Vector Analysis – Mathematics Optional PYQs (2008–2026)

Here are all 89 Vector Analysis questions asked in UPSC Mathematics Optional Paper I from 2008 to 2026, arranged under the 3 topics of the official syllabus. Within each topic the latest questions come first.

Each question ends with a code such as [2024/10m/1a] – the year, the marks and the question number in the paper.

6.1] Vector differentiation

34 questions

1. For a scalar point function $\varphi$ and a vector point function $\vec{F}$, prove that $\nabla \times \left(\varphi \vec{F}\right)=\left(\nabla \varphi \right)\times \vec{F}+\varphi (\nabla \times \vec{F})$ If $\vec{r}=x\hat{i}+y\hat{j}+z\hat{k}$ and $r=\left|\vec{r}\right|$, then verify the above identity for $\varphi =\frac{1}{r}$ and $\vec{F}=\vec{r}$. [2026/10m/8c(i)]

2. If $\vec{r}=acost\hat{i}+asint\hat{j}+bt\hat{k}$, then show that ${|\vec{r}\prime \times \vec{r}″|}^2=a^2(a^2+b^2)\mathrm{\text{and}}\left[\vec{r}\prime \vec{r}″\vec{r}‴\right]=a^{2}b$ [2026/10m/8c(ii)]

3. If $u=x+y+z$, $v=x^2+y^2+z^2$ and $w=xy+yz+zx$, then show that $gradu$, $gradv$ and $gradw$ are coplanar. [2025/10m/5e]

4. Find the absolute value of the directional derivative of $\varphi (x,y,z)=x^2{y}^2{z}^2$ at the point $(1,1,-1)$ in the direction of the tangent to the curve $x=e^t,y=2sint+1,z=t-cost$, at $t=0$. [2025/10m/6c(i)]

5. If $\nabla \cdot \vec{E}=0$, $\nabla \cdot \vec{H}=0$, $\nabla \times \vec{E}=-\frac{\partial \vec{H}}{\partial t}$ and $\nabla \times \vec{H}=\frac{\partial \vec{E}}{\partial t}$, then show that ${\nabla }^2\vec{H}=\frac{{\partial }^2\vec{H}}{\partial t^2}$ and ${\nabla }^2\vec{E}=\frac{{\partial }^2\vec{E}}{\partial t^2}$. [2025/10m/6c(ii)]

6. At any time $t$ (in seconds), the coterminous edges of a variable parallelepiped are represented by the vectors $\begin{array}{cc}\hfill \bar{\alpha }\hfill & \hfill =t\hat{i}+(t+1)\hat{j}+(2t+1)\hat{k}\hfill \\ \hfill \bar{\beta }\hfill & \hfill =2t\hat{i}+(3t-1)\hat{j}+t\hat{k}\hfill \\ \hfill \bar{\gamma }\hfill & \hfill =\hat{i}+3t\hat{j}+\hat{k}\hfill \end{array}$ What is the rate of change of the vectorial area of the parallelogram, whose coterminous edges are $\bar{\alpha }$ and $\bar{\gamma }$? Also find the rate of change of the volume of the parallelepiped at $t=1$ second. [2024/10m/5c]

7. If $\begin{array}{cc}\hfill \vec{a}\hfill & \hfill =sin\theta \hat{i}+cos\theta \hat{j}+\theta \hat{k}\hfill \\ \hfill \vec{b}\hfill & \hfill =cos\theta \hat{i}-sin\theta \hat{j}-3\hat{k}\hfill \\ \hfill \vec{c}\hfill & \hfill =2\hat{i}+3\hat{j}-3\hat{k}\hfill \end{array}$ then find the values of the derivative of the vector function $\vec{a}\times (\vec{b}\times \vec{c})$ w.r.t. $\theta$ at $\theta =\frac{\pi }{2}$ and $\theta =\pi$. [2023/10m/5e]

8. For a scalar point function $\varphi$ and vector point function $\vec{f}$, prove the identity $\nabla \cdot \left(\varphi \vec{f}\right)=\nabla \varphi \cdot \vec{f}+\varphi (\nabla \cdot \vec{f})$. Also find the value of $\nabla \cdot \left(\frac{f\left(r\right)}{r}\vec{r}\right)$ and then verify stated identity. [2023/15m/8c]

9. Show that $\vec{A}=(6xy+z^3)\hat{i}+(3x^2-z)\hat{j}+(3x{z}^2-y)\hat{k}$ is irrotational. Also find $\varphi$ such that $\vec{A}=\nabla \varphi$. [2022/10m/5e]

10. Show that ${\nabla }^2[\nabla \cdot \left(\frac{\vec{r}}{r}\right)]=\frac{2}{r^4}$, where $\vec{r}=x\hat{i}+y\hat{j}+z\hat{k}$. [2021/10m/5e]

11. For what value of $a$, $b$, $c$ is the vector field $\overline{V}=(-4x-3y+az)\hat{i}+(bx+3y+5z)\hat{j}+(4x+cy+3z)\hat{k}$ irrotational? Hence, express $\overline{V}$ as the gradient of a scalar function $\varphi$. Determine $\varphi$. [2020/10m/5c]

12. Find the directional derivative of the function $x{y}^2+y{z}^2+z{x}^2$ along the tangent to the curve $x=t$, $y=t^2$, $z=t^3$ at the point $(1,1,1)$. [2019/10m/5e]

13. Let $\vec{v}=v_1\vec{i}+v_2\vec{j}+v_3\vec{k}$. Show that $curl\left(curl\vec{v}\right)=grad\left(div\vec{v}\right)-{\nabla }^2\vec{v}$. [2018/12m/8a]

14. For what values of the constants $a$, $b$ and $c$ the vector $\bar{V}=(x+y+az)\hat{i}+(bx+2y-z)\hat{j}+(-x+cy+2z)\hat{k}$ is irrotational. Find the divergence in cylindrical coordinates of this vector with these values. [2017/10m/5d]

15. The position vector of a moving point at time $t$ is $\bar{r}=sint\hat{i}+cos2t\hat{j}+(t^2+2t)\hat{k}$. Find the components of acceleration $\bar{a}$ in the directions parallel to the velocity vector $\bar{v}$ and perpendicular to the plane of $\bar{r}$ and $\bar{v}$ at time $t=0$. [2017/10m/5e]

16. Prove that the vectors $\vec{a}=3\hat{i}+\hat{j}-2\hat{k}$, $\vec{b}=-\hat{i}+3\hat{j}+4\hat{k}$, $\vec{c}=4\hat{i}-2\hat{j}-6\hat{k}$ can form the sides of a triangle. Find the lengths of the medians of the triangle. [2016/10m/5b]

17. Find $f\left(r\right)$ such that $\nabla f=\frac{\vec{r}}{r^5}$ and $f\left(1\right)=0$. [2016/10m/8a]

18. Find the angle between the surfaces $x^2+y^2+z^2-9=0$ and $z=x^2+y^2-3$ at $(2,-1,2)$. [2015/10m/5e]

19. Find the value of $\lambda$ and $\mu$ so that the surfaces $\lambda x^2-\mu yz=(\lambda +2)x$ and $4x^{2}y+z^3=4$ may intersect orthogonally at $(1,-1,2)$. [2015/12m/6c]

20. A vector field is given by $\vec{F}=(x^2+x{y}^2)\hat{i}+(y^2+x^{2}y)\hat{j}$ Verify that the field $\vec{F}$ is irrotational or not. Find the scalar potential. [2015/12m/7c]

21. Calculate ${\nabla }^2\left(r^n\right)$ and find its expression in terms of $r$ and $n$, $r$ being the distance of any point $(x,y,z)$ from the origin, $n$ being a constant and ${\nabla }^2$ being the Laplace operator. [2013/10m/8a]

22. If $\begin{array}{cc}\hfill \vec{A}\hfill & \hfill =x^{2}yz\vec{i}-2x{z}^3\vec{j}+x{z}^2\vec{k}\hfill \\ \hfill \vec{B}\hfill & \hfill =2z\vec{i}+y\vec{j}-x^2\vec{k}\hfill \end{array}$ find the value of $\frac{{\partial }^2}{\partial x\partial y}(\vec{A}\times \vec{B})$ at $(1,0,-2)$. [2012/12m/5e]

23. For two vectors $\vec{a}$ and $\vec{b}$ given respectively by $\vec{a}=5t^2\hat{i}+t\hat{j}-t^3\hat{k}$ and $\vec{b}=sint\hat{i}-cost\hat{j}$ determine : (i) $\frac{d}{dt}(\vec{a}\cdot \vec{b})$ and (ii) $\frac{d}{dt}(\vec{a}\times \vec{b})$. [2011/10m/5e]

24. If $u$ and $v$ are two scalar fields and $\vec{f}$ is a vector field, such that $u\vec{f}=gradv,$ find the value of $\vec{f}\cdot curl\vec{f}$ [2011/10m/5f]

25. Examine whether the vectors $\nabla u$, $\nabla v$ and $\nabla w$ are coplanar, where $u$, $v$ and $w$ are the scalar functions defined by : $\begin{array}{cc}\hfill u\hfill & \hfill =x+y+z,\hfill \\ \hfill v\hfill & \hfill =x^2+y^2+z^2\hfill \\ \hfill \mathrm{\text{and}}w\hfill & \hfill =yz+zx+xy.\hfill \end{array}$ [2011/15m/8a]

26. If $\vec{r}$ be the position vector of a point, find the value(s) of $n$ for which the vector $r^n\vec{r}$ is (i) irrotational, (ii) solenoidal. [2011/15m/8c]

27. Find the directional derivative of $f(x,y)=x^2{y}^3+xy$ at the point $(2,1)$ in the direction of a unit vector which makes an angle of $\pi /3$ with the $x$-axis. [2010/12m/5e]

28. Show that the vector field defined by the vector function $\vec{V}=xyz(yz\vec{i}+xz\vec{j}+xy\vec{k})$ is conservative. [2010/12m/5f]

29. Prove that $div\left(f\vec{V}\right)=f\left(div\vec{V}\right)+\left(gradf\right)\cdot \vec{V}$ where $f$ is a scalar function. [2010/20m/6c]

30. Show that $div\left(grad{r}^n\right)=n(n+1)r^{n-2}$ where $r=\sqrt{x^2+y^2+z^2}$. [2009/12m/5e]

31. Find the directional derivative of— (i) $4x{z}^3-3x^2{y}^2{z}^2$ at $(2,-1,2)$ along $z$-axis; (ii) $x^{2}yz+4x{z}^2$ at $(1,-2,1)$ in the direction of $2\hat{i}-\hat{j}-2\hat{k}$. [2009/12m/5f]

32. Find the constant $a$ and $b$ so that the surface $a{x}^2-byz=(a+2)x$ will be orthogonal to the surface $4x^{2}y+z^3=4$ at the point $(1,-1,2)$. [2008/12m/5e]

33. Show that $\vec{F}=(2xy+z^3)\hat{i}+x^2\hat{j}+3x{z}^2\hat{k}$ is a conservative force field. Find the scalar potential for $\vec{F}$ and the work done in moving an object in this field from $(1,-2,1)$ to $(3,1,4)$. [2008/12m/5f]

34. Prove that ${\nabla }^{2}f\left(r\right)=\frac{d^{2}f}{d{r}^2}+\frac{2}{r}\frac{df}{dr}$ where $r={(x^2+y^2+z^2)}^{1/2}$. Hence find $f\left(r\right)$ such that ${\nabla }^{2}f\left(r\right)=0$. [2008/12m/8a]

6.2] Curves in space

14 questions

1. Determine $\hat{T}$, $\hat{N}$, $\hat{B}$, $\kappa$ and $\tau$ for the parabola $y^2=4ax$, where $a$ is a constant. Here $\hat{T}$, $\hat{N}$, $\hat{B}$, $\kappa$ and $\tau$ denote unit tangent vector, unit principal normal vector, unit binormal vector, curvature and torsion respectively. [2026/15m/6c]

2. Let $C$ be a plane curve $\bar{r}\left(t\right)=f\left(t\right)\hat{i}+g\left(t\right)\hat{j}$, where $f$ and $g$ have second-order derivatives. Show that the curvature at a point is given by $\kappa =\frac{|f\prime \left(t\right)g″\left(t\right)-g\prime \left(t\right)f″\left(t\right)|}{{({\left[f\prime \left(t\right)\right]}^2+{\left[g\prime \left(t\right)\right]}^2)}^{3/2}}$ What is the value of torsion $\tau$ at any point of this curve? [2024/5m/5e(i)]

3. Show that the principal normals at two consecutive points of a curve do not intersect unless torsion $\tau$ is zero. [2024/5m/5e(ii)]

4. If the tangent to a curve makes a constant angle $\theta$ with a fixed line, then prove that the ratio of radius of torsion to radius of curvature is proportional to $tan\theta$. Further prove that if this ratio is constant, then the tangent makes a constant angle with a fixed direction. [2023/15m/7c]

5. Find the radius of curvature and radius of torsion of the helix $x=acosu$, $y=asinu$, $z=autan\alpha$. [2019/15m/7b]

6. Find the angle between the tangent at a general point of the curve whose equations are $x=3t$, $y=3t^2$, $z=3t^3$ and the line $y=z-x=0$. [2018/10m/5b]

7. Find the curvature and torsion of the curve $\vec{r}=a(u-sinu)\vec{i}+a(1-cosu)\vec{j}+bu\vec{k}$ [2018/12m/7b]

8. Find the curvature vector and its magnitude at any point $\bar{r}=\left(\theta \right)$ of the curve $\bar{r}=(acos\theta ,asin\theta ,a\theta )$. Show that the locus of the feet of the perpendicular from the origin to the tangent is a curve that completely lies on the hyperboloid $x^2+y^2-z^2=a^2$. [2017/16m/7a]

9. Find the curvature vector at any point of the curve $\bar{r}\left(t\right)=tcost\hat{i}+tsint\hat{j}$, $0\le t\le 2\pi$. Give its magnitude also. [2014/10m/5e]

10. Show that the curve $\vec{x}\left(t\right)=t\hat{i}+\left(\frac{1+t}{t}\right)\hat{j}+\left(\frac{1-t^2}{t}\right)\hat{k}$ lies in a plane. [2013/10m/5e]

11. A curve in space is defined by the vector equation $\vec{r}=t^2\hat{i}+2t\hat{j}-t^3\hat{k}$. Determine the angle between the tangents to this curve at the points $t=+1$ and $t=-1$. [2013/10m/8b]

12. Derive the Frenet-Serret formulae. Define the curvature and torsion for a space curve. Compute them for the space curve $x=t,y=t^2,z=\frac{2}{3}{t}^3$ Show that the curvature and torsion are equal for this curve. [2012/20m/8a]

13. Find $\kappa /\tau$ for the curve $\vec{r}\left(t\right)=acost\vec{i}+asint\vec{j}+bt\vec{k}$ [2010/12m/5c]

14. Show that for the space curve $x=t,y=t^2,z=\frac{2}{3}{t}^3,$ the curvature and torsion are same at every point. [2008/15m/8b]

6.3] Integral theorems

41 questions

1. Given $\vec{F}=(2x+3y)\hat{i}-4z\hat{j}-5x\hat{k}$ and $S$ is the surface $2x+4y+4z=5$, bounded by $x=0$, $x=2$, $y=0$ and $y=3$. Evaluate ${\iint }_S^{}(\nabla \times \vec{F})\cdot \hat{n}dS$. [2026/10m/5e]

2. If $\vec{F}=4xz\hat{i}-y^2\hat{j}+yz\hat{k}$, then evaluate ${\iint }_S^{}\vec{F}\cdot \hat{n}dS$, where $S$ is the surface of a unit cube with two opposite corners at $(0,0,0)$ and $(1,1,1)$ respectively. Hence verify the divergence theorem. [2026/15m/7b]

3. Verify Green’s theorem in the plane for ${\oint }_C^{}[(xy+y^2)dx+x^{2}dy]$, where $C$ is the boundary of the region bounded by the curves $y=x$ and $y=x^2$. [2025/15m/7b]

4. Verify Gauss’s divergence theorem for $\vec{F}=[(x^2-yz)\hat{i}+(y^2-zx)\hat{j}+(z^2-xy)\hat{k}]$, taken over the rectangular parallelopiped $0\le x\le a,0\le y\le b,0\le z\le c$. [2025/15m/8b]

5. State Stokes’ theorem and verify it for the vector field $\vec{F}=xy\hat{i}+yz\hat{j}+zx\hat{k}$ over the surface $S$, which is the upwardly oriented part of the cylinder $z=1-x^2$, for $0\le x\le 1$, $-2\le y\le 2$. [2024/20m/7c]

6. Using Gauss divergence theorem, evaluate the integral ${\iint }_S^{}(y^2\hat{i}+x{z}^3\hat{j}+{(z-1)}^2\hat{k})\cdot \hat{n}dS$ over the region bounded by the cylinder $x^2+y^2=16$ and the planes $z=1$ and $z=5$. [2024/15m/8b]

7. Evaluate the integral ${\iint }_S^{}(3y^2{z}^2\hat{i}+4z^2{x}^2\hat{j}+z^2{y}^2\hat{k})\cdot \hat{n}dS$, where $S$ is the upper part of the surface $4x^2+4y^2+4z^2=1$ above the plane $z=0$ and bounded by the $xy$-plane. Hence, verify Gauss-Divergence theorem. [2023/20m/6c]

8. Verify Green’s theorem in the plane for ${\oint }_C^{}(3x^2-8y^2)dx+(4y-6xy)dy$, where $C$ is the boundary curve of the region defined by $x=0$, $y=0$, $x+y=1$. [2022/15m/6c]

9. Verify Stokes’ theorem for $\vec{F}=x\hat{i}+z^2\hat{j}+y^2\hat{k}$ over the plane surface : $x+y+z=1$ lying in the first octant. [2022/20m/7a]

10. Using Gauss’ divergence theorem, evaluate ${\iint }_S^{}\vec{F}\cdot \vec{n}dS$, where $\vec{F}=x\hat{i}-y\hat{j}+(z^2-1)\hat{k}$ and $S$ is the cylinder formed by the surfaces $z=0$, $z=1$, $x^2+y^2=4$. [2022/15m/8c]

11. Evaluate ${\int }_C^{}\vec{F}\cdot d\vec{r}$, where $C$ is an arbitrary closed curve in the $xy$-plane and $\vec{F}=\frac{-y\hat{i}+x\hat{j}}{x^2+y^2}$. [2021/15m/6c]

12. Verify Gauss divergence theorem for $\vec{F}=2x^{2}y\hat{i}-y^2\hat{j}+4x{z}^2\hat{k}$ taken over the region in the first octant bounded by $y^2+z^2=9$ and $x=2$. [2021/20m/7a]

13. Using Stokes’ theorem, evaluate ${\iint }_S^{}(\nabla \times \vec{F})\cdot \hat{n}dS$, where $\vec{F}=(x^2+y-4)\hat{i}+3xy\hat{j}+(2xy+z^2)\hat{k}$ and $S$ is the surface of the paraboloid $z=4-(x^2+y^2)$ above the $xy$-plane. Here, $\hat{n}$ is the unit outward normal vector on $S$. [2021/15m/8c]

14. For the vector function $\overline{A}$, where $\overline{A}=(3x^2+6y)\hat{i}-14yz\hat{j}+20x{z}^2\hat{k}$, calculate ${\int }_C^{}\overline{A}\cdot d\overline{r}$ from $(0,0,0)$ to $(1,1,1)$ along the following paths : (i) $x=t,y=t^2,z=t^3$ (ii) Straight lines joining $(0,0,0)$ to $(1,0,0)$, then to $(1,1,0)$ and then to $(1,1,1)$ (iii) Straight line joining $(0,0,0)$ to $(1,1,1)$ Is the result same in all the cases? Explain the reason. [2020/15m/6b]

15. Verify the Stokes’ theorem for the vector field $\overline{F}=xy\hat{i}+yz\hat{j}+xz\hat{k}$ on the surface $S$ which is the part of the cylinder $z=1-x^2$ for $0\le x\le 1$, $-2\le y\le 2$; $S$ is oriented upwards. [2020/20m/7a]

16. Evaluate the surface integral ${\iint }_S^{}\nabla \times \overline{F}\cdot \hat{n}dS$ for $\overline{F}=y\hat{i}+(x-2xz)\hat{j}-xy\hat{k}$ and $S$ is the surface of the sphere $x^2+y^2+z^2=a^2$ above the $xy$-plane. [2020/15m/8b]

17. Find the circulation of $\vec{F}$ round the curve $C$, where $\vec{F}=(2x+y^2)\hat{i}+(3y-4x)\hat{j}$ and $C$ is the curve $y=x^2$ from $(0,0)$ to $(1,1)$ and the curve $y^2=x$ from $(1,1)$ to $(0,0)$. [2019/15m/6b]

18. State Gauss divergence theorem. Verify this theorem for $\vec{F}=4x\hat{i}-2y^2\hat{j}+z^2\hat{k}$, taken over the region bounded by $x^2+y^2=4$, $z=0$ and $z=3$. [2019/15m/8c(i)]

19. Evaluate by Stokes’ theorem ${\oint }_C^{}{e}^{x}dx+2ydy-dz$, where $C$ is the curve $x^2+y^2=4$, $z=2$. [2019/5m/8c(ii)]

20. If $S$ is the surface of the sphere $x^2+y^2+z^2=a^2$, then evaluate ${\iint }_S^{}[(x+z)dydz+(y+z)dzdx+(x+y)dxdy]$ using Gauss’ divergence theorem. [2018/12m/6d]

21. Evaluate the line integral ${\int }_C^{}-y^{3}dx+x^{3}dy+z^{3}dz$ using Stokes’ theorem. Here $C$ is the intersection of the cylinder $x^2+y^2=1$ and the plane $x+y+z=1$. The orientation on $C$ corresponds to counterclockwise motion in the $xy$-plane. [2018/13m/8b]

22. Let $\vec{F}=x{y}^2\vec{i}+(y+x)\vec{j}$. Integrate $(\nabla \times \vec{F})\cdot \vec{k}$ over the region in the first quadrant bounded by the curves $y=x^2$ and $y=x$ using Green’s theorem. [2018/13m/8c]

23. Evaluate the integral : ${\iint }_S^{}\bar{F}\cdot \hat{n}ds$ where $\bar{F}=3x{y}^2\hat{i}+(y{x}^2-y^3)\hat{j}+3z{x}^2\hat{k}$ and $S$ is a surface of the cylinder $y^2+z^2\le 4$, $-3\le x\le 3$, using divergence theorem. [2017/9m/8c(i)]

24. Using Green’s theorem, evaluate the ${\int }_C^{}F\left(\bar{r}\right)\cdot d\bar{r}$ counterclockwise where $F\left(\bar{r}\right)=(x^2+y^2)\hat{i}+(x^2-y^2)\hat{j}$ and $d\bar{r}=dx\hat{i}+dy\hat{j}$ and the curve $C$ is the boundary of the region $R=\{(x,y)\mid 1\le y\le 2-x^2\}$. [2017/8m/8c(ii)]

25. Prove that ${\oint }_C^{}fd\vec{r}={\iint }_S^{}d\vec{S}\times \nabla f$ [2016/10m/8b]

26. Evaluate ${\int }_C^{}{e}^{-x}(sinydx+cosydy)$, where $C$ is the rectangle with vertices $(0,0)$, $(\pi ,0)$, $(\pi ,\frac{\pi }{2})$, $(0,\frac{\pi }{2})$. [2015/12m/8c]

27. Evaluate by Stokes’ theorem ${\int }_{\Gamma }^{}(ydx+zdy+xdz)$ where $\Gamma$ is the curve given by $x^2+y^2+z^2-2ax-2ay=0$, $x+y=2a$, starting from $(2a,0,0)$ and then going below the $z$-plane. [2014/20m/6c]

28. By using Divergence Theorem of Gauss, evaluate the surface integral $\iint {(a^2{x}^2+b^2{y}^2+c^2{z}^2)}^{-\frac{1}{2}}dS$, where $S$ is the surface of the ellipsoid $a{x}^2+b{y}^2+c{z}^2=1$, $a$, $b$ and $c$ being all positive constants. [2013/15m/8c]

29. Use Stokes’ theorem to evaluate the line integral ${\int }_C^{}(-y^{3}dx+x^{3}dy-z^{3}dz)$, where $C$ is the intersection of the cylinder $x^2+y^2=1$ and the plane $x+y+z=1$. [2013/15m/8d]

30. Verify Green’s theorem in the plane for ${\oint }_C^{}[(xy+y^2)dx+x^{2}dy]$ where $C$ is the closed curve of the region bounded by $y=x$ and $y=x^2$. [2012/20m/8b]

31. If $\vec{F}=y\vec{i}+(x-2xz)\vec{j}-xy\vec{k}$, evaluate ${\iint }_S^{}(\vec{\nabla }\times \vec{F})\cdot \vec{n}d\vec{s}$ where $S$ is the surface of the sphere $x^2+y^2+z^2=a^2$ above the $xy$-plane. [2012/20m/8c]

32. If $\vec{u}=4y\hat{i}+x\hat{j}+2z\hat{k}$, calculate the double integral $\iint (\nabla \times \vec{u})\cdot d\vec{s}$ over the hemisphere given by $x^2+y^2+z^2=a^2,z\ge 0.$ [2011/15m/8b]

33. Verify Gauss’ Divergence Theorem for the vector $\vec{v}=x^2\hat{i}+y^2\hat{j}+z^2\hat{k}$ taken over the cube $0\le x,y,z\le 1.$ [2011/15m/8d]

34. Use the divergence theorem to evaluate ${\iint }_S^{}\vec{V}\cdot \vec{n}dA$ where $\vec{V}=x^{2}z\vec{i}+y\vec{j}-x{z}^2\vec{k}$ and $S$ is the boundary of the region bounded by the paraboloid $z=x^2+y^2$ and the plane $z=4y$. [2010/20m/7c]

35. Verify Green’s theorem for $e^{-x}sinydx+e^{-x}cosydy$ the path of integration being the boundary of the square whose vertices are $(0,0)$, $(\pi /2,0)$, $(\pi /2,\pi /2)$ and $(0,\pi /2)$. [2010/20m/8c]

36. Evaluate $I={\iint }_S^{}xdydz+dzdx+x{z}^{2}dxdy$ where $S$ is the outer side of the part of the sphere $x^2+y^2+z^2=1$ in the first octant. [2009/20m/4b]

37. Find the work done in moving the particle once round the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$, $z=0$ under the field of force given by $\vec{F}=(2x-y+z)\hat{i}+(x+y-z^2)\hat{j}+(3x-2y+4z)\hat{k}$ [2009/20m/8a]

38. Using divergence theorem, evaluate ${\iint }_S^{}\vec{A}\cdot d\vec{S}$ where $\vec{A}=x^3\hat{i}+y^3\hat{j}+z^3\hat{k}$ and $S$ is the surface of the sphere $x^2+y^2+z^2=a^2$. [2009/20m/8b]

39. Find the value of ${\iint }_S^{}(\vec{\nabla }\times \vec{F})\cdot d\vec{S}$ taken over the upper portion of the surface $x^2+y^2-2ax+az=0$ and the bounding curve lies in the plane $z=0$, when $\vec{F}=(y^2+z^2-x^2)\hat{i}+(z^2+x^2-y^2)\hat{j}+(x^2+y^2-z^2)\hat{k}$ [2009/20m/8c]

40. Evaluate ${\int }_C^{}\vec{A}\cdot d\vec{r}$ along the curve $x^2+y^2=1$, $z=1$ from $(0,1,1)$ to $(1,0,1)$ if $\vec{A}=(yz+2x)\hat{i}+xz\hat{j}+(xy+2z)\hat{k}$ [2008/15m/8c]

41. Evaluate ${\iint }_S^{}\vec{F}\cdot \hat{n}dS$ where $\vec{F}=4x\hat{i}-2y^2\hat{j}+z^2\hat{k}$ and $S$ is the surface of the cylinder bounded by $x^2+y^2=4$, $z=0$ and $z=3$. [2008/15m/8d]

« Previous: Dynamics and Statics  |  Next: Algebra »

Back to: Mathematics Optional PYQs – Topic-wise | Also see: Mathematics Optional year-wise question papers

Last updated: 30 October 2026