HPAS Maths Optional PYQs: Vector Calculus

On that page, you will find year-wise and question-wise solutions.

The sources show that Vector Calculus is a consistently tested topic in the HPAS Maths Optional Paper-1 examinations, covering fundamental operators (gradient, divergence, curl), important theorems (Gauss, Green’s, Stokes’), directional derivatives, and vector identities. Here are the year-wise questions pertaining to Vector Calculus:

HPAS 2024 Vector Calculus Questions

Question 1(d): Show that the vector field defined by the vector function \(\vec{V} = xyz(yz\mathbf{i} + xz\mathbf{j} + xy\mathbf{k})\) is conservative.

Question 6(a): Find the work done by the force \(\vec{F}=(x^2-y^2)\mathbf{i}+(x+y)\mathbf{j}\) in moving a particle along the closed path C containing the curves \(x+y=0\), \(x^2+y^2=16\), and \(y=x\) in the first and the fourth quadrants (Line Integral/Green’s Theorem application).

Question 6(b): Evaluate the surface integral \(\iint_{S} \vec{F} \cdot \mathbf{n} \,dA\) where \(\vec{F} = z^2\mathbf{i} + xy\mathbf{j} – y^2\mathbf{k}\) and S is the portion of the surface of the cylinder \(x^2+y^2=36\) for \(0 \le z \le 4\) included in the first quadrant.

HPAS 2023 Vector Calculus Questions

Question 1(c): For what values of \(a\) and \(b\) is the vector field \(\vec{F} = (x+z)\mathbf{i} + a(y+z)\mathbf{j} + b(x+y)\mathbf{k}\) a conservative field?.

Question 6(a): Given \(\vec{F} = y\mathbf{i} – z^3\mathbf{j} + x^2\mathbf{k}\), use Stokes’s theorem to evaluate the line integral \(\int_C \vec{F} \cdot d\vec{r}\), where C is the boundary of the area S formed by the part of the plane \(x+4y+z=4\) that lies in the first octant.

Question 6(b): Find the directional derivative of \(f(x,y,z) = x^2 + 3y^2 + 2z^2\) in the direction of the vector \(2\mathbf{i} – \mathbf{j} – 2\mathbf{k}\) and determine its value at the point (1, -3, 2).

HPAS 2021 Vector Calculus Questions

Question 1(b): Determine the directional derivative of the function \(f(x,y,z) = 4e^{2x-y+z}\) at the point (1, 1, 1) in the direction towards the point (-3, 5, 6).

Question 4(a): Find the directional derivative of \(f(x,y) = x^2y^3 + xy\) at the point (2,1) in the direction of a unit vector which makes an angle of \(\pi/3\) with the x-axis.

Question 6(b): Evaluate the line integral \(\int_C (x+y)dx – x^2dy + (y+z)dz\), where C is the curve defined by \(x^2=4y\), \(z=x\), and \(0 \le x \le 2\).

Question 6(c): Verify Stokes’s theorem for the vector field \(\vec{v} = (3x-y)\mathbf{i} – 2yz^2\mathbf{j} – 2y^2z\mathbf{k}\), where S is the surface of the sphere \(x^2+y^2+z^2=16\) and \(z>0\).

HPAS 2020 Vector Calculus Questions

Question 6(a): State the statement of Stokes’ theorem and verify it for the line integral \(\oint_C [(x+y)dx + (2x-z)dy + (y+z)dz]\), where C is the boundary of the triangle with vertices (2,0,0), (0,3,0), and (0,0,6).

HPAS 2018 Vector Calculus Questions

Question 1(d): Find the unit outward normal vector at the point \((\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 0)\) for the surface \(x^2+y^2+z^2=1\).

Question 3(a): The direction of a vector A is radially outward from the origin, and its magnitude is \(|\vec{A}| = kr^n\), where \(r^2 = x^2+y^2+z^2\). Find the value of \(n\) for which \(\nabla \cdot \vec{A} = 0\) (Divergence).

HPAS 2017 Vector Calculus Questions

Question 1(d): If \(\vec{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}\), then show that the vector \(\vec{r}\) is an irrotational vector (Curl).

Question 6(a): Obtain the Serret-Frenet formulas (Differential Geometry of Curves).

Question 6(b): Verify Stokes’ theorem for the function \(\vec{F} = z\mathbf{i} + x\mathbf{j} + y\mathbf{k}\), where C is the unit circle in the xy-plane bounding the hemisphere \(z = \sqrt{1-x^2-y^2}\).

HPAS 2016 Vector Calculus Questions

Question 1(d): Find the angle between the surfaces \(x \log z = y^2 – 1\) and \(x^2y = 2 – z\) at point (1, 1, 1) (Gradient application).

Question 6(a): Find the directional derivative of the scalar function \(\phi = xy^2 + yz^3\) at point (2, -1, 1) in the direction of the normal to the surface \(x \log z – y^2 = -4\) at point (-1, 2, 1).

Question 6(b): Show that the field of force given by \(\vec{F} = (y^2 \cos x + z^3)\mathbf{i} + (2y \sin x – 4)\mathbf{j} + (3xz^2 + 2)\mathbf{k}\) is conservative and find the work done in moving the particle in the field from a point A (0, 1, 1) to a point B (\(\pi/2\), -1, 2).

HPAS 2015 Vector Calculus Questions

Question 1(e): If \(u = x + y + z\), \(v = x^2 + y^2 + z^2\) and \(w=xy+yz+zx\), then find: \((\mathbf{grad}~u)\cdot{(\mathbf{grad}~v)\times(\mathbf{grad}~w)}\).

Question 7(a): If \(f\) and \(g\) are two scalar point functions, prove the identity: \(\mathbf{div}(f\nabla g)=f\nabla^{2}g+\nabla f\cdot\nabla g\).

Question 7(b): Prove the identity: \(\nabla\times(\nabla\times a)=\nabla(\nabla\cdot a)-\nabla^{2}a\).

Question 7(c): Evaluate by Green’s theorem: \(\int_{C}[(\cos~x\sin~y-xy)dx+\sin~x\cos~y~dy]\), where C is the circle \(x^{2}+y^{2}=1\).

HPAS 2014 Vector Calculus Questions

Question 7(b): If \(\alpha=\sin~\theta i+\cos~\theta j+\theta k\), \(b=\cos~\theta i-\sin~\theta j-3k\) and \(c=2i+3j-k\), then evaluate \(\frac{d}{d\theta}[a\times(b\times c)]\) for \(\theta=0\) (Vector differentiation).

Question 7(c): Apply Gauss theorem to evaluate the surface integral: \(\int_{s}{(x^{3}-yz)dydz-2x^{2}y~dzdx+zdxdy}\) over the surface of a cube bounded by the coordinate planes and the planes \(x=y=z=a\).