Engineering Mathematics-I PYQ 2019-20 AKTU Question Paper

AKTU · BTECH · Semester 1 · Engineering Mathematics-I · Session 2019-20 · PYQ

AKTU Engineering Mathematics-I previous year question paper 2019-20 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 2019-20

  1. Q1(a). Show that vectors $(1,6,4)$, $(0,2,3)$ and $(0,1,2)$ are linearly independent. (2 marks, 2019-20)
  2. Q1(b). Define Lagrange's mean value theorem. (2 marks, 2019-20)
  3. Q1(c). If $u=x(1-y)$ and $v=xy$, find $\frac{\partial(u,v)}{\partial(x,y)}$. (2 marks, 2019-20)
  4. Q1(d). Show that $\vec V=(x+3y)\hat i+(y-3z)\hat j+(x-2z)\hat k$ is solenoidal. (2 marks, 2019-20)
  5. Q1(e). Find the value of $b$ so that the rank of $A=\begin{bmatrix}2&4&2\\3&1&2\\1&0&b\end{bmatrix}$ is 2. (2 marks, 2019-20)
  6. Q1(f). Evaluate $\int_0^2\int_0^1(x^2+3y^2)\,dy\,dx$. (2 marks, 2019-20)
  7. Q1(g). Find $\operatorname{grad}\phi$ at $(2,1,3)$, where $\phi=x^2+yz$. (2 marks, 2019-20)
  8. Q1(h). If $u=\cos^{-1}\!\left(\frac{x+y}{\sqrt x+\sqrt y}\right)$, find $x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}$. (2 marks, 2019-20)
  9. Q1(i). Find $\frac{du}{dt}$ if $u=x^3+y^3$, $x=a\cos t$, $y=b\sin t$. (2 marks, 2019-20)
  10. Q1(j). Find the area lying between the parabola $y=4x-x^2$ and above the line $y=x$. (2 marks, 2019-20)
  11. Q2(a). Verify Cayley-Hamilton theorem for $A=\begin{bmatrix}4&0&1\\0&1&2\\1&0&1\end{bmatrix}$ and hence find $A^{-1}$. (10 marks, 2019-20)
  12. Q2(b). If $y=e^{m\cos^{-1}x}$, prove that $(1-x^2)y_{n+2}-(2n+1)xy_{n+1}-(n^2+m^2)y_n=0$. Hence find $y_n$ when $x=0$. (10 marks, 2019-20)
  13. Q2(c). If $u^3+v^3+w^3=x+y+z$, $u^2+v^2+w^2=x^3+y^3+z^3$, and $u+v+w=x^2+y^2+z^2$, show that $\frac{\partial(u,v,w)}{\partial(x,y,z)}=\frac{(x-y)(y-z)(z-x)}{(u-v)(v-w)(w-u)}$. (10 marks, 2019-20)
  14. Q2(d). Evaluate the integral by changing the order of integration: $I=\int_0^1\int_{x^2}^{2-x}xy\,dy\,dx$. (10 marks, 2019-20)
  15. Q2(e). Verify Stoke's theorem for $\vec F=(x^2-y^2)\hat i+2xy\hat j$, integrated round the rectangle in the plane $z=0$ bounded by $x=0$, $y=0$, $x=a$, $y=b$. (10 marks, 2019-20)
  16. Q3(a). For what values of $\lambda$ and $\mu$ does the system $x+y+z=6$, $x+2y+5z=10$, $2x+3y+\lambda z=\mu$ have (i) a unique solution, (ii) no solution, and (iii) infinitely many solutions? Also find the solution for $\lambda=2$ and $\mu=8$. (10 marks, 2019-20)
  17. Q3(b). Find the rank of $A=\begin{bmatrix}1&3&4&2\\2&-1&3&2\\3&-5&2&2\\6&-3&8&6\end{bmatrix}$ by reducing it to normal form. (10 marks, 2019-20)
  18. Q4(a). Verify Cauchy's mean value theorem for $e^x$ and $e^{-x}$ on $[a,b]$. Also show that $c$ is the arithmetic mean of $a$ and $b$. (10 marks, 2019-20)
  19. Q4(b). Trace the curve $r^2=a^2\cos2\theta$. (10 marks, 2019-20)
  20. Q5(a). If $u=f(2x-3y,3y-4z,4z-2x)$, prove that $\frac12\frac{\partial u}{\partial x}+\frac13\frac{\partial u}{\partial y}+\frac14\frac{\partial u}{\partial z}=0$. (10 marks, 2019-20)
  21. Q5(b). Find the volume of the largest rectangular parallelepiped that can be inscribed in $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$. (10 marks, 2019-20)
  22. Q6(a). Evaluate $\iint_R(x+y)^2\,dx\,dy$, where $R$ is the parallelogram with vertices $(1,0)$, $(3,1)$, $(2,2)$, $(0,1)$, using $u=x+y$, $v=x-2y$. (10 marks, 2019-20)
  23. Q6(b). Find the volume of the region bounded by $y=x^2$, $x=y^2$ and the planes $z=0$, $z=3$. (10 marks, 2019-20)
  24. Q7(a). Verify the divergence theorem for $\vec F=4xz\hat i-y^2\hat j+yz\hat k$ over the rectangular parallelepiped $0\le x\le1$, $0\le y\le1$, $0\le z\le1$. (10 marks, 2019-20)
  25. Q7(b). Find the directional derivative of $\phi(x,y,z)=x^2yz+4xz^2$ at $(1,-2,1)$ in the direction $2\hat i-\hat j-2\hat k$. Find also the greatest rate of increase of $\phi$. (10 marks, 2019-20)

AKTU paper codes: BAS103, KAS103, KAS103T

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