Engineering Mathematics-I PYQ 2025-26 AKTU Question Paper

AKTU · BTECH · Semester 1 · Engineering Mathematics-I · Session 2025-26 · PYQ

AKTU Engineering Mathematics-I previous year question paper 2025-26 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 2025-26

  1. Q1(a). In the matrix equation $\begin{bmatrix} 3 & -1 \\ 2 & 5 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 4 \\ -3 \end{bmatrix}$, what are the values of x and y? (2 marks, 2025-26)
  2. Q1(b). Find the Rank of the following Matrix A: $A = \begin{bmatrix} 2 & -4 & 6 \\ -1 & 2 & -3 \\ 3 & -6 & 9 \end{bmatrix}$. (2 marks, 2025-26)
  3. Q1(c). If $y = x^3 \log_e x$, prove that $\frac{d^4y}{dx^4} = \frac{6}{x}$. (2 marks, 2025-26)
  4. Q1(d). If $u(x, y) = (\sqrt{x} + \sqrt{y})^5$, then find the degree of the function $u(x, y)$. (2 marks, 2025-26)
  5. Q1(e). If $x=r\cos\theta$, $y=r\sin\theta$, find $\frac{\partial(r,\theta)}{\partial(x,y)}$. (2 marks, 2025-26)
  6. Q1(f). Evaluate the following integral using the Beta-Gamma function: $\int_{0}^{2} x(8-x^3)^{1/3} dx$. (2 marks, 2025-26)
  7. Q1(g). Show that gradient field describing a motion is irrotational. (2 marks, 2025-26)
  8. Q2(a). Determine the values of $\lambda$ and $\mu$ such that the system $2x-5y+2z=8$, $2x+4y+6z=5$, $x+2y+\lambda z=\mu$ has (i) no solution, (ii) a unique solution, and (iii) infinitely many solutions. (7 marks, 2025-26)
  9. Q2(b). Find the $n^{th}$ derivative of $\frac{2x+1}{(2x-1)(2x+3)}$. (7 marks, 2025-26)
  10. Q2(c). Expand $x^2y + 3y - 2$ in powers of $(x-1)$ and $(y+2)$ up to 3rd degree using Taylor’s series. (7 marks, 2025-26)
  11. Q2(d). Change the order of integration and hence evaluate $\int_0^a\int_{\sqrt{ax}}^a\frac{y^2}{\sqrt{y^4-a^2x^2}}\,dy\,dx$. (7 marks, 2025-26)
  12. Q2(e). Find the directional derivative of the divergence of $f(x, y, z) = xyi + xy^2j + z^2k$ at the point (2, 1, 2) in the direction of the outer normal to the sphere $x^2 + y^2 + z^2 = 9$. (7 marks, 2025-26)
  13. Q3(a). Find the eigen values and eigen vectors of the following Matrix A: $A = \begin{bmatrix} 6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3 \end{bmatrix}$. (7 marks, 2025-26)
  14. Q3(b). If $X_1 = \begin{bmatrix} 3 \\ 1 \\ -4 \end{bmatrix}, X_2 = \begin{bmatrix} 2 \\ 2 \\ -3 \end{bmatrix}, X_3 = \begin{bmatrix} 0 \\ -4 \\ 1 \end{bmatrix}$, then show that the vectors $X_1, X_2, X_3$ are linearly dependent. (7 marks, 2025-26)
  15. Q4(a). If $z = x^4y^2 \sin^{-1}(\frac{x}{y}) + \log x - \log y$, show that $x \frac{\partial z}{\partial x} + y \frac{\partial z}{\partial y} = 6x^4y^2 \sin^{-1}(\frac{x}{y})$. (7 marks, 2025-26)
  16. Q4(b). Trace the curve: $y^2(a+x) = x^2(3a-x)$. (7 marks, 2025-26)
  17. Q5(a). Find the minimum distance from the point (1,2,0) to the cone $z^2 = x^2 + y^2$. (7 marks, 2025-26)
  18. Q5(b). Find approximate value of $[(0.98)^2 + (2.01)^2 + (1.94)^2]^{1/2}$. (7 marks, 2025-26)
  19. Q6(a). Prove that the area in the positive quadrant bounded by the curves $y^2 = 4ax, y^2 = 4bx, xy = c^2$ and $xy = d^2$ is $\frac{1}{3}(d^2-c^2) \log(\frac{b}{a}); d > c, b > a$. (7 marks, 2025-26)
  20. Q6(b). Find the volume of the solid bounded by the co-ordinate planes and the surface $\sqrt{\frac{x}{a}} + \sqrt{\frac{y}{b}} + \sqrt{\frac{z}{c}} = 1$. (7 marks, 2025-26)
  21. Q7(a). Apply Green’s theorem to evaluate $\int_{C} [(y - \sin x) dx + \cos x dy]$, where C is the triangle formed by $y = 0, x = \frac{\pi}{2}, y = \frac{2}{\pi}x$. (7 marks, 2025-26)
  22. Q7(b). Apply Gauss’s divergence theorem to evaluate $\iint_{S} \vec{F} \cdot \hat{n} ds$, where $\vec{F} = (2x^2 - 3z)i - 2xyj - 4xk$ and S is the Surface bounded by the planes $x = 0, y = 0, z = 0$ and $2x + 2y + z = 4$. (7 marks, 2025-26)

AKTU paper codes: BAS103, KAS103, KAS103T

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