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Engineering Mathematics-I — Minor 1 2023 question paper

MMMUT Chemical Engineering previous year question paper for Engineering Mathematics-I (BSM-110), semester 1, Minor 1 2023. All 15 questions are listed below, each with a written answer on Nexsus.

Paper details

  • Subject: Engineering Mathematics-I (BSM-110)
  • Branch: Chemical Engineering
  • Semester: 1
  • Exam: Minor 1 2023
  • Questions: 15

Questions asked in Engineering Mathematics-I Minor 1 2023

  1. Q1(a). If $u, v, w$ are the roots of the cubic equation $(\lambda - x)^3 + (\lambda - y)^3 + (\lambda - z)^3 = 0$ in $\lambda$, then find $\frac{\partial(u, v, w)}{\partial(x, y, z)}$. [4 marks]
  2. Q1(b). If $\frac{x^2}{a^2+u} + \frac{y^2}{b^2+u} + \frac{z^2}{c^2+u} = 1$, then prove that$\left(\frac{\partial u}{\partial x}\right)^2 + \left(\frac{\partial u}{\partial y}\right)^2 + \left(\frac{\partial u}{\partial z}\right)^2 = 2\left(x\frac{\partial u}{\partial x} + y\frac{\partial u}{\partial y} + z\frac{\partial u}{\partial z}\right).$ [4 marks]
  3. Q1(c). Find the inverse of the matrix$A = \begin{bmatrix} 0 & 2 & 1 & 3 \\ 1 & 1 & -1 & -2 \\ 1 & 2 & 0 & 1 \\ -1 & 1 & 2 & 6 \end{bmatrix}$ [4 marks]
  4. Q1(d). Let $A = \begin{bmatrix} 6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3 \end{bmatrix}$. Find matrix $P$ such that $P^{-1}AP$ is a diagonal matrix. [4 marks]
  5. Q2(a). The Temperature $T$ at any point $(x, y, z)$ in space is $T(x, y, z) = kxyz^2$, $k$ is positive constant. Find the highest temperature on the surface of sphere $x^2 + y^2 + z^2 = a^2$. [3 marks]
  6. Q2(b). If $u$ be the homogeneous function of degree $n$ in $x$ and $y$, then show that$x^2 \frac{\partial^2 u}{\partial x^2} + 2xy \frac{\partial^2 u}{\partial x \partial y} + y^2 \frac{\partial^2 u}{\partial y^2} = n(n - 1)u$ [3 marks]
  7. Q2(c). Expand $f(x,y) = \tan^{-1}(xy)$ in powers of $(x - 1)$ and $(y - 1)$ up to the second-degree term. Hence compute $f(0.9, 1.1)$ approximately. [3 marks]
  8. Q2(d). If $\log y = \tan^{-1} x$, show that$(1 + x^2)y_{n+2} + \{2(n + 1)x - 1\}y_{n+1} + n(n + 1)y_n = 0.$ [3 marks]
  9. Q3(a). Find the rank of the matrix $A$ by changing it into normal form.$A = \begin{bmatrix} 2 & 3 & -1 & -1 \\ 1 & -1 & -2 & -4 \\ 3 & 1 & 3 & -2 \\ 6 & 3 & 0 & -7 \end{bmatrix}$ [4 marks]
  10. Q3(b). Find the value of $a$ and $b$ for which the system of equations$3x - 2y + z = b,\ 5x - 8y + 9z = 3,\ 2x + y + az = -1$has (i) unique solution (ii) no solution (iii) infinitely many solutions. [4 marks]
  11. Q3(c). State the Cayley-Hamilton theorem. Verify this theorem for the matrix $A = \begin{bmatrix} 1 & 2 \\ -1 & 3 \end{bmatrix}$. Hence express $A^6 - 4A^5 + 8A^4 - 12A^3 + 14A^2$ as linear polynomial in $A$. [3 marks]
  12. Q3(d). Find the value of $k$ such that the system of equations$x + ky + 3z = 0,\ 4x + 3y + kz = 0,\ 2x + y + 2z = 0$has nontrivial solution. [3 marks]
  13. QQ1. Attempt any three parts of the following.
  14. QQ2. Attempt any three parts of the following.
  15. QQ3. Attempt any three parts of the following.