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

Instructions

• Attempt seven question in all.
• Q. 1 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 calculators is allowed.

Questions

Questions

• Q1. Verify whether the following statements are true or false with justifications in each case: $(1 \times 10 = 10)$ (i) The series $\sum \frac{1}{n^2}$ is divergent. (ii) The function $f(x) = 1 + |x|$ is an odd function in $(-2, 2)$. (iii) The point $x = 0$ is an ordinary point of the differential equation $xy'' + 2y' + y = 0$. (iv) The general solution of $r - 6s + 9t = 0$ is $z = f(y + 3x) + x \cdot g(y + 3x)$. (v) The complete solution of $DD'Z = e^x$ is $z = f(y) + g(y) + xe^x$. (vi) If $A = \begin{bmatrix} 2 & 4 \\ -1 & -2 \end{bmatrix}$, then $A^2 = 0$. (vii) A subset of a linearly independent set of vectors is linearly independent. (viii) The mapping $T: R \to R^2$ defined as $T(x) = (x, x^2)$ is a linear transformation, where $R$ is the set of all reals. (ix) If $S$ and $T$ are linear transformations on $R^2$ such that the composite map $SoT = O$, then $ToS = O$. (x) The eigen…

UNIT - 1

• Q2.(a). Discuss the convergence of the series: $1^p + \left(\frac{1}{2}\right)^p + \left(\frac{1.3}{2.4}\right)^p + \left(\frac{1.3.5}{2.4.6}\right)^p + \dots$ [8 marks]
• Q2.(b). Show that the series $\sum u_n$ is divergent, where $u_n = \frac{n(n + 1)}{(n + 2)(n + 3)(n + 4)}$ [7 marks]
• Q3.(a). Define the terms: Absolute convergence and conditional convergence. Also verify whether the following series is conditional convergent or not? $1 - \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} - \frac{1}{\sqrt{4}} + \dots$ [8 marks]
• Q3.(b). Show that the following series is convergent: $\sum \frac{1.3.5 \dots (2n - 1)}{2.4.6 \dots 2n} \cdot \frac{1}{n}$ [7 marks]

UNIT - II

• Q4.(a). Draw the graph of the series: $f(x) = \begin{cases} -x, & -2 \le x < -1 \\ 2 - x^2, & -1 \le x \le 1 \\ x, & 1 < x \le 2 \end{cases}$ Is it even or odd? Also expand $f(x)$ as a Fourier series in $(-2, 2)$. [8 marks]
• Q4.(b). Using power series method, solve the legendre's differential equation of order 3. [7 marks]
• Q5.(a). State Parseval's identity for Fourier sine series of a periodic function in $(0, l)$. Also apply it to find the value of $\frac{1}{1^4} + \frac{1}{3^4} + \frac{1}{5^4} + \frac{1}{7^4} + \dots$, where $f(x) = \begin{cases} x, & 0 < x \le \pi/2 \\ \pi - 2, & \pi/2 < x < \pi \end{cases}$ [8 marks]
• Q5.(b). Find two power series solution of Bessel's differential equation of order $1/2$. [7 marks]

UNIT - III

• Q6.(a). Solve the following p.d.e. : (i) $z^2 (p^2y^2 + q^2x^2) = x^4y^2$ (ii) $(x - y)y^2p + (y - x)x^2q = z(x^2 + y^2)$ [10 marks]
• Q6.(b). Obtain p.d.e. by eliminating arbitrary function "f" from the relation: $z = (x - y) \cdot f(x + y)$ [5 marks]
• Q7.(a). Find the partial differential equation of all spheres whose centres lies on the z-axis. [5 marks]
• Q7.(b). Solve : $(p^2 + q^2)x = pz$. [5 marks]
• Q7.(c). Solve : $px + qy = p + q$. [5 marks]

UNIT - 4

• Q8.(a). Find P.I. of $(D^2D' - DD'^2 - 2D'^3)Z = \sin(2x + y)$. [5 marks]
• Q8.(b). Solve: $(D^2 + D'^2 - 2DD' - 3D + 3D' + 2)z = 0$. [5 marks]
• Q8.(c). Explain the method of solution of one dimensional Heat Flow equation. [5 marks]
• Q9.(a). Find P.I. of $(D^2 - D'^2 - 2D + 2D')z = xy$. [5 marks]
• Q9.(b). Solve : $(D^4 - D'^4)z = 4e^{x+y}$. [5 marks]
• Q9.(c). Explain briefly the solution of Laplace equation. [5 marks]

UNIT - V

• Q10.(a). Find the rank of the following matrix for various values of k: $\begin{bmatrix} 1 & k & -1 & 2 \\ 2 & -1 & k & 5 \\ 1 & 10 & -6 & 1 \end{bmatrix}$ [5 marks]
• Q10.(b). Find all the eigen values of $A - 3I$, where $A = \begin{bmatrix} 2 & 0 & 1 \\ 0 & 2 & 0 \\ 1 & 0 & 2 \end{bmatrix}$ [5 marks]
• Q10.(c). Using Cayley - Hamilton theorem, find the matrix represented by $2A^6 - 12A^5 - 2A^4 - 3A^3 + 17A^2 + 9A + I$ where $A = \begin{bmatrix} 2 & 3 \\ 3 & 4 \end{bmatrix}$ [5 marks]
• Q11.(a). Find all the eigen vectors of the matrix $A$ corresponding to eigen values $1, 5$; where $A = \begin{bmatrix} 2 & 2 & 1 \\ 1 & 3 & 1 \\ 1 & 2 & 2 \end{bmatrix}$ [5 marks]
• Q11.(b). Using Gauss Elimination method, solve the system of equations: $x + y + z = 1, 3x + 5y + 6z = 4, 9x + 26y + 36z = 16$. [5 marks]
• Q11.(c). Using elementary row transformations only, find the inverse of the matrix. $\begin{bmatrix} 7 & -3 & -3 \\ -1 & 1 & 0 \\ -1 & 0 & 1 \end{bmatrix}$ [5 marks]

UNIT - VI

• Q12.(a). Define a linear transformation and it kernel. Also show that the kernel of a linear transformation is vector subspace of the vector space. [8 marks]
• Q12.(b). Define $T$ on $R^3$ as $T(x, y, z) = (2y + z, x - 4y, 3x)$ for any $x, y, z \in R$ where $R$ is the set of all reals. Then show that $T$ is linear and also find the matrix associated with $T$ relative to the basis set $B = \{(1, 1, 1), (1, 1, 0), (1, 0, 0)\}$. [7 marks]
• Q13.(a). Let $T: R^2 \to R^3$ be defined as $T(x, y) = (x + y, x - y, y)$ for any $x, y \in R$. Then show that $T$ is linear. Also find a basis and dimension of its range space and null space. [8 marks]
• Q13.(b). Show that a linear transformation is invertible iff it is one one and onto. [7 marks]

Related pages

• Engineering Mathematics-II syllabus
• All Engineering Mathematics-II papers from 2018

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