Engineering Mathematics-II 2018 Question Paper — Government College o…

Instructions

• Attempt Seven quesitons in all.
• Q1 is compulsory and is of objective type carrying 10 marks.
• There will be two questions in each unit. Attempt one question from each unit. All are of 15 marks.
• Use of Calculator is allowed.

Questions

Questions

• Q1. Verify whether the following statements are true or false with justification in each case: i) The series $\sum \sin\frac{1}{n}$ is convergent. ii) The function $f(x) = x + x^2$ is even is $(-2, 2)$. iii) The function $f(x) = \cos x$ is analytic for all real $x$. iv) One power series solution of the Legendre's differential equation of order 2 is a second degree polynomial. v) $x^2 p^2 + y^2 q^2 = z^2$ is a linear p.d.e. of first order. vi) C.F. of $(D^2 - DD')z = 0$ is $z = f(x) + g(x+y)$. vii) The vectors $(1, 0, 0), (0, 2, 0)$ and $(0, 0, -3)$ are linearly independent. viii) Eigen values of a square matrix are real numbers. ix) The map $T: R^2 \to R$ defined as $T(x,y) = xy$ is linear, where R denotes set of all reals. x) There is a unique basis set of vectors for a vector space. [10 marks]

UNIT - I

• Q2. a) Show that the series $\sum \left[\sqrt{(n^3+1)} - \sqrt{n^3}\right]$ is convergent. b) Discuss the convergence of the series $\sum \frac{1}{(\log n)^p}$. c) State Liebnitz test on alternating series. [15 marks]
• Q3. a) Find the interval of convergence for the series: $x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \dots$. b) Prove that the series $\sum_{n=1}^\infty \frac{1}{n^p}$ is convergent when $p > 1$. c) If the series $\sum_{n=1}^\infty a_n$ is convergent, show that $\lim_{n\to\infty} a_n = 0$. [15 marks]

UNIT - II

• Q4. a) Draw the graph of the function $$f(x) = \begin{cases} 0, & -\pi \le x \le 0 \\ x^2, & 0 < x \le \pi \end{cases}$$ Is it even or odd? Also expand it as a Fourier series in $(-\pi, \pi)$. b) Find two power series solution of Bessel's differential equation of order 2. [15 marks]
• Q5. a) Using Parseval's Identity for Fourier series of $f(x) = x^2$ in $(-\pi, \pi)$ prove that $\sum \frac{1}{n^4} = \frac{\pi^4}{90}$. b) Using power series method, solve Legendre's differential equation of order 3. [15 marks]

UNIT - III

• Q6. a) Find the partial differential equation of all spheres whose centres lies on the z-axis. b) Solve: $y^2(x-y)p + x^2(y-x)q = (x^2+y^2)z$. c) Solve: $p^3 + q^3 = 3pqz$. [15 marks]
• Q7. a) Solve: $x^2 p^2 = z(z - qy)$. b) Solve: $(y-z)yzp + (z-x)zxq = (x-y)xy$ c) Obtain p.d.e. by eliminating 'f and g' from $z = f(2x+y) + g(x-3y)$. [15 marks]

UNIT - IV

• Q8. a) Solve: $r - 2s + t = \sin(x-y)$. b) Find P.I. of $(D^2 - D'^2 - 2D + 2D')z = xy$. c) Using the method of separation of variables derive the solution for one dimensional Heat flow equation. [15 marks]
• Q9. a) Derive the solution for wave equation of a stretched flexible string of length L. b) Solve: $(D^2 + 2D'^2 + 3DD' - 4D - 5D' + 3)z = 0$. c) Find P.I. of $(D^2 + 3DD' + 2D'^2)z = \cos(2x-y)$. [15 marks]

UNIT - V

• Q10. a) Using elementary column transformation, compute the inverse of $\begin{bmatrix} 1 & 2 & 5 \\ 3 & 1 & 4 \\ 1 & 1 & 2 \end{bmatrix}$. b) Find all the eigen values of $A^3$ and $A+2I$, where $A = \begin{bmatrix} 2 & 2 & 1 \\ 1 & 3 & 1 \\ 1 & 2 & 2 \end{bmatrix}$. c) If $\lambda$ is an eigen value of a square matrix, show that $\lambda^3$ is an eigen value of $A^3$. [15 marks]
• Q11. a) Test the following system of equations for consistency: $$2x - 3y + 7z = 5$$ $$3x + y - 3z = 13$$ $$2x + 19y - 47z = 32$$ b) Reduce the matrix A to normal form and also find its rank, where $A = \begin{bmatrix} 24 & 19 & 36 & 72 \\ 49 & 40 & 73 & 147 \\ 73 & 59 & 98 & 219 \\ 47 & 36 & 71 & 141 \end{bmatrix}$. c) Find the eigen vectors for matrix A corresponding to eigen values $\lambda = 1, 2$; where $A = \begin{bmatrix} 4 & 6 & 6 \\ 1 & 3 & 2 \\ -1 & -5 & -2 \end{bmatrix}$. [15 marks]

UNIT - VI

• Q12. a) Prove that a super set of a linearly dependent set in a vector space in linearly dependent. b) Verify Rank-Nullity theorem for the linear transformation $T : R^2 \to R^2$ defined as $T(x,y) = (x+y, x-y)$ for any $x, y \in R$. [15 marks]
• Q13. a) Define $T : R^2 \to R^2$ as $T(x,y) = (2x - 3y, x+y)$ for any $x,y \in R$. Then show that T is linear and also find the matrix associated with T relative to basis set $B = [e_1, e_2]$, where $e_1 = (1, 0)$ and $e_2 = (0, 1)$. b) Let T be a linear transformation from a vector space V into itself then show that T is invertible if and only if T is one one and onto. [15 marks]

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• Engineering Mathematics-II syllabus
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