A curve is defined by the parametric equations $$\[ x=3 t-2 \sin t, \quad y=5 t+4 \cos t, \]$$ where $$$0 \leqslant t \leqslant 2 \pi$$$. At each of the points $$$P$$$ and $$$Q$$$ on the curve, the gradient of the curve is $$$\frac{5}{2}$$$. Show that the values of $$$t$$$ at $$$P$$$ and $$$Q$$$ satisfy the equation $$$10 \cos t-8 \sin t=5$$$. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mathematics

IGCSE&ALevel

CAIE

Exam No：9709_w20_qp_23 Year：2020 Question No：7(a)

Answer:

Obtain $\frac{\mathrm{d} x}{\mathrm{~d} t}=3-2 \cos t$ and $\frac{\mathrm{d} y}{\mathrm{~d} t}=5-4 \sin t$
Equate expression for $\frac{\mathrm{d} y}{\mathrm{~d} x}$ to $\frac{5}{2}$
Obtain $10 \cos t-8 \sin t=5$

Knowledge points:

2.4.1 use the derivatives of ,ln x,sin x,cos x,tan x, together with constant multiples, sums, differences and composites

2.4.3 find and use the first derivative of a function which is defined parametrically or implicitly.

Answer:

Knowledge points:

Solution:

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