MATHEMATICS-III (Differential Calculus) - B.Tech 3rd Semester Exam., 2020 (New Course)

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

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

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

1. \( f(x) \) and \( g(x) \) both are continuous at \( x=0 \)
2. \( f(x) \) and \( g(x) \) both are differentiable at \( x=0 \)
3. \( f(x) \) is differentiable but \( g(x) \) is not differentiable at \( x=0 \)
4. \( f(x) \) and \( g(x) \) both are not differentiable at \( x=0 \)

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

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

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

1. \( -(\hat{j}-2\hat{k}) \)
2. \( -(\hat{i}-3\hat{k}) \)
3. \( -(\hat{i}-2\hat{k}) \)
4. \( (\hat{i}-2\hat{j}-\hat{k}) \)

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

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

1. \( x^2-y^2=0 \)
2. \( 2x-y=0 \)
3. \( x-2y=0 \)
4. None of the above

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

1. (1, 1)
2. (1, 2)
3. (2, 1)
4. (2, 2)

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

1. \( y = AJ_8(x) + BJ_{-8}(x) \), where A and B are arbitrary constants
2. \( y = AJ_8(x) + BY_{-8}(x) \), where A and B are arbitrary constants
3. \( y = AJ_8(x) + J_{-8}(x) \), where A is arbitrary constant
4. \( y = J_{3/4}(x) + Y_{3/4}(x) \)

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

1. 1
2. 2
3. 0
4. None of the above

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. \( \sin x / \sin y = \phi(\sin y, \sin z) \)
4. \( \sin x / \sin y = \varphi(\sin y \cdot \sin z) \)

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