Mathematics-1 (Calculus & Linear Algebra) - B.Tech 1st Semester Exam-2022

Lagrange's mean value theorem can be proved for a function f(x) by applying Roll's mean value theorem

1. \( \varphi(x) = f(x) + kx^2 \)
2. \( \varphi(x) = f(x) - kx^2 \)
3. \( \varphi(x) = f(x) + kx \)
4. \( \varphi(x) = \{f(x)\}^2 + kx^2 \)

The function \( f(x) = x(x + 3) e^{-\frac{x}{2}} \) satisfy all conditions of Roll's mean value theorem in the interval [-3, 0]. Then the value of c is:

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

if \( \begin{bmatrix} 4/3 \\ 1 \end{bmatrix} \) is an eigenvector of \( \begin{bmatrix} 3 & 4 \\ 3 & 1 \end{bmatrix} \). What is the associated eigenvalue?

1. 4/3
2. 5
3. -2
4. None of the above

if f is continuous on [-2.354, 2.354] then

1. \( \int_{-2.354}^{2.354} f(\cos x) dx = 0 \)
2. \( \int_{-2.354}^{2.354} f(\cos x) dx = 2 \int_{0}^{2.354} f(\cos x) dx \)
3. 2.354
4. none of the above

let \( f(x) = |x|^{\frac{3}{2}}, x \in R \) then

1. f is uniformly continuous
2. f is continuous, but not differentiable at \( x = 0 \)
3. f is differentiable, and derivative of f is continuous
4. f is differentiable, but derivative of \( x \) is discontinuous at \( x = 0 \)

if \( A(2) = 2i - j + 2k, A(3) = 4i - 2j + 3k \), then \( \int_{2}^{3} A \cdot \frac{dA}{dt} dt \) is

1. 5
2. 10
3. 15
4. 20

if \( \nabla \times \vec{F} \). Then it is called.

1. solenoidal
2. Rotational
3. irrotational
4. None

if \( 3x + 2y + z = 0, x + 4y + z = 0, 2x + y + 4z = 0 \) 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 coefficients is zero
4. None of these

If f(x) is continuous in the closed interval [a,b] and f(x) exists in (a,b) and f(a) = f(b), then there exist at least one value \( c(a < c < b) \) Such that \( f'(c)=0 \) is called

1. Taylor's theorem
2. Mac Laurin's theorem
3. Rolle's theorem
4. Lagrange's mean value theorem

The rank of the matrix \( \begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{bmatrix} \) is

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

show that : \( \int_{0}^{\infty} x^2 e^{-x^4} dx \times \int_{\infty}^{\infty} e^{-x^4} dx = \frac{\pi}{8\sqrt{2}} \)

Use L' Hospital rule to find the following limits. \( \lim_{x \to 0} \frac{\cos - \ln (1+x) - 1 + xy^x}{\sin^2 x} \)

Test the convergence of \( 1 + \frac{3}{7} x + \frac{3.6}{7.10} x^2 + \frac{3.6.9}{7.10.13} x^3 + \frac{3.6.9.12}{7.10.13.16} x^4 + \ldots \)

Examine the convergence of the series of which the general term is \( 2^2 4^2 6^2 \cdots \frac{(2n-2)^2}{3.4.5 \cdots (2n-1)2n} x^{2n} \)

Obtain the fourth-degree Taylor's polynomial approximation to \( f(x) = e^{2x} \) about \( x = 0 \). Find the maximum error when \( 0 \leq x \leq 0.5 \)

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

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

if \( f(x) = \log (1 + x), x > 0 \) using Maclaurin's theorem, show that for \( 0 < \theta < 1 \), \( \log (1 + x)=x - \frac{x^2}{2} + \frac{x^3}{3(1+\theta x)^3} \). Deduce that \( \log (1+x)=< x - \frac{x^2}{2} + \frac{x^3}{3} \) for \( x> 0 \)

Using Taylor's theorem, express the polynomial \( 2x^3 + 7x^2 + x - 6 \) in powers of (x - 1).

Find the values of a,b,c if A = \( \begin{bmatrix} 0 & 2b & c \\ a & b & -c \\ a & -b & c \end{bmatrix} \) is orthogonal?

Verify Cayley-Hamilton theorem for the matrix A = \( \begin{bmatrix} 1 & 4 \\ 2 & 3 \end{bmatrix} \) and find its inverse. Also express \( A^5 - 4A^4 - 7A^3 + 11A^2 - A - 10I \) as a linear polynomial in A.

Expand \( \ln x \) in power of \( (x - 1) \) by Taylor's theorem.

By using Beta function evaluate \( \int_{0}^{1} x^5 (1 + x^3)^{10} dx \).

Find the Fourier series to represent the function defined as \( f(x) = \begin{cases} x + \frac{\pi}{2}, & -\pi < x < 0 \\ \frac{\pi}{2} - x, & 0 < x < \pi \end{cases} \)

Evaluate: (i) div F and Curl F, where F= grad \( (x^3 + y^3 + z^3 - 3xyz) \). (ii) If \( F = (x + y + z) i + j - (x + y)k \) show that \( F \cdot \text{Curl} F = 0 \).