and 1 member reply

### Top Member

VJesus12 Master | Next Rank: 500 Posts
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#### If x â‰  0, what is the value of (x^w/x^y)^4?

Thu Mar 01, 2018 5:32 am
If x â‰  0, what is the value of (x^w/x^y)^4?

(1) w = y

(2) x^2 = 1

The OA is D.

Experts, may you help me here? I need some help. I don't know how to prove that each statement alone is sufficient.

### GMAT/MBA Expert

Jay@ManhattanReview GMAT Instructor
Joined
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Posted:
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Wed Mar 07, 2018 9:42 pm
VJesus12 wrote:
If x â‰  0, what is the value of (x^w/x^y)^4?

(1) w = y

(2) x^2 = 1

The OA is D.

Experts, may you help me here? I need some help. I don't know how to prove that each statement alone is sufficient.
Given: x â‰  0

We have to get the value of (x^w/x^y)^4.

Let's take each statement one by one.

(1) w = y

(x^w/x^y)^4 => (x^w/x^w)^4; replacing the value of w as y.

(x^w/x^w)^4 = (1/1)^4 = 1. Sufficient.

(2) x^2 = 1

=> x = Â±1

Case 1: Say x = 1, then (x^w/x^y)^4 = (1^w/1^y)^4 = 1/1)^4 = 1.
Case 2: Say x = -1, then (x^w/x^y)^4 = ((-1)^w/(-1)^y)^4 = [Â±(1/1)]^4 = Â±1^4 = 1. Sufficient.

Note that -1 raised to even integer is positive.

The correct answer: D

Hope this helps!

-Jay
_________________
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### Top Member

Vincen Legendary Member
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Sat Mar 03, 2018 2:19 am
VJesus12 wrote:
If x â‰  0, what is the value of (x^w/x^y)^4?

(1) w = y

(2) x^2 = 1

The OA is D.

Experts, may you help me here? I need some help. I don't know how to prove that each statement alone is sufficient.
Hello.

I think I can help you. We need to find the value of $$\left(\frac{x^w}{x^y}\right)^4$$ but this expression is equal to $$\left(x^{w-y}\right)^4=x^{4w-4y}$$ Now:

If w=y then 4w-4y=0 and then $$x^{4w-4y}=x^0=1.$$ (1) is sufficient.

On the other hand, if $$x^2=1\ \leftrightarrow\ \ \ x=\pm1$$ Now, no matter the sign of x, we have that $$\frac{x^w}{x^y}=\pm1\ \leftrightarrow\ \ \ \left(\pm1\right)^4=1$$ and therefore (2) is sufficient.

The correct answer is D .

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text:
# How to fit an object of constant size based on measurements to known points

I'm looking for a mathematical solution for solving where the base of a camera crane (ie a constant square or rectangle of known dimensions) is with measurements to known points. This seems to be a relatively simple geometry problem but I'm having a tough time with it... The workflow is as follows:

1-In 3d software, I have very accurate positions of points in the environment. 2-The crane base moves, and I need to make my computer crane in the identical position. 3-I go to each corner, and use a laser measuring tool to get the distance from that corner to a known survey point. 4-Currently, I'm visualizing the measurements with circles and by hand moving/rotating the box until each corner touches part of the circle, hence the only solution for where those four corners could be based on the measurements....

I've been researching Euclidean plane movements and vector algebra, but any push in the right direction would be much appreciated as this seems like it should be fairly straightforward....The attached image perhaps does a better job of explaining than I do....Oops nevermind I need a 10 reputation or something to post it....Thanks! Casey

-
Do your fixed survey points and your movable rectangle all lie in a plane? Or, saying it another way, is this a 2D problem or a 3D problem? –  bubba Mar 29 '13 at 4:45
Hi bubba thanks for responding ... The survey points are very much 3d as they represent the environment and have lots of height variation...However the crane/rectangle will always be on the floor, hence taking one of the three axis out of the problem... It's as if 4 strings were each attached to one corner of a square and then all 4 were pulled taught. We know how long the strings are, where the points holding them are, and the size of the square. –  casey Mar 29 '13 at 5:41
My guess is that this problem usually doesn't have a solution. If you had three survey points, you'd be OK. Having 4 of them over-constrains the problem. Here's a way to think about it: –  bubba Mar 29 '13 at 10:07
Ignore the previous comment. Typing before thinking. Here's a way to think about the problem: at each survey point, you have a distance, r (the length of the attached "string"). Construct a sphere of radius r centered at the survey point. You know that one corner of the plate has to lie on this sphere. Similarly, the other three corners of the plate have to lie on three other spheres. If you had three survey points and the plate was infinitely small, this would just be the standard triangulation problem, which can be solved just by intersecting 3 spheres. Your problem is a lot harder, though. –  bubba Mar 29 '13 at 10:15
I'm still not sure I understand the role of the "floor". But anyway ... suppose you intersect the four spheres with the "floor" plane. You get four circles. The corners of the plate have to lie on these 4 circles. That problem seems over-constrained -- it usually won't have a solution. –  bubba Mar 29 '13 at 10:20

Following the remarks above, let's assume that we have only three survey points, instead of four. At each survey point, we construct a sphere, using the known "string length" distance as the radius. We intersect the three spheres with the "floor" plane to get three circles. Let's say that the $i$-th circle has center point $C_i = (x_i, y_i)$ and radius $r_i$ for $i=1,2,3$.

We can represent the location of the square base plate by a corner point $P_1 = (x,y)$ and an orientation angle $\theta$. Here, $x$, $y$ and $\theta$ are the unknowns that we need to calculate. Suppose the plate has side length $k$. Then the corners of the plate adjacent to $P_1$ are $$P_2 = (x + k\cos\theta, y + k\sin\theta)$$ $$P_3 = (x - k\sin\theta, y + k\cos\theta)$$

After some re-ordering, if neccessary, we can express the known constraints by the equations $$\text{dist}(P_i, C_i) = r_i \quad \text{for } i=1,2,3$$. These equations just say that the point $P_i$ lies on the circle centered at $C_i$. We can then write these equations out in full, in terms of the unknowns $x$, $y$ and $\theta$. We get: $$(x - x_1)^2 + (y - y_1)^2 = r_1^2$$ $$(x + k\cos\theta - x_2)^2 + (y + k\sin\theta - y_2)^2 = r_2^2$$ $$(x - k\sin\theta - x_3)^2 + (y + k\cos\theta - y_3)^2 = r_3^2$$ These three equations can be solved to get $x$, $y$ and $\theta$. This is easier said than done, though. How to do it ... I don't see any clever algebraic approach, but maybe someone else can suggest one, or maybe a computer algebra system like Mathematica or Maple could find one. Alternatively, if you have a good CAD system that has a "sketch constraint solver", you can probably get it to solve this problem for you. If neither of those approaches work, then you'd need to use a numerical equation solver.

Here's a solution produced by using the constraint solving capabilities of a CAD system:

And here's another solution (with different survey points) that just uses plain distance constraints instead of circles:

-
Bubba this is great.. I am using a software called Maya that is largely used in the VFX industry but is very capable of CAD style work . It also has a programming language similar to C. I'll code this on Monday and post the results if all turns out well. Thanks! –  casey Mar 30 '13 at 3:47
I don't think Maya has any constraint solving capability, but you can probably drag the plate around until it looks about right. I used a big-time CAD system that does support constraint solving to make the picture shown. –  bubba Mar 30 '13 at 6:15


text:
# Inverse Square Law for light

## Main Question or Discussion Point

Does the inverse square law work for electromagnetic radiation?

It should only work if we strictly looks at electromagnetic radiation as 'waves'. But the photon particle isn't a wave, so how would you explain how it works with the photon? I may have misconceptet the photon, but I think that a photon is a particle.

Related Other Physics Topics News on Phys.org
Hootenanny
Staff Emeritus
Gold Member
jtbell
Mentor
If you're analyzing light as a classical electromagnetic wave, the inverse square law describes the energy flux density: the energy carried by the wave per unit area that it passes through, per second.

If you're analyzing light as a collection of many many photons, the inverse square law describes the photon flux density: the number of photons that pass through a unit area, per second.

If you're analyzing (very weak) light as a collection of a few photons, the inverse square law describes a probability density: the probability that a photon passes through a unit area, per second.

I understand, so if we are analalyzing light as a collcetion of many photons, will the amount of photons that are 'shot' out be equal in every direction?
ehm,
Is there be a randomly selected 'amount' of photons that are shot in each way, just that it seems like they are equal in every direction?

I got a bit of problem of describing my question, I hope you understand. Because if photons are not stricly waves I wonder how they works...

And if we go at an extremely long distance away from the emitting atom, where the probability is 0.1 per square meter. Would this mean that there is 1 photon per tenth meter at average?

DaveC426913
Gold Member
I understand, so if we are analalyzing light as a collcetion of many photons, will the amount of photons that are 'shot' out be equal in every direction?
ehm,
Is there be a randomly selected 'amount' of photons that are shot in each way, just that it seems like they are equal in every direction?

I got a bit of problem of describing my question, I hope you understand. Because if photons are not stricly waves I wonder how they works...

And if we go at an extremely long distance away from the emitting atom, where the probability is 0.1 per square meter. Would this mean that there is 1 photon per tenth meter at average?
You can dim down a light to the point where you are effectively counting single photons. This is how we look at quantum effects in such things as the double slit experiment.

Yes, the formula will represent the probability or frequency of photons at any given point.

(BTW, your numbers are wrong. You divided when you shoulod have multiplied. A probability of 0.1 per square meter works out to one photon for every ten square meters, not tenth of a square metre.)

Yeah, I MEANT every ten square meter, I'm just not familiar with the english definition of that..

By the way, does this mean that photons are particles? In my science book it says that it has wave and particle properties, but they went with the wave model... What is the correct one? I understand that this won't have a single answer, but I want to know what makes the scientists uncertain of this.

jtbell
Mentor
Photons are neither classical waves nor classical particles. They are objects that are fully described only by quantum field theory. They have both wavelike characteristics and particle-like characteristics.

jtbell
Mentor
I understand, so if we are analalyzing light as a collcetion of many photons, will the amount of photons that are 'shot' out be equal in every direction`?
ehm,
Is there be a randomly selected 'amount' of photons that are shot in each way, just that it seems like they are equal in every direction?
To make an analogy, it's like when you toss a coin multiple times (or multiple coins at once). You expect equal numbers to come up "heads" and "tails". However, with only a few tosses (or coins), it's very possible for them all to come up "heads." As the number of tosses (or coins) becomes larger, the percentages of "heads" and "tails" both approach 50%.

Similarly, with a light source that radiates "uniformly" in all directions, if it's very very weak, it's possible for the photons to come out more or less bunched in one direction. But if you have bazillions and bazillions of photons, as is the case with everyday light sources, you get very nearly the same number in all directions.

Last edited:
Oh, ok. Do you have a link to the explanation?

DaveC426913
Gold Member
Just Google or Wiki "wave-particle duality of light".

kk, thanks

I1 x d12=I2 x d22
where:
I=intensity d2=distance per sq. unit

Hope this works for you...that is, if the thread is still active.

jtbell
Mentor
that is, if the thread is still active.
After nearly three years, well, uh... :uhh:


text:
# Projectile motion ball hit

#### MrWarlock616

1. Homework Statement

These are the exact words:

A player is ready to throw a ball with max. possible velocity of 15 m/s from a height of 1.5 m above the ground. He wants to hit the wall 16 m away from him at its highest point. If the ceiling height of the room is 8 m, find the angle of projection of the ball and the height of the point where the ball hits the wall.

2. Homework Equations

v=u+at
s=ut+0.5t^2
v^2=u^2+2as

3. The Attempt at a Solution

The attached file is the diagram.

To find α, I did this:

$15\cos(\alpha)=\frac{16}{t}$

$0=15\sin(\alpha)-9.81t$

$\therefore 15\sin(\alpha)=\frac{9.81\times16}{15\cos(\alpha)}$

$\therefore 15^2\sin(\alpha)\cos(\alpha)=9.8\times16$

$\therefore \sin(2\alpha)=\frac{2\times9.81\times16}{15^2}$

$\therefore \sin(2\alpha)=1.3952 > 1$ <-- WHAT

Did I do something wrong?

#### Attachments

• 19.5 KB Views: 233
Related Introductory Physics Homework Help News on Phys.org

#### frogjg2003

This is much easier if you don't plug numbers in first.

The reason you're getting the impossible answer is because you're assuming the peak of the curve is at the wall. You were asked to find the highest point the player could hit. You have to write an equation for the height of the ball as a function of angle and initial velocity, then find where it is highest.

#### NasuSama

You need to split into two equations - horizontal and vertical. This way, you are able to solve this problem easily [takes some steps to solve the problem]

First thing first.

You need to determine the vertical component and the horizontal component of the path the ball travels, using the first formula you just indicated.

Which component of the path does gravity take place?
Is there acceleration in horizontal, vertical or both?

#### MrWarlock616

You need to split into two equations - horizontal and vertical. This way, you are able to solve this problem easily [takes some steps to solve the problem]

First thing first.

You need to determine the vertical component and the horizontal component of the path the ball travels, using the first formula you just indicated.

Which component of the path does gravity take place?
Is there acceleration in horizontal, vertical or both?
Gravity is in the vertical component of velocity, and acceleration is ZERO in the horizontal motion.

This is much easier if you don't plug numbers in first.

The reason you're getting the impossible answer is because you're assuming the peak of the curve is at the wall. You were asked to find the highest point the player could hit. You have to write an equation for the height of the ball as a function of angle and initial velocity, then find where it is highest.
Actually I was absent when my teacher solved this. I asked my friend and he did it using v^2=u^2+2as like this:

$0^2=15^2(\sin(\alpha))^2-2\times9.81\times6.5$
$\therefore \alpha=48.84$ which is the correct answer.
Why did they take s=6.5 m?

#### NasuSama

Good. You said that there is no acceleration in the horizontal and gravity for the vertical component.

Then, set up the expression, using this form...

s = s_0 + vt + 0.5at^2 [where s_0 is your initial height]

You get two equations - horizontal and vertical.

#### MrWarlock616

Good. You said that there is no acceleration in the horizontal and gravity for the vertical component.

Then, set up the expression, using this form...

s = s_0 + vt + 0.5at^2 [where s_0 is your initial height]

You get two equations - horizontal and vertical.
Oh yeah I got it. frogjg was right, my assumption was wrong. I used s=ut+0.5at^2 and I got it!! Thank you!!

#### NasuSama

Oh yeah I got it. frogjg was right, my assumption was wrong. I used s=ut+0.5at^2 and I got it!! Thank you!!
v_f² = v_i² + 2as makes no use of what you are trying to show here.

#### MrWarlock616

v_f² = v_i² + 2as makes no use of what you are trying to show here.
But my teacher used that and got the correct angle xD

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text:
Choosing point uniformly on surface of sphere

I saw the following statement posted by Fermat's Library on Twitter:

The way to correctly generate a random point on the surface of a unit sphere is not to pick uniform distributions $$\theta$$ in $$[0,2\pi)$$ and $$\phi$$ in $$[0,\pi)$$.

Instead, choose $$u$$ and $$v$$ from uniform distributions on $$[0,1)$$. Then,

• $$\phi = \cos^{-1}(2v-1)$$
• $$\theta = 2\pi u$$

I can see why uniformly choosing $$\theta$$ in $$[0,2\pi)$$ and $$\phi$$ in $$[0,\pi)$$ would lead to points near the poles being more likely to be chosen than points near the equator.

But I can't think of an intuitive explanation for why the given solution correctly gives a uniform distribution. Is there a good intuitive explanation for this?

• it's the famous theorem of Archimedes saying that the projection of a sphere to a tangent cylinder preserves the area (a small piece of the sphere gets stretched in one direction and squeezed in the orthogonal direction, by the same amount). Dec 19, 2020 at 7:26

The length of the ring at latitude $$\phi$$ is proportional to $$\cos\phi$$; this is also proportional to the pdf of the random variable that is the latitude of a point uniformly distributed on a sphere.

The integral of the pdf is the cdf; since the pdf expression contains $$\cos\phi$$ its integral should also be a sine wave. If we use the angular difference from a pole as variable, this wave should go from $$0$$ at $$0$$ to $$1$$ at $$\pi$$.

The inverse of the cdf is the quantile function, which is what we really want. Since the cdf was a sine wave, the quantile function should be the $$\cos^{-1}$$ or $$\sin^{-1}$$ of something; looking at the required values ($$0$$ at $$0$$, $$\pi$$ at $$1$$) we infer that the quantile function is $$\cos^{-1}(1-2x)$$. To generate the latitude for a uniform random point on the sphere, $$x$$ should be a uniform $$U(0,1)$$ variate; it is easy to see that both $$1-2x$$ and $$2x-1$$ are uniformly distributed over $$[-1,1]$$, so we have $$\phi=\cos^{-1}(2x-1)$$.


text:
### 12th Standard CBSE Maths Study material & Free Online Practice Tests - View and download Sample Question Papers with Solutions for Class 12 Session 2019 - 2020 CBSE

#### 12th CBSE Mathematics Differential Equations Model Question Paper - by Shalini Sharma - Udaipur Aug 12, 2019 - View & Download

• 1)

Write the degree of the differential equation: $5x{ \left( \frac { dy }{ dx } \right) }^{ 2 }-\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } =0.$

• 2)

Write the sum of the order and degree of the differential equation $1+\left( \frac { dy }{ dx } \right) ^{ 4 }=7\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) ^{ 3 }$

• 3)

Write the degree of the differential equation ${ x }^{ 3 }\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) ^{ 2 }+x\left( \frac { dy }{ dx } \right) ^{ 4 }=0$

• 4)

Find the differential equation representing the family of curves $V=\frac { A }{ r } +B$ , where A and B arbitrary constants.

• 5)

Find the integrating factor of the differential equation $\left( \frac { { e }^{ -2\sqrt { x } } }{ \sqrt { x } } -\frac { y }{ \sqrt { x } } \right) \frac { dx }{ dy } =1$ .

#### 12th Standard CBSE Mathematics Unit 8 Application of Integrals Model Question Paper - by Shalini Sharma - Udaipur Aug 05, 2019 - View & Download

• 1)

Find the area of the region by the curve $y=\frac { 1 }{ x }$ , X-axis and between X = 1, X = 4.

• 2)

Find the area of the region bounded by the curve y2 = x and the lines x = 1 , x = 4 and the x -  axis.

• 3)

The area between x = y2 and x = 4 is divided into two equal parts by the line x = a, find the value of a.

• 4)

Using integration, find the area of the triangular region whose sides have the equations:
y = 2x + 1, y = 3x + 1and x = 4.

• 5)

Smaller area enclosed by the circle x2 + y2 = 4 and the line x + y = 2 is :
(A) $2(\pi -2)$
(B) $\pi -2$
(C) $2\pi -1$
(D) $2(\pi +2)$

#### 12th Standard CBSE Mathematics Unit 7 Integrals Model Question Paper - by Shalini Sharma - Udaipur Aug 03, 2019 - View & Download

• 1)

Evaluate the integral: $\int { {x^2\over1+x^3}dx. }$

• 2)

Evaluate the integral: $\int {(ax\ +\ b)^3}dx$

• 3)

$\int cos^2x\ cosec^2x\ dx.$

• 4)

$\int {x\over \sqrt {x+2}}dx.$

• 5)

Given $\int { { e }^{ x }(tanx+1)secxdx={ e }^{ x }f(x)+c }$. Write f(x) satisfying the above.

#### 12th CBSE Mathematics Unit 6 Application of Derivatives Model Question Paper - by Shalini Sharma - Udaipur Jul 31, 2019 - View & Download

• 1)

The amount of pollution content added in air in a city due to X diesel vehicles is given by P(x)=0.005x3+0.02x2 +30x.Find the marginal increase   in pollution content when 3 diesel vehicles are added and write which value is indicated in the above question?

• 2)

For what value of m,The function f(x)=mx+c,is decreasing for x $\epsilon$ R.

• 3)

Show that f(x)=(x-1) ex+1 is an increasing function for x > 0.

• 4)

For the function y=x3, if x=5 and $\Delta$x=0.01,find $\Delta$y

• 5)

f(x)=9x2+12x+2

#### 12th CBSE Mathematics Unit 5 Continuity and Differentiability Important Question Paper - by Shalini Sharma - Udaipur Jul 29, 2019 - View & Download

• 1)

Examine the continuity of the function f (x)=$\frac { 1 }{ x+3 } ,\quad x\quad \varepsilon \quad R$.

• 2)

Give an example of a function which is continuous at x=1, but not differentiable at x=1.

• 3)

State the points of discountinuity for the function $f(x)= [x]$  in $-3 < x < 3.$

• 4)

If y= sec-1 $\left( \frac { \sqrt { x } +1 }{ \sqrt { x } -1 } \right) +\quad sin^{ -1 }\left( \frac { \sqrt { x } -1 }{ \sqrt { x } +1 } \right) ,\quad find\frac { dy }{ dx } .$

• 5)

Differentiate the following w.r.t. x, or find $\frac { dy }{ dx }$.

$y={ e }^{ x }+{ e }^{ { x }^{ 2 } }+{ e }^{ { x }^{ 3 } }+{ e }^{ { x }^{ 4 } }+{ e }^{ { x }^{ 5 } }.$

#### 12th CBSE Mathematics Unit 4 Determinants Important Question Paper - by Shalini Sharma - Udaipur Jul 27, 2019 - View & Download

• 1)

If A is a square matrix of order 3 and |3A|=k|A|, then write the value of k.

• 2)

What positive value of x makes the following pair of determinants equal?

$\begin{vmatrix} 2x & 3 \\ 5 & x \end{vmatrix},\begin{vmatrix} 16 & 3 \\ 5 & 2 \end{vmatrix}$

• 3)

Write the adjoint of the following matrix $\begin{bmatrix} 2 & -1 \\ 4 & 3 \end{bmatrix}$

• 4)

Evaluate $\begin{vmatrix} cos15^{ o } & sin15^{ o } \\ sin75^{ o } & cos75^{ o } \end{vmatrix}$

• 5)

Given determinant $\begin{vmatrix} a_{ 11 } & a_{ 12 } & a_{ 13 } \\ a_{ 21 } & a_{ 22 } & a_{ 23 } \\ a_{ 31 } & a_{ 32 } & a_{ 33 } \end{vmatrix}$.Find the value of a11A21+a12A22+a13A23, where Aij is cofactor of element aij

#### 12th Standard CBSE Mathematics Unit 3 Matrices Important Question Paper - by Shalini Sharma - Udaipur Jul 26, 2019 - View & Download

• 1)

If $\left[ \begin{matrix} x & +3y & y \\ 7 & -x & 4 \end{matrix} \right]$=$\begin{bmatrix} 4 & -1 \\ 0 & 4 \end{bmatrix}$, find the values of x and y.

• 2)

If matrix A=$[\begin{matrix} 1 & 2 & 3 \end{matrix}]$ write AA' , where A' is the transpose of matrix A.

• 3)

If $\left[ \begin{matrix} y & +2x & 5 \\ & -x & 3 \end{matrix} \right] =\begin{bmatrix} 7 & 5 \\ -2 & 3 \end{bmatrix}$, find the value of y.

• 4)

If $A=\left[ { a }_{ ij } \right] =\left[ \begin{matrix} 2 & 3 & -5 \\ 1 & 4 & 9 \\ 0 & 7 & -2 \end{matrix} \right] andb=\left[ { b }_{ ij } \right] =\left[ \begin{matrix} 2 & 1 & -1 \\ -3 & 4 & 4 \\ 1 & 5 & 2 \end{matrix} \right] ,then\quad find\quad { a }_{ 22 }+{ b }_{ 21 }.$

• 5)

If $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\begin{bmatrix} 3 & 1 \\ 2 & 5 \end{bmatrix}=\begin{bmatrix} 7 & 11 \\ k & 23 \end{bmatrix}$, find the value of k.

#### 12th CBSE Maths Unit 2 Inverse Trigonometric Functions Important Question Paper - by Shalini Sharma - Udaipur Jul 24, 2019 - View & Download

• 1)

Evaluate  $sin\left[ \frac { \pi }{ 3 } -{ sin }^{ -1 }\left( -\frac { 1 }{ 2 } \right) \right]$ .

• 2)

Using principal value, evaluate the following:  ${ cos }^{ -1 }\left( cos\frac { 2\pi }{ 3 } \right) +{ sin }^{ -1 }\left( sin\frac { 2\pi }{ 3 } \right)$

• 3)

Show that ${ sin }^{ -1 }\left( 2X\sqrt { 1-{ X }^{ 2 } } \right) =2{ sin }^{ -1 }X$

• 4)

Solve for X, $\quad { tan }^{ -1 }\frac { 1-X }{ 1+X } =\frac { 1 }{ 2 } { tan }^{ -1 }X,\quad X>0.$

• 5)

Write the principal values of the following: ${ cos }^{ -1 }\left( cos\frac { 7\pi }{ 6 } \right)$

#### CBSE 12th Standard Mathematics Unit 1 Important Questions - by Shalini Sharma - Udaipur Jul 18, 2019 - View & Download

• 1)

If f(X)=X+7 and g(X)=X-7, $X\in R,$ find fog(7). ?

• 2)

If the binary operation * on the set of integers Z is defined by a*b=a+3bthen find the value of 2*4.

• 3)

Let * be a binary operation on N given by a*b=HCF(a,b), $a,b\in N$. Write the value of 22*4.

• 4)

If the binary operation * defined on Q is defined as a*b=2a+b-ab, for all $a,b\in Q,$ find the value of 3*4.

• 5)

If f:$R\to R$ be defined by $f(X)=(3-X^3)^{1\over3}$ then find fof(X).

#### Maths CBSE 12 Class Revision Test paper - by Bala Dec 29, 2018 - View & Download

• 1)

For the set A={1,2,3} define a relation R in the set A is follows:

R={(1,1).(2,2),(3,3),(1,3)}. Write the ordered pairs to be added to R to make it the smallest equivalence relation.

• 2)

If matrix A=$[\begin{matrix} 1 & 2 & 3 \end{matrix}]$ write AA' , where A' is the transpose of matrix A.

• 3)

Evaluate : $\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { e }^{ x }Isinx+cosx) } dx$

• 4)

Find $\lambda$, if the vectors
$\overrightarrow a=\overset\wedge i+3\overset\wedge j+\overset\wedge k,\overrightarrow b=2\overset\wedge i-\overset\wedge j-\overset\wedge k$ and $\overrightarrow c=\lambda \overset\wedge j+3\overset\wedge k$ are coplanar.

• 5)

How many equivalence relations on the set {1,2, 3} containing (I, 2) and (2, 1) are there in all ? Justify your answer.

#### CBSE Maths First Full Test Class 12 - by Bala Dec 29, 2018 - View & Download

• 1)

Let * be a binary operation on N given by a * b = LCM(a,b) for all a,b $\in$ N. Find 5*7

• 2)

Use elementary column operation ${ C }_{ 2 }\rightarrow { C }_{ 2 }+2{ C }_{ 1 }$ in the following matrix equation :
$\left( \begin{matrix} 4 & 2 \\ 3 & 3 \end{matrix} \right) =\left( \begin{matrix} 1 & 2 \\ 0 & 3 \end{matrix} \right) \left( \begin{matrix} 2 & 0 \\ 1 & 1 \end{matrix} \right)$

• 3)

Evaluate the integral: $\int {(1-x)\sqrt x\ dx}$

• 4)

Find $\lambda$ when the projection of $\overrightarrow { a } =\lambda \hat { i } +\hat { j } +4\hat { k } \quad on\quad \overrightarrow { b } =2\hat { i } +6\hat { j } +3\hat { k }$ is 4 units.

• 5)

Define Reflexive.Give one example.

#### CBSE Class 12 Mathematics Important Question Paper - by Bala Dec 29, 2018 - View & Download

• 1)

Show that the relation R:{1,2,3}$\rightarrow${1,2,3} given by R={(1,1),(2,2),(3,3),(1,2),(2,3)} is reflexive but neither symmetric nor transitive.

• 2)

If $A=\begin{bmatrix} cos\alpha - & sin\alpha \\ sin\alpha & cos\alpha \end{bmatrix}$, then for what value of $\alpha$, A is an identify matrix?

• 3)

Evaluate the integral: $\int {(1-x)\sqrt x\ dx}$

• 4)

Write the value of the area of the parallelogram determined by the vectors $2\hat { i } \quad and\quad 3\hat { j }$

• 5)

Let f and g be real function be $f(x)=\sqrt { x+4 } ,x\ge 4$ find the function fg, $\frac { f }{ g }$

#### Vector Algebra Important Questions CBSE 12th Mathematics - by Bala Dec 14, 2018 - View & Download

• 1)

Find a unit vector in the direction of $\overrightarrow { a } =3\overrightarrow { i } -2\overrightarrow { j } +6\overrightarrow { k }$

• 2)

If p(1,5,4) and Q(4,1,-2) find the direction ratios of $\overrightarrow { PQ }$

• 3)

Write the direction cosines of the vector $-2\hat { i } +\hat { j } -5\hat { k }$

• 4)

If $\overrightarrow { a }$ is a unit vector and $(\overrightarrow { x } -\overrightarrow { a } ).(\overrightarrow { x } +\overrightarrow { a } )=8\quad find\quad \left| \overrightarrow { x } \right|$

• 5)

Find the sum of the vectors $\overset\rightarrow a=\overset\wedge i-2\overset\wedge j+\overset\wedge k,\overset\rightarrow b=-2\overset\wedge i+4\overset\wedge j+5\overset\wedge k,\overset\rightarrow c=\overset\wedge i-6\overset\wedge j-7\overset\wedge k.$

#### 12th CBSE Mathematics Three Dimensional Geometry Important Question Paper - by Bala Dec 10, 2018 - View & Download

• 1)

Find the direction ratios of the line $\frac { x+2 }{ 1 } =\frac { 2y-1 }{ 3 } =\frac { 3-z }{ 5 } .$

• 2)

Find the angle between the line $\vec { r } =(2\hat { i } -\hat { j } +3\hat { k } )+\lambda (3\hat { i } -\hat { j } +2\hat { k } )$  and the plane $\vec { r } .(\hat { i } +\hat { j } +\hat { k } )=3.$

• 3)

What is the distance of the point (p,q,r) from the x-axis?

• 4)

Find the distance of a point (2, 5, - 3) from the plane $\overset { \rightarrow }{ r } (6\hat { i } +3\hat { j+6\hat { 2k } )=4 }$

• 5)

Find the length of the perpendicular drawrt from the origin to the plane 2x - 3y + 6z + 21 = 0

#### 12th CBSE Mathematics Linear Programming Model Question Paper - by Bala Dec 01, 2018 - View & Download

• 1)

The objective function is maximum or minimum, which lies on the boundary of the feasible region.

• 2)

Solve the following problem graphically:
Minimise and Maximise Z=3x+9y subject to the constraints:
$x+3y\le 60,x+y\ge 10,x\le y,x\ge 0,y\ge 0.$

• 3)

Maximise Z=3x+2y
subject to $x+2y\le 10,3x+y\le 15,x,y\ge 0.$

• 4)

Reshma wishes to mix two types of food P and Q in such a way that the vitamin contents of the mixture contain at least 8 units of vitamin A and 11 units of vitamin B. Food P costs RS. 60/kg and Food Q costs RS. 80/kg. Food P contains 3 units/kg of Vitamin A AND 4 units/kg of vitamin B. Food Q contains 5 units/kg Vitamin A and 2 units/kg of vitamin B. Determine the minimum cost of the mixture.

• 5)

A farmer mixes two brands P and Q of cattle feed. Brand P, costing RS.250 per bag, contains 3 units of nutritional element A, 2.5 units of element B and 2 units of vitamin C. Brand Q costing RS.200 per bag contains 1.5 units of nutritional element A, 11.25 units of element B, and 3 units of element C. The minimum requirements of nutrients A,B and C are 18 units, 45 units and 24 units respectively. Determine the number of bags of each brand which should be mixed in order to produce a mixture having a minimum cost per bag? What is the minimum cost of the mixture per bag?

#### 12th CBSE Mathematics Probability Important Question Paper - by Bala Nov 30, 2018 - View & Download

• 1)

If P(A)=0.4, P(B)=p and $P(A\cup B)=0.7$ find the value of p, if A and B are independent events.

• 2)

Given P(A)=0.4, P(B)=0.7 and P(B/A)=0.6, Find $P(A\cup B)$

• 3)

A bag contain 2 red, 6 black and 8 green balls. A ball is drawn at random from the bag. Find the probabilty:
(a) a red ball
(b) a black ball
(c) a green ball
(d) a non-red ball

• 4)

If P(E) =$\frac { 6 }{ 11 }$, P(F) =$\frac { 5 }{ 11 }$ and P(E$\cup$F)=$\frac { 7 }{ 11 }$ then find (a) P(E/F), (b) P(F/E)

• 5)

If P(F) = 0.35 and P(E$\cup$F) = 0.85 and E and F are independent events. Find P(E).

#### 12th Standard Mathematics Revision Model Question Paper 1 - by Bala Nov 29, 2018 - View & Download

• 1)

State the reason for the relation R in the set {1,2,3} given by R={(1,2),(2,1),} not to be transitive.

• 2)

Find the principal value of ${ sin }^{ -1 }\left( \frac { \sqrt { 3 } }{ 2 } \right)$

• 3)

If ${ X }_{ m\times 3 }{ Y }_{ p\times 4 }={ Z }_{ 2\times b }$, for three matrices X,Y and Z, find the values of m,p and b.

• 4)

$\int {x^2-1\over x^2+1}dx.$

• 5)

Let f:$X\rightarrow Y$ be a function Define a relation R on X given be R=[(a,b) ; (f(b)] Show that R is an equivalence relation ?

#### 12th Standard CBSE Mathematics Term Test Question Paper - by Bala Sep 15, 2018 - View & Download

• 1)

If the binary operation * on the set of integers Z is defined by a*b=a+3bthen find the value of 2*4.

• 2)

Let * be a binary operation on N given by a*b=HCF(a,b), $a,b\in N$. Write the value of 22*4.

• 3)

If the binary operation * defined on Q is defined as a*b=2a+b-ab, for all $a,b\in Q,$ find the value of 3*4.

• 4)

Let f:$R\rightarrow R$ is defined by f(x)=|x|. Is function f onto? Give reasons.

• 5)

Let R be a relation in the set of natural numbers N defined by R={(a,b)$\in$NXN;a

#### Probability - Important Questions Model Paper In 12th Maths - by Bala Aug 01, 2018 - View & Download

• 1)

Given P(A)=$1\over2$,P(B)=$1\over3$ and $P(A\cap B)={1\over6}$  Are the events A and B independent?

• 2)

Given P(A)=0.2, P(B)=0.3 and $P(A\cap B)=0.3$ Find P(A/B)

• 3)

Given P(A)=0.4, P(B)=0.7 and P(B/A)=0.6, Find $P(A\cup B)$

• 4)

Events E and F are given to be independent. Find P(F) if it is given that P(E)=0.60 and P(E$\cap$F)=0.35

• 5)

Does the following represent a probability distribution? Give reasons.

 X 0 1 2 P(x) 1/3 1/3 1/6

#### Linear Programming - Important Questions Model Question Paper 1 In 12th Maths - by Bala Aug 01, 2018 - View & Download

• 1)

Solve the following linear programming problem graphically:
Maximise Z=4x+y subject to the constraints:
$\\ x+y\le 50,3x+y\le 90,x\ge 0,y\ge 0.$

• 2)

Solve the following problem graphically:
Minimise and Maximise Z=3x+9y subject to the constraints:
$x+3y\le 60,x+y\ge 10,x\le y,x\ge 0,y\ge 0.$

• 3)

Determine the minimum value of Z=3x+2y (if any), if the feasible region for an LLP is shown in the figure:

• 4)

Solve the following LLP graphically:
Maximise Z=2x+3y, subject to $x+y\le 4,x\ge 0,y\ge 0.$

• 5)

Minimize and Maximize Z=5x+2y, subject to the following constraints:
$x-2y\le 2,3x+2y\le 12,-3x+2y\le 3,x\ge 0,y\ge 0.$

#### Three Dimensional Geometry - Important Questions Model Question Paper 1 In 12th Maths - by Bala Aug 01, 2018 - View & Download

• 1)

Find the distance of the point (2,3,4) from the plane $\overrightarrow { r } .(3\acute { i } -6\acute { j } +2\acute { k } )=-11$.

• 2)

Write the Cartesian equation of the following line given in vector form:

$\overrightarrow { r } =2\hat { i } +\hat { j } +4\hat { k } +\lambda (\hat { i } +\hat { j } -\hat { k } )$

• 3)

Write the vector equation of a line whose Cartesian equation is $\frac { x+3 }{ 2 } =\frac { y-1 }{ 4 } =\frac { z+1 }{ 5 }$.

• 4)

Find the vector normal to the plane $\vec { r } .(3\hat { i } -7\hat { k } )+5=0$.

• 5)

Find the coordinates of a point, where the line $\frac { x+2 }{ 1 } =\frac { y-5 }{ 3 } =\frac { z+1 }{ 5 }$ cuts yz-plane.

#### Vector Algebra - Important Questions Model Question Paper 1 In 12th Maths - by Bala Jul 31, 2018 - View & Download

• 1)

Find the angle between the vectors $\overrightarrow { a } =\overrightarrow { i } -\overrightarrow { j } +\overrightarrow { k } and\quad \overrightarrow { b } =\overrightarrow { i } +\overrightarrow { j } -\overrightarrow { k }$

• 2)

Find a vector in the direction of $\overrightarrow { a } =\overrightarrow { i } -2\overrightarrow { j }$ whose magnitude is 7.

• 3)

Vectors $\overrightarrow { a } \quad and\quad \overrightarrow { b }$ are such that $\left| \overrightarrow { a } \right| =\sqrt { 3 } ,\left| \overrightarrow { b } \right| =\frac { 2 }{ 3 } and\quad (\overrightarrow { a } \times \overrightarrow { b } )$ is a unit vector. write the angle between $\overrightarrow { a } \quad and\quad \overrightarrow { b }$?

• 4)

If $\overrightarrow { a } =x\hat { i } +2\hat { j } -z\hat { k } \quad and\quad \overrightarrow { b } =3\hat { i } -y\hat { j } +\hat { k }$ are two equal vectors then write the value of x+y+z.

• 5)

If $\left| \overrightarrow { a } \right| =3$ interpret the following:

(i)2$\overrightarrow { a }$   (ii)-5$\overrightarrow { a }$

#### Differential Equations - Important Questions Model Question Paper 1 In 12th Maths - by Bala Jul 31, 2018 - View & Download

• 1)

What is the degree of the following differential equation?

${ { 5x\left( \frac { dy }{ dx } \right) } }^{ 2 }-\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } -6y=logx$

• 2)

Write the order and degree of the differential equation $(\frac{{d}^{2}y}{{dx}^{2}})^{3}$ -5$\frac{dy}{dx}$+6=0.

• 3)

Find the differential equation of the family of lines passing through the origin.

• 4)

Write the differential equation representing the curve y2 = 4ax, where a is an arbitrary constant.

• 5)

Find the integrating factor of the differential equation $\left( \frac { { e }^{ -2\sqrt { x } } }{ \sqrt { x } } -\frac { y }{ \sqrt { x } } \right) \frac { dx }{ dy } =1$ .

#### Application Of Integrals - Important Questions Model Question Paper 1 In 12th Maths - by Bala Jul 31, 2018 - View & Download

• 1)

Find the area of the region bounded by y2 = 9x, x = 2, x = 4 and the x - axis in the first quadrant.

• 2)

Find the area of the region in the first quadrant enclosed by x - axis and $x=\sqrt { 3 } y$ by the circle ${ x }^{ 2 }+{ y }^{ 2 }=4$.

• 3)

The area between x = y2 and x = 4 is divided into two equal parts by the line x = a, find the value of a.

• 4)

Find the area bounded by the curve x2 = 4y and the line x = 4y - 2.

• 5)

Area lying between the curves y2 = 4x and y = 2x is :
(A) $\frac { 2 }{ 3 }$
(B) $\frac { 1 }{ 3 }$
(C) $\frac { 1 }{ 4 }$
(D) $\frac { 3 }{ 4 }$

#### Integrals - Important Questions Model Question Paper 1 In 12th Maths - by Bala Jul 31, 2018 - View & Download

• 1)

Evaluate the integral: $\int { {x^2\over1+x^3}dx. }$

• 2)

$\int {1+tan\ x\over 1-tan\ x}dx$.

• 3)

$\int {dx\over x\ cos^2 (1+log\ x)}$.

• 4)

$\int {x^2-1\over x^2+1}dx.$

• 5)

$\int {e^x\over 1+e^x}dx.$

#### Application Of Derivatives - Important Questions Model Question Paper 1 In 12th Maths - by Bala Jul 30, 2018 - View & Download

• 1)

The amount of pollution content added in air in a city due to X diesel vehicles is given by P(x)=0.005x3+0.02x2 +30x.Find the marginal increase   in pollution content when 3 diesel vehicles are added and write which value is indicated in the above question?

• 2)

The money to be spent for the welfare of the employees of a firm is proportional to the rate of change of its total revenue(marginal revenue).If the total revenue(in rupees)received from the sale of x units of a product is given by R9x)=3x2+36+5,find the marginal revenue,when=5,and write which value does the question indicate?

• 3)

The total cost C(x) associated with provision of free mid-day meals to x students of a school in primary classes is given by

C(x)=0.005x3-0.02x2+30x+50

If the marginal cost is given by rate of change $dC\over dX$of total cost,write the marginal cost of food for 300students.What value is shown here?

• 4)

The total revenue received from the sale of x souvenirs in connection with 'PEACE DAY'is given by R(x)=3x2+40x+10.Find the marginal revenue when 100souvenirs were sold.What is the importance of celebrating Peace Day in our life?

• 5)

A balloon which always remains spherical has a variable diameter ${3\over2}(2x+1)$.Find the rate of change of its volume with respect to x.

#### Continuity And Differentiability - Important Questions Model Paper In 12th Maths - by Bala Jul 30, 2018 - View & Download

• 1)

Examine the continuity of the function f (x) = x2+5 at x=-1

• 2)

Examine the continuity of the function f (x)=$\frac { 1 }{ x+3 } ,\quad x\quad \varepsilon \quad R$.

• 3)

Give an example of a function which is continuous at x=1, but not differentiable at x=1.

• 4)

State the points of discountinuity for the function $f(x)= [x]$  in $-3 < x < 3.$

• 5)

Show that the function f (x)= $\begin{cases} { x }^{ 3 }+3\quad ,\quad if\quad x\neq 0 \\ 1\quad \quad \quad ,\quad if\quad x=0 \end{cases}$ is not continuous at x=0.

#### Determinants - Important Questions Model Question Paper 1 In 12th Maths - by Bala Jul 30, 2018 - View & Download

• 1)

Find the cofactor of a12 in the following.$\begin{vmatrix} 2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7 \end{vmatrix}$

• 2)

Write the adjoint of the following matrix $\begin{bmatrix} 2 & -1 \\ 4 & 3 \end{bmatrix}$

• 3)

If $\begin{vmatrix} x+1 & x-1 \\ x-3 & x+2 \end{vmatrix}=\begin{vmatrix} 4 & -1 \\ 1 & 3 \end{vmatrix}$then write the value of x.

• 4)

A is a non-singular matrix of order 3 and |A|=-4. Find |adj A|

• 5)

In the interval  $\frac { \pi }{ 2 } , find the value of x for which that matrix \(\left[ \begin{matrix} 2sinx \\ 1 \end{matrix}\begin{matrix} 3 \\ 2sinx \end{matrix} \right]$ is singular.

#### Matrices - Important Questions Model Question Paper 1 In 12th Maths - by Bala Jul 30, 2018 - View & Download

• 1)

If $\left[ \begin{matrix} x & +3y & y \\ 7 & -x & 4 \end{matrix} \right]$=$\begin{bmatrix} 4 & -1 \\ 0 & 4 \end{bmatrix}$, find the values of x and y.

• 2)

Write the value of x-y+z from the following equation :

$\left[ \begin{matrix} x+y+z \\ x+z \\ y+z \end{matrix} \right] =\left[ \begin{matrix} 9 \\ 5 \\ 7 \end{matrix} \right]$

• 3)

If ${ A }^{ T }=\left[ \begin{matrix} 3 & 4 \\ -1 & 2 \\ 0 & 1 \end{matrix} \right]$and $B=\left[ \begin{matrix} -1 & 2 & 1 \\ 1 & 2 & 3 \end{matrix} \right]$, then find ${ A }^{ T }-{ B }^{ T }$.

• 4)

If matrix $A=\begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix}$ and ${ A }^{ 2 }=kA$, then write the value of k.

• 5)

A matrix has 18 elements. Write the possible orders of a matrix

#### 12th Maths Model Question Paper 5 - by Bala Jul 27, 2018 - View & Download

• 1)

Show that ${ sin }^{ -1 }\left( 2X\sqrt { 1-{ X }^{ 2 } } \right) =2{ sin }^{ -1 }X$

• 2)

Solve for X, $\quad { tan }^{ -1 }\frac { 1-X }{ 1+X } =\frac { 1 }{ 2 } { tan }^{ -1 }X,\quad X>0.$

• 3)

Write the principal values of the following: sec-1(-2).

• 4)

Write the principal values of the following: ${ cot }^{ -1 }\left( -\sqrt { 3 } \right)$

• 5)

Find the principal value of ${ tan }^{ -1 }\sqrt { 3 } -{ sec }^{ -1 }\left( -2 \right)$

#### 12th Maths Model Question Paper 1 - by Bala Jul 27, 2018 - View & Download

• 1)

If the binary operation * on the set of integers Z is defined by a*b=a+3bthen find the value of 2*4.

• 2)

Let * be a binary operation on N given by a*b=HCF(a,b), $a,b\in N$. Write the value of 22*4.

• 3)

If f:$R\to R$ and g:$R\to R$are given by f(X)=sin x and g(x)=5x2 find gof(x).

• 4)

State the reason for the relation R in the set {1,2,3} given by R={(1,2),(2,1),} not to be transitive.

• 5)

Let f:$R\rightarrow R$ is defined by f(x)=x2. Is f one-one?

#### Inverse Trigonometric Functions - Important Four Marks Questions Model Question Paper In 12th Maths - by AJAY KAKKAR Jun 09, 2018 - View & Download

• 1)

Write in the simplest form: $({ tan }^{ -1 }\left[ \frac { \sqrt { 1+sin\quad x } +{ \sqrt { 1-sin\quad x } }\quad }{ \sqrt { 1+sin\quad x } +{ \sqrt { 1-sin\quad x } } } \right] ,0<x<\frac { \pi }{ 2 }$

• 2)

Solve for X, 2tan-1(sinX)=tan-1(2secX),$X\neq \frac { \pi }{ 2 }$

• 3)

Solve the following equations:

${ cot }^{ -1 }X-{ cot }^{ -1 }\left( X+2 \right) =\frac { \pi }{ 12 }$

• 4)

Solve the following equations:

${ tan }^{ -1 }\frac { X+1 }{ X-1 } +{ tan }^{ -1 }\frac { X-1 }{ X } ={ tan }^{ -1 }\left( -7 \right)$

• 5)

Solve the following equations in term of  $\beta$${ tan }^{ -1 }\left[ \frac { \sqrt { 1+{ x }^{ 2 } } -\sqrt { 1-{ x }^{ 2 } } }{ \sqrt { 1+{ x }^{ 2 } } +\sqrt { 1-{ x }^{ 2 } } } \right] =\beta$

#### Inverse Trigonometric Functions - Important Two Marks Questions Model Question Paper In 12th Maths - by AJAY KAKKAR Jun 09, 2018 - View & Download

• 1)

Write in the simplest form : ${ tan }^{ -1 }\left[ \frac { cos\quad x }{ 1+sin\quad x } \right] ,x\left[ -\frac { \pi }{ 2 } ,\frac { \pi }{ 2 } \right]$

• 2)

Show that : ${ tan }^{ -1 }\frac { 3 }{ 4 } +{ tan }^{ -1 }\frac { 3 }{ 5 } -{ tan }^{ -1 }\frac { 8 }{ 19 } =\frac { \pi }{ 4 }$

• 3)

Show that ${ tan }^{ -1 }\frac { x }{ y } -{ tan }^{ -1 }\frac { x-y }{ x+y } =\frac { \pi }{ 4 }$

• 4)

Show that ${ sin }^{ -1 }\frac { 5 }{ 13 } +{ cos }^{ -1 }\frac { 3 }{ 5 } ={ tan }^{ -1 }\frac { 63 }{ 16 }$

• 5)

Evaluate : $4 { tan }^{ -1 }\frac { 1 }{ 5 }$

#### Continuity And Differentiability - Important Two Marks Questions Model Question Paper In 12th Maths - by AJAY KAKKAR Jun 09, 2018 - View & Download

• 1)

Show that the function f (x)= $\begin{cases} { x }^{ 3 }+3\quad ,\quad if\quad x\neq 0 \\ 1\quad \quad \quad ,\quad if\quad x=0 \end{cases}$ is not continuous at x=0.

• 2)

Check the continuity of the function f given by:
f(x)=2x+3 at x=1.

• 3)

Examine whether the function f given by:
f(x)=${ x }^{ 2 }$ is continuous at x=0.

• 4)

Discuss the continuity of the function f given by:
$f(x)=\left| x \right| at\quad x=0.$

• 5)

Show that the function f given by:
$f(x)=\begin{cases} { x }^{ 3 }+3,\quad \quad \quad if\quad x\neq 0 \\ 1,\quad \quad \quad \quad \quad \quad if\quad x=0 \end{cases}$
is not continuous at x=0.

### CBSE Education Study Materials

#### Revision Plan to score full Marks in CBSE Class 12 Mathematics Bala Jan 23, 2019

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#### Tips to crack CBSE Class 12 Mathematics with 100/100 Bala Jan 23, 2019

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Bulletin 34 | Matthew Edwards

COVID-19 mortality is associated with risk factors such as obesity and diabetes that imply higher ‘normal’ (all-cause) mortality. In this Bulletin, we consider how this may lead overall mortality of the post-pandemic ‘survivor pool’ to be lighter than it was pre-pandemic. We conclude that the effect will be low other than in subgroups with a high proportion of COVID-19 deaths (which is what you would expect). This is the first of three bulletins looking at 'post-pandemic' mortality - next we look at mortality analyses post-pandemic, and then the impact on mortality improvements.

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text:
# Inverse Fourier Transform involving a Bessel Function, Exponential, and Power

I'm interested in this integral as a function of $r$ for various spectral densities $S(s)$: $\frac{2 \pi}{r^{p/2}-1} \int_{0}^{\infty} S(s) J_{p/2-1}(2 \pi r s) s^{p/2} ds$, where $J_{p/2-1}$ is a Bessel function of order $(p/2-1)$.

For example, if $S(s)$ is a Gaussian centred at the origin, we can use the integral identity $\int_{0}^{\infty} x^{\nu+1} e^{-\alpha x^2} J_{\nu}(\beta x) dx = \frac{\beta^\nu}{(2\alpha)^{\nu+1}} \exp(-\frac{\beta^2}{4 \alpha}) \qquad \text{Re}(\alpha) > 0 \,, \quad \text{Re}(\nu) > -1$
to recover a zero mean Gaussian in $r$ centred at 0.

However, suppose $S(s)$ is a Gaussian with mean $\mu$, in which case I want to analytically solve $\frac{2 \pi}{r^{p/2}-1} \int_{0}^{\infty} \exp[-\frac{(s-\mu)^2}{2 \sigma^2}] J_{p/2-1}(2 \pi r s) s^{p/2} ds$. I am not sure how I should best do this integral. I found the possibly helpful identities $\int_{0}^{\infty} x^{\nu+1} e^{-\alpha x} J_{\nu}(\beta x) dx = \frac{2 \alpha (2 \beta)^{\nu} \Gamma(\nu+\frac{3}{2})}{\sqrt{\pi}(\alpha^2 + \beta^2)^{\nu+\frac{3}{2}}} \qquad \text{Re}(\nu) > -1, \text{Re}(\alpha) > |\text{Im}(\beta)|$

$\int_{0}^{\infty} x^{m+1} e^{-\alpha x} J_{\nu}(\beta x) dx = (-1)^{m+1} \beta^{-\nu} \frac{d^{m+1}}{d\alpha^{m+1}} [\frac{(\sqrt{\alpha^2 + \beta^2} - \alpha)^\nu}{\sqrt{\alpha^2+\beta^2}}] \,, \beta > 0, \text{Re}(\nu) > -m -2 \,.$

It doesn't seem I can simply do a change of variables $z = s-\mu$. I feel there should be a nice clean solution to this problem.

I would greatly appreciate your help.

-
I am still really hoping to answer this question. I am happy to clarify, or expand on the question, so that it will be of more general interest. – Gordon Jul 26 '13 at 17:02


text:
# Miscellaneous Definite Integrals

## Homework Statement

show that

$\int^{∞}_{0}\frac{sin^{2}x}{x^{2}}dx= \frac{\pi}{2}$

## Homework Equations

consider

$\oint_{C}\frac{1-e^{i2z}}{z^{2}}dz$

where C is a semi circle of radius R, about 0,0 with an indent (another semi circle) excluding 0,0.

## The Attempt at a Solution

curve splits into
C1, line segment from -R to -ε
C2, semi circle z=εe θ goes from ∏ to 0
C3, line segment from ε to R
C4, semi circle z=Re θ goes from 0 to ∏

function is holomorphic in/on C. so integral =0

for C4, applying limit R > infinity integral = 0

C3. really stuck here

I sub in the value of z

to get

$\int^{0}_{∏}\frac{1-e^{i2e^{iθ}}}{ε^{2}e^{2iθ}}εe^{iθ}$

but I can't take the limit ε -> 0 of this since theres an ε on the denominator

Related Calculus and Beyond Homework Help News on Phys.org
vela
Staff Emeritus
Homework Helper
You forgot an ε in the exponent and a factor of i.
$$\int^0_\pi \frac{1-e^{i2\varepsilon e^{i\theta}}}{\varepsilon^2 e^{2i\theta}} i\varepsilon e^{i\theta}\,d\theta$$

You forgot an ε in the exponent and a factor of i.
$$\int^0_\pi \frac{1-e^{i2\varepsilon e^{i\theta}}}{\varepsilon^2 e^{2i\theta}} i\varepsilon e^{i\theta}\,d\theta$$
Thank you. I was half awake when I wrote that.

I just don't know how to get that ε out from the denominator, I cannot take the limit otherwise.

vela
Staff Emeritus
Homework Helper
Try expanding the exponential and the numerator as a series.

The teacher gave me a hint, use L'hopitals Rule, which isn't clear to be possible since i forgot to add in the limit

vela
Staff Emeritus
Homework Helper
I'm not sure what you mean by that. In any case, either way should work. The numerator is first-order in ε. Along with the factor of ε from dz, it cancels the ε2 in the denominator, so you will get a finite result.

I mean, differentiating the top and bottom before taking the limit.

as in a taylor series?

vela
Staff Emeritus


text:
# Flownet

A flownet is a graphical representation of two-dimensional steady-state groundwater flow through aquifers.

Construction of a flownet is often used for solving groundwater flow problems where the geometry makes analytical solutions impractical. The method is often used in civil engineering, hydrogeology or soil mechanics as a first check for problems of flow under hydraulic structures like dams or sheet pile walls. As such, a grid obtained by drawing a series of equipotential lines is called a flownet. The flownet is an important tool in analysing two-dimensional irrotational flow problems. Flow net technique is a graphical representation method.

## Basic method

The method consists of filling the flow area with stream and equipotential lines, which are everywhere perpendicular to each other, making a curvilinear grid. Typically there are two surfaces (boundaries) which are at constant values of potential or hydraulic head (upstream and downstream ends), and the other surfaces are no-flow boundaries (i.e., impermeable; for example the bottom of the dam and the top of an impermeable bedrock layer), which define the sides of the outermost streamtubes (see figure 1 for a stereotypical flownet example).

Mathematically, the process of constructing a flownet consists of contouring the two harmonic or analytic functions of potential and stream function. These functions both satisfy the Laplace equation and the contour lines represent lines of constant head (equipotentials) and lines tangent to flowpaths (streamlines). Together, the potential function and the stream function form the complex potential, where the potential is the real part, and the stream function is the imaginary part.

The construction of a flownet provides an approximate solution to the flow problem, but it can be quite good even for problems with complex geometries by following a few simple rules (initially developed by Philipp Forchheimer around 1900, and later formalized by Arthur Casagrande in 1937) and a little practice:

• streamlines and equipotentials meet at right angles (including the boundaries),
• diagonals drawn between the cornerpoints of a flownet will meet each other at right angles (useful when near singularities),
• streamtubes and drops in equipotential can be halved and should still make squares (useful when squares get very large at the ends),
• flownets often have areas which consist of nearly parallel lines, which produce true squares; start in these areas — working towards areas with complex geometry,
• many problems have some symmetry (e.g., radial flow to a well); only a section of the flownet needs to be constructed,
• the sizes of the squares should change gradually; transitions are smooth and the curved paths should be roughly elliptical or parabolic in shape.

## Example flownets

The first flownet pictured here (modified from Craig, 1997) illustrates and quantifies the flow which occurs under the dam (flow is assumed to be invariant along the axis of the dam — valid near the middle of the dam); from the pool behind the dam (on the right) to the tailwater downstream from the dam (on the left).

There are 16 green equipotential lines (15 equal drops in hydraulic head) between the 5 m upstream head to the 1m downstream head (4 m / 15 head drops = 0.267 m head drop between each green line). The blue streamlines (equal changes in the streamfunction between the two no-flow boundaries) show the flowpath taken by water as it moves through the system; the streamlines are everywhere tangent to the flow velocity.

Example flownet 2, click to view full-size.

The second flownet pictured here (modified from Ferris, et al., 1962) shows a flownet being used to analyze map-view flow (invariant in the vertical direction), rather than a cross-section. Note that this problem has symmetry, and only the left or right portions of it needed to have been done. To create a flownet to a point sink (a singularity), there must be a recharge boundary nearby to provide water and allow a steady-state flowfield to develop.

## Flownet results

Darcy's law describes the flow of water through the flownet. Since the head drops are uniform by construction, the gradient is inversely proportional to the size of the blocks. Big blocks mean there is a low gradient, and therefore low discharge (hydraulic conductivity is assumed constant here).

An equivalent amount of flow is passing through each streamtube (defined by two adjacent blue lines in diagram), therefore narrow streamtubes are located where there is more flow. The smallest squares in a flownet are located at points where the flow is concentrated (in this diagram they are near the tip of the cutoff wall, used to reduce dam underflow), and high flow at the land surface is often what the civil engineer is trying to avoid, being concerned about piping or dam failure.

## Singularities

Irregular points (also called singularities) in the flow field occur when streamlines have kinks in them (the derivative doesn't exist at a point). This can happen where the bend is outward (e.g., the bottom of the cutoff wall in the figure above), and there is infinite flux at a point, or where the bend is inward (e.g., the corner just above and to the left of the cutoff wall in the figure above) where the flux is zero.

The second flownet illustrates a well, which is typically represented mathematically as a point source (the well shrinks to zero radius); this is a singularity because the flow is converging to a point, at that point the Laplace equation is not satisfied.

These points are mathematical artifacts of the equation used to solve the real-world problem, and do not actually mean that there is infinite or no flux at points in the subsurface. These types of points often do make other types of solutions (especially numeric) to these problems difficult, while the simple graphical technique handles them nicely.

## Extensions to standard flownets

Typically flownets are constructed for homogeneous, isotropic porous media experiencing saturated flow to known boundaries. There are extensions to the basic method to allow some of these other cases to be solved:

• inhomogeneous aquifer: matching conditions at boundaries between properties
• anisotropic aquifer: drawing the flownet in a transformed domain, then scaling the results differently in the principle hydraulic conductivity directions, to return the solution
• one boundary is a seepage face: iteratively solving for both the boundary condition and the solution throughout the domain

Although the method is commonly used for these types of groundwater flow problems, it can be used for any problem which is described by the Laplace equation (${\displaystyle \nabla ^{2}\phi =0}$), for example electric current flow through the earth.

## References

• Casagrande, A., 1937. Seepage through dams, Journal of New England Water Works, 51, 295-336 (also listed as: Harvard Graduate School Eng. Pub. 209)
• Cedergren, Harry R. (1977), Seepage, Drainage, and Flow Nets, Wiley. ISBN 0-471-14179-8
• Chanson, H. (2009). Applied Hydrodynamics: An Introduction to Ideal and Real Fluid Flows. CRC Press, Taylor & Francis Group, Leiden, The Netherlands, 478 pages. ISBN 978-0-415-49271-3.
• Knappett, Jonathan and R.F. Craig, 2012. Craig's Soil Mechanics 8th edition, Spon Press. ISBN 978-0-415-56126-6
• Ferris, J.G., D.B. Knowles, R.H. Brown & R.W. Stallman, 1962. Theory of Aquifer Tests. US Geological Survey Water-Supply Paper 1536-E. (available from the USGS website as a pdf)
• Harr, M.E., 1962. Groundwater and Seepage, Dover. ISBN 0-486-66881-9 — mathematical treatment of 2D groundwater flow, classic work on flownets.