MCQs of Vector Calculus (Mathematics - II - 3110015)

Mathematics - II (3110015) MCQs

MCQs of Vector Calculus

Showing 41 to 50 out of 79 Questions

41.

Let

r = (x (t), y (t), z (t))

be the position vector on a curve

C

in space at any point

t

. Then the arc length of curve

C

form point

a

b

L

=

(a)	$\int_{a}^{b} r' (t) dt$
(b)	$\int_{a}^{b} {[r' (t)]}^{2} dt$
(c)	$\int_{a}^{b} r (t) dt$
(d)	$\int_{a}^{b} \|r' (t)\| dt$

42.

The arc length of curve of the portion of circular helix

r = \cos (t) \hat{i} + \sin (t) \hat{j} + t \hat{k}

from

t = 0

t = π

(a)	$2 π$
(b)	$- 2 π$
(c)	$\sqrt{2} π$
(d)	$\sqrt{2 π}$

43.

The length of the astroid

x = a {(\cos (θ))}^{3}

y = a {(\sin (θ))}^{3}

in the first quadrant is

(a)	$0$
(b)	$\frac{3}{4} a$
(c)	$3 a$
(d)	$\frac{3}{2} a$

44.

Consider the curve

y = f (x)

, then the arc length from

x = a

x = b

can be expressed as

(a)	$\int_{a}^{b} \sqrt{1 + (\frac{d y}{d x})} dx$
(b)	$\int_{a}^{b} \sqrt{1 - {(\frac{d y}{d x})}^{2}} dx$
(c)	$\int_{a}^{b} \sqrt{1 + {(\frac{d y}{d x})}^{2}} dx$
(d)	$\int_{a}^{b} \sqrt{{[1 + (\frac{d y}{d x})]}^{2}} dx$

45.

The length of the curve

y = \log (\sec (x))

from

x = 0

x = \frac{π}{3}

(a)	$\log (2 + \sqrt{3})$
(b)	$\log (2 - \sqrt{3})$
(c)	$\log (2 + \frac{1}{\sqrt{3}})$
(d)	$\log (2 - \frac{1}{\sqrt{3}})$

46.

Consider the curve

r = f (θ)

, then the arc length from

θ = θ_{1}

θ_{2}

can be expressed as

(a)	$\int_{θ_{1}}^{θ_{2}} \sqrt{1 + (\frac{d r}{d θ})} dθ$
(b)	$\int_{θ_{1}}^{θ_{2}} \sqrt{r^{2} + {(\frac{d r}{d θ})}^{2}} dθ$
(c)	$\int_{θ_{1}}^{θ_{2}} \sqrt{r^{2} + (\frac{d r}{d θ})} dθ$
(d)	$\int_{θ_{1}}^{θ_{2}} \sqrt{1 + {(\frac{d r}{d θ})}^{2}} dθ$

47.

The length of the spiral

r = e^{2 θ}

from

θ = 0

θ = 2 π

(a)	$\frac{\sqrt{5}}{2} e^{4} π$
(b)	$\frac{\sqrt{5}}{2} (e^{4 π} - 1)$
(c)	$\frac{\sqrt{5}}{2} e^{4 π}$
(d)	$\frac{\sqrt{5}}{2} (e^{4} π - 1)$

48.

Any integral which is to be evaluated along a curve is called

(a)	Double integral
(b)	Triple integral
(c)	Line integral
(d)	Solenoidal

49.

Let

C : r = x (t) \hat{i} + y (t) \hat{j} + z (t) \hat{k}

be an oriented curve. Then the line integration of scalar function

f (x, y, x)

along the curve

C

from point

a

b

is defined as

(a)	$\int_{t = a}^{t = b} f (x (t), y (t), z (t)) \cdot r (t) dt$
(b)	$\int_{t = a}^{t = b} f (x (t), y (t), z (t)) \cdot r^{'} (t) dt$
(c)	$\int_{t = a}^{t = b} f (x (t), y (t), z (t)) \cdot \|r^{'} (t)\| dt$
(d)	$\int_{t = a}^{t = b} f (x (t), y (t), z (t)) \cdot \|r (t)\| dt$

50.

Let

C

be the straight line joining the points

A (0, 0, 0)

B (1, 1, 1)

. Then the line integration of

f (x, y, z) = x^{2} + y^{2} + z^{2}

over the curve

C

(a)	$3 \sqrt{3}$
(b)	$\sqrt{3}$
(c)	$0$
(d)	$- 3 \sqrt{3}$

Showing 41 to 50 out of 79 Questions