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# STPM 2016 Mathematics (T) Term 1 Assignment

## Introduction

Parametric equations express a set of related quantities as explicit functions of an independent variable, known as a parameter. An equation, relating variables x and y in Cartesian coordinates, can be expressed by parametric equations which describe a position on the curve.
Find information about parametric equations yourself.

## Sample Question

Parametric equations express a set of related quantities as explicit functions of an independent variable, known as a parameter. An equation, relating variables $x$ and $y$ in Cartesian coordinates, can be expressed by parametric equations which describe a position on the curve. 1  The parametric equations of a plane curve are defined by $x = e^t, y = t^2-1, -2 \leq t \leq 2$. Tabulate the values $t$, $x$ and $y$, and plot the curve. 2 (a)  Find three sets of parametric equations for the curve whose equation is $(y - 1)^2 = x - 49$. (b)  Is it possible to choose $x =-t^2$ as the parametric equation for $x$? Can you start with any choice for the parametric equation for $x$? (c)  Can you start with any choice for the parametric equation for $y$? 3  Suppose that the position of a particle at time $t$ is given by

$x_1 = 2 \sin t, y_1 = 3 \cos t, 0\leq t\leq 2\pi$,

and the position of another particle is given by

$x_2 = \cos t - 2, y_2 = 1 + \sin t, 0 \leq t \leq 2\pi$,

(a)  Sketch the paths of the particles on the same coordinate axes. (b)  How many points of intersection are there? (c)  Determine whether there is any point where the particles collide.
##### Sample Solution
Scroll down for the sample solution 🙂 The solution below is sample solution and not the accurate answer. 🙂 Please refer to your school teacher if you want to get the correct answer.

## Question 2

(a) Three sets of parametric equations.

$x=t^2+49,y=t+1,t\in R$

$x=49\sec^2 \theta, y=7\tan\theta, \frac{\pi}{2}<\theta<\frac{\pi}{2}$

$x=4t^2+49, y=2t+1, t\in R$

Let me know if you know more.
(b) $x=-t^2$ is impossible to be chosen as the parametric equation for $x$ because $x-49$ must be always positive as $(y-1)^2 \geq 0$ for every values of $y$. We can start with any choice for the parametric equation for $x$ as long as $x-49$ is not negative.
(c) Basically, we can start with any choice for the parametric equation for $y$ because $x=(y-1)^2+49$.

## Question 3

(a) The curve in pink is $2\sin t, y=3\cos t$. And the curve in black is $x=\cos t -2, y=1+\sin t$.
(b) Based on the graph above, there are two points of intersection.
(c) To determine whether the particles collide, we need to solve the simultaneous equations from the parametric equations. Please remember the particles does not collide even you are able to solve the equations. You must verify the values. Collision happens only when positions are the same with same $t$.

### Verification

Hence, the particles collide at $t=\frac{3\pi}{2}$.
Hence, the particles do not collide. Please do yourself for $y=0$ and $y=\frac{9}{5}$.   The particles collide at only 1 point.

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#### KK LEE

KK LEE has been a STPM Mathematics tuition teacher since June 2006, but his love of Maths dates back to at least 1999 when he was Form 4. KK LEE started teaching in 2006 at Pusat Tuisyen Kasturi. He was known as "LK" when he was teaching in PTK. After teaching STPM Mathematics for 8 years in PTK, he joined Ai Tuition Centre in 2014. Over the years he has taught Mathematics (T), Mathematics (S), Mathematics (M), Additional Mathematics.

1. Teacher may I ask a question…

Is number 2a) the equation for hyperbola or parabola…And if it’s hyperbola why is it so…
Thank you very much

• 2a is parabola

• Thank you sir…I really appreciate it..

2. Sir, I would like to ask on the equation x=e^t, did you eliminate the variable e or you just ignore it? If you eliminate it, how did you do it?

• Substitute the values of t into the equation of x

• HUH IF I DO SO. THEN GOT 2 UNKNOWNS LOR.

3. Teacher, how can i find the intersection points in 3(b)?

• Question is asking “HOW MANY points of intersection”. So based on the graph, there are two points of intersection. DONE.

• Any equations that satisfy the Cartesian equation.

5. can i know how to get the third parametric equation for question 2a ?I tried so many times and still din get that ans….sir,can u give me some clues..

• Refer the above comments, any parametric equations that satisfy the Cartesian equation given.

6. can i know for the 2Qb)for the choice is based on the equation on the question 2a? ? thanks ^^

• No

• that means i can create my own parametric equation for x without links to the parabola equation for question2a) ?

• Question is asking can or not. And not asking what is the parametric equation. so if you are going to create your own. Ask your teacher and good luck.

7. Sir , can your kindly write the introduction , methodology and conclusion ? Thx a lot for your kindness

• No

• Can you show me some guidance on doing the introduction , methodology and conclusion of this assignment .

• but teachers of my school just leave us to deal with it …. So I hope to get some information from you sir …

• Hi. Sorry. Cant help on this. 🙂

8. Sir…. for the parametric eqn right…. can this? :
1. y=sinϑ +1 ; x=50-(cosϑ)2 &
2. y=tanϑ +1 ; x=(secϑ)2-48……….

• yes. any equations

• How you get that parametic eqn? steps?

• With the understanding of parametric equations, you can obtain the three equation easily.

9. question 3C) on the verification part, after u sub the x value, why are they so many t value if use x=-6/5 but all prove it does not collide, but using x= -2 , many t value as well but only 1 prove it collide at that time. If t value we found prove it dont collide, then they are many t value outside which we dont find on the equation said it don collide.

question 3b)u sub x1 and x2 into sin^2 t +cos ^2 t = 1 and y value as well, and u found the point of intersection, but referring to the point we found on the graph , one of the point which is x=-6/5 does not seem to be logic, based on the graph it should -1.5 something, which clearly shown on the graph. does that mean it is wrong?

• For 3c. It is given that t lies between 0 and $latex 2\pi$.

For 3b. I NEVER substitute any values in my calculation. Thanks

• sry maybe i dont really understand, may i know for the first part of 3c), why do u sub x1/2 = sin t and x2 + 2 = cos t equation into sin^2 t + cos^2 t = 1?

• To solve the simultaneous equations. You can use any other ways to solve the simultaneous equations to find the points where the particles collide.

• ok, so we found the point of intersection using simultaneous equations, which have two value for both x and y , but one of the x value which is x= -6/5 does not seem to be logic ,can u explain this?? because graphical method also can be used to find the point of intersection right? so if i use graphical method then my x coordinate for the point of intersection is not the same although y coordinate correct.

• To find points of intersection of the curve and to find the point where the particles collide are not the same. Please ask your teacher or attend my class. 😛

• oic, then that makes clear that i dont really understand haha, cuz i thought u solve the simultaneous equation to find the point of intersection first, and assuming one of the particles be at that point then traces back t value and find coordinate of another particle.If x1=x2 and y1=y2 then it collide, So now your working is to find the point where particles collide? and may i know how to find point of intersection by using simultaneous equation?

• Ways to find the points of intersection will only be shown to my students. Thanks

• haha alright then, tq so much for your time ! appreciate it !

10. Is that q2 c is using b^2-4ac to solve ? And then we need to list up the possible parametric equation ?

• It depends on your teacher.

11. Is there any other way to find the collision of points for 3(c) ?

• There are many methods to find the points.

12. Question 2 how do start ? I don’t understand…

13. Sir, are you sure that t= 3.785 & 5.690 when sin t= -0.6? When I use the calculator, I got a different answer.

• Yes. It is correct. Change your calculator or change your hand.

• Its correct. When you press your calculator,you shouldnt insert the negative in. Just insert the 0.6. After that you will get 36.8698… The negative sign shows that it’s actually in the third and fourth quadrant.For the third quadrant, its (180+36.8698), after that change the ans in term of rad. Which means you should take [(180+36.87)xπ]/180 = 3.786
Do the same thing for the fourth quadrant (360-36.87) and you will get 5.64

You’re welcome 🙂

• Haha.. A long explanation. But, u dint utilise the power of your calculator. 🙁

• is 5.64 or 5.69？

• 5.639 so it is 5.64

• I have a different ans too..

14. can we use graph(method)to solve ques 2

• Any methods. It depends on your teacher. 🙂

15. My teacher says that the answer for 2c is wrong.If you substitute your y equals to t² ,my graph will be limited to positive only.But according to the graph my y has negative values.

16. How to write a suitable title?

• “Race between turtle and rabbit using parametric equation”

• Race between turtle and rabbit? is tht real?

• Why not?

• excuse me, but anybody here have intro, methodology and others for this assignment? idk how to do.

or sir, can you please write an entry abt how to write all this stuffs? thank you

17. Sir it seems that there is more than 1 graph fr various valuess of t

• What you mean?

18. Sir,Mat i know how to do the intro??

19. Sir for the question 3c
I think you made a mistake
Its 5x^2 + 16x + 15
Not 5x^2 + 16x + 12

• I think 5x^2+16x+12 is correct.

20. Hello sir. How to calculate the position of both particles collide in Q3 c）when t=3.785 and 5.640?

• Subs into the equation of x and y

• Which equation of x and y？

• Please refer to the question. It is stated there

21. Sir, how do i explain the part for 3(c) in my methodology, whereby i change the sin^-1(-0.6) to 3.785 and 5.64?

• “By calculator”

22. Sir….. Regarding no.2(a) … If I let x=t^2 , y=+-(surd t^2-49)+1…. In this case, I’ll have 3 parametric equations. One x and two y. So which of the y parametric eqn should I choose to get a set??

23. Sir , how to find three sets of parametric equation like question 2a ?

• Any equations that satisfy the Cartesian equation

24. Sir,may i ask y did u chg the ans to rad when using x=-6/5 in verification step…thx in advance

• It is in radian from the beginning. I dint change anything.

25. Sir,at verification step, if i substitute x=-2 into x=cos t – 2, then cos t would be 0 and i will get 2 answers, so i just have to conclude that the particles would not collide when t=pi or is it that i am only supposed to use x=2sin t

• Use ALL equations to verify.

26. sir, do you have any example or tips on how to do the methodology and conclusion?

• Nope

27. Sir, why the 2nd parametric equation in 2(a)is y=7tanθ, but not y=7tanθ+1? I thought it should be y=7tanθ+1…

• May I know How do you find the parametric eqn for y=7tan + 1 ??

• Make sure you can get all the points.

28. sir…may i ask you one quest…for the answer of Q2 n Q3…should i write on methodology or results?

• result

29. another quest,,,,,where did you get t=3/2 pi….for Q3(c) verification,,,,when i press on the calculator,,,the answer is diffrence from yours

• Make sure you are using the rad mode. Else change your calculator.

30. sir, i notice sumthing…… according to u….
when x=-6/5 the 2nd t= 5.690
but t=5.640 when u are relating the t to both curves……
i already tried using calculator…. t=5.640 is more reasonable……

• I cannot understand what you asking. I hope you have solved your question.

31. sir,how to do the methodology for those question?

32. sir,,,why did you used sin^2t-cos^2t=1 for quest 3 C).
why did you used alphbet t??can we use any aplphbets other than t?

• Because there is t in the question.

• thanks a lot sir

33. sir,may i know,,where did you get the coordinates for question 3 C)?did you subsitute the values of t in the equation of x=2sint,x=cost-2,y=3cost,y=1+sint?

• yes

34. Sry sir…it’s tht can,if I just gv the definition of parametric equ n some example for intro.^^

• Ya. It depends on your teacher.

35. sir can i know how to do the conclusion. what point must we put in the conclusion

• Thank you sir. Sir why x-49 must be in positive?

• Because $latex \sqrt(x-49)$ is defined only if $latex x-49>=0$.

36. Sir , can i know how to draw the graph using computer

37. Sir, does t represent the time when the particles collide?

• t represents the time, any time when the particle is moving

• kk, tq sir

38. Sir, can give me some idea how to write the title

39. Sir can i know how to find the range of the second for question 2(a)?

• You have to test one by one to get the complete curve.

40. Sir what means of this “$latex \sqrt(x-49)$ is defined only if $latex x-49>=0$.”

• x-49 must be positive, else the square root function cannot be defined.

41. For question 3c, x=-6/5 and -2,then to find t,I have to use equation x=2sin t right?Should I use another equation of x ,(x=cos t-2)to find t?

• You can use any equations.

42. For the 2a) question right, x= 4t^2+49 y= 4t+1
Sir am i right?

• x= 4t^2+49 y= 2t+1

• NO sir. i mean x= (4t)^2+49 y= 4t+1
So sir. am i right?

• can

43. sir, for the question 2a) can i just substitute the x=e^t into the eq 2a as my parametric equation?

• Can. If you can find the corresponding y

44. Teacher,can you tell me how to get 3 parametric equation in 2a.please?

45. Teacher,can you give me same tips on how to write methology and conclusion because our maths t teacher already retire and want to wait until next year new teacher come and we don’t have any teacher guide us.Thanks you for your reply~

• For methodology, try to write about the methods/theorem/ways you used to solve the questions.

For conclusion, you can rewrite again all your important result from each part and write one or two example that related.

46. Sir,can you tell me how to get 3 parametric equation in Q2a,thx.

• It can be obtained directly if you know what is parametric equation. I can’t explain further here.

47. Hi ?I want to ask that the questions 2)a) are we need to form 3 parametric equation form that cartesian equation or just sub the x or y value that you give into that cartesian equation??? Thank you ?

• You need to form 3 equations.

• Ok, thank you very much.

49. Sir, for question 2 a) can I let x=t, x=2t and x=t^2 and in each case express y in terms of t. [x=t, y=+/- square root(x-49)-1] is a set?

• For me, No. For your teacher, i don’t know.

50. Can i use y=t,y=(t^2)-1 and y=-t^2 to prove that it can used for y in 2(c)?
As my teacher want us to show the equation….

• Can. If your teacher accept.

51. Sir,what should i write for an introduction for parametric equation?

52. Sir, is it 2(a) need to plot the curve??because the question have say “….the curve whose equation is…..” so need to sketh a parabola graph or not??

53. hi sir , 🙂 can i ask about question 2(b) why cannnot ?

• Hi. Reason is stated there.

54. Sir,for question 2(a),How to find the other 2 sets of parametric equations?For set 1,I used the formula x=at^2 and y=2at…but how about the other 2 sets?How can i get the other 2 sets?I have no idea…Just use back the same formula or there is another way to solve this?

• Does it mean any parametric equations of curve as long as it satisfy the Cartesian equation?So,If that is true,can I use the parametric equation of hyperbola?

• You can use any equation that satisfy the original equation.

• How to meant by satisfy the original equation?

• Ohh…I think maybe i get what you mean…Thanks a lot!

55. Hi Sir, to find if the particles collide in 3c, can I do a table of t(0, pi/2, pi, 3pi/2, 2pi values), x1, x2, y1 & y2? and subs t values into the original equations in the questions to find the (x,y)coordinates that are the same? The answer will still be 3pi/2 as point where they collide.

• For me, NO. For your teacher, i don’t know.

56. Can you explain more from 3.C Verification? Only subs the value of x on x= 2 sin t?

• Substitute into all the original equations to verify

57. Hi sir, may i know that for the question 3, should i use the same value as you had used for the simultaneous equation and verification or should i use a different set of values?

• Same

58. Hello, sir. How do I get the values of t for Q3?

• Solve the simultaneous equations

59. 2a can use standard form of parabola to create the equation?

• For me NO. For your teacher i dont know

60. that helps alot.thanks

61. Sir, can you please give me some example of the application of parametric equation in our daily life??? Because I cannot find it in the internet……..

62. Hi sir
For 2a
Can I use x=50-sin^2€ and y=cos€+1 and the range of € is between 0 and pi?
I have calculated the values using the range but it seems to be a graph with y-axis as major axis but the other two equations above are graphs with x-axis as major axis..

• You can try to switch the x and y

63. Sir,may I know how to get 3.785 and 5.640 when x=-1.2

• Using calculator

64. Sir , for x=-2 and y=0 , I’ve got the value of t that is (3π/2) , so is it that the coordinates of the ellipse and circle for x=-2 AND y=0 are the same ?

• Yes

65. Sir,how to write the conclusion and methodology?

66. Excuse me teacher? I would like to ask about what should i write for any comment or statement after i plot the curve(question 1)?Can you guide me some suggestions?

• You can explain the shape.

67. sir for question 2a ,may i make the original equation into this form ? x²/36² – y²/6² = 1 ? thn my parametric eqaution will be like this,x = 36 sec ,y = 6 tan

• For 2a. It is a parabola, which means it cant be x²/36² – y²/6² = 1

68. u know what , u r rude, n i read all your replies to all the comments.. u kinda reply in a sarcastic way.. just chill on , cause we r new to this kinda no guide assignment.. we need some help bcoz all of our school teacher dont give a damn.. so the point is have some courtesy to share knowledge… hope u get me, stingy 🙂

• All right. Thanks of your comment.

1. I am not a school teacher. And i never attend any meeting regarding the assignment. So do not expect me to know all the school teachers requirement. I have tried my best to give the information that i know. I have students from different schools. That’s why i know different school teachers have different requirement on the particular questions. This is also the reason i never answer all the questions that are not related to maths.

2. I am posting all maths related solutions for your reference and please do not expect me to post 100% answers.

3. I have tried my best to help and this will be my last posting on solutions of assignments. Thanks.

• U r so disgusting u know?people give u a sample to refer,u never say tanks, determine, but u still complain that the ans is not acceptable in your school?so ridiculous,no one is a must to provide amethyst is needed by your teacher,ok???I feel pity on u….

69. sir, do you have math (T) TERM 2 assignment solution?

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