Engineering Mathematics-I PYQ 2018-19 AKTU Question Paper

AKTU · BTECH · Semester 1 · Engineering Mathematics-I · Session 2018-19 · PYQ

AKTU Engineering Mathematics-I previous year question paper 2018-19 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 2018-19

  1. Q1(a). Find the rank of the matrix $\begin{bmatrix} 2 & 2 & 2 \\ 2 & 2 & 2 \\ 2 & 2 & 2 \end{bmatrix}$ (2 marks, 2018-19)
  2. Q1(b). Find the stationary point of $f(x,y)=x^3+y^3+3axy, a>0$ (2 marks, 2018-19)
  3. Q1(c). If $x=r\cos\theta, y=r\sin\theta, z=z$ then find $\frac{\partial(r,\theta,z)}{\partial(x,y,z)}$ (2 marks, 2018-19)
  4. Q1(d). Define del $\nabla$ operator and gradient. (2 marks, 2018-19)
  5. Q1(e). If $\phi=3x^2y-y^3z^2$, find $\text{grad } \phi$ at point $(2, 0, -2)$. (2 marks, 2018-19)
  6. Q1(f). Evaluate $\int_0^1\int_0^{x^2}e^{y/x}\,dx\,dy$. (2 marks, 2018-19)
  7. Q1(g). If the eigen values of matrix A are 1, 1, 1, then find the eigen values of $A^2+2A+3I$. (2 marks, 2018-19)
  8. Q1(h). Define Rolle's Theorem (2 marks, 2018-19)
  9. Q1(j). In $RI=E$, possible errors in $E$ and $I$ are 20% and 10% respectively; find the error in $R$. (2 marks, 2018-19)
  10. Q1(k). State the Taylor's Theorem for two variables. (2 marks, 2018-19)
  11. Q2(b). If $y=\sin(m\sin^{-1}x)$, prove that : $(1-x^2)y_{n+2}-(2n+1)x y_{n+1}-(n^2-m^2)y_n=0$ and find $y_n$ at $x=0$. (10 marks, 2018-19)
  12. Q2(c). If $u, v, w$ are the roots of the equation $(x-a)^3+(x-b)^3+(x-c)^3=0$, then find $\frac{\partial(u,v,w)}{\partial(a,b,c)}$ (10 marks, 2018-19)
  13. Q2(d). Evaluate $\int_{0}^{\infty} \int_{0}^{\infty} e^{-(x^2+y^2)} dx dy$ by changing to polar coordinates. Hence show that $\int_{0}^{\infty} e^{-x^2} dx = \frac{\sqrt{\pi}}{2}$ (10 marks, 2018-19)
  14. Q2(e). Verify the divergence theorem for $\vec{F}=(x^3-yz)\hat{i}+(y^3-zx)\hat{j}+(z^3-xy)\hat{k}$, taken over the cube bounded by planes $x=0, y=0, z=0, x=1, y=1, z=1$. (10 marks, 2018-19)
  15. Q3(a). Find inverse employing elementary transformation $A=\begin{bmatrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{bmatrix}$ (10 marks, 2018-19)
  16. Q3(b). Reduce the matrix A to its normal form when $A=\begin{bmatrix} 1 & 2 & -1 & 4 \\ 2 & 4 & 3 & 4 \\ 1 & 2 & 3 & 4 \\ -1 & -2 & 6 & -7 \end{bmatrix}$. Hence find the rank of A. (10 marks, 2018-19)
  17. Q4(a). If $\sin^{-1}y=2\log(x+1)$ show that $(x+1)^2 y_{n+2}+(2n+1)(x+1)y_{n+1}+(n^2+4)y_n=0$ (10 marks, 2018-19)
  18. Q4(b). Verify Lagrange's Mean value Theorem for the function $f(x)=x^3$ in $[-2, 2]$ (10 marks, 2018-19)
  19. Q5(a). Find the maximum or minimum distance of the point (1, 2, -1) from the sphere $x^2+y^2+z^2=24$. (10 marks, 2018-19)
  20. Q5(b). If $u=\cos^{-1}(\frac{x+y}{\sqrt{x}+\sqrt{y}})$ then show that $x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}+\frac{1}{2}\cot u=0$ (10 marks, 2018-19)
  21. Q6(a). Change the order of integration and then evaluate $\int_0^2\int_{x^2/4}^{3-x}xy\,dy\,dx$. (10 marks, 2018-19)
  22. Q6(b). Calculate the volume of the solid bounded by the surface $x=0, y=0, x+y+z=1$ & $z=0$. (10 marks, 2018-19)
  23. Q7(a). Prove that $(y^2-z^2+3yz-2x)\hat{i}+(3xz+2xy)\hat{j}+(3xy-2xz+2z)\hat{k}$ is both Solenoidal and Irrotational. (10 marks, 2018-19)
  24. Q7(b). Find the directional derivative of $\Phi=5x^2y-5y^2z+\frac{5}{2}z^2x$ at the point $P(1, 1, 1)$ in the direction of the line $\frac{x-1}{2}=\frac{y-3}{-2}=\frac{z}{1}$. (10 marks, 2018-19)

AKTU paper codes: BAS103, KAS103, KAS103T

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