Engineering Mathematics-I Unit 5 – Vector Calculus: AKTU previous year questions
12 AKTU questions from Unit 5 (Vector Calculus) asked in 2024–2026, tagged by marks, year and topic. In short: about 29 marks of every paper come from this unit; the most asked topic is Gradient, Curl, Divergence and Their Physical Interpretations (15 times); 12 questions came back in a later year. The latest ones are listed below; on the page you can filter them by 2-mark or long questions, topic and repeats.
Unit 5 previous year questions
- Show that gradient field describing a motion is irrotational. (2 marks, 2026, Gradient, Curl, Divergence and Their Physical Interpretations)
- 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, 2026, Directional Derivatives)
- 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, 2026, Green's Theorem (Without Proof) and Applications)
- 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, 2026, Gauss' Divergence Theorem (Without Proof) and Applications)
- Prove that $curl \vec{r} = 0$. (2 marks, 2025, Gradient, Curl, Divergence and Their Physical Interpretations)
- Show that the vector $f(r)\vec{r}$ is irrotational. Where $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$. (7 marks, 2025, Gradient, Curl, Divergence and Their Physical Interpretations)
- Show that $div (grad r^n) = n(n + 1)r^{n-2}$. Where $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$. (7 marks, 2025, Laplacian and Radial Scalar Fields)
- Verify Stokes theorem for $\vec{F} = (x^2 + y^2)\hat{i} - 2xy\hat{j}$ taken round the rectangle bounded by the lines $x = 0, x = a, y = 0, y = b$. (7 marks, 2025, Stokes' Theorem (Without Proof) and Applications)
- Prove that $\vec A=(6xy+z^3)\hat i+(3x^2-z)\hat j+(3xz^2-y)\hat k$ is irrotational. (2 marks, 2024, Gradient, Curl, Divergence and Their Physical Interpretations)
- Find a unit normal vector to the surface $xy^3z^2=4$ at the point $(-1,-1,2)$. (2 marks, 2024, Gradient, Curl, Divergence and Their Physical Interpretations)
- Find the directional derivative of $f(x,y,z)=e^{2x}\cos yz$ at $(0,0,0)$ in the direction of the tangent to the curve $x=a\sin\theta,\ y=a\cos\theta,\ z=a\theta$ at $\theta=\pi/4$. (7 marks, 2024, Directional Derivatives)
- Applying Gauss Divergence theorem, evaluate $ \iint_S[e^x\,dy\,dz-ye^x\,dz\,dx+3z\,dx\,dy], $ where $S$ is the surface of the cylinder $x^2+y^2=c^2$, $0\le z\le h$. (7 marks, 2024, Gauss' Divergence Theorem (Without Proof) and Applications)
Most asked Unit 5 topics
- Gradient, Curl, Divergence and Their Physical Interpretations – asked 15 times
- Directional Derivatives – asked 6 times
- Gauss' Divergence Theorem (Without Proof) and Applications – asked 6 times
Unit 5 question pattern
- 2-mark questions: 4 asked in 2024–2026
- 7-mark questions: 8 asked in 2024–2026
- About 29 marks from Unit 5 in every paper
- 12 questions were asked again in a later year
Unit 5 questions year by year
- 2026: 4 Unit 5 questions asked (23 marks across that year's papers)
- 2025: 4 Unit 5 questions asked (23 marks across that year's papers)
- 2024: 4 Unit 5 questions asked (18 marks across that year's papers)