Instructions Input the value of \( \mathrm{x}, \mathrm{y} \), and z in the applet above. Based on the output, answer the questions given below. 1. \( \mathrm{x}= \) cost, \( \mathrm{y}=\operatorname{sint}, \mathrm{z}=\mathrm{t} / 5 \) 2. \( \mathrm{x}=\mathrm{t} \operatorname{cost}, \mathrm{y}=\mathrm{t} \operatorname{sint}, \mathrm{z}=\mathrm{t} \) 3. \( \mathrm{x}=\mathrm{t}^{2}, \mathrm{y}=\mathrm{t}^{2}+3, \mathrm{z}=\mathrm{t}^{2}-1 \) 4. \( \mathrm{x}=2 \operatorname{sint}, y=0, z=2 \operatorname{cost} \) Question 1 Which of the following space curve represent helix? तलका मध्ये कुनचाही space curve ले helix (कलमको spring जस्तो curve) जनाँउदछ? Tick all that apply A \( \square \mathrm{x}= \) hcost, \( \mathrm{y}=\operatorname{sint}, \mathrm{z}=\mathrm{t} / 5 \) B \( \square \mathrm{x}=\mathrm{t} \operatorname{cost}, \mathrm{y}=\mathrm{t} \operatorname{sint}, \mathrm{z}=\mathrm{t} \) C \( \square \mathrm{x}=\mathrm{t}^{2}, \mathrm{y}=\mathrm{t}^{2}+3, \mathrm{z}=\mathrm{t}^{2}-1 \) D \( \square x=2 \operatorname{sint}, y=0, z=2 \operatorname{cost} \)

1. In option A, we have \[ x=\cos t,\quad y=\sin t,\quad z=\frac{t}{5}. \] The \(x\) and \(y\) components define a circle of constant radius (since \(\cos^2 t+\sin^2 t=1\)), while the \(z\) component is a linear function of \(t\). This is the standard parameterization of a helix. 2. In option B, we have \[ x=t\cos t,\quad y=t\sin t,\quad z=t. \] Here, the \(x\) and \(y\) components yield a radius of \[ \sqrt{(t\cos t)^2+(t\sin t)^2}=\sqrt{t^2}=|t|, \] which is not constant but increases with \(t\). Thus, this curve is not a traditional helix with constant radius but rather an outward spiral. 3. In option C, the functions are quadratic in \(t\) (i.e., \[ x=t^2,\quad y=t^2+3,\quad z=t^2-1, \] ), so the path does not involve circular motion in any plane. It does not represent a helix. 4. In option D, we have \[ x=2\sin t,\quad y=0,\quad z=2\cos t. \] This parameterization describes a circle in the \(xz\)-plane with constant radius \(2\) and does not have a linear component along any axis. Hence, it is a planar circle, not a helix. Thus, the only space curve that represents a helix is: - Option A.