Vector Analysis-Curves in Space, Curvature and Torsion [IAS (Admin.) Mains Mathematics]: Questions 1 - 7 of 7

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Question 1

Appeared in Year: 2007

Describe in Detail Subjective▾

For any constant vector show that the vector represented by curl is always parallel to the vector being the position vector of a point measured from the origin. (Paper 1)

Edit

Explanation

Let

Also given is the position vector of a point measured from the origin.

… (110 more equations) …

Question 2

Appeared in Year: 2007

Describe in Detail Subjective▾

Find curvature and torsion at any point of the curve (Paper 1)

Edit

Explanation

Let be the position vector of any point on the given curve.

… (592 more equations) …

Question 3

Appeared in Year: 2019

Describe in Detail Subjective▾

Find the radius of curvature and radius of torsion of helix

(Paper-1)

Edit

Explanation

Curvature ; radius of curvature

… (115 more equations) …

Question 4

Appeared in Year: 2022

Describe in Detail Subjective▾

Trace the curve , where a is real constant (Paper 1)

Edit

Explanation

Symmetry: the curve is symmetrical about both the axes.

Origin does not lie on the curve

Points of intersection: put , we get

So, the curve passes through the points and

Asymptotes: the asymptotes parallel to th…

… (49 more equations, 6 figures) …

Question 5

Appeared in Year: 2022 (IFS)

Describe in Detail Subjective▾

If a curve in a space is represented by then derive expressions of its torsion and curvature in terms of . Find the curvature and torsion of the curve given by

.

(Paper -1)

Edit

Explanation

Let

… (109 more equations) …

Question 6

Appeared in Year: 2023

Describe in Detail Subjective▾

Trace the curve (Paper 1)

Edit

Explanation

The curve is symmetrical about the x-axis

It does not pass through the origin

Illustration: 2023-Mathematics-I, Paper 1

The curve meets the x-axis at the point and the y-axis at the point

… (28 more equations, 6 figures) …

Question 7

Appeared in Year: 2023

Describe in Detail Subjective▾

If the tangent to a curve makes a constant angle with a fixed line, then prove that the ratio of radius of torsion to radius of curvature is proportional to . Further prove that if this ratio is constant, then the tangent makes a constant angle with a fixed direction. (Paper 1)

Edit

Explanation

We know that in calculus of space curve, we calculate the quantities curvature (k) and torsion both have inverse-length as units, so their reciprocals and have units of length, and are called radius of curvature and radius of torsion.

Let e, be the unit vector parallel to the g…

… (350 more equations) …