Mathematics-III (Differential Calculus)

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

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

The maximum value of $ \sin x + \cos x $ is

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

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

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

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

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

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

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

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.

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

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

The maximum value of $ \sin x + \cos x $ is

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

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

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

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

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

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

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

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.

The value of $ \lim_{x\rightarrow0}\left(\frac{\sin x}{x}\right)^{1/x} $ is

Let $ f(x)=|x| $ and $ g(x)=|x^3| $, then

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

If $ \vec{v} = xy^2\hat{i} - 2x^2yz\hat{j} - 3yz^2\hat{k} $, then the value of curl $ \vec{v} $ at (1,1,1) is equal to

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

The solution of the boundary value problem $ (x-y^2x)dx - (x^2y-y)dy = 0 $, $ y(0)=0 $ is

Let $ P_n(x) $ be the Legendre polynomial of degree $ n \ge 0 $. If $ \int_{-1}^1 P_{n-1}^2(x)dx = \frac{2}{4n-1} $ then the value of (k, l) is

The general solution of Bessel differential equation $ x^2y''(x) + xy'(x) + (x^2-64)y(x) = 0 $ is

The equation $ p \tan y + q \tan x = \sec^2 z $ is of order

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

If $ y = (\sin^{-1}x)^2 $, then show that $ (1-x^2)y_{n+2} - (2n+1)xy_{n+1} - n^2y_n = 0 $. Hence find $ (y_n)_0 $.

Find the value of $ \lim_{x\rightarrow0} \left(\frac{\tan x}{x}\right)^{1/x^2} $.

Discuss the continuity of the following function $ f(x,y) $ at point (0, 0):
$ f(x,y) = \begin{cases} \frac{\sin\sqrt{|xy|} - \sqrt{|xy|}}{\sqrt{x^2+y^2}}, & (x,y) \neq (0,0) \\ 0, & (x,y) = (0,0) \end{cases} $

For the function $ f(x,y) = \begin{cases} \frac{xy(2x^2-3y^2)}{x^2+y^2}, & (x,y) \neq (0,0) \\ 0, & (x,y) = (0,0) \end{cases} $ check whether $ f_{xy}(0,0) $ and $ f_{yx}(0,0) $ are equal or not.

Find the minimum value of $ x^2+y^2+z^2 $ subject to the condition $ xyz=a^3 $.

Obtain the second-order Taylor's series approximation to the function $ f(x,y) = xy^2 + y \cos(x-y) $ about the point (1, 1).

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^2dy] $ where C is bounded by $ y=x $, $ y=x^2 $.

Find the value of n for which the vector $ r^n\vec{r} $ solenoidal, where $ \vec{r} = x\hat{i} + y\hat{j} + z\hat{k} $.

Solve the differential equation $ (y^4+2y)dx + (xy^3+2y^4-4x)dy = 0 $.

Solve $ p = \sin(y-xp) $. Also find its singular solution.

Solve $ x^2\frac{d^2y}{dx^2} - 3x\frac{dy}{dx} + 5y = x \log x $.

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) $.

Solve $ x^2p + y^2q = (x+y)z $.

Solve $ (x+y)(p+q)^2 + (x-y)(p-q)^2 = 1 $.

The value of $ \lim_{x\rightarrow0}\left(\frac{\sin x}{x}\right)^{1/x} $ is

Let $ f(x)=|x| $ and $ g(x)=|x^3| $, then

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

If $ \vec{v} = xy^2\hat{i} - 2x^2yz\hat{j} - 3yz^2\hat{k} $, then the value of curl $ \vec{v} $ at (1,1,1) is equal to

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

The solution of the boundary value problem $ (x-y^2x)dx - (x^2y-y)dy = 0 $, $ y(0)=0 $ is

Let $ P_n(x) $ be the Legendre polynomial of degree $ n \ge 0 $. If $ \int_{-1}^1 P_{n-1}^2(x)dx = \frac{2}{4n-1} $ then the value of (k, l) is

The general solution of Bessel differential equation $ x^2y''(x) + xy'(x) + (x^2-64)y(x) = 0 $ is

The equation $ p \tan y + q \tan x = \sec^2 z $ is of order

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

If $ y = (\sin^{-1}x)^2 $, then show that $ (1-x^2)y_{n+2} - (2n+1)xy_{n+1} - n^2y_n = 0 $. Hence find $ (y_n)_0 $.

Find the value of $ \lim_{x\rightarrow0} \left(\frac{\tan x}{x}\right)^{1/x^2} $.

Discuss the continuity of the following function $ f(x,y) $ at point (0, 0):
$ f(x,y) = \begin{cases} \frac{\sin\sqrt{|xy|} - \sqrt{|xy|}}{\sqrt{x^2+y^2}}, & (x,y) \neq (0,0) \\ 0, & (x,y) = (0,0) \end{cases} $

For the function $ f(x,y) = \begin{cases} \frac{xy(2x^2-3y^2)}{x^2+y^2}, & (x,y) \neq (0,0) \\ 0, & (x,y) = (0,0) \end{cases} $ check whether $ f_{xy}(0,0) $ and $ f_{yx}(0,0) $ are equal or not.

Find the minimum value of $ x^2+y^2+z^2 $ subject to the condition $ xyz=a^3 $.

Obtain the second-order Taylor's series approximation to the function $ f(x,y) = xy^2 + y \cos(x-y) $ about the point (1, 1).

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^2dy] $ where C is bounded by $ y=x $, $ y=x^2 $.

Find the value of n for which the vector $ r^n\vec{r} $ solenoidal, where $ \vec{r} = x\hat{i} + y\hat{j} + z\hat{k} $.

Solve the differential equation $ (y^4+2y)dx + (xy^3+2y^4-4x)dy = 0 $.

Solve $ p = \sin(y-xp) $. Also find its singular solution.

Solve $ x^2\frac{d^2y}{dx^2} - 3x\frac{dy}{dx} + 5y = x \log x $.

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) $.

Solve $ x^2p + y^2q = (x+y)z $.

Solve $ (x+y)(p+q)^2 + (x-y)(p-q)^2 = 1 $.