Mathematics—I (Calculus and Linear Algebra) - B.Tech 1st Semester Special Exam., 2020

If \( Y = \int_0^{\infty} \frac{x^a}{a^x} \, dx, \, a > 1 \), then the value of \( Y \) is

1. \( \frac{\Gamma(a)}{[\log_e a]^a} \)
2. \( \frac{\Gamma(a + 1)}{[\log_e a]^a} \)
3. \( \frac{\Gamma(a + 1)}{[\log_e a]^{a+1}} \)
4. \( \frac{\Gamma(a)}{[\log_e a]^{a+1}} \)

The area bounded by the axis of \( x \), and the curve and ordinates \( y = \cosh \frac{x}{c} \) from \( x = 0 \) to \( x = a \) is

1. \( \cosh \frac{a}{c} \)
2. \( \sinh \frac{a}{c} \)
3. \( \sinh \frac{a}{c} \)
4. None of the above

Consider the following functions :
1. \( y = x \sin \frac{1}{x}, \, x \neq 0; \, \text{and } y = 0 \, \text{if } x = 0 \)
2. \( y = x^2 \sin \frac{1}{x}, \, x \neq 0; \, \text{and } y = 0 \, \text{if } x = 0 \)
3. \( y = x^2 \cos \frac{1}{x}, \, x \neq 0; \, \text{and } y = 0 \, \text{if } x = 0 \)
4. \( y = x \cos \frac{1}{x}, \, x \neq 0; \, \text{and } y = 0 \, \text{if } x = 0 \)
The functions, differentiable at \( x = 0 \), are

1. 1 and 2
2. 2 and 3
3. 3 and 4
4. 1 and 4

For a positive term series \( \sum a_n \), the ratio test states that

1. the series converges, if \( \lim_{n \to \infty} \frac{a_{n+1}}{a_n} > 1 \)
2. the series converges, if \( \lim_{n \to \infty} \frac{a_{n+1}}{a_n} < 1 \)
3. the series converges, if \( \lim_{n \to \infty} \frac{a_{n+1}}{a_n} = 1 \)
4. None of the above

If \( \lim_{x \to \infty} \frac{\sin 2x + a \sin x}{x^3} = b \) where \( b \) is finite, then the values of \( a \) and \( b \) respectively will be

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

The expansion of \( \tan x \) in powers of \( x \) by Maclaurin's theorem is valid in the interval

1. \( (-\infty, \infty) \)
2. \( \left(-\frac{3\pi}{2}, \frac{3\pi}{2}\right) \)
3. \( (-\pi, \pi) \)
4. \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \)

The value of \( \lim_{(x, y) \to (k, 0)} \left(1 + \frac{x}{y}\right)^y \) is

1. 1
2. \( e^{-k} \)
3. \( e^k \)
4. Does not exist

The gradient of the function \( f(x, y, z) = \sin(xyz) \), at \( (1, -1, \pi) \), is

1. \( \pi(i - j + k) \)
2. \( \pi(i + j + k) \)
3. \( (i + j + k) \)
4. \( (\pi i - \pi j + k) \)

If det(A) = 7, where \( A = \begin{bmatrix} a & b & c \\ 1 & 1 & g \\ g & \omega & 1 \end{bmatrix} \), then det(2A)^{-1} is equal to

1. \( \frac{14}{49} \)
2. \( \frac{1}{56} \)
3. \( \frac{7}{2} \)

If \( 3x + 2y + z = 0 \), \( x + 4y + z = 0 \) and \( 2x + y + 4z = 0 \) be a system of equations, then

1. it is inconsistent
2. it has only the trivial solution (0, 0, 0)
3. it can be reduced to a single equation and so a solution does not exist
4. the determinant of the matrix of coefficients is zero

Evaluate \( \int_{0}^{\infty} \log \left( x + \frac{1}{x} \right) \frac{dx}{1 + x^2} \)

Find the volume of the solid generated by rotating completely about the x-axis where the area enclosed between \( y^2 = x^3 + 5x \) and the line \( x = 2 \) and \( x = 4 \) about its major axis.

Find the maximum value of the function \( f(x) = \frac{x}{1 + x \tan x} \)

It is given that Rolle's theorem holds for the function \( f(x) = x^3 + bx^2 + cx \), \( 1 \leq x \leq 2 \) at the point \( x = \frac{4}{3} \). Find the values of b and c.

Discuss the convergence of the sequence whose n-th term is \( \alpha_n = \frac{(-1)^n}{n} + 1 \)

Test the convergence of the following series : \( x^2 + \frac{2^2 x^4}{3.4} + \frac{2^2 4^2 x^6}{3.4.5.6} + \frac{2^2 4^2 6^2 x^8}{3.4.5.6.7.8} ... \)

Find the Fourier series expansion of the function \( f(x) = \{x^2, -2 \leq x \leq 2\} \). Hence deduce that \( \frac{\pi^2}{6} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + ... \)

Find the Fourier cosine series and Fourier sine series of the following function in given interval : \( f(x) = \begin{cases} x, & 0 < x < 2 \\ 2, & 2 \leq x < 4 \end{cases} \)

Discuss continuity of the following function at the point \( (0, 0) \) : \( f(x, y) = \begin{cases} \frac{x^2 y^2}{\left( x^3 + y^3 \right)} , & (x, y) \neq (0, 0) \\ 0, & (x, y) = (0, 0) \end{cases} \)

Find the maximum value of \( xyz \) under the constraints \( x^2 + z^2 = 1 \) and \( y - x = 0 \).

Find the value of \( \lim_{x \to {\infty}} \left( \frac{x+4}{x+2} \right)^{x+3} \)

Find the equation of the tangent plane to the surface \( x^2 - 3y^2 - z^2 = 2 \), at the point \( (3, 1, 2) \).

Find the eigenvalues and eigenvectors of the following matrix : \( \begin{bmatrix} 6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3 \end{bmatrix} \)

Verify Cayley-Hamilton theorem for the matrix \( \begin{bmatrix} 0 & 0 & 1 \\ 3 & 1 & 0 \\ -2 & 1 & 4 \end{bmatrix} \)

Determine the range of the following linear transformation. Also find the rank of \( T \), where it exists. \( T: V_2 \to V_3 \) defined by \( T(x_1, x_2) = (x_1, x_1 + x_2, x_2) \)