Mathematics-1 (Calculus and Linear Algebra) - B.Tech. 1st Semester Examination, 2023

The value of \( \int_0^1 x^3 (1 - x^2)^{5/2} dx \) is

1. \( \frac{\pi}{2} \)
2. \( \frac{2}{63} \)
3. 1
4. None of the above

The area \( \iint_R \sin(x + y) \, dxdy \) over R \( \{(x, y); 0 \leq x \leq \frac{\pi}{2}, 0 \leq x \leq \frac{\pi}{2}\} \)

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

The matrix \( \begin{bmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{bmatrix} \) is

1. Invertible but not orthogonal
2. Invertible and orthogonal
3. Neither invertible nor orthogonal
4. None of the above

The sequence \( \{a_n\} \), where \( a_n = \sqrt[n]{n} \) is

1. Is not convergent.
2. Convergent and \( \lim_{n \to \infty} a_n = 0 \)
3. Convergent and \( \lim_{n \to \infty} a_n = 1 \)
4. None of the above

Let \( f(x) = \cos(x^2), X \in R \) then

1. \( f \) is uniformly continuous
2. \( f \) is continuous, but not uniformly continuous
3. \( f \) continuous but unbounded
4. \( f \) is Lipschitz continuous

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

1. Does not exists
2. Exists and the value is 2
3. Exists and the value is 3
4. None of the above

\( A = 2i + \alpha j + k, B = i + 3j - 8k \). Then \( A \) and \( B \) are orthogonal if

1. \( \alpha = -2 \)
2. \( \alpha = 2 \)
3. \( \alpha = -1 \)
4. \( \alpha = 1 \)

Let \( x + 2y + z = 0, 3x + 4y + z = 0, x - z = 0 \) be a system of equations. Then

1. It is inconsistent
2. It has only the trivial solution \( x = 0, y = 0, z = 0 \)
3. Determinant of the matrix of coefficient is zero
4. None of these

The value at which the Rolle's theorem is applicable for \( f(x) = \cos \frac{x}{2} \) in \( |\pi, 3\pi| \) is

1. \( \frac{5\pi}{2} \)
2. \( \frac{3\pi}{2} \)
3. \( 2\pi \)
4. none of the above

The rank of the matrix \( \begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 7 \\ 3 & 6 & 10 \end{bmatrix} \) is

1. 3
2. 2
3. 1
4. none of these

Show that: \( \int_{0}^{\frac{\pi}{2}} \log(\tan x + \cot x) \, dx = \pi \log 2 \)

Find the following limit: \( \lim_{n \to \infty} \left[ t \ln \left( 1 + \frac{3}{t} \right) \right] \)

Test the convergence of \( x^2 + \frac{2^2}{3.4} x^4 + \frac{2^2 4^2}{3.4,5.6,7.8} x^8 + \cdots \)

Examine the convergence of the series of which the general term is \( \sum_{n=1}^{\infty} \frac{1}{n(n+1)} \)

Prove that \( \log (1+x) = \frac{x}{1+\theta x} \), where \( 0 < \theta < 1 \). Hence deduce, \( \frac{x}{1+x} < \log(1+x) < x \)

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

Evaluate \( \int_{0}^{\pi/2} \frac{1}{a\cos^2 x + b\sin^2 x} \, dx \)

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

Show that the equation of the evolute of the curve \( x^{2/3} + y^{2/3} = a^{2/3} \) is \( (x+y)^{2/3} + (x-y)^{2/3} = 2a^{2/3} \).

Find the areas of the regions that lies inside the circle \( r = a \cos \theta \) and outside the cardioid \( r = a(1 - \cos \theta) \).

Find the solution of the following system of equations:
\( x_1 + 2x_2 + 2x_3 = 2 \)
\( x_1 + 8x_3 + 5x_4 = -6 \)
\( x_1 + x_2 + 5x_3 + 5x_4 = 3 \)
Also find the basis and dimension of the solution space.

Verify Cayley-Hamilton theorem for the matrix \( A = \begin{bmatrix} 2 & 4 \\ 2 & 5 \end{bmatrix} \) and find its inverse. Also express \( A^5 - 4A^4 - 7A^3 + 11A^2 - A - 10I \) as a linear polynomial in \( A \).

Expand \( \sin x \) in Taylor's series about \( x = \frac{\pi}{2} \).

By using Beta function evaluate \( \int_{0}^{1} x^4 (1-\sqrt{x})^5 \, dx \).

Develop the Fourier series on \( -\pi < x < \pi \) to represent the function defined as:
\( f(x) = \begin{cases} 0, & -\pi < x < 0 \\ \pi, & 0 < x < \pi \end{cases} \)

(i) Evaluate div \( \vec{F} \) and Curl \( \vec{F} \), where \( \vec{F} = \text{grad} (x^2 + y^2 + z^2) e^{-\sqrt{x^2 + y^2 + z^2}} \).
(ii) Let \( \vec{F} = (-4x - 3y + az) i + (bx + 3y + 5z) j \)
+ (\(4x + cy + 3z) k \). Find the values of \( a \), \( b \) and \( c \) such that \( F \) is irrotational.