Discussion – 

175

Discussion – 

175

STPM 2016 Mathematics (T) Term 1 Assignment

STPM 2016 Term 1 Mathematics (T) 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 1

Question 1 Table
Question 1 Graph

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.
Question 3a
(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.
Simultaneous equation for 3c

Verification

Question 3c Checking
Hence, the particles collide at t=\frac{3\pi}{2}.
Checking again
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.

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