Which of the following space curve represent spiral curve? तलका मध्ये कुनचाही space curve ले spiral (भुरुङुको डोरी जस्तो curve) जनाँउदछ Tick all that apply A \( \square \mathrm{x}=\operatorname{cost}, \mathrm{y}=\sin \mathrm{t}, \mathrm{z}=\mathrm{t} / 5 \) 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 \( \square \mathrm{x}=2 \sin t, y=0, z=2 \operatorname{cost} \)

We analyze each space curve to see if it represents a spiral (a curve that winds around an axis while moving away from or along that axis). **Option A:** The parametric equations are \[ x = \cos t,\quad y = \sin t,\quad z = \frac{t}{5}. \] The \(x\) and \(y\) components form a circle (since \(\cos^2 t + \sin^2 t = 1\)). The \(z\) component increases linearly with \(t\), so the curve wraps around the \(z\)-axis while moving upward. This is a helix, which is a type of spiral curve. **Option B:** The parametric equations are \[ x = t\cos t,\quad y = t\sin t,\quad z = t. \] Here, the \(x\) and \(y\) components describe a circle whose radius increases with \(t\) (since the radius is \(t\)). With \(z = t\), the curve moves upward as \(t\) increases while expanding radially. This is an expanding spiral (or conical spiral). **Option C:** The parametric equations are \[ x = t^2,\quad y = t^2+3,\quad z = t^2-1. \] In this case, all components are functions of \(t^2\), meaning that the curve is confined to a plane since there is a linear relation between the coordinates (for example, \(y = x + 3\) and \(z = x - 1\)). There is no circular or winding motion. Thus, it does not represent a spiral. **Option D:** The parametric equations are \[ x = 2\sin t,\quad y = 0,\quad z = 2\cos t. \] The \(x\) and \(z\) components describe a circle of radius 2 in the \(xz\)-plane, while \(y=0\) is constant. This is simply a circular curve, not a spiral, because it does not move along an axis. Thus, the curves that represent spiral curves are Options A and B. Final Answer: A, B.