Mathematics-III (Differential Calculus) - End Semester Examination - 2022

The value of \( \lim_{x \to 0} \left(\frac{\sin x}{x}\right)^{1/x} \) is

1. 0
2. 1
3. e
4. \( 1/e \)

The value of the integral \( \int_c \{yz dx + (xz+1)dy + xy dz\} \) where C is any path from (1, 0, 0) to (2, 1, 4) is

1. 6
2. 7
3. 8
4. 9

The maximum value of \( \sin x + \cos x \) is

1. 1
2. 2
3. \( \sqrt{2} \)
4. 0

The value of \( \nabla^2 [(1-x)(1-2x)] \) is equal to

1. 2
2. 3
3. 4
4. 6

The degree of the differential equation \( y\frac{dx}{dy} - \left(\frac{dx}{dy}\right)^2 - \sin y \left(\frac{dx}{dy}\right)^3 - \cos x = 0 \) is

1. 0
2. 1
3. 2
4. Cannot be determined

If \( f = \tan^{-1}\left(\frac{y}{x}\right) \), then div (grad \( f \)) is equal to

1. 1
2. -1
3. 0
4. 2

If \( P_n \) is the Legendre polynomial of first kind, then the value of \( \int_{-1}^1 x P_n P_n' dx \) is

1. \( \frac{2}{(2n+1)} \)
2. \( \frac{2n}{(2n+1)} \)
3. \( \frac{2}{(2n+3)} \)
4. \( \frac{2n}{(2n+3)} \)

If \( J_n \) is the Bessel's function of first kind, then the value of \( J_{-1/2} \) is

1. \( \sqrt{\frac{2}{\pi x}}\left(\frac{\cos x}{x} - \sin x\right) \)
2. \( \sqrt{\frac{2}{\pi x}}\left(\frac{\sin x}{x} - \cos x\right) \)
3. \( \sqrt{\frac{2}{\pi x}}\sin x \)
4. \( \sqrt{\frac{2}{\pi x}}\cos x \)

The solution of \( p \tan x + q \tan y = \tan z \) is

1. \( \sin x / \sin y = \varphi(\sin y/\sin z) \)
2. \( \sin x \cdot \sin y = \varphi(\sin y/\sin z) \)
3. \( \sin x / \sin y = \varphi(\sin y, \sin z) \)
4. \( \sin x / \sin y = \varphi(\sin y \cdot \sin z) \)

The vector \( \vec{v} = e^x \sin y \hat{i} + e^x \cos y \hat{j} \) is

1. Solenoidal
2. irrational
3. rotational
4. cannot be found

Form the partial differential equation \( (x-a)^2 + (y-b)^2 + z^2 = 1 \).

Solve \( xp + yq = 3z \).

Find the directional derivative of \( \phi = z^2yz + 4xz^2 \) at the point (1, -2, 1) in the direction of the vector \( 2\hat{i} - \hat{j} - 2\hat{k} \).

Find a unit vector normal to the surface \( x^3 + y^3 + 3xyz = 3 \) at the point (1, 2, -1).

Solve partial differential equation \( \frac{y^2 z}{x}p + xzq = y^2 \).

Show that the function \( f(x,y) = \begin{cases} \frac{xy}{\sqrt{x^2+y^2}}, & (x,y) \neq (0,0) \\ 0, & (x,y) = (0,0) \end{cases} \) is continuous at origin.

If \( f = (x^2+y^2+z^2)^{-n} \) then find div grad \( f \) and determine n, if div grad \( f = 0 \).

Verify Green's theorem for \( \int_C \{(xy+y^2)dx + x^2 dy\} \) Where C is bounded by \( y=x \), \( y=x^2 \).

Evaluate the integral by changing the order of integration \( \int_0^\infty \int_0^x x e^{-\frac{x^2}{y}} dy dx \).

Solve the differential equation \( (x^2+y^2+x)dx - (2x^2+2y^2-y)dy = 0 \).

Verify the stokes' theorem for \( \vec{A} = (y-z+2)\hat{i} + (yz+4)\hat{j} - xz\hat{k} \) Where S is the surface of the cube \( x=0, y=0, z=0, x=2, y=2 \) and \( z=2 \) above the xy-plane.

Prove that \( 2nJ_n(x) = x(J_{n-1}(x) + J_{n+1}(x)) \).

Prove that \( \sum_{n=0}^\infty \frac{x^{n+1}}{n+1} P_n(1) = \frac{1}{2} \log\left(\frac{1+x}{1-x}\right) \).

Using Green's theorem, evaluate \( \int_C [(y-\sin x)dx + \cos x dy] \) where C is the plane triangle enclosed by the lines \( y=0, x=\frac{\pi}{2} \) and \( y=\frac{2x}{\pi} \).

Prove that div\( (r^n \vec{r}) = (n+3)r^n \). Hence show that div \( \left(\frac{\vec{r}}{r^3}\right) \) is solenoidal.