Engineering Mathematics-I PYQ 2021-22 AKTU Question Paper

AKTU · BTECH · Semester 1 · Engineering Mathematics-I · Session 2021-22 · PYQ

AKTU Engineering Mathematics-I previous year question paper 2021-22 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 2021-22

  1. Q1(a). If $A=\begin{pmatrix}-1&2&3\\0&3&5\\0&0&-2\end{pmatrix}$, find the eigen values of $A^3+5A+8I$. (2 marks, 2021-22)
  2. Q1(b). Reduce the matrix $\begin{pmatrix}1&1&1\\3&1&1\end{pmatrix}$ into normal form and find its rank. (2 marks, 2021-22)
  3. Q1(c). Find the envelope of the family of straight line $y = mx + \frac{a}{m}$, where m is a parameter. (2 marks, 2021-22)
  4. Q1(d). Can mean value theorem be applied to $f(x) = \tan x$ in $[0, \pi]$. (2 marks, 2021-22)
  5. Q1(e). State Euler’s Theorem on homogeneous function. (2 marks, 2021-22)
  6. Q1(f). Find the critical points of the function $f(x, y) = x^3 + y^3 - 3axy$. (2 marks, 2021-22)
  7. Q1(g). Find the area bounded by curve $y^2 = x$ and $x^2 = y$. (2 marks, 2021-22)
  8. Q1(h). Find the value of $\int_0^1\int_0^x\int_0^{x+y}\,dx\,dy\,dz$. (2 marks, 2021-22)
  9. Q1(i). Find a unit normal vector to the surface $z^2 = x^2 + y^2$ at the point $(1, 0, -1)$. (2 marks, 2021-22)
  10. Q1(j). State Stoke’s Theorem. (2 marks, 2021-22)
  11. Q2(a). Find the characteristic equation of the matrix A = \begin{pmatrix} 2 & 1 & 1 \\ 0 & 1 & 0 \\ 1 & 1 & 2 \end{pmatrix}, compute $A^{-1}$ and prove that $A^8 - 5A^7 + 7A^6 - 3A^5 + A^4 - 5A^3 + 8A^2 - 2A + I = \begin{pmatrix} 8 & 5 & 5 \\ 0 & 3 & 0 \\ 5 & 5 & 8 \end{pmatrix}$. (10 marks, 2021-22)
  12. Q2(b). State Rolle’s theorem and verify Rolle’s theorem for the function $f(x) = \frac{\sin x}{e^x}$ in $[0, \pi]$. (10 marks, 2021-22)
  13. Q2(c). If u, v and w are the roots of $(\lambda-x)^3 + (\lambda-y)^3 + (\lambda-z)^3 = 0$, cubic in $\lambda$, find $\frac{\partial(u,v,w)}{\partial(x,y,z)}$. (10 marks, 2021-22)
  14. Q2(d). Find the volume bounded by the cylinder $x^2 + y^2 = 4$ and the plane $y + z = 4$ and $z = 0$. (10 marks, 2021-22)
  15. Q2(e). Apply Green’s theorem to evaluate $\int_C [(2x^2 - y^2)dx + (x^2 + y^2)dy]$, where C is the boundary of the area enclosed by the x-axis and the upper half of the circle $x^2 + y^2 = a^2$. (10 marks, 2021-22)
  16. Q3(a). Find the value of $k$ for which the system of equations $(3k - 8)x + 3y + 3z = 0$, $3x + (3k - 8)y + 3z = 0$, $3x + 3y + (3k - 8)z = 0$ has a non-trivial solution. (10 marks, 2021-22)
  17. Q3(b). Find the eigen values and eigen vectors of matrix A = \begin{pmatrix} 2 & 1 & 1 \\ 2 & 3 & 2 \\ 3 & 3 & 4 \end{pmatrix}. (10 marks, 2021-22)
  18. Q4(a). If $f(x) = \frac{x}{1+e^{1/x}}$; $x \neq 0$ and $f(0) = 0$, then show that the function is continuous but not differentiable at $x = 0$. (10 marks, 2021-22)
  19. Q4(b). If $y = (x + \sqrt{1+x^2})^m$, find $y_n(0)$. (10 marks, 2021-22)
  20. Q5(a). Expand $x^y$ in powers of $(x - 1)$ and $(y - 1)$ up to the third-degree terms and hence evaluate $(1.1)^{1.02}$. (10 marks, 2021-22)
  21. Q5(b). A rectangular box which is open at the top having capacity 32c.c. Find the dimension of the box such that the least material is required for its constructions. (10 marks, 2021-22)
  22. Q6(a). Change the order of integration in $I = \int_0^1 \int_{x^2}^{2-x} xydydx$ and hence evaluate the same. (10 marks, 2021-22)
  23. Q6(b). Find the position of the C.G. of a semicircular lamina of radius, a if its density varies as the square of the distance from the diameter. (10 marks, 2021-22)
  24. Q7(a). Find the directional derivative of $\nabla(\nabla f)$ at the point $(1, -2, 1)$ in the direction of the normal to the surface $xy^2z = 3x + z^2$ where $f = 2x^3y^2z^4$. (10 marks, 2021-22)
  25. Q7(b). Find the constants $a, b, c$ so that $\vec{F} = (x + 2y + az)\hat{i} + (bx - 3y - z)\hat{j} + (4x + cy + 2z)\hat{k}$ is irrotational and hence find function $\phi$ such that $\vec{F} = \nabla\phi$. (10 marks, 2021-22)

AKTU paper codes: BAS103, KAS103, KAS103T

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