Engineering Mathematics-I PYQ 2020-21 AKTU Question Paper

AKTU · BTECH · Semester 1 · Engineering Mathematics-I · Session 2020-21 · PYQ

AKTU Engineering Mathematics-I previous year question paper 2020-21 for B.Tech Semester 1. Covers Matrices, Differential Calculus-I, Differential…

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Engineering Mathematics-I AKTU syllabus

  1. Unit 1: Matrices
  2. Unit 2: Differential Calculus-I
  3. Unit 3: Differential Calculus-II
  4. Unit 4: Multiple Integration
  5. Unit 5: Vector Calculus

Questions in Engineering Mathematics-I AKTU PYQ 2020-21

  1. Q1(a). Prove that the matrix $\frac{1}{\sqrt{3}} \begin{pmatrix} 1 & 1+i \\ 1-i & -1 \end{pmatrix}$ is unitary. (2 marks, 2020-21)
  2. Q1(b). State Rank-Nullity Theorem. (2 marks, 2020-21)
  3. Q1(c). State Rolle’s Theorem. (2 marks, 2020-21)
  4. Q1(d). Discuss all the symmetry of the curve $x^2y^2 = x^2 - a^2$ (2 marks, 2020-21)
  5. Q1(e). If $u = f(y - z, z - x, x - y)$, prove that $\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} + \frac{\partial u}{\partial z} = 0$ (2 marks, 2020-21)
  6. Q1(f). If $x = e^v \sec u, y = e^v \tan u$, then evaluate $\frac{\partial(x, y)}{\partial(u, v)}$. (2 marks, 2020-21)
  7. Q1(g). Evaluate $\int_0^1\int_0^{x^2}e^{y/x}\,dy\,dx$. (2 marks, 2020-21)
  8. Q1(h). Calculate the volume of the solid bounded by the surface x = 0, y = 0, x+y+z=1 and z=0. (2 marks, 2020-21)
  9. Q1(i). Show that the vector $\vec{V} = (x + 3y)\hat{i} + (y - 3z)\hat{j} + (x - 2z) \hat{k}$ is solenoidal. (2 marks, 2020-21)
  10. Q1(j). State Green’s theorem. (2 marks, 2020-21)
  11. Q2(a). Find the inverse of the matrix $A = \begin{pmatrix} 2 & 3 & 4 \\ 4 & 3 & 1 \\ 1 & 2 & 4 \end{pmatrix}$ (10 marks, 2020-21)
  12. Q2(b). If $y = e^{\tan^{-1}x}$, prove that $(1 + x^2)y_{n+2} + [(2n+2)x - 1]y_{n+1} + n(n+1)y_n = 0$. (10 marks, 2020-21)
  13. Q2(c). If $u^3 + v + w = x + y^2 + z^2$, $u + v^3 + w = x^2 + y + z^2$, $u + v + w^3 = x^2 + y^2 + z$, Show that: $\frac{\partial(u, v, w)}{\partial(x, y, z)} = \frac{1 - 4xy(xy + yz + zx) + 16xyz}{2 - 3(u^2 + v^2 + w^2) + 27u^2v^2w^2}$ (10 marks, 2020-21)
  14. Q2(d). Evaluate by changing the variables, $\iint_R (x+y)^2 dx dy$ where R is the region bounded by the parallelogram x+y=0, x+y =2, 3x-2y=0 and 3x-2y = 3. (10 marks, 2020-21)
  15. Q2(e). Use divergence theorem to evaluate the surface integral $\iint_S (x dy dz + y dz dx + z dx dy)$ where S is the portion of the plane x+2y+3z=6 which lies in the first octant. (10 marks, 2020-21)
  16. Q3(a). Find non-singular matrices P and Q such that PAQ is normal form. $\begin{pmatrix} 1 & 1 & 2 \\ 1 & 2 & 3 \\ 0 & 1 & 1 \end{pmatrix}$ (10 marks, 2020-21)
  17. Q3(b). Find the eigen values and the corresponding eigen vectors of the following matrix. $A = \begin{pmatrix} 2 & 0 & 1 \\ 0 & 3 & 0 \\ 1 & 0 & 2 \end{pmatrix}$. (10 marks, 2020-21)
  18. Q4(a). Find the envelope of the family of lines $\frac{x}{a} + \frac{y}{b} = 1$, where $a$ and $b$ are connected by the relation $a^n + b^n = c^n$ (10 marks, 2020-21)
  19. Q4(b). If y = sin (m sin⁻¹x), find the value of yn at x= 0. (10 marks, 2020-21)
  20. Q5(a). Divide 24 into three parts such that continued product of first, square of second and cube of third is a maximum. (10 marks, 2020-21)
  21. Q5(b). If $u = \sec^{-1} \left( \frac{x^3 - y^3}{x + y} \right)$, prove that $x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = 2 \cot u$. Also evaluate $x^2 \frac{\partial^2 u}{\partial x^2} + 2xy \frac{\partial^2 u}{\partial x \partial y} + y^2 \frac{\partial^2 u}{\partial y^2}$. (10 marks, 2020-21)
  22. Q6(a). Evaluate the following integral by changing the order of integration $\int_0^\infty \int_x^\infty \frac{e^{-y}}{y} dy dx$. (10 marks, 2020-21)
  23. Q6(b). A triangular thin plate with vertices (0,0),(2,0) and (2,4) has density $\rho = 1+x+y$. Then find: (i) The mass of the plate. (ii) The position of its centre of gravity G. (10 marks, 2020-21)
  24. Q7(a). A fluid motion is given by $\vec{v} = (y \sin z - \sin x)\hat{i} + (x \sin z + 2yz)\hat{j} + (xy \cos z + y^2)\hat{k}$. Is the motion irrotational? If so, find the velocity potential. (10 marks, 2020-21)
  25. Q7(b). Verify Stoke’s theorem for the function $\vec{F} = x^2\hat{i} + xy\hat{j}$ integrated round the square whose sides are x=0,y=0,x=a,y=a in the plane z=0. (10 marks, 2020-21)

AKTU paper codes: BAS103, KAS103, KAS103T

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