Differential and Integral Calculus Problems focuses on “Evolutes”.
1. The name of the evolute of an ellipse is ___________
a) Centroid
b) Astroid
c) Asteroid
d) Cycloid
Answer: b
Explanation: Astroid is known as the evolute of ellipse whereas centroid is associated with triangle. Asteroids are small planet like structures found in space. Cycloid is the curve traced by a point on the circumference of a circle.
2. The curvature of a curve is equal to ____________
a) Reciprocal of radius of curvature
b) Radius of Curvature
c) Twice the radius of curvature
d) One
Answer: a
Explanation: Curvature is the property of a curve by which the curve deviates from that of a straight line. It is equal to the reciprocal of the radius of curvature of the curve. The radius of curvature is equal to the radius of the curve by magnitude.
3. Involute is also known as ___________
a) Evolute
b) Evolvent
c) Envelope
d) Tangent
Answer: b
Explanation: The curve itself is the involute of the evolute of the curve with a known starting point. The other name for involute is evolvent.
4. What is the evolute of parabola called?
a) Cycloid
b) Spiral
c) Congruent parabola
d) Semicubical parabola
Answer: d
Explanation: The evolute of parabola is called semicubical parabola. It is defined parametrically as
x=t2 and y=at3.
5. Definition of evolute of a curve is ___________
a) locus of centre of the given curve
b) locus of centre of tangential curve
c) locus of circumferential point on the curve
d) locus of tangent to the curve
Answer: a
Explanation: The evolute of a curve is defined as the locus of centre of the given curve. It is the path traced by the centre of the curve.
6. What is the curvature of the curve x2 + y2 = 25?
a) 5
b) 25
c) 0.5
d) 0.2
Answer: d
Explanation: From the equation, it is clear that the radius of the curve (circle) is 5 units. The curvature is equal to the reciprocal of radius. Hence, curvature = 1/5 = 0.2.
7. What is the curvature of straight line?
a) infinity
b) one
c) zero
d) length of the straight line
Answer: c
Explanation: A straight line is a curve of infinite radius. Curvature = 1/radius. Hence, the curvature of straight line is 1/infinity which is equal to zero.
8. Number of possible evolutes for a curve is ____________
a) Two
b) Equal to radius
c) One
d) Infinity
Answer: c
Explanation: A curve possesses only a single evolute. It can have infinite number of involutes.
9. The curvature of a plane curve at K is _________
a) one
b) d ψ/ds
c) zero
d) infinity
Answer: b
Explanation: The curvature of a plane curve is given as d ψ/ds. The curvature of straight line is zero.
10. What is the radius of curvature of the curve xy = c2 at (c,c)?
a) c
b) 2
c) 2c
d) √2 c
Answer: d
Explanation: y = c2/x
Integrating,
(y_1 = frac{-c^2}{x^2} = frac{-c^2}{c^2} = -1)
(y_2 = frac{2c^2}{x^3} = frac{2c^2}{c^3} = frac{2}{c})
Radius of curvature = (frac{left(1+y_{1}^{2}right)^frac{3}{2}}{y_2} = frac{left(1+1right)^frac{3}{2}}{frac{2}{c}} = frac{c.2^{frac{3}{2}}}{2} = sqrt{2}c.)
Global Education & Learning Series – Differential and Integral Calculus.
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