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Calculus and Linear Algebra — Major (End Semester) 2023 question paper

MMMUT Chemical Engineering previous year question paper for Calculus and Linear Algebra (BSM-101), semester 1, Major (End Semester) 2023. All 24 questions are listed below, each with a written answer on Nexsus.

Paper details

  • Subject: Calculus and Linear Algebra (BSM-101)
  • Branch: Chemical Engineering
  • Semester: 1
  • Exam: Major (End Semester) 2023
  • Questions: 24

Questions asked in Calculus and Linear Algebra Major (End Semester) 2023

  1. Q1(a). (a) Find the n th derivative of (x−2) 5 x ​ [2 marks]
  2. Q1(b). If x=u(1−v),y=v−u then compute ∂(u,v) ∂(x,y) ​ and ∂(x,y) ∂(u,v) ​ and prove that ∂(u,v) ∂(x,y) ​ ⋅ ∂(x,y) ∂(u,v) ​ =1 [2 marks]
  3. Q1(c). Find the sum and product of the eigenvalues of the matrix A= ​ 1 1 −2 ​ 2 0 1 ​ −2 3 3 ​ ​ [2 marks]
  4. Q1(d). If α and β are the eigenvalues of [ 3 −1 ​ −1 5 ​ ] then form the matrix whose eigenvalues are α 3 and β 3 . [2 marks]
  5. Q1(e). Show that ∇(r n+1 )=(n+1)r n [2 marks]
  6. Q1(f). Evaluate ∫ 1−x 4 ​ x 2 dx ​ ×∫ 1+x 2 ​ dx ​ [2 marks]
  7. Q1(g). Find the constants a and b such that the surface 3x 2 −2y 2 −3z 2 +8=0 is orthogonal to x 2 +9y 2 =bz−(1,2,1) [2 marks]
  8. Q2(a). If z is a function of x and y, and x=ucosα−vsinα,y=usinα+vcosα then show that ∂x 2 ∂ 2 z ​ + ∂y 2 ∂ 2 z ​ = ∂u 2 ∂ 2 z ​ + ∂v 2 ∂ 2 z ​ [5 marks]
  9. Q2(b). Examine f(x,y)=x 5 +y 5 −12x−3y+20 for its extreme points. Hence find its extreme value. [5 marks]
  10. Q2(c). Suppose f(x,y)=tan −1 ( y x ​ ) Use Taylor’s series expansion to compute an approximate value of f(0.9,−1.2) [5 marks]
  11. Q3(a). (a) Find the inverse of the following matrix using elementary operations: A= ​ 0 1 1 −1 ​ 2 1 2 1 ​ 1 −1 0 2 ​ 3 −2 1 6 ​ ​ [5 marks]
  12. Q3(b). Use Cayley–Hamilton theorem to find the matrix A 5 −5A 4 +7A 3 −3A 2 +8A−5A 4 +8A 2 −2A+I if A= ​ 0 1 1 ​ 1 1 1 ​ 1 1 2 ​ ​ [5 marks]
  13. Q3(c). Check the following system of equations for its consistency. If possible, solve it.$3x + 3y + 2z = 1, \quad x + 2y = 4, \quad 10y + 3z = -2, \quad 2x - 3y - z = 5$ [5 marks]
  14. Q4(a). a) * (i) Evaluate $\int_{0}^{\infty} \int_{0}^{x} x e^{-x^2/y} \, dy dx$ by changing the order of integration.(ii) Evaluate $\iint xy \, dxdy$ over the region bounded by $x = 2a$ and the curve $x^2 = 4ay$. [5 marks]
  15. Q4(b). b) Find the volume and mass contained in the solid region of the positive octant of the surface $\left(\frac{x}{a}\right)^p + \left(\frac{y}{b}\right)^q + \left(\frac{z}{c}\right)^r = 1$, where $p, q, \& \ r > 0$, given that density at any point $\rho(x,y,z) = k\sqrt{xyz}$. [5 marks]
  16. Q4(c) . Evaluate(i) $\int_{0}^{1} \log \Gamma(x) \, dx$(ii) $\int_{-\infty}^{\infty} \cos \frac{\pi}{2} x^2 \, dx$ [5 marks]
  17. Q5(a) . a) * (i) In what direction from the point $(1, 1, -2)$, the directional derivative of $\phi = x^2 - 2y^2 + 4z^2$ is maximum. Also find maximum directional derivative.(ii) Show that the vector field $\vec{F} = \frac{\vec{r}}{r^3}$ is irrotational as well as solenoidal. Find the scalar potential. Here $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ and $r = |\vec{r}|$. [5 marks]
  18. Q5(b) . b) Verify Gauss divergence theorem for $\vec{F} = 4xz\hat{i} - y^2\hat{j} + yz\hat{k}$ taken over the cube bounded by $x = 0, x = 1, y = 0, y = 1, z = 0$ and $z = 1$. [5 marks]
  19. Q5(c) . State and verify Stoke's theorem for A =(x 2 +y 2 ) i ^ −2xy j ^ ​ taken round the rectangle bounded by the lines x=±a,y=0,y=b. [5 marks]
  20. QQ1. Attempt any five parts of the following
  21. QQ2. Attempt any two parts of the following [2 marks]
  22. QQ3. Attempt any two parts of the following
  23. QQ4. Attempt any Two parts of the following.
  24. QQ5. Attempt any Two parts of the following.