Mathematics—I (Calculus and Linear Algebra) - B.Tech 1st Semester Exam., 2021

If \( Y = \int_0^a t^4 e^{-2t^2} dt \), then the value of \( Y \) is

1. \( \frac{\sqrt{2\pi}}{64} \)
2. \( \frac{3\sqrt{\pi}}{64} \)
3. \( \frac{5\sqrt{2\pi}}{64} \)
4. \( \frac{3\sqrt{2\pi}}{64} \)

The maximum area of a sector whose perimeter is \( k \) is given by

1. \( \frac{k^2}{16} \)
2. \( \frac{k^2}{8} \)
3. \( \frac{k^2}{4} \)
4. \( \frac{k^2}{1} \)

The function \( f(x) = \begin{cases} x^2, & 0 \leq x \leq \frac{1}{2} \\ x, & \frac{1}{2} < x \leq 1 \end{cases} \), then \( \lim_{x \to \frac{1}{2}} f(x) \)

1. has the value \( \frac{1}{2} \)
2. has the value 1
3. has the value 2
4. does not exist

If \( \alpha, \beta \) are the roots of the equation \( ax^2 + bx + c = 0 \), then \( \lim_{x \to \alpha} \frac{1 - \cos(\alpha x^2 + bx + c)}{(x - \alpha)^2} \) is equal to

1. \( \frac{a^2}{2} (\alpha - \beta)^2 \)
2. \( -\frac{a^2}{2} (\alpha - \beta)^2 \)
3. \( \frac{a^2}{2} (\alpha - \beta)^2 \)
4. 0

The series \( x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + ... \) converges for

1. \( (-1 < x \leq 1) \)
2. \( |x| < 1 \) only
3. \( |x| \leq 1 \)
4. all real values of \( x \)

If \( u_n = \sqrt{n+1} - \sqrt{n}, \quad v_n = \sqrt{n^4 + 1} - n^2 \), then

1. \( \sum_{n=1}^{\infty} u_n \) converges but \( \sum_{n=1}^{\infty} v_n \) diverges
2. \( \sum_{n=1}^{\infty} u_n \) diverges but \( \sum_{n=1}^{\infty} v_n \) converges
3. \( \sum_{n=1}^{\infty} u_n \) and \( \sum_{n=1}^{\infty} v_n \) both converge
4. \( \sum_{n=1}^{\infty} u_n \) and \( \sum_{n=1}^{\infty} v_n \) both diverge

\( \lim_{(x, y) \to (0, 0)} \frac{1 - x - y}{x^2 + y^2} \) is

1. 0
2. 1
3. -1
4. Does not exist

The gradient of the function \( f(x, y, z) = \log (x^2 + y^2 + z^2) \) at \( (3, -4, 5) \) is

1. \( \frac{1}{\sqrt{3}} (i + j + k) \)
2. \( \frac{1}{3} (i + j + k) \)
3. \( \frac{1}{25} (3i^2 - 4j + 5k) \)
4. \( \frac{1}{25} (3i^2 + 4j + 5k) \)

If \( A \) and \( B \) are any two square matrices of order \( 2 \times 2 \), then \( (A + B)^2 \) is equal to

1. \( A^2 + 2AB + B^2 \)
2. \( A^2 + AB - BA + B^2 \)
3. \( A^2 + AB + BA + B^2 \)
4. \( A^2 + 2BA + B^2 \)

The system of equations \( x - y + 3z = 4, x + z = 2, x + y - z = 0 \) has

1. infinitely many solutions
2. finitely many solutions
3. a unique solution
4. no solution

Evaluate \( \int_{0}^{\infty} x^{m} e^{-ax^n} dx \).

Find the surface of the solid formed by the revolution about the axis of \( y \), of the part of the curve \( ay^2 = x^3 \) from \( x = 0 \) to \( x = 4a \), which is above the \( x \)-axis.

Verify the Rolle's theorem for the function \( f(x) = 2x^3 + x^2 - 4x - 2 \).

Find \( \lim_{x \to \frac{\pi}{2}} (\sin x)^{\tan x} \).

Discuss the convergence of the sequence whose \( n \) th term is \( a_n = \frac{1}{\log n} \).

Test the convergence of the following series : \( 1 + \frac{x}{1} + \frac{1}{2} \cdot \frac{x^3}{3} + \frac{1 \cdot 3 \cdot x^5}{2 \cdot 4 \cdot 5} + \frac{1 \cdot 3 \cdot 5 \cdot x^7}{2 \cdot 4 \cdot 5 \cdot 6 \cdot 7} + \cdots \)

Find the Fourier series expansion of the function \( f(x) = \{ \pi + x \}, -\pi \leq x \leq \pi \). Hence deduce that \( \frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots \)

Find the Fourier cosine series and Fourier sine series of the following function in given interval : \( f(x) = \{ x + x^2 \}, 0 < x < 1 \)

Discuss continuity of the following function at the point \( (-1, c) \) : \( f(x, y) = \begin{cases} \frac{x^2 y}{(1 + x)}, & x \neq -1 \\ y, & (x, y) = (-1, c) \end{cases} \)

Find the extreme value of \( xyz \), when \( x + y + z = a, a > 0 \).

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

Find the equation of the tangent plane to the surface \( xy + yz + zx = -1 \), at the point \( (1, -1, 2) \).

Find the eigenvalues and eigenvectors of the following matrix : \( \begin{bmatrix} -2 & 2 & -3 \\ 2 & 1 & -6 \\ -1 & -2 & 0 \end{bmatrix} \)

Verify Cayley-Hamilton theorem for the matrix \( \begin{bmatrix} 0 & c & b \\ -c & 0 & a \\ b & -a & 0 \end{bmatrix} \)

Determine the range of the following linear transformation. Also find the rank of \( T \), where it exists, \( T: V_3 \rightarrow V_3 \), defined by \( T(x_1, x_2, x_3) = \left( \frac{1}{2} x_1 + x_2 + x_3, \, x_1 - \frac{1}{3} x_2, \, x_3 \right) \)