Length of a Plane Curve-Previous year questions

Previous Year Questions on Length of a Plane Curve

Length of a Plane Curve is a critical concept in Mathematics. Historically, the computation of arc lengths has enabled the development of engineering designs and advanced scientific models. Its importance lies in applications spanning physics, computer graphics, and Integral Calculus. Explore Mathematics questions from various universities to deepen understanding and enhance problem-solving skills.

Vidyasagar University

2023-24 (NEP)

For a curve, if \(x\sin{\theta}+y\cos{\theta}=f^{\prime}(\theta)\) and \(x\cos{\theta}-y\sin{\theta}=f^{\prime\prime}(\theta)\), prove that \(s=f(\theta)+f^{\prime\prime}(\theta)+k \), where \(k\) is a constant. [2]
Find the length of the curve \(x=e^{\theta}\sin{\theta}\) and \(y=e^{\theta}\cos{\theta}\) between \(\theta=0\) to \(\theta=\frac{\pi}{2}\). [2]

2023-24 (CBCS)

2022-23 (CBCS)

2021-22 (CBCS)

Show that the arc of the upper half of the cardiode \(r=a(1-\cos{\theta}) \) is bisected at \(\theta=\frac{2}{3}\pi \). Find also the perimeter of the curve. [4]
Compute the length of the curve \(x=2\cos{\theta},~y=\sin{2\theta},~0\leq \theta \leq \pi \) [3]

2020-21 (CBCS)

Determine the length of one arc of the cycloid \( x=a\big(\theta- \sin{\theta}\big),~y=a\big(1- \cos{\theta}\big) \). [4]
Find the whole length of the loop of the curve \( 9ay^{2}=\big(x-2a \big)\big(x-5a \big)^{2}\). [6]

2019-20 (CBCS)

Find the length of the curve \(x=e^{\theta}\sin{\theta}\) and \(y=e^{\theta}\cos{\theta}\) between \(\theta=0\) to \(\theta=\frac{\pi}{2}\). [2]

2018-19 (CBCS)

Show that in the astroid \(x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}\); \(s\) being measured from the point for which \(x=0\). [2]
Show that the arcs of the curves \(x=f(t)-\phi^{\prime}(t),~y=\phi(t)+f^{\prime}(t)\) and \(x=f^{\prime}(t)\sin{t}-\phi^{\prime}(t)\cos{t},~y=f^{\prime}(t)\cos{t}+\phi^{\prime}(t)\sin{t}\) corresponding to same interval of variation of \(t\) have equal lengths. [5]

2017-18 (CBCS)

If \(s\) be the length of the arc \(3ay^{2}=x\big( x-a\big)^{2}\) measured from the origin to any point \((x,y) \), show that \begin{align*} 3s^{2}=4x^{2}+3y^{2}~~[5] \end{align*}
Find the arc length parameter along the curve \begin{align*} C:\vec{r}(t)=\big(1+2t \big)\hat{i}+\big(1+3t \big)\hat{j}+6\big(1-t \big)\hat{z} \end{align*} from the point , where \(t=0\). [5]

FAQs

What is the formula to find the Length of a Plane Curve?
The formula integrates the arc length by using Integral Calculus.
Why is the Length of a Plane Curve important in real-life applications?
It is used in fields like engineering and physics for designing curves and calculating distances.
Where can I find solved examples for this topic?
Solved examples are available in Mathematics questions sections.
How is this topic connected to Differential Calculus?
Differential Calculus provides the foundational formulas for calculating derivatives essential in arc length computation.
What are some typical questions from university exams?
Common questions involve solving problems using parametric or explicit equations.
How is this topic tested in Integral Calculus?
It is tested through integration techniques taught in Integral Calculus.
What are parametric curves?
Parametric curves describe the plane curve using parameters instead of explicit equations.
What role does this concept play in geometry?
It helps measure the exact arc lengths of geometric curves.
How are university syllabi aligned with this topic?
Most universities include it under Mathematics courses in Differential and Integral Calculus.
What additional resources are recommended?
Textbooks listed above and university question papers can supplement your learning.