hpas MATHS p1 - Banogyani

Paper-I Topics Index

Linear Algebra
Calculus
Analytic Geometry
Ordinary Diff. Eqns
Vector Analysis
Statics
Dynamics

Linear Algebra

Syllabus:

Matrices, row and column reductions, echelon forms. Eigenvalues, eigenvectors and characteristic equation of a matrix.
Cayley-Hamilton theorem and its applications, rank of a matrix.
Applications of matrices to solve a system of linear homogeneous /non-homogeneous equations.
Vector space, linear dependence and independence, Subspaces, Bases, dimensions. Finite dimensional vector spaces.
Linear transformations, the algebra of linear transformations, isomorphism, representation of transformations by Matrices, linear functionals.
The double dual and the transpose of a linear transformation.
Inner product spaces. Cauchy-Schwarz inequality. Orthogonal vectors. Orthogonal complements.
Orthonormal sets and orthonormal bases. Bessel’s inequality for finite dimensional spaces. Gram-Schmidt orthogonalization process.
Linear functionals and adjoints.

HPAS 2024

Question 1(a)

Show that similar matrices have the same minimal polynomial.

Question 2(a)

Show that the inner product space is a normed vector space but the converse is not true.

Question 4(a)

Show that every non-zero finite-dimensional inner product space has an orthonormal basis.

Question 4(b)

Using the concept of diagonalizability, determine $A^5$ where

A = \begin{pmatrix} 0 & 0 & -2 \\ 1 & 2 & 1 \\ 1 & 0 & 3 \end{pmatrix}

Question 8(a)

Show that every real n-dimensional vector space is isomorphic to $\mathbb{R}^n$ .

HPAS 2023

Question 1(d)

Give an example of a diagonalizable matrix that does not have distinct eigen values.

Question 2(a)

Using the Cauchy-Schwarz inequality, show that the cosine of an angle has an absolute value of at most 1.

Question 2(b)

Let V be a finite-dimensional vector space and W be a subspace of V. Show that $\dim A(W) = \dim V - \dim W$ , where $A(W)$ is the annihilator of W.

Question 3(c)

Let $T: V \to W$ be a linear transformation. Then show that $\text{Rank}(T) + \text{Nullity}(T) = \dim(V)$ .

Question 7(a)

Let V be the vector space of real-valued functions $y=f(x)$ satisfying $\frac{d^3y}{dx^3} - 6\frac{d^2y}{dx^2} + 11\frac{dy}{dx} - 6y = 0$ . Then show that V is a 3-dimensional vector space over $\mathbb{R}$ .

HPAS 2021

Question 1(a)

Show that two finite-dimensional vector spaces over a field $F$ are isomorphic if and only if they have the same dimension.

Question 2(a)

Let V be a non-zero inner product space of dimension $n$ . Then show that V has an orthonormal basis.

Question 2(b)

Let V be the space of all real-valued continuous functions. Define a linear operator $T: V \to V$ by $(Tf)(x) = \int_{0}^{x} f(t) dt$ . Show that T has no eigenvalues.

HPAS 2020

Question 1(b)

Find a linear transformation $T:\mathbb{R}^{3}\to\mathbb{R}^{3}$ such that its image space is the plane $x+y+z=0$ .

Question 2(a)

Define a basis of a vector space. Find a basis of the subspace of the vector space $\mathbb{R}^4(\mathbb{R})$ generated by the subset: $\{(1,1,0,-1), (2,4,6,0), (-2,-3,-3,1), (-1,-2,-2,2), (4,6,4,-6)\}$ .

Question 2(b)

Let $A = \begin{pmatrix} 2 & 2 \\ 1 & 3 \end{pmatrix}$ . Find an invertible $2 \times 2$ matrix P such that $PAP^{-1}$ is a diagonal matrix.

HPAS 2018

Question 1(b)

How many solutions does the following system of linear equations have?

\begin{cases} -x+5y = -1 \\ x-y = 2 \\ x+3y = 3 \end{cases}

Question 5(a)

Let U and V be vector spaces and $T:U \to V$ be a surjective linear mapping. If $\dim U=6$ and $\dim V=3$ , find $\dim(\text{Ker } T)$ .

Question 5(b)

Find the rank of the linear transformation $T: \mathbb{R}^3 \to \mathbb{R}^3$ defined by $T(x,y,z)=(y,0,z)$ .

Question 6(a)

Let A be a $3\times3$ square matrix with eigenvalues 1, -1, and 0. Find the value of $\det(I+A^{100})$ .

Question 6(b)

Let $\mathbb{R}^4(\mathbb{R})$ be a vector space and let S be its subspace spanned by the vectors (1,2,3,0), (2,3,0,1), and (3,0,1,2). Find the dimension of the quotient space $\mathbb{R}^4/S$ .

Question 7(a)

In a finite-dimensional inner product space V, let $\{w_1, w_2, \dots, w_n\}$ be an orthonormal subset of V such that $\sum_{i=1}^{n} |\langle w_i, v \rangle|^2 = ||v||^2$ for all $v \in V$ . Find a basis of V.

Question 7(c)

Let $V=\mathbb{R}^2$ be a finite-dimensional standard inner product space. Prove that $\{(-1, 0), (0, -1)\}$ forms an orthonormal basis of V.

HPAS 2017

Question 1(a)

If A is a nilpotent matrix of index 2, show that $A(I \pm nA) = A$ for any positive integer n.

Question 2(a)

Find the matrix representation of a linear transformation t on $V_3(\mathbb{R})$ defined as $t(x,y,z)=(2y+z, x-4y, 3x)$ corresponding to the basis $B = \{(1,0,0), (0,1,0), (0,0,1)\}$ .

Question 2(b)

State and prove the Cauchy-Schwarz inequality.

HPAS 2016

Question 1(a)

Determine all values of d for which rank of the matrix

\begin{pmatrix} d & -1 & 0 & 0 \\ 0 & d & -1 & 0 \\ 0 & 0 & d & -1 \\ -6 & 11 & -6 & 1 \end{pmatrix}

is equal to 3.

Question 1(b)

Show that the vectors $v_1=(1,1,2,4)$ , $v_2 = (2, -1, -5, 2)$ , $v_3=(1,-1,-4,0)$ and $v_4=(2,1,1,6)$ are linearly dependent in $\mathbb{R}^4$ .

Question 2(a)

Let T be a linear operator on $\mathbb{R}^3$ defined by $T(x,y,z)=(3x, x-y, 2x+y+z)$ . Show that T is invertible and determine $T^{-1}$ .

Question 2(b)

Determine the value of a and b so that the system of equations:

\begin{cases} 2x+3y+5z=9 \\ 7x+3y-2z=8 \\ 2x+3y+az=b \end{cases}

has: (i) no solution (ii) a unique solution (iii) an infinite number of solutions.

Question 8(a)

Let V be a finite dimensional inner product space over field F, and let $g: V \to F$ be a linear transformation. Then show that there exists a unique vector $y \in V$ such that $g(x) = \langle x,y \rangle$ for all $x \in V$ .

HPAS 2014

Question 1(a)

Find characteristic equation and roots of the matrix:

A=\begin{bmatrix}8&-6&2\\ -6&7&-4\\ 2&-4&3\end{bmatrix}

Question 1(b)

Prove that the set S = $\{(1, 2, 1), (2, 1, 0), (1, -1, 2)\}$ forms a basis of vector space $V_{3}(R)$ .

Question 2(a)

Apply matrix theory to solve the following system of equations:

\begin{cases} x+y+z=6 \\ x-y+z=2 \\ 2x+2y-z=1 \end{cases}

Question 2(b)

Prove that the mapping $f:V_{3}(R)\rightarrow V_{2}(R)$ defined by $f(u_{1},u_{2},u_{3})=(u_{1}-u_{2},u_{1}-u_{3})$ is a linear transformation.

Question 2(c)

State and prove Cauchy-Schwarz inequality.

HPAS 2013

Question 1(a)

Find rank of the matrix:

A=\begin{bmatrix}1&1&1&-1\\ 1&2&3&4\\ 3&4&5&2\end{bmatrix}

Question 1(b)

Show that the set: $W=\{(a,b,c):a-3b+4c=0; a,b,c \in R\}$ , is a subspace of the vector space $V_{3}(R)$ .

Question 2(a)

Find the characteristic roots and characteristic vectors of the following matrix:

\begin{bmatrix}1&2&3\\ 0&-4&2\\ 0&0&7\end{bmatrix}

Question 2(b)

Solve the following equations using matrix method:

\begin{cases} 2x_{1}+3x_{2}+x_{3}=9 \\ x_{1}+2x_{2}+3x_{3}=6 \\ 3x_{1}+x_{2}+2x_{3}=8 \end{cases}

Question 2(c)

Prove that the mapping: $t:V_{2}(R)\rightarrow V_{3}(R)$ , which is defined by $t(x,y)=(x,y,0)$ is a linear transformation from vector space $V_{2}(R)$ to the vector space $V_{3}(R)$ .

Calculus

Syllabus:

Real numbers, limits, continuity, differentiability, mean-value theorems. Taylor’s theorem with remainders.
Indeterminate forms, maxima and minima, asymptotes.
Curvature, Concavity, Convexity, Points of inflexion and tracing of curves.
Functions of two variables: continuity, differentiability, partial derivatives, Euler’s theorem for homogeneous functions, Jacobian, maxima and minima.
Lagrange’s method of multipliers.
Riemann’s definition of definite integrals. Indefinite integrals, infinite and improper integrals, beta and gamma functions.
Double and triple integrals. Areas, surface and volumes.

HPAS 2024

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

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

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

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

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

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

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 2014

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: $\lim_{x\rightarrow0}\left(\frac{1}{x^{2}}-\cot^{2}x\right)$ (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: $x^2 + y^2 = z^2$ , $z=1$ (Triple Integration).

Question 4(c)

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

HPAS 2013

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 $u=\tan^{-1}\left(\frac{x^{3}+y^{3}}{x-y}\right)$ 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 $f^{\prime\prime}(x)\ne0$ .

Analytic Geometry

Syllabus:

Cartesian and polar coordinates in two and three dimensions, second degree equations in two and three dimensions, reduction to canonical forms.
Straight lines, shortest distance between two skew lines.
Plane, sphere, cone, cylinder, paraboloid, ellipsoid, hyperboloid of one and two sheets and their properties.

HPAS 2024

Question 3(b)

Find the equation of the plane passing through the line of intersection of the planes $a_1x+b_1y+c_1z+d_1=0$ and $a_2x+b_2y+c_2z+d_2=0$ and perpendicular to the xy-plane.

HPAS 2023

Question 2(c)

Find the equations of the lines in which the plane $2x+y-z=0$ cuts the cone $4x^2 - y^2 + 3z^2 = 0$ .

Question 4(b)

Find the magnitude and the equations of the shortest distance between the lines:

\frac{x}{2} = \frac{-y}{3} = \frac{z}{1} \quad \text{and} \quad \frac{x-2}{3} = \frac{y-1}{-5} = \frac{z+2}{2}

HPAS 2021

Question 1(c)

Show that the tangent planes at the extremities of any diameter of an ellipsoid are parallel.

Question 3(a)

Show that the equation $ax^2+by^2+cz^2+2ux+2vy+2wz+d=0$ represents a cone if:

\frac{u^2}{a} + \frac{v^2}{b} + \frac{w^2}{c} = d

Question 7(a)

Determine the center and radius of the circle in which the sphere $x^2+y^2+z^2+2x-2y-4z-19=0$ is cut by the plane $x+2y+2z+7=0$ .

HPAS 2020

Question 3(a)

Find the locus of the point of intersection of three mutually perpendicular tangent planes to $ax^2+by^2+cz^2=1$ .

Question 4(c)

Reduce the following equation to the standard form:

3x^2+5y^2+3z^2+2yz+2zx+2xy-4x-8z+5=0

Find the nature of the conicoid, its center, and the equations of its axes.

HPAS 2018

Question 3(b)

If P, Q, and R are three points having Cartesian coordinates (3, -2, -1), (1, 3, 4), and (2, 1, -2) respectively, find the distance from point P to the plane OQR, where O is the origin.

HPAS 2017

Question 4(a)

Find the condition that the straight line $\frac{l}{r} = A\cos\theta + B\sin\theta$ may touch the circle $r = 2a\cos\theta$ (in polar coordinates).

Question 4(b)

Find the equation of a sphere which passes through the points (1, 0, 0), (0, 1, 0), and (0, 0, 1) and has the smallest possible radius.

Question 7(b)

Two forces act, one along the line $y=0, z=0$ and the other along the line $x=0, z=c$ . Show that the surface generated by the central axis of their equivalent wrench is $(x^2+y^2)z = cy^2$ .

HPAS 2016

Question 7(a)

Show that the equation: $2x^2 - 6y^2 - 12z^2 + 18yz + 2zx + xy = 0$ represents a pair of planes and find the angle between them.

Question 7(b)

Find the equations of the tangent planes to the hyperboloid $2x^2 - 6y^2 + 3z^2 = 5$ which pass through the lines $3x-3y+6z-5=0$ and $x+9y-3z=0$ .

HPAS 2014

Question 5(a)

Reduce the equation $11y^{2}+14yz+8zx+14xy-6x-16y+2z-2=0$ to canonical form and state the nature of the surface (Conicoid).

Question 5(b)

Find the equation of the sphere having the circle: $x^{2}+y^{2}+z^{2}+10y-4z=8$ , $x+y+z=3$ as a great circle.

Question 5(c)

Find the equation of the cylinder whose generators are parallel to the line: $\frac{x}{1}=\frac{y}{-2}=\frac{z}{3}$ and whose guiding curve is the ellipse: $x^2 + 2y^2 = 1, z=0$ .

HPAS 2013

Question 1(d)

Find the surface represented by the equation: $x^{2}+4y^{2}+z^{2}-4yz+2zx-4xy -2x+4y-2z-3=0$ .

Question 5(a)

Find the equation of the sphere which passes through the points: (1, 0, 0); (0, 1, 0) and (0, 0, 1), and has its radius as small as possible.

Question 5(b)

What conic does the equation: $13x^{2}-18xy+37y^{2}+2x+14y-2=0$ represent? Find its centre and the equation to the conic referred to the centre as origin (2D Conics).

Question 5(c)

Find length and the equations to the shortest distance between the following lines:

\frac{x-3}{3}=\frac{y-8}{-1}=\frac{z-3}{1} \quad \text{and} \quad \frac{x+3}{-3}=\frac{y+7}{2}=\frac{z-6}{4}

Ordinary Differential Equations

Syllabus:

Formulation of differential equations, order and degree, equations of first order and first degree, integrating factor.
Equations of first order but not of first degree, Clairaut’s equation, singular solution.
Higher order linear equations with constant coefficients, complementary function and particular integral, general solution, Euler-Cauchy equation.
Second order linear equations with variable coefficients, determination of complete solution when one solution is known, method of variation of parameters.
Solution by Power series method and its basis, solution of Bessel and Legendre’s equations, properties of Bessel and Legendre functions.

HPAS 2024

Question 1(c)

Compute the general solution of the non-linear differential equation $y = xy' + (y')^2$ where $y' = \frac{dy}{dx}$ . (This is Clairaut’s form).

Question 5(a)

Solve the differential equation: $x^2\frac{d^2y}{dx^2} - 2x\frac{dy}{dx} + 2y = x + x^2\log x + x^3$ . (This is a Cauchy-Euler type equation with a non-homogeneous term).

Question 5(b)

Determine the power series solution about the origin of the differential equation: $(1-x^2)\frac{d^2y}{dx^2} - 4x\frac{dy}{dx} + 2y = 0$ .

HPAS 2023

Question 5(a)

Determine the general and singular solutions of the differential equation $9p^2(2-y)^2 = 4(3-y)$ , where $p = \frac{dy}{dx}$ .

Question 5(b)

Solve the differential equation $x\frac{d^2y}{dx^2} + 2\frac{dy}{dx} + \frac{xy}{2} = 0$ in terms of Bessel functions.

Question 5(c)

Using the method of variation of parameters, solve the differential equation $(D^2 - 2D + 2)y = e^x \tan x$ , where $D = \frac{d}{dx}$ .

Question 7(a)

HPAS 2021

Question 1(d)

Solve the ordinary differential equation $xp^2 - yp - y = 0$ , where $p = dy/dx$ .

Question 5(a)

Show that the smallest root of the equation $J_0(x)=0$ lies in the interval $(2, \sqrt{8})$ , where $J_0(x)$ is the Bessel’s function of order zero.

Question 5(b)

Solve the ordinary differential equation:

\{x^2D^2 - (2m-1)xD + (m^2+n^2)\}y = n^2x^m \log x

where $D=d/dx$ .

Question 5(c)

Find the series solution near $x=0$ of the differential equation:

x(1-x)\frac{d^2y}{dx^2} + (1-x)\frac{dy}{dx} - y = 0

HPAS 2020

Question 1(a)

Obtain the general solution of the following differential equation: $x\frac{dy}{dx}+(2-x)y=e^{3x}$ , for $x > 0$ .

Question 7(b)

Solve the following differential equation: $x\frac{d^2y}{dx^2} - \frac{dy}{dx} = x^2e^x$ , for $x > 0$ .

Question 8(b)

Apply the method of power series to solve the following differential equation:

(1-x^2)\frac{d^2y}{dx^2} - 2x\frac{dy}{dx} + 12y = 0

HPAS 2018

Question 1(a)

Find the order and degree of the differential equation whose general solution is $y^2 = 2c(x+\sqrt{c})$ , where $c$ is a positive parameter.

Question 2(a)

Two solutions of the ordinary differential equation $y'' - 2y' + y = 0$ are $e^x$ and $5e^x$ . Is $y = Ae^x + B(5e^x)$ the general solution of the differential equation?

Question 2(b)

If the integrating factor of the differential equation $(x^7y^2+3y)dx + (3x^8y-x)dy=0$ is $x^m y^n$ , then find the values of $m$ and $n$ .

HPAS 2017

Question 1(c)

Find the degree and order of the following differential equation:

\left|1+\left(\frac{dy}{dx}\right)^2\right|^{2/3} = \rho\frac{d^2y}{dx^2}

Question 5(a)

Solve the following differential equation by the method of variation of parameters:

(1-x)\frac{d^2y}{dx^2} + x\frac{dy}{dx} - y = (1-x)^2

Question 5(b)

For Bessel’s function $J_n(x)$ , show that: $2nJ_n(x) = x[J_{n-1}(x) + J_{n+1}(x)]$ .

HPAS 2016

Question 1(e)

Show that the Legendre polynomial $P_n(x)$ satisfies $P_n(-x)=(-1)^n P_n(x)$ .

Question 5(a)

Determine the general and singular solution of the non-linear differential equation: $y = xy' + (y')^2$ .

Question 5(b)

Solve the differential equation: $(D^2 - 2D + 2)y = e^x \tan x$ , where $D = \frac{d}{dx}$ .

HPAS 2014

Question 1(d)

Solve: $(D^{3}-7D-6)y = e^{2x}(1+x)$ , where $D=\frac{d}{dx}$ . (Linear DE with constant coefficients).

Question 6(a)

Discuss the solutions of the following equation: $p^{2}(2-3y)^{2}=4(1-y)$ .

Question 6(b)

Solve: $x^{2}\frac{d^{2}y}{dx^{2}}-(x^{2}+2x)\frac{dy}{dx}+(x+2)y=x^{3}e^{x}$ .

Question 6(c)

For Bessel function, prove that: $2n J_{n}(x)=x[J_{n-1}(x)+J_{n+1}(x)]$ .

HPAS 2013

Question 1(e)

Solve: $(D^{4}+2D^{3}-3D^{2})y=x^{3}+2 \sin x$ , where $D\equiv\frac{d}{dx}$ . (Linear DE with constant coefficients).

Question 6(a)

Discuss the solutions of the following equation: $x^{3}p^{2}+x^{2}yp+a^{3}=0$ , where $p\equiv\frac{d}{dx}$ .

Question 6(b)

Solve by the method of variation of parameters: $x^{2}\frac{d^{2}y}{dx^{2}}+x\frac{dy}{dx}-y=x^{2}e^{x}$ .

Question 6(c)

For Bessel function, prove that: $x J_{n}^{\prime}(x)=n J_{n}(x)-x J_{n+1}(x)$ .

Vector Analysis

Syllabus:

Scalar and vector fields, triple products, differentiation of vector function of a scalar variable.
Gradient, divergence and curl in Cartesian, cylindrical and spherical coordinates and their physical interpretations.
Higher order derivatives, vector identities and vector equations.
Applications to Geometry: curves in space, curvature and torsion. Serret-Frenet’s formulae.
Gauss’ and Stokes’ theorems, Green’s identities.

HPAS 2024

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

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

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

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

Question 1(d)

Find the unit outward normal vector at the point $\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 0\right)$ 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

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

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 2014

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 2013

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 $x=y=z=a$ .

Statics

Syllabus:

Analytical conditions of equilibrium of coplanar forces, virtual work.
Forces in three dimensions, Poinsot’s central axis, Wrenches, Null lines and planes, Stable and unstable equilibrium.

HPAS 2024

Question 7(b)

$R$ is the resultant of forces $P$ and $Q$ acting on a particle. If $P$ is reversed, with $Q$ remaining the same, the resultant becomes $R'$ . If $R$ and $R'$ are perpendicular to each other, show that P=Q.

HPAS 2023

Question 7(b)

Three forces $P$ , $Q$ , and $R$ act on a particle and keep it in equilibrium. If the angle between $P$ and $Q$ , and between $Q$ and $R$ , is $120^\circ$ each, then show that P=Q=R.

HPAS 2021

Question 7(b)

Three forces act perpendicular to the sides of a triangle at their middle points and are proportional to the sides. Show that they are in equilibrium.

Question 8(a)

The resultant of two forces acts along a line perpendicular to one force and is in magnitude half the other. Compute the angle between the forces.

HPAS 2020

Question 6(b)

Define a wrench of a system of forces. Three forces $P$ , $Q$ , and $R$ act along the three straight lines $x=0, y-z=a$ ; $y=0, z-x=a$ ; and $z=0, x-y=a$ respectively. Find the vector symmetrical equation of the central axis and the pitch of the equivalent wrench.

Question 8(a)

A heavy uniform rod rests with one end against a smooth vertical wall and with a point in its length resting on a smooth peg. Find the position of equilibrium, and show that it is unstable.

HPAS 2018

Question 8(b)

Three forces $P$ , $Q$ , and $R$ act along the sides of the triangle taken in order, formed by the lines $x+y=1$ , $y-x=1$ , and $y=2$ . Find the equation of the line of action of their resultant.

HPAS 2017

Question 7(a)

Five weightless rods of equal length are jointed together to form a rhombus $ABCD$ with one diagonal $BD$ . If a weight $W$ is attached to $C$ and the system is suspended from $A$ , show that there is a thrust in BD equal to $\frac{W}{\sqrt{3}}$ .

Question 7(b)

Two forces act, one along the line $y=0, z=0$ and the other along the line $x=0, z=c$ . As the forces vary, show that the surface generated by the central axis of their equivalent wrench is $(x^2+y^2)z = cy^2$ .

HPAS 2014

Question 1(f)

Forces 13, 10 and 5 kg weight act along the sides $BC$ , $CA$ and $AB$ of an equilateral triangle $ABC$ . Find the direction and the magnitude of their resultant.

Question 8(a)

Six equal heavy rods, freely hinged at their ends form a regular hexagon $ABCDEF$ which when hung-up by the point $A$ is kept from altering its shape by two light rods $BF$ and $CE$ . Find the thrusts of these rods.

HPAS 2013

Question 1(f)

Two rods, each of length $2a$ , have their ends united at an angle $\alpha$ , and are placed in a vertical plane on a sphere of radius $r$ . Prove that the equilibrium is stable or unstable according as $\sin \alpha > \text{ or } < \frac{2r}{a}$ .

Question 8(a)

The moments of a given system of coplanar forces about three points: $(2,0)$ , $(0, 2)$ and $(2, 2)$ in their plane are 3, 4 and 10 units respectively. Find the magnitude of the resultant force and the equation of its line of action.

Dynamics

Syllabus:

Simple harmonic motion, motion on rough curve, tangential & normal accelerations, motion in a resisting medium, motion when the mass varies.
Velocity along radial and transverse directions, central orbits.
Kepler’s laws of motion, motion of a particle in three dimensions, acceleration in terms of Polar and Cartesian co-ordinate systems.

HPAS 2024

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.

Question 7(a)

A particle is projected in a plane with velocity $\sqrt{\frac{\mu}{3a^6}}$ at a distance $a$ from the center of force, attracting according to the law $\frac{\mu}{r^7}$ , in a direction inclined at $30^\circ$ to the radius vector. Show that the orbit is $r^2 = 2a^2 \cos(2\theta)$ .

Question 8(b)

The amplitude of a simple harmonic oscillator is doubled. How does this affect the time period, total energy, and maximum velocity of the oscillator?

HPAS 2023

Question 1(e)

A particle executes Simple Harmonic Motion with a period of 10 seconds and an amplitude of 5 cm. Calculate the maximum velocity.

Question 8(a)

A particle performing Simple Harmonic Motion has a mass of 2.5 gm and a frequency of vibration of 10 Hz. It is oscillating with an amplitude of 2 cm. Calculate the total energy of the particle.

Question 8(b)

The motion of a particle under the influence of a central force is described by $r = a \sin\theta$ . Find an expression for the force.

HPAS 2021

Question 7(c)

Show that the only law for a central attraction, for which the velocity in a circle at any distance is equal to the velocity acquired in falling from infinity to that distance, is that of the inverse cube.

Question 8(b)

A particle of mass $m$ is attached to a light wire which is stretched tightly between two fixed points with a tension $T$ . If $a$ and $b$ are the distances of the particle from the two ends, then show that the period of the small transverse oscillation of $m$ is $2\pi\sqrt{\frac{mab}{T(a+b)}}$ .

HPAS 2020

Question 1(d)

Find the velocity of a particle moving on the surface of a right circular cylinder of radius $b$ .

Question 5(b)

A particle of mass $m$ moves with a central attractive force $\mu(r^5-c^4r)$ towards the origin. It is projected from an apse at distance $c$ with velocity $\sqrt{\frac{2\mu}{3}}c^3$ . Show that the equation of the central orbit is $x^4+y^4=c^4$ .

Question 7(a)

The tangential acceleration of a particle moving along a circle of radius $a$ is $\lambda$ times the normal acceleration. If its speed at a certain time is $u$ , then prove that it will return to the same point after a time $\frac{a}{\lambda u}(1-e^{-2\pi\lambda})$ .

HPAS 2018

Question 8(a)

With usual notations, prove that the angular acceleration in the direction of motion of a point moving in a plane is $\frac{v}{\rho}\frac{dv}{ds} - \frac{v^2}{\rho^2}\frac{d\rho}{ds}$ .

HPAS 2017

Question 8(a)

A particle moves in a curve such that its tangential and normal accelerations are equal and the angular velocity of the tangent is constant. Find the path.

Question 8(b)

A particle describes an ellipse under a force $\frac{\mu}{(\text{distance})^2}$ towards a focus. If it was projected with velocity $V$ from a point at a distance $r$ from the center of force, show that its periodic time is

\frac{2\pi}{\sqrt{\mu}} \left[ \frac{2}{r} - \frac{V^2}{\mu} \right]^{-3/2}

HPAS 2016

Question 8(b)

A particle moves in a plane in such a manner that its tangential and normal accelerations are always equal and its velocity varies as $e^{\tan^{-1}(s/c)}$ , $s$ being the length of the arc of the curve measured from a fixed point on the curve. Find the path.

HPAS 2014

Question 8(b)

A particle is moving vertically downwards from rest through a medium whose resistance varying as velocity, discuss its motion.

Question 8(c)

The greatest and least velocities of a certain planet in its orbit round the sun are 30 and 29.2 km/sec. Find the eccentricity of the orbit.

HPAS 2013

Question 8(b)

A particle moves with simple harmonic motion in a straight line. If in the first second after starting from rest it travels a distance $a$ and in the next second it travels a distance $b$ in the same direction, then find the amplitude and period of the motion.

Question 8(c)

A particle of mass $m$ is falling under gravity through a medium whose resistance is $\mu$ times the velocity. If the particle is released from rest, show that the distance fallen through in time $t$ is $\frac{gm^{2}}{\mu^{2}}\left(e^{\frac{\mu t}{m}}-1-\frac{\mu t}{m}\right)$ .