The curve $C$ is defined by the parametric equations \(x = \cos t + \frac{1}{2}\cos 5t, \quad y =

Question

The curve $C$ is defined by the parametric equations $x = \cos t + \frac{1}{2}\cos 5t, \quad y = \sin t + \frac{1}{2}\sin 5t \quad \text{for } 0 \le t \le \frac{\pi}{2}.$ Find the coordinates of the point on the curve $C$ corresponding to $t=0$.[1] Another curve $D$ has the following parametric equations. $x = 2 + \sqrt{k}\cos\theta, \quad y = \frac{3}{2}\sin\theta \quad \text{for } 0 \le \theta \le 2\pi \text{ and } k > 0.$ Find the cartesian equation for curve $D$.[2] Sketch the curves $C$ and $D$ on the same diagram.[2] Hence determine the range of values of $k$ such that the equation $\frac{(\cos t + 0.5\cos 5t - 2)^2}{k} + \frac{4(\sin t + 0.5\sin 5t)^2}{9} = 1$ has no real solutions.[1]

Accepted Answer

(a)coordinates : $(\frac{3}{2},0)$ (b)$\frac{(x-2)^2}{k} +\frac{4y^2}{9} =1$ (d)k is positive, $0<$ $k<\frac{1}{4}$

2025 ASRJC P2 Q4

2025 ASRJC P2 Q4