Calculus and Linear Algebra — Minor 2 2023 question paper

Paper details

• Subject: Calculus and Linear Algebra (BSM - 101)
• Branch: Chemical Engineering
• Semester: 1
• Exam: Minor 2 2023
• Questions: 8

Questions asked in Calculus and Linear Algebra Minor 2 2023

1. Q1(a). (i) Change the order of integration in$\int_{0}^{a} \int_{\sqrt{a^2-x^2}}^{x+2a} dy dx$Hence evaluate it.(ii) Show that$\int_{0}^{\infty} \frac{x^c}{c^x} dx = \frac{\Gamma(c+1)}{(\log c)^{c+1}}, \quad c > 1.$(Note: Labelled as (i) in the paper) [5 marks]
2. Q1(b). Evaluate$\int_{0}^{a} \int_{0}^{x} \int_{0}^{x+y} e^{x+y+z} dx dy dz$ [3 marks]
3. Q1(c). c)Evaluate $\iint_{R} (x-y)^4 e^{x+y} dx dy$, where $R$ is the square with vertices at $(1,0)$, $(2,1)$, $(1,2)$ and $(0,1)$. [2 marks]
4. Q2(a). (i) Find $\iint_{S} \vec{F} \cdot \hat{n} \, dS$, where $\vec{F} = (2x + 3z^2y)\hat{i} - (x^2z + 5y)\hat{j} + (y^2x + 10z)\hat{k}$, and $S$ is the surface of the sphere having centre at $(3.5, 2)$ and radius 3.(ii) Find the directional derivative of a function $\phi = xy^2 + yz^3$ at the point $(2,-1,1)$ in the direction of normal to the surface $x \log z - y^2 + 4 = 0$ at the point $(-1,2,1)$. [4 marks]
5. Q2(b). Evaluate the line integral $\int_{C} y^2 dx - x^2 dy$ around the triangle whose vertices are $(1,0)$, $(0,1)$ and $(-1,0)$ in the positive sense. [3 marks]
6. Q2(c). Prove that $\text{div}(\text{grad } r^n) = \nabla^2(r^n) = n(n+1)r^{n-2}$, where $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$. [3 marks]
7. QQ1. .Attempt any Two parts of the following. Q. 1 (a) is compulsory.
8. QQ2. Attempt any Two parts of the following. Q. 2 (a) is compulsory.