Q105 / 160 · TS EAMCET 2023 (14 May Shift 2) · Mathematics

Match the values of $\frac{d y}{d x}$ at $x=\frac{\pi}{3}$ for the following system of curves in parametric form given in List-I with those of the items in List-II Figure

  • Correct answer
    A
    (i) $\rightarrow$ C, (ii) $\rightarrow$ D, (iii) $\rightarrow$ B, (iv) $\rightarrow$ A
  • B
    (i) $\rightarrow$ C, (ii) $\rightarrow \mathrm{E}$, (iii) $\rightarrow \mathrm{D}$, (iv) $\rightarrow \mathrm{A}$
  • C
    (i) $\rightarrow \mathrm{D}$, (ii) $\rightarrow$ C, (iii) $\rightarrow \mathrm{B}$, (iv) $\rightarrow \mathrm{A}$
  • D
    (i) $\rightarrow$ D, (ii) $\rightarrow$ C, (iii) $\rightarrow$ E, (iv) $\rightarrow$ B

Solution

(i) $x=a(\theta-\sin \theta)$ $\begin{aligned} & \frac{d x}{d \theta}=a(1-\cos \theta)=a\left(2 \sin ^2 \frac{\theta}{2}\right) \\ & y=a(1-\cos \theta) \\ & \frac{d y}{d \theta}=a \sin 6=2 a \sin \frac{\theta}{2} \cos \frac{\theta}{2}\end{aligned}$ $\begin{aligned} & \therefore \quad \frac{d y}{d x}=\frac{\frac{d y}{d \theta}}{\frac{d x}{d \theta}}=\frac{2 a \sin \frac{\theta}{2} \cos \frac{\theta}{2}}{2 a \sin ^2 \frac{\theta}{2}}=\cot \frac{\theta}{2} \\ & \left.\therefore \quad \frac{d y}{d x}\right|_{x=\frac{\pi}{3}}=\sqrt{3} .\end{aligned}$ (ii) $x=3 \cos \theta-2 \cos ^3 \theta$ $\begin{aligned} & \frac{d x}{d \theta}=-3 \sin \theta+6 \cos ^2 \theta \sin \theta \\ & =3 \sin \theta\left(2 \cos ^2 \theta-1\right)=3 \sin \theta \cos 2 \theta \\ & y=3 \sin \theta-2 \sin ^3 \theta \\ & \frac{d y}{d \theta}=3 \cos \theta-6 \sin ^2 \theta \cos \theta \\ & =3 \cos \theta\left(1-2 \sin ^2 \theta\right)=3 \cos \theta \cos 2 \theta \\ & \frac{d y}{d x}=\frac{\frac{d y}{d \theta}}{\frac{d x}{d \theta}}=\frac{3 \cos \theta \cos 2 \theta}{3 \sin \theta \cos 2 \theta}=\cot \theta \\ & \left.\therefore \quad \frac{d y}{d x}\right|_{\pi / 3}=\frac{1}{\sqrt{3}}\end{aligned}$ Clearly (i) $\rightarrow C$ and (ii) $\rightarrow D$.

Hence, the correct answer is option: 'A'.

Relevant Questions