Calculus and Linear Algebra — Minor 1 2023 question paper

Paper details

• Subject: Calculus and Linear Algebra (BSM - 101)
• Branch: Chemical Engineering
• Semester: 1
• Exam: Minor 1 2023
• Questions: 7

Questions asked in Calculus and Linear Algebra Minor 1 2023

1. Q1(a). i. If $y = \sqrt{\frac{1+x}{1-x}}$, prove that $(1 - x^2)y_n - [2(n - 1)x + 1]y_{n-1} - (n - 1)(n - 2)y_{n-2} = 0$ii. If $y = x^{n-1} \log x$, prove that $y_n = \frac{(n-1)!}{x}$. Also, if $I_n = D^n(x^n \log x)$ prove that $I_n = n I_{n-1} + (n - 1)!$ [5 marks]
2. Q1(b). If $u = \sin^{-1} \left[ \frac{x^{1/3} + y^{1/3}}{x^{1/2} + y^{1/2}} \right]^{\frac{1}{2}}$, then find $x^2 \frac{\partial^2 u}{\partial x^2} + 2xy \frac{\partial^2 u}{\partial x \partial y} + y^2 \frac{\partial^2 u}{\partial y^2}$. [3 marks]
3. Q1(c). If $u^3 = xyz$, $\frac{1}{v} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ and $w^2 = x^2 + y^2 + z^2$, then evaluate $\frac{\partial(u,v,w)}{\partial(x,y,z)}$. [3 marks]
4. Q2(a). i. Show that diagonals of skew symmetric matrix are either zero or pure imaginary. Further, find rank of the matrix$\begin{bmatrix} 6 & 1 & 3 & 8 \\ 4 & 2 & 6 & -1 \\ 10 & 3 & 9 & 7 \\ 16 & 4 & 12 & 15 \end{bmatrix}$ ii. Find the values of $k$ for which the equations $x + y + z = 1$, $2x + y + 4z = k$, $4x + y + 10z = k^2$ has solution?4 [4 marks]
5. Q2(b). Find the eigen values and eigen vectors of the matrix$A = \begin{bmatrix} 3 & 1 & 4 \\ 0 & 2 & 6 \\ 0 & 0 & 5 \end{bmatrix}$ [3 marks]
6. Q2(c). Find the characteristic equation of the matrix$A = \begin{bmatrix} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{bmatrix}$and verify that it is satisfied by $A$ and hence obtain $A^{-1}$. [3 marks]
7. QQ1. . Attempt any Two parts of the following. Q 1(a) is compulsory.