HPAS Maths Optional PYQs: Calculus

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

This revised list focuses solely on Calculus questions (Differential, Integral, Real Analysis, Multivariable), distinguishing them from Vector Calculus.

HPAS 2024 Calculus Questions

Question 1(b): Find the asymptotes of the curve: $x^{3}+x^{2}y+xy^{2}+y^{3}+2x^{2}+3xy-4y^{2}+7x+2y=0$.

Question 2(b): If the plane $x+y+z=1$ cuts the cylinder $x^2+y^2=1$ in an ellipse, then determine the points on the ellipse that lie closest to and farthest from the origin.

Question 3(a): Find the volume of the solid enclosed between the surfaces $x^2+y^2=a^2$ and $x^2+z^2=a^2$ (Multiple Integration).

HPAS 2023 Calculus Questions

Question 1(b): Let the function $f$ be continuous on the real line $\mathbb{R}$. Then show that the set $A = \{x : f(x) = 0\}$ is closed (Real Analysis).

Question 3(a): Show that the radius of curvature of the lemniscate $(x^2+y^2)^2 = a^2(x^2-y^2)$ at any point where the tangent is parallel to the x-axis, is $\frac{\sqrt{2}a}{3}$.

Question 3(b): Evaluate $\lim_{x\to0} x^m (\log_e x)^n$, where $m$ and $n$ are positive integers (Limits/L’Hopital’s Rule).

Question 4(a): If $u(x,y) = \sin^{-1}\left(\left(\frac{x^{1/3}+y^{1/3}}{x^{1/2}+y^{1/2}}\right)^{1/2}\right)$, then show that $x^2\frac{\partial^2 u}{\partial x^2} + 2xy\frac{\partial^2 u}{\partial x \partial y} + y^2\frac{\partial^2 u}{\partial y^2} = \frac{1}{144}\tan u(13 + \tan^2 u)$ (Partial Derivatives).

HPAS 2021 Calculus Questions

Question 1(e): Determine the value of the integral: $\int_{0}^{2a} \int_{0}^{\sqrt{2ay-y^2}} dx \,dy$ (Double Integration).

Question 3(b): Find the maximum and minimum values of $u^2+v^2+w^2$ subject to the conditions $\frac{u^2}{4} + \frac{v^2}{5} + \frac{w^2}{25} = 1$ and $w=u+v$ (Multivariable Optimization).

Question 3(c): Using the concept of Gamma and Beta functions, show that: $\int_{0}^{\pi/2} \sqrt{\tan x} \,dx = \frac{\pi}{\sqrt{2}}$.

Question 4(b): Show that a function $f$ defined on the real line $\mathbb{R}$ is continuous if and only if for each open set $G$ in $\mathbb{R}$, $f^{-1}(G)$ is an open set in $\mathbb{R}$ (Real Analysis/Topology).

Question 4(c): Test the convergence of the integral: $\int_{0}^{4} \frac{\sin^2 x}{\sqrt{x}(x-1)} dx$ (Improper Integrals).

Question 6(a): Show that the radius of curvature R at any point $(r, \theta)$ on the curve $r^2 = a^2 \sec(2\theta)$ is proportional to $r^3$.

HPAS 2020 Calculus Questions

Question 3(b): Show that $\lim_{x\to0} \cos\frac{1}{x}$ does not exist.

Question 3(c): Evaluate $\lim_{x\to\infty}(\sqrt{x^2+3x}-x)$ and $\lim_{x\to0^{+}}(1-\sin x)^{1/x}$ (Limits/Indeterminate forms).

Question 4(a): State a set of sufficient conditions for a local maximum or minimum at a point for a twice continuously differentiable function $f(x,y)$. Test the function $f(x,y)=x^3+y^3-9xy+1$ for local maximum or minimum (Multivariable Extremes).

Question 4(b): Let $f:\mathbb{R}^2 \to \mathbb{R}^2$ be defined by $f(x,y)=(x^2+y^2, xy)$. Compute the total derivative of f at the point (1,2).

Question 5(a): Find the volume of the solid region that is interior to both the sphere $x^2+y^2+z^2=4$ and the cylinder $(x-1)^2+y^2=1$ (Multiple Integration).

HPAS 2018 Calculus Questions

Question 1(c): Find the intervals in which the function $f(x) = 10 – 6x – 2x^2$ is strictly increasing or strictly decreasing.

Question 4(a): Suppose a function $f(x)$ satisfies the conditions: (i) $f(0)=2$, $f(1)=1$; (ii) f has a minimum value at $x=5/2$; and (iii) $f'(x) = 2ax+b$. Determine the constants $a, b$ and the function $f(x)$.

Question 4(b): Change the order of integration in the integral $\iint f(x,y)dxdy$. The area of integration is enclosed by the curves $y = x \tan\alpha$, $y = \sqrt{a^2-x^2}$, $x=0$ and $x = a\cos\alpha$ (Double Integration).

HPAS 2017 Calculus Questions

Question 1(b): Examine the curve $x=6t^2$, $y=4t^3-3t$ for concavity and convexity.

Question 3(b): Prove that a bounded function is not necessarily Riemann integrable (Real Analysis).

HPAS 2016 Calculus Questions

Question 1(c): Evaluate: $\lim_{x\to 0} \left(\frac{\tan x}{x}\right)^{1/x^2}$ (Limits/Indeterminate forms).

Question 3(a): Show that the function f defined on $\mathbb{R}$ by: $$ f(x) = \begin{cases} x, & \text{if x is irrational} \\ -x, & \text{if x is rational} \end{cases} $$ is continuous only at $x=0$ (Real Analysis).

Question 3(b): Find the asymptotes of the curve: $2x^3 – 5x^2y + 4xy^2 – y^3 + 6x^2 – 7xy + y^2 – x + 5y – 3 = 0$.

Question 4(a): Find the extreme values of $f(x,y,z)=2x+3y+z$ such that $x^2+y^2=5$ and $x+z=1$ (Optimization/Lagrange Multipliers).

Question 4(b): Evaluate the integral: $\int_{0}^{2}\int_{0}^{y^2/2}\frac{y}{\sqrt{x^2+y^2+1}}dxdy$ (Double Integration).

HPAS 2015 Calculus Questions

Question 1(c): If $f(x)=\sin x$ and $g(x)=\cos x$, $\forall x\in[0,\frac{\pi}{2}]$, then find the value of $c$ with the help of Cauchy’s mean value theorem.

Question 3(a): Evaluate: $\lim_{x\rightarrow0}(\frac{1}{x^{2}}-\cot^{2}x)$ (Limits/Indeterminate forms).

Question 3(b): Prove that the radius of curvature at any point $(x, y)$ on the Astroid: $x^{2/3}+y^{2/3}=a^{2/3}$ is three times the length of the perpendicular from the origin on the tangent at that point.

Question 3(c): If: $u=x~\phi(y/x)+\psi(y/x)$, then prove that: $x^{2}\frac{\partial^{2}u}{\partial x^{2}}+2xy\frac{\partial^{2}u}{\partial x\partial y}+y^{2}\frac{\partial^{2}u}{{\partial y^{2}}}=0$ (Partial Derivatives/Homogeneous Functions).

Question 4(a): If: $u^{3}+v^{3}=x+y$ and $u^{2}+v^{2}=x^{3}+y^{3}$, then find the value of: $\frac{\partial(u,v)}{\partial(x,y)}$ (Jacobian).

Question 4(b): Evaluate: $\iint_{V}2z~dxdydz$ where V is a cone enclosed by the surface: $x² + y² = z²$, $z=1$ (Triple Integration).

Question 4(c): Prove that a rectangular solid of maximum volume within a sphere is a cube (Optimization).

HPAS 2014 Calculus Questions

Question 1(c): Examine for continuity of the following function at $x=0$: $$ f(x)=\begin{cases}\frac{x-|x|}{x}&,&x\ne0\\ 1&,&x=0\end{cases} $$ Is it differentiable?.

Question 3(a): Given the sum of the perimeters of a square and a circle, show that the sum of their areas is least when the side of the square is equal to the diameter of the circle (Optimization).

Question 4(a): Evaluate: $\iint_{A}(xy)(x+y)dxdy$ where region A is the area between the parabola $y=x^{2}$ and the line $y=x$ (Double Integration).

Question 4(b): If $u=tan^{-1}(\frac{x^{3}+y^{3}}{x-y})$ then prove that: $x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}=\sin~2u$ (Partial Derivatives/Euler’s Theorem).

Question 4(c): Prove that the limit of $\theta$ used in Lagrange’s mean value theorem tends to $\frac{1}{2}$ when $h\rightarrow0$, provided $f^{\prime\prime}(x)$ is continuous and $f^{\prime\Windows\ne0$.