math 2109 pyq

Here is the systematic categorization of all the questions from the provided past papers, strictly aligned with your criteria.

Topic 1: Matrix

Subtopic	Exact Question	Year(s) of Appearance	Figure / Reference
Matrix Definitions & Properties	Define a rank of a matrix. (03)	2025	None
	Define rank of a matrix. Find the rank of $A=\left[\begin{array}{cccc}2& 3& 4& 5\\ 4& 6& 8& 10\\ 4& 6& 7& 10\end{array}\right]$. (10)	2019	None
	Define Hermitian and Skew-Hermitian matrix with examples. (04)	2025	None
	Compare between Hermetian and skew Hermetian matrices. Find the skew-hermetian part of the matrix. $A=\left[\begin{array}{ccccccc}1+i& 3i& 2-3i-4+2i& -6i& 5-i2-2i& -3+6i& -8-2i\end{array}\right]$ (10)	2021	None
	Express A as the sum of a Hermitian and a skew Hermitian matrices, where $A=\left[\begin{array}{ccccccc}2+i& 3& 4-i2i& 3+i& 43& 5& -7\end{array}\right]$. (10)	2019	None
	Define Unitary matrix, Hermitian matrix and Elementary matrix. Give an example and at least one property of each of them. (10)	2023	None
	Define symmetric and skew-symmetric matrices. Show that the diagonal elements of skew-symmetric matrices are zero. (12)	2019	None
	Define the following matrices with example: (i) Upper triangular matrix (ii) Equivalent matrix (iii) Skew-Hermitian matrix. (12)	2015	None
	Define nilpotent matrix. Show that $A=\left[\begin{array}{ccccccc}1& 2& 31& 2& 3-1& -2& -3\end{array}\right]$ is a nilpotent matrix of order 2. (10)	2015	None
	Define the following terms with example: (i) Skew symmetric matrix, ii) Elementary matrix, iii) Submatrices of a matrix, and iv) Hermitian matrix. (12)	2016	None
	Define with examples: (i) Square Matrix ; (ii) Skew-symmetric matrix ; (iii) Hermitian matrix; (iv) singular matrix. (12)	2017	None
	Define Hermitian and skew-Hermitian matrices. Express $\left[\begin{array}{ccccccc}-3& -4& 62& 0& 88& 12& 1\end{array}\right]$ as the sum of a symmetric and a skew-symmetric matrices. (12)	2018	None
	Express the following matrix as the sum of a symmetric matrix and a skew symmetric matrix. $\left[\begin{array}{ccccccc}1& 0& 22& 1& 31& -2& 1\end{array}\right]$ (10)	2024	None
	Find the symmetric and skew-symmetric parts of the matrix $A=\left[\begin{array}{ccccccc}1& 2& 46& 8& 13& 5& 7\end{array}\right]$. (06)	2015	None
	Find the Hermitian and Skew-Hermitian part of $A=\left[\begin{array}{ccccccc}1+i& 2i& 33-2i& 2& 4+i-i& 0& -2i\end{array}\right]$. (06)	2025	None
	What is the relationship between $tr(A)$ and $tr(A^{-1})$ for a $2\times 2$ invertible matrix $A=\left[\begin{array}{ccc}a& bc& d\end{array}\right]$? (05)	2023	None
	For which three numbers of c, the matrix B is not invertible, and why not? $B=\left[\begin{array}{ccccccc}2& c& cc& c& c8& 7& c\end{array}\right]$ (10)	2023	None
	Matrix A and B are such that $3A-2B=\left[\begin{array}{ccc}2& 1-2& -1\end{array}\right]$ and $-4A+B=\left[\begin{array}{ccc}-1& 2-4& 3\end{array}\right]$. (10)	2017	None
	Let $A=\left[\begin{array}{ccc}3& -6-1& 2\end{array}\right]$. Construct a $2\times 2$ matrix B such that AB is a zero matrix, where B has two different non-zero columns. (08)	2018	None
	If $A=\left[\begin{array}{ccc}1& 00& -1\end{array}\right]$ and $B=\left[\begin{array}{ccc}0& 11& 0\end{array}\right]$, then verify which of the following statements are true. (show detail calculations) i) A and B are commute, ii) A and B are periodic with period 3, iii) A and B are nilpotent, iv) A and B are involutory, and v) A and B are anti-commute. (15)	2018	None
	If A and B are square matrices of the same order and A and B anti-commute then prove that $(A+B{)}^2=A^2+B^2$. (05)	2016	None
	Verify Cayley Hamilton theorem for the matrix $A=\left[\begin{array}{ccc}1& 23& 4\end{array}\right]$. (09)	2016	None
	Show that the matrix $A=1/3\left[\begin{array}{ccccccc}1& 2& 22& -1& 22& 2& -1\end{array}\right]$ is orthogonal. (05)	2015	None
	If $A=\left[\begin{array}{ccc}0& -tan(\theta /2)tan(\theta /2)& 0\end{array}\right]$ and I is $2\times 2$ unit matrix, show that $(I+A)(I-A{)}^{-1}=\left[\begin{array}{ccc}cos\theta & -sin\theta sin\theta & cos\theta \end{array}\right]$. (08)	2020	None
	Define vector space. Can $X_1=(2,3,-1{)}^t,X_2=(1,-1,2{)}^t$, and $X_3=(-1,1,3{)}^t$ form a vector space? (13)	2019	None
	Write the vector $v=(2,-5,3)$ as a linear combination of the vectors $v_1=(1,-3,2),v_2=(2,-4,-1)$ and $v_3=(1,-5,7)$. (09)	2015	None
Inverse & Echelon Form	Utilize row canonical form to evaluate inverse of the matrix $A=\left[\begin{array}{ccccccc}3& 4& -11& 0& 32& 5& -4\end{array}\right]$. (13)	2025	None
	Using row elementary operation, find the inverse of $A=\left[\begin{array}{ccccccc}1& -1& 12& -1& 01& 0& 0\end{array}\right]$. (11)	2024	None
	Explain whether a square matrix with rank less than its order has its inverse or not. Using elementary row operation, evaluate $A^{-1}$ (if exist). $A=\left[\begin{array}{ccccccc}3& -2& -1-4& 1& -12& 0& 1\end{array}\right]$ (13)	2022	None
	Define inverse of a matrix. Test whether the matrix $A=\left[\begin{array}{ccccccc}1& -3& 2-3& 3& -12& -1& 0\end{array}\right]$ has an inverse. If so find the inverse of A using row transformation only. Show that inverse of a matrix if exists, is unique. (18)	2016	None
	Define inverse of a matrix. Find the inverse of the matrix $A=\left[\begin{array}{ccccccc}1& 2& 32& 5& 7-2& -4& -5\end{array}\right]$, if possible, by the method of elementary transformations. (12)	2015	None
	When does a matrix have its inverse? If possible find the inverse of $\left[\begin{array}{ccccccc}1& 0& 22& -1& 34& 1& 8\end{array}\right]$. (13)	2017	None
	Test the possibility of existing the inverse of the matrix $A=\left[\begin{array}{ccccccc}1& 2& 32& 3& 00& 1& 2\end{array}\right]$ and if possible find its inverse by elementary transformation. (13)	2018	None
	Reduce the matrix $A=\left[\begin{array}{cccccccccc}2& 3& -1& 1-1& 2& 0& 61& 5& 4& 3\end{array}\right]$ to echelon form and find its rank. (09)	2025	None
	Find the rank of the following matrix. $\left[\begin{array}{ccccccccccccc}3& -2& 0& -10& 2& 2& -11& -2& -3& 20& 1& 2& 1\end{array}\right]$ (10)	2024	None
	Determine the rank of the matrix, $C=\left[\begin{array}{ccccccccccccccccc}2& 5& -3& -4& 84& 7& -4& -3& 96& 9& -5& 2& 40& -9& 6& 5& -6\end{array}\right]$. (12)	2023	None
	What could be the maximum and minimum value of rank of the matrix $A_{4\times 3}$? Find the rank of $B=\left[\begin{array}{ccccccccc}1& 2& 31& -2& 5-1& 6& -73& 8& 8\end{array}\right]$ by reducing it to row echelon form. (10)	2022	None
	Reduce the following matrix ‘A’ to its echelon form, canonical form and finally normal form by using elementary transformation. Also find its rank. $A=\left[\begin{array}{cccccccccc}0& 2& -3& 12& -1& 2& 02& 1& -1& 1\end{array}\right]$ (13)	2021	None
	Reduce the matrix $B=\left[\begin{array}{ccccccccccccc}1& -2& -3& 22& -5& 1& 23& -8& 5& 25& -12& -1& 6\end{array}\right]$ to echelon form and then to canonical and hence find its rank. Is the echelon form of a matrix unique? (14)	2016	None
	Define the normal form of a matrix. Reduce the matrix A to its row canonical form and then normal form and hence find the rank of A, where $A=\left[\begin{array}{ccccccccccccc}1& 1& 2& -34& 1& 0& 20& 3& 0& 10& 1& 0& 2\end{array}\right]$. (16)	2015	None
	Reduce the matrix A into its canonical form then to normal form and also find its rank where, $A=\left[\begin{array}{ccccccc}1& -2& 31& 2& 53& -8& -4\end{array}\right]$. (12)	2017	None
	Find the row canonical form of the matrix $A=\left[\begin{array}{cccccccccccccccc}0& 3& 6& 6& 4& -53& -7& 8& -5& 8& 93& -9& 12& -9& 6& 15\end{array}\right]$. What is its rank? (12)	2018	None
System of Linear Equations	Define a homogeneous system of linear equations. (03)	2025	None
	Find unique, many, or no solution for the system of equations $a_{1}x+b_{1}y+c_1=0$, $a_{2}x+b_{2}y+c_2=0$. Also, analyze your results geometrically with appropriate figures. (08)	2025	None
	For what value of $\lambda$, the following linear equations may have solutions or no solution, $x+y+z=1$, $x+2y+4z=\lambda$, $x+4y+10z={\lambda }^2$. (11)	2025	None
	For what value of $\lambda$ and $\mu$ the following system of liner equations has (i) no solution, (ii) many solutions, (iii) unique solution. $x+y+z=6$; $x+2y+3z=10$; $x+2y+\lambda z=\mu$. (12)	2024	None
	Solve the system: $2x+y+12z=1$, $x+2y+9z=-1$. Hence interpret the solution geometrically. (10)	2023	None
	Apply Gaussian elimination method to solve the values of x, y, z so that the following matrix be symmetric. $\left[\begin{array}{ccccccc}2& x+y+2z& 2x+4y-3z9& -3& 3x+6y-5z1& 0& 4\end{array}\right]$ (12)	2022	None
	Find the value of $\mu$ for which the following system of linear equation may have (i) no solution (ii) unique solution, and (iii) many solutions, where $2\mu x+3y=5$, $x+2y=\mu$. (10)	2021	None
	Determine the values of k for which the system of equations: $x-ky+z=0$, $kx+3y-kz=0$, $3x+y-z=0$ has (i) only trivial solution (ii) non-trivial solution. Hence obtain the non-trivial solution for the largest value of k. (14)	2020	None
	When a system of nonhomogeneous linear equations is said to be consistent? Calculate the currents in each wires of the following circuits: (10)	2019	Figure 5(a)
	Apply rank test to solve the following system of linear equations: $2x_1+3x_2+x_3=9$, $x_1+2x_2+3x_3=6$, $3x_1+x_2+2x_3=8$. Also, if consistent, then find its solutions. (15)	2019	None
	Discuss the consistency of the following system of equations: $2x+3y+4z=11$, $x+5y+7z=15$, $3x+11y+13z=25$. (08)	2017	None
	Solve the following system of linear equations: $x_1+x_2-x_3-x_4=-1$, $x_1+x_2+x_3+x_4=3$, $x_1-x_2+x_3-x_4=3$, $4x_1+x_2-2x_3+x_4=0$, If Possible. (13)	2017	None
	Check whether the system of homogeneous equations possesses non-trivial solution and find it, if exist: $x_1-2x_2-x_3+3x_4=0$, $-2x_1+4x_2+5x_3-5x_4=0$, $3x_1-6x_2-6x_3+8x_4=0$. (10)	2018	None
	Construct the matrix form for the system of m simultaneous linear equations in m unknowns. Solve the following system of linear equations by the matrix method: $x+2y+3z=14$, $3x+y+2z=11$, $2x+3y+z=11$. (13)	2015	None
Eigenvalues & Diagonalization	Define eigen value and eigen vector. (03)	2025	None
	Find the eigen values and eigen vector corresponding to the largest eigen value of the matrix $A=\left[\begin{array}{ccc}5& 29& 2\end{array}\right]$. (10)	2025	None
	Determine the matrix P such that $P^{-1}AP$ is diagonal, where $A=\left[\begin{array}{ccccccc}4& 6& 61& 3& 2-1& -4& -3\end{array}\right]$. (12)	2025	None
	Evaluate minimal polynomial for the matrix, $A=\left[\begin{array}{ccccccccccccc}-1& 0& 0& -50& -1& 0& 60& 0& 2& 00& 0& 0& 2\end{array}\right]$. (10)	2025	None
	Find the matrix P that diagonalize the matrix $A=\left[\begin{array}{ccccccc}4& 6& 61& 3& 2-1& -4& -3\end{array}\right]$ and justify if $P^{-1}AP$ is diagonal or not. (14)	2024	None
	Define eigenvalues and eigenvectors. Find bases for the eigenspaces of the matrix $D=\left[\begin{array}{ccc}-1& 32& 0\end{array}\right]$. (09)	2023	None
	Find the spectral radius modal matrix for $E=\left[\begin{array}{ccccccc}2& 1& -13& 2& -33& 1& -2\end{array}\right]$. Find $E^5$ using diagonalization process. (13)	2023	None
	Find the spectral and modal matrices for $B=\left[\begin{array}{ccccccc}5& -6& 11& 1& 03& 0& 1\end{array}\right]$. Hence find $B^{100}$ using those matrices. (15)	2022	None
	Define eigenvalue and eigenvector of matrix. Determine, whether $x=[111{]}^T$ is an eigenvector of $A=\left[\begin{array}{ccccccc}2& -1& -1-1& 2& -1-1& -1& 2\end{array}\right]$. Also, find characteristic equation of A. (12)	2021	None
	Examine whether the matrix $A=\left[\begin{array}{ccccccc}-2& 2& -32& 1& -6-1& -2& 0\end{array}\right]$ is diagonalizable. If so, obtain the matrix P such that $P^{-1}AP$ is a diagonal matrix. (14)	2020	None
	Find the eigen values and the corresponding eigen vectors of $A=\left[\begin{array}{ccc}2& -13& 4\end{array}\right]$. (13)	2019	None
	Write down three properties of Eigen value. (06)	2016	None
	Find the eigenvalues and eigenvector corresponding to the largest eigenvalue for the matrix $A=\left[\begin{array}{ccccccc}8& -6& 2-6& 7& -42& -4& 3\end{array}\right]$. (14)	2015	None
	Define Eigen value and Eigen vector. Find the Eigen values and Eigen vectors for the matrix: $A=\left[\begin{array}{ccccccc}2& 2& 11& 3& 11& 2& 2\end{array}\right]$. (15)	2017	None
	Define eigen value and eigen vector. Check whether the vectors $X_1=\left[\begin{array}{c}6-5\end{array}\right]$ and $X_2=\left[\begin{array}{c}3-2\end{array}\right]$ are the eigen vectors of the matrix $A=\left[\begin{array}{ccc}1& 65& 2\end{array}\right]$ or not. If not, find the eigen vectors of A and the invertible matrix P such that $P^{-1}AP$ is a diagonal matrix. (18)	2018	None
Matrix Applications	Find the current in the following circuit: (13)	2025	Figure of Q. 7(d)
	Construct a system of linear equations from the given circuit to find $i_1$, $i_2$, and $i_3$. (13)	2024	Figure of Q 7.c
	Determine the currents $I_1$, $I_2$ and $I_3$ for the electrical network shown in the figure 2(c). (13)	2023	Figure 2(c)
	Find a cubic polynomial whose graph passes through the points $(-1,-1)$, $(0,1)$, $(1,3)$, and $(4,-1)$. (10)	2023	None
	Write a matrix equation that determines the loop (see Figure 4.(a)) currents and hence solve it to find the loop currents. (13)	2022	Figure of Q.4(a)
	Make use of Gaussian elimination method to determine the currents $I_1$, $I_2$ and $I_3$ for the electrical network shown in the figure 5 (b). (12)	2021	Fig. 5(b)
	The figure below shows known flow rates and direction of flow through certain branches of a network. i) Set up a linear system whose solution provides the unknown flow rates. ii) Solve the system for unknown flow rates. iii) If $x_4=100$, find the flow rates and direction of flow. (16)	2020	Figure of Q. 2(a)
	In an electric network the following equations were obtained for the currents $i_1$, $i_2$, and $i_3$: $i_1-3i_2-8i_3+10=0$, $3i_1+i_2-4=0$, $2i_1+5i_2+6i_3-13=0$. Find $i_1$, $i_2$, and $i_3$. (12)	2016	None
	The steady state temperature distribution of a thin uniform metal plate is shown in the following figure. If the temperature at the four interior nodes of the mesh are $T_1$, $T_2$, $T_3$ and $T_4$, and the temperature of these nodes are approximately equal to the average of the four nearest nodes, determine the temperature at each node. (17)	2018	Figure

Topic 2: Vector Calculus

Subtopic	Exact Question	Year(s) of Appearance	Figure / Reference
Vector Definitions & Basic Properties	Define gradient, solenoidal vector, and conservative vector field. (03)	2025	None
	Write the physical significance of gradient, divergence, and curl. (06)	2024	None
	Define scalar point function and vector point function with example. (05)	2021	None
	Point out some physical outcomes of gradient of a scalar point function and divergence of a vector point function. (05)	2021	None
	Define solinoidal vector. Do the vector $\bar{A}=x^{2}yz\bar{i}+(2yz-x+4)\bar{j}+(y^2-z^2-xy{z}^2)\bar{k}$ solinoidal? (09)	2019	None
Vector Differentiation	Show that $grad\varphi$ is a vector perpendicular to the surface $\varphi (x,y,z)=c$, where c is a constant. (06)	2025	None
	Determine an equation of tangent plane to the surface $z=x^2+y^2$ at the point $(1,-1,2)$. (10)	2025	None
	Find the constants 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 $(1,-1,2)$. (13)	2025	None
	Find the directional derivative of $\varphi =x^{2}yz+4x{z}^2$ at $(1,-2,-1)$ in the direction $2\bar{i}-\bar{j}-2\bar{k}$. (06)	2024	None
	Find an equation for the tangent plane and the parametric equation of the normal line to the surface $48z=2x^2+3y^2$ at $(3,2,5/6)$. (14)	2024	None
	Evaluate $\nabla \cdot [\bar{r}\nabla (\frac{1}{r^3})]$. (09)	2024	None
	Find the arc length of the curve traced out by the endpoint of the vector-valued function $\bar{r}(t)=<t^2,cost+tsint,sint-tcost>$, for $0\le t\le \pi$. (09)	2023	None
	In a three dimensional region, the temperature in a certain medium is given by $T(x,y,z)=T_0(1+ax+by)e^{cz}$, where a, b, c and $T_0(>0)$ are constants. At the origin, find the direction along which the temperature changes most rapidly. (10)	2023	None
	In a three dimensional region, the temperature in a certain medium is given by $T=T_0(1+ax+by)e^{cz}$, where a, b, c and $T_0(>0)$ are constants. At the origin, find the direction along which the temperature changes most rapidly. (10)	2017	None
	Find the value of the constant ‘a’ so that the vector field, $\bar{F}=(axy-z^3)\bar{i}+(a-2)x^2\bar{j}+(1-a)x{z}^2\bar{k}$ is irrotational. In such case, is $\bar{F}$ solenoidal? Why or why not? (12)	2023	None
	Interpret the physical significance of divergence graphically. (04)	2023	None
	Define gradient of a scalar function. Sketch the gradient field of $f(x,y)=x^2-y^2$. (08)	2022	None
	A particle moves in a force field along the path $x=3cost$, $y=3sint$ and $z=t$. i. Find directional velocity, curvature, tangential and normal components of acceleration at $t=0$. ii. Find normal vector and osculating plane at $t=1$, iii. Sketch the path (roughly) and comment about the movement of the particle by considering above information. (30)	2021	None
	By applying appropriate vector differential operator: i. Find the angle between the two surface $S1:xyz=1$ and $S2:x^2+y^2+z^2=3$ at $(1,1,1)$, ii. Find the tangential plane of the surface S2 at the point $(1,1,1)$, iii. Also, find the maximum rate of change of heat of the level surface S1 of temperature and the directional derivative along $\bar{i}+\bar{j}$ direction. (16)	2021	None
	Find the constants a, b, c such that the vector field defined by $\bar{F}=(4xy+a{z}^3)\bar{i}+(b{x}^2+3z)\bar{j}+(6x{z}^2+cy)\bar{k}$ is irrotational. With these values of a, b, c determine a scalar function $\varphi (x,y,z)$ such that $\bar{F}=\nabla \varphi$. (15)	2020	None
	Prove that $\nabla f(r)=\frac{f^{\prime }(r)}{r}\bar{r}$. Hence prove that ${\nabla }^{2}f(r)=\frac{d^{2}f}{d{r}^2}+\frac{2}{r}\frac{df}{dr}$. (12)	2019	None
	Show that $\bar{n} = \frac{\nabla\phi}{	\nabla\phi	}$isunitnormaltothesurface$\phi(x,y,z)=c$. (07)
	The acceleration $\bar{a}$ of a rocket at any time $t\ge 0$ is given by $\bar{a}=e^{-t}\bar{i}-6(t+1)\bar{j}+3sint\bar{k}$. If the velocity $\bar{v}$ and displacement $\bar{r}$ are zero at $t=0$, then find $\bar{v}$ and $\bar{r}$ at any time. (10)	2019	None
	A particle moves so that its position vector is given by $\bar{r}=cos(\omega t)\bar{i}+sin(\omega t)\bar{j}$, where $\omega$ is a constant. Show that i) the velocity $\bar{v}$ of the particle is perpendicular to $\bar{r}$, ii) the acceleration $\bar{a}$ is directed toward the origin and has magnitude proportional to the distance from the origin, and iii) $\bar{r}\times \bar{v}=a$ constant vector. (13)	2018	None
	Evaluate ${\nabla }^2(\frac{1}{r})$, where r is the magnitude of the position vector $\bar{r}$ at $(x,y,z)$. (09)	2018	None
	Find the value of a if the vector $\bar{A}=(a{x}^{2}y+yz)\bar{i}+(x{y}^2-x{z}^2)\bar{j}+(2xyz-2x^2{y}^2)\bar{k}$ has zero divergence. Find the curl of the above vector which has zero divergence. (12)	2018	None
	Find equations for the tangent plane and normal line to the surface $4z=x^2-y^2$ at the point $(3,1,2)$. (13)	2018	None
	Show that $\nabla \varphi$ is a vector perpendicular to the surface $\varphi (x,y,z)=c$ where c is a constant. (09)	2018	None
	Is $\bar{F}=(y^2+2x{z}^2-1)\bar{i}+2xy\bar{j}+2x{z}^2\bar{k}$ irrotational? If so, find u such that $\bar{F}=gradu$. (12)	2017	None
	Find the value of the constant a so that the vector $\bar{F}=(axy-z^3)\bar{i}+(a-2)x^2\bar{j}+(1-a)x{z}^2\bar{k}$ is irrotational. (09)	2016	None
	Define directional derivative. Find the directional derivative of $f(x,y,z)=x{y}^2+y{z}^3$ at the point $(2,-1,1)$ in the direction of the vector $\bar{i}+2\bar{j}+2\bar{k}$. (11)	2016	None
	A particle moves along the curve $x=2t^2$, $y=t^2-4t$, $z=3t-5$, where t is the time. Find the components of its velocity and acceleration at time $t=1$ in the direction $\bar{i}-3\bar{j}+2\bar{k}$. (10)	2015	None
	What is meant by the gradient? What is the physical meaning of the gradient? Find the values of the a, b, c so that the directional derivative of $\varphi =ax{y}^2+byz+c{z}^2{x}^3$ at $(1,2,-1)$ has a maximum of magnitude 64 in a direction parallel to the z-axis. (15)	2015	None
	If $\bar{A}=x^{2}yz\bar{i}-2x{z}^3\bar{j}+x{z}^2\bar{k}$ and $\bar{B}=2z\bar{i}+y\bar{j}-x^2\bar{k}$, Find $\frac{{\partial }^2}{\partial x\partial y}(\bar{A}\times \bar{B})$ at $(1,0,-2)$. (10)	2015	None
	If $\nabla \cdot \bar{E}=0$, $\nabla \cdot \bar{H}=0$, $\nabla \times \bar{E}=-\frac{\partial \bar{H}}{\partial t}$, $\nabla \times \bar{H}=\frac{\partial \bar{E}}{\partial t}$, where E and H represent the electric and magnetic fields, then show that E and H satisfy the wave equation ${\nabla }^{2}u=\frac{{\partial }^{2}u}{\partial t^2}$. (10)	2015	None
	For the particle moving in a plane, find the position $\bar{r}(t)$ and the velocity $\bar{v}(t)$ given that the acceleration $\bar{a}(t)=6t\bar{i}+12t^2\bar{j}$ and the position and velocity at time $t=0$ are $\bar{r}(0)=\bar{i}+\bar{j}$ and $\bar{v}(0)=0$. (10)	2015	None
Line Integral & Work Done	Show that the vector field $\bar{A}=(6xy+z^3)\bar{i}+(3x^2-z)\bar{j}+(3x{z}^2-y)\bar{k}$ is conservative. Find its scalar potential and find the work done in moving an object in this field from $(1,-2,1)$ to $(3,1,4)$. (16)	2025	None
	If $\bar{F}=(2x+y)\bar{i}+(3y-x)\bar{j}$, evaluate ${\int }_c\bar{F}\cdot d\bar{r}$ where c is the curve in the xy-plane consisting of the straight lines from $(0,0)$ to $(2,0)$ and then to $(3,2)$. (11)	2025	None
	If $\bar{F}=(2x+y^2)\bar{i}+(3y-4x)\bar{j}$, calculate $\oint \bar{F}\cdot d\bar{r}$ around the triangle, (11)	2024	Figure of Q 2(a)
	Find the work done in moving a particle in the force field $\bar{F}=3x^2\bar{i}+(2xz-y)\bar{j}+z\bar{k}$ along (i) the straight line from $(0,0,0)$ to $(2,1,3)$; (ii) the curve defined by $x^2=4y$, $3x^3=8z$ from $x=0$ to $x=2$. (12)	2024	None
	Find the line integral of the field $\bar{F}=(x+y)\bar{i}+xy\bar{j}-z^2\bar{k}$ along the path C consisting of the line segments from $(0,0,0)$ to $(1,3,1)$ and then to $(2,-1,4)$. (11)	2023	None
	A fluid motion is given by $\bar{v}=2x(y^2+z^3)\bar{i}+2x^{2}y\bar{j}+3x^2{z}^2\bar{k}$. Determine whether the vector field is conservative or not. Also, find the work done in moving a particle from $(-1,2,1)$ to $(2,3,4)$. (12)	2022	None
	Find the line integral of the vector field $\bar{F}(x,y,z)=(x+y)\bar{i}+xy\bar{j}-z^2\bar{k}$ along the path C: line segments from $(0,0,0)$ to $(1,3,1)$ to $(2,-1,4)$. (10)	2022	None
	Examine, whether $\bar{F}=r^2\bar{r}$ is conservative vector field or not. If conservative then determine its scalar potential $\varphi (x,y,z)$, where $\bar{r}=x\bar{i}+y\bar{j}+z\bar{k}$ and $r =	\bar{r}	$. (14)
	Find the work done of a moving particle in a force field $\bar{F}$, which moves from $(0,0,1)$ to $(1,1,1)$ along the path $y=x^2,z=1$; where $\bar{F}=x^{2}yz\bar{i}+x{z}^2\bar{j}+z\bar{k}$. (09)	2021	None
	Show that $\bar{E}=(y+z-2xy{z}^2+4)\bar{i}+(x-x^2{z}^2+2y)\bar{j}+(x-2x^{2}yz+3)\bar{k}$ can represent an electric field. Hence find the potential $\varphi (x,y,z)$, if $\varphi (2,3,-1)=6$. (14)	2019	None
	Show that if $\bar{A}=\nabla \varphi$ then ${\int }_{p_1}^{p_2}\bar{A}\cdot d\bar{r}$ is independent of the path joining $p_1$ and $p_2$. (07)	2019	None
	Evaluate ${\int }_A^B\bar{F}\cdot d\bar{r}$ along the straight line joining $A(2,-1,3)$ and $B(3,4,1)$. where $\bar{F}=y^2\bar{i}+xy\bar{j}+yz\bar{k}$. (08)	2019	None
	Find the work done in moving a particle once around a circle C in the xy plane, if the circle has center at the origin and radius 3, and if the force field is given by $\bar{F}=(2x-y+z)\bar{i}+(x+y-z^2)\bar{j}-(3x-2y+4z)\bar{k}$. (10)	2018	None
	If $\bar{A}=(3x^2+6y)\bar{i}-14yz\bar{j}-20x{z}^2\bar{k}$, evaluate ${\int }_C\bar{A}\cdot d\bar{r}$ from $(0,0,0)$ to $(2,2,2)$ along the path C as $x=2t,y=t^2,z=t^3$. (10)	2017	None
	Calculate $\int \bar{F}\cdot d\bar{r}$, where c is the part of the spiral $\bar{r}=(acos\theta ,asin\theta ,a\theta )$ corresponding to $0\le \theta \le \pi /2$ and $\bar{F}=r^2\bar{r}$. (10)	2016	None
	Evaluate: $\int \bar{F}\times d\bar{r}$ where $\bar{F}=xy\bar{i}-z\bar{j}+x^2\bar{k}$ and C is the curve given by $x=t^2,y=2t,z=t^3$ from $t=0$ to $t=1$. (15)	2016	None
	What is meant by the conservative force field? If $\bar{E}=r\bar{r}$ is an electric filed, then show that $\bar{E}$ is conservative. Also determine the scalar potential $\varphi$ such that, $\bar{E}=-\nabla \varphi$. (12)	2015	None
Surface & Volume Integral	Evaluate ${\iiint }_v(2x+y)dv$, where v is the closed region bounded by the cylinder $z=4-x^2$ and the planes $x=0$, $y=0$, $y=2$ and $z=0$. (11)	2025	None
	Evaluate ${\iint }_S\bar{F}\cdot \bar{n}ds$, by using the law of projection, where s is bounded by the surface of the plane $x+y+z=1$ in the first octant. $\bar{F}=2y\bar{i}-z\bar{j}+x^2\bar{k}$ (12)	2024	None
	The vector field $\bar{F}=x^2\bar{i}+z\bar{j}+yz\bar{k}$ is defined over the volume of the cuboid given by $0\le x\le a,0\le y\le b,0\le z\le c$, enclosing the surface S. Evaluate the surface integral ${\iint }_S\bar{F}\cdot d\bar{S}$. (12)	2023	None
	Stating significance of surface integral, evaluate ${\iint }_S\bar{F}\cdot \hat{n}ds$ where $\bar{F}=2y\bar{i}-3\bar{j}+x^2\bar{k}$ and s is the surface ot the parabolic cylinder $y^2=8x$ in the first octant bounded by the planes $y=4$ and $z=6$. (13)	2022	None
	Evaluate ${\iint }_S\bar{A}\cdot \bar{n}ds$ where $\bar{A}=(x+y^2)\bar{i}-2x\bar{j}+2yz\bar{k}$ and S is the surface of the plane $2x+y+2z=6$ in the first octant. (13)	2021	None
	Let $T(x,y,z)=x^2+y^2+z^2$ represent temperature and let the “flow” of the heat be given by the vector field $\bar{F}=-\nabla T$. Find the flux of heat out of the sphere $x^2+y^2+z^2=a^2$. (15)	2020	None
	Evaluate ${\iint }_S\bar{A}\cdot \bar{n}ds$, where S is the $1^{st}$ octant portion of the paraboloid $y=9-x^2-z^2$. where $\bar{A}=(xy+2)\bar{i}+xz\bar{j}+(x^2-x)\bar{k}$ (13)	2019	None
	Evaluate ${\iint }_S\bar{A}\cdot \bar{n}ds$ where S is the entire surface of the region $x=0,y=0,z=0,x^2+y^2+z^2=16$ and $\bar{A}=(x+y)\bar{i}+(y^2+xz)\bar{j}+x^2{y}^2\bar{k}$. (15)	2019	None
	Evaluate ${\iint }_S\bar{A}\cdot \bar{n}ds$, where $\bar{A}=z\bar{i}+x\bar{j}-3y^{2}z\bar{k}$ and S is the surface of the cylinder $x^2+y^2=16$ included in the first octant between $z=0$ and $z=5$. (13)	2018	None
	Evaluate ${\iint }_S\bar{F}\cdot \bar{n}ds$, where $\bar{F}=2xy\bar{i}+y{z}^2\bar{j}+xz\bar{k}$ and S is the surface of the parallelepiped bounded by $x=0,y=0,z=0,x=2,y=1$ and $z=3$. (11)	2018	None
	Evaluate ${\iint }_S\bar{A}\cdot \bar{n}ds$, where $\bar{A}=z\bar{i}+x\bar{j}-3y^{2}z\bar{k}$ and S is the surface of the cylinder $x^2+y^2=16$ included in the first octant between $z=0$ and $z=4$. (13)	2017	None
	Evaluate ${\iint }_S\bar{A}\cdot \bar{n}ds$, where $\bar{A}=18z\bar{i}-12\bar{j}+3y\bar{k}$ and s is that part of the plane $2x+3y+6z=12$ which is located in the first octant. (13)	2015	None
Vector Theorems	Write the statements of divergence theorem, Stoke’s theorem, and Green’s theorem. (06)	2025	None
	By using the divergence theorem, evaluate ${\iint }_s\bar{A}\cdot \bar{n}ds$ over the entire surface s of the region bounded by the cylinder $x^2+z^2=9,x=0,y=0,z=0$ and $y=8$, if $\bar{A}=6z\bar{i}+(2x+y)\bar{j}-x\bar{k}$. (11)	2025	None
	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 $y=x^2$. (14)	2024	None
	Apply the surface integral part of divergence theorem to evaluate ${\iint }_S\bar{A}\cdot \bar{n}ds$ for $\bar{A}=4x\bar{i}-2y^2\bar{j}+z^2\bar{k}$ taken over the region bounded by $x^2+y^2=4,z=0,z=3$. (14)	2024	None
	State Stoke’s theorem. (02)	2023, 2021	None
	Apply Stoke’s theorem to evaluate ${\oint }_C\bar{F}\cdot d\bar{r}$, where $\bar{F}=y^2\bar{i}+x^2\bar{j}-(x+z)\bar{k}$ and C is the boundary of the triangle with vertices at $(0,0,0),(1,0,0)$ and $(1,1,0)$. (10)	2023	None
	If $\bar{F}=(y-z+2)\bar{i}+(yz+4)\bar{j}-xz\bar{k}$, S is the surface of a cube $x=0,y=0,z=0,x=2,y=2,z=2$. Verify the theorem considering open the bottom of the surface. (15)	2022	None
	State Green’s theorem. Apply the theorem to evaluate ${\oint }_{C}x{y}^{6}dx+(3x^2{y}^5+6x)dy$ where c is the ellipse $x^2+4y^2=4$. (12)	2022	None
	By using Gauss’s divergence theorem, find the total flux of the vector field $\bar{F}$ in the region bounded by the surfaces $x=0,x=4,y=0,y=6,z=3$ and $z=9$; where $\bar{F}=xy\bar{i}+yz\bar{j}+x^2\bar{k}$. [Hints: Total flux = ${\iint }_S\bar{F}\cdot \hat{n}ds$] (11)	2021	None
	Use the Stoke’s theorem to evaluate ${\oint }_C\bar{F}\cdot d\bar{r}$. Where $\bar{F}=xy\bar{i}+2yz\bar{j}+xz\bar{k}$ and C is the boundary of the plane $z=1-y$ as shown in the following figure. (15)	2020	Figure of Q. 5(a)
	Use Green’s theorem for the plane to evaluate ${\oint }_C(e^{tan{h}^{-1}x}-xy)dx+(log(tany)+x^2)dy$, where C consists of the lines $y=0$, $x+2y=4$ and $x=0$. (10)	2019	None
	State Green’s theorem for the plane. Evaluate ${\oint }_C(y-sinx)dx+cosxdy$, where C is the triangle of the adjoining figure. (15)	2018	Figure
	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 $x^2=y$. (12)	2017	None
	Using Divergence theorem find the flux of the electric field $\bar{E}=4xz\bar{i}-y^2\bar{j}+2yz\bar{k}$ over the closed surface S consisting of the planes $x=0$, $y=0$, $z=0$, and $2x+2y+z=4$. (20)	2016	None
	Evaluate ${\oint }_C[(xy+y^2)dx+x^{2}dy]$, where C is the closed curve formed by $y=1/2x^2$ and $y=x$ by using Green’s theorem. (10)	2016	None
	Use Gauss divergence theorem to evaluate ${\iint }_S\bar{F}\cdot \bar{n}ds$, where $\bar{F}=2xy\bar{i}+y{z}^2\bar{j}+xz\bar{k}$ and s is the surface of the parallelepiped bounded by $x=0,y=0,z=0,x=2,y=1$ and $z=3$. (10)	2015	None
	State Green’s theorem. 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 $y=x^2$. (15)	2015	None

Topic 3: Laplace Transform

Subtopic	Exact Question	Year(s) of Appearance	Figure / Reference
Transform Definitions & Properties	Find the Laplace transform of $f(t)=sint;0<t<\pi ,0;\pi <t<2\pi$. (10)	2025	None
	State the first shifting, second shifting, and convolution theorem of Laplace transformation. (10)	2024	None
	Sketch the graph of $f(t)$, where $f(t)=2,0\le t\le 2;0,t>2$ and find $Lf(2t)$. Also examine your result by diagram of $f(2t)$ and $Lf(2t)$. (12)	2024	None
	Define Laplace transform. (02)	2023	None
	Find the Laplace transform of the periodic function $f(t)=sin2t,2\pi <t<4\pi ;0,otherwise$. (08)	2023	None
	State the sufficient conditions for existence of Laplace Transform of a function. Does $f(t)=1/\sqrt{t}$ satisfy this/these condition(s)? Why or why not? (10)	2022	None
	Sketch the graph of $f(t)$ and evaluate $Lf(t)$ where $f(t)=t,0\le t\le 1;2-t,1\le t\le 2f(t+2)=f(t)$. (12)	2022	None
	Find the Laplace Transform of $g(t)=e^t\cdot t\cdot erf(\sqrt{t})$ by showing your details calculation. (11)	2022	None
	Define unit step function. Express the following function in terms of unit step function. Hence evaluate its Laplace transform. (12)	2021	Graph
	Prove that $Lterf(2\sqrt{t})=\frac{3s+8}{s^2(s+4{)}^{3/2}}$. (13)	2021	None
	Find the Laplace transform of the periodic function $F(t)$ with period $2\pi$; $F(t)=sint,0<t<\pi ;0,\pi <t<2\pi$. (10)	2021	None
	Write the existence conditions of Laplace transform. Show that $Lsin(\sqrt{t})=\frac{\sqrt{\pi }}{2s^{3/2}}{e}^{-1/4s}$. (10)	2021	None
	Use definition to find the Laplace transform of $f(t)=t^{7/2}$, given that $\Gamma (1/2)=\sqrt{\pi }$. (08)	2020	None
	Find the Laplace transform of $\frac{1-e^{2t}}{t}$ using suitable properties. (08)	2020	None
	Is there any function $f(t)$ which has the Laplace transform $F(s)=\frac{s^2}{s^2+9}$? Why or why not? (10)	2020	None
	Define Laplace transform. Find Laplace transform of $F(t)=0,0\le t<1;t-1,t\ge 1$ if it exists. (08)	2019	None
	Define periodic function with an example. Find $LF(t)$ of the following function given in figure 7(b). (12)	2019	Figure 7(b)
	Find the Laplace transform of the piecewise continuous function $f(t)$ given by $f(t)=e^t,0<t<1;0,t>1$. (08)	2018	None
	Find the Laplace transform of the following functions: (i) $tcos2t{e}^{-2t}$ ; (ii) $\frac{e^{-at}-e^{-bt}}{t}$. (15)	2017	None
	Define periodic function. Find the Laplace transform of the function, $F(t)=sint,0<t<\pi ;0,\pi <t<2\pi$ where $f(t+2\pi )=f(t)$. (10)	2017	None
	Evaluate ${\int }_0^{\infty }t{e}^{-2t}costdt$ using Laplace transform. (09)	2016	None
	If $L{f}^{\prime \prime }(t)=ta{n}^{-1}(1/s)$, $f(0)=2$ and $f^{\prime }(0)=-1$, find $Lf(t)$. (10)	2016	None
	Define the Laplace transform. Draw the graph of $f(t)$, where $f(t)=sint,0<t<\pi ;0,\pi <t<2\pi$ Extended periodically with period $2\pi$. Also find $Lf(t)$. (10)	2015	None
	Construct the function whose graph is given below. Determine what kind of function is this. Hence evaluate its Laplace transform. (13)	2021	Graph
Inverse Laplace & Convolution	Define inverse Laplace transformation. (02)	2025	None
	Find the inverse Laplace transform of $F(s)=\frac{s}{(s^2+4)(s^2+9)}$, by using convolution theorem. (10)	2025	None
	Using the convolution theorem, find the following inverse Laplace transform, $L^{-1}\frac{1}{s^2}\cdot \frac{1}{s-a}$. (13)	2024	None
	Find the inverse Laplace transform of $\frac{s}{(s^2+a^2{)}^2}$. (09)	2024	None
	Using convolution theorem find $L^{-1}\frac{1}{s^2(s+1{)}^2}$. Hence find $L^{-1}\frac{e^{-s}}{s^2(s+1{)}^2}$. (10)	2023	None
	Using convolution theorem for inverse Laplace Transform find $L^{-1}\frac{s{e}^{-\pi s}}{(s^2+a^2{)}^2}$. (11)	2022	None
	State convolution theorem. Use the theorem to evaluate $L^{-1}\frac{3e^{-s}}{s^2(s+4)}$. (12)	2021	None
	Write convolution theorem for inverse Laplace transform. Evaluate ${\int }_0^t\frac{cos6t-cos4t}{t}dt$ with the help of Laplace transform. (12)	2019	None
	Evaluate $L^{-1}\frac{(-3s+2)e^{-2s}}{s^2-2s+6}$. (10)	2019	None
	Find the inverse Laplace transform of $\frac{4}{s^2(s-4)}$ by using convolution property. (10)	2018	None
	Apply the convolution theorem for inverse Laplace transform find $L^{-1}\frac{s}{(s^2+a^2{)}^2}$. (10)	2017	None
	Find $L^{-1}\frac{2s^2-4}{(s+1)(s-2)(s-3)}$. (10)	2017	None
	Find $L^{-1}\frac{s^2}{(s^2+4{)}^2}$. (10)	2016	None
	State the convolution theorem for the inverse Laplace transform. Use this theorem to evaluate $L^{-1}\frac{1}{s^2(s+1{)}^2}$. (10)	2015	None
Applications (ODEs & Circuits)	A circuit has in series an electromotive force given by $E(t)=5sin100t$ V, a resistor of $10\Omega$, an inductor of $0.005$ H, and a capacitor of $2\times 10^{-4}$ Farads. If the initial current and charge on the capacitor are both zero. Find the charge at any time $t>0$ by using Laplace transform. (13)	2025	None
	Solve the differential equation $Y^{\prime \prime }-2Y^{\prime }-8Y=0$ using the Laplace transform subject to the condition $Y(0)=3$, $Y^{\prime }(0)=6$. (12)	2024	None
	In a series RLC circuit, the sum of voltage in the circuit expressed as, $L\frac{d{q}^2}{d{t}^2}+R\frac{dq}{dt}+\frac{1}{C}q(t)=v(t)$, $q(0)=0$, $q^{\prime }(0)=-1$ and given $L=3$ H, $R=24\Omega$, $C=1/75$ F, $v(t)=40cos5t$. Find the charge $q(t)$. (14)	2024	None
	Solve the following differential equation using Laplace Transform: $Y^{\prime \prime }+9Y=cos2t$; $Y(0)=1$, $Y(\pi /2)=-1$. (13)	2023	None
	The differential equation for the current $i(t)$ in a single-loop LR series circuit is $L\frac{di}{dt}+Ri=E(t)$. Determine the current $i(t)$ when $i(0)=0$. $E(t)$ is the square wave function shown in the figure 4(c). (15)	2023	Figure 4(c)
	Determine the current $i(t)$ in a single-loop LRC circuit when $L=0.1$ h, $R=2\Omega$, $C=0.1$ f, $i(0)=0$ and the impressed voltage is $E(t)=120t-120tU(t-1)$. (13)	2022	None
	Using Laplace Transform solve the IVP: $y^{\prime \prime }+2y^{\prime }+5y=e^{-t}sint$ subjected to $y(0)=0$, $y^{\prime }(0)=1$. (13)	2022	None
	An electromotive force $E(t)=120,0\le t\le 20;0,t>20$ is applied to an RL series circuit in which the inductance (L) is $20$ henry and the resistance (R) is $2$ ohms. Use Laplace transform to find the current $i(t)$, if $i(0)=0$. (13)	2021	None
	Derive the expression for the current in the circuit shown in the figure below when switch S is closed at $t=0$. (14)	2020	Figure of Q. 3(c)
	Find $I(t)$ for the following circuit. Consider that initially there was no flow of current and the capacitor was uncharged. (15)	2019	Figure 7(c)
	In an electrical circuit with electromagnetic force $E(t)$, resistance R and inductance L, the current i build up at the rate given by $L(di/dt)+Ri=E(t)$. If the switch is connected at $t=0$ and disconnected at $t=3$, find the current i at any instant (use Laplace transform method). (17)	2018	None
	Solve by using Laplace transformation technique: $\frac{d^{2}y}{d{t}^2}-6\frac{dy}{dt}+9y=t^2{e}^{3t}$; subject to the condition $y(0)=2$, $y^{\prime }(0)=6$. (12)	2017	None
	An inductor of 1 henry, a resistor of $6\Omega$ and a capacitor of $1/9$ farad are connected in series with an e.m.f of E volts. At time $t=0$, the charge on the capacitor and current in the circuit are zero. Find the charge and current at any time $t>0$ if $E=sint$ volts. Draw the circuit also. (13)	2017	None
	Solve $y^{\prime \prime }+9y=cos2t$ if $y(0)=1$, and $y(\pi /2)=-1$ by the method of Laplace transform. (14)	2016	None
	An inductor of 2 henrys, a resistor of 16 ohms and a capacitor of 0.02 farads are connected in series with an e.m.f of 300 volts. At $t=0$, the charge on capacitor and current in the circuit are zero. Find the charge and current at any time $t>0$ by using the Laplace transform. Also draw this circuit. (15)	2015	None

Topic 4: Fourier Series and Transform

Subtopic	Exact Question	Year(s) of Appearance	Figure / Reference
Fourier Series Expansion	Derive Fourier series for $f(x)=xcosx$ defined in the interval $-\pi \le x\le \pi$. (12)	2025	None
	Write the Dirichlet’s conditions. (03)	2025	None
	Deduce the Fourier series of the function $f(x)=0$, when $-\pi <x<0$; $f(x)=\frac{\pi x}{4}$, when $0<x<\pi$. (14)	2025	None
	Write the existence condition of Fourier series. (07)	2024	None
	Find the Fourier series expansion of the function $f(x)=-\pi ,-\pi <x<0;x,0<x<\pi$, hence evaluate ${\sum }_{n=1}^{\infty }\frac{1}{(2n-1{)}^2}$. (14)	2024	None
	Expand the function $f(x)=x$, $0\le x\le \pi$ in a half range Fourier Sine series. (06)	2024	None
	Write down the conditions for existence of Fourier Series. (05)	2023	None
	A tightly stretched flexible uniform string has it’s ends fixed at the points $x=0$ and $x=l$. The midpoint of the string is displaced a distance ‘a’ as shown in figure 7(b). If $f(x)$ denotes the displacement profile of the string, express $f(x)$ as a Fourier series expansion consisting only sine terms. (14)	2023	Figure 7(b)
	The following table gives the variation of a periodic current over a period $\pi$. Show that there is a direct current part of 0.75 amp in the variable current. Also, obtain the amplitude of the second harmonic. (16)	2023	Table provided
	Define Fourier series. State the conditions to represent a function in Fourier series. (08)	2022	None
	Construct the function whose graph is given below. Hence find the series of sines and cosines of multiples of x which represent $f(x)$ in the given interval. (12)	2022	Figure of Q.5(b)
	Build the relation among a function and its Fourier co-efficients for the interval $(-c,c)$. Hence applying it deduce the value of ${\sum }_{n=1}^{\infty }\frac{1}{n^4}$ from Fourier cosine series of $x(\pi -x)$ in the interval $(0,\pi )$. (15)	2022	None
	Define odd and even functions precisely. Identify $h(t)=\Vert sint\Vert$ as odd or even or neither. (10)
	Define Fourier series. Write the Dirichlet’s conditions for a Fourier series. (07)	2021	None
	Represent the following function as a Fourier series, where $f(x)=x,-2<x<2$ and $f(x+4)=f(x)$. (15)	2021	None
	Sketch the following periodic wave $f(t)$ and find the frequency spectrum of $f(t)$: $f(t)=cosx,0<x<\pi /2;0,\pi /2<x<\pi$; $f(t+\pi )=f(t)$. (15)	2020	None
	Find the Fourier series of the periodic function $f(x)$ with period 4 represent by the graph below: (16)	2020	Figure of Q. 6(a)
	Write the form of Fourier series for the function $f(x)$ with $T=6$ defined in the interval $-2<x<4$, along with the Fourier coefficients. (06)	2019	None
	Find the amplitude of the third harmonic of $f(x)=x,0<x<2;2,2<x<4$. (17)	2019	None
	Find the Fourier series of the half wave rectified sinusoidal wave defined by $i(t)=sint,0<t<\pi ;0,\pi <t<2\pi$; $i(t)=i(t+2\pi )$ (12)	2019	None
	Show that an even function can have no sine term in its Fourier series. (10)	2018	None
	Find the Fourier series corresponding to the function $f(x)=0,-5<x<0;3,0<x<5$ where $f(x+10)=f(x)$. (12)	2018	None
	Find the Fourier series for the function $f(t)=\pi -\Vert t\Vert$ ; $(-\pi \le t\le \pi )$, defined over a single period. (10)
	Find the complex form of Fourier series for the function $g(t)=e^{-t}$ ; $(-1<x<1)$. (10)	2017	None
	An alternating current after passing through a rectifier, has the form $I(x)=I_{0}sinx,0\le x\le \pi ;0,\pi <x<2\pi$. Where $I_0$ is maximum current and the period is $2\pi$. Determine the Fourier series expression of $I(x)$. Also sktech the graph of $I(x)$. (15)	2017	None
	Suppose a periodic function $f(t)$ has its period T. Now write the formula for Fourier coefficients as well as its Fourier series. (xx)	2016	None
	Find the Fourier series of the function: $f(x)=2,for-2\le x<0;x,for0<x<2$ Where $f(x+4)=f(x)$. (xx)	2016	None
	An alternating current, after passing through a rectifier, has the form: $i=I_{0}sinx,for0\le x\le \pi ;0,for\pi <x\le 2\pi$. Where $I_0$ is the maximum current and the period is $2\pi$. Sketch the graph of the given function in the interval $[-2\pi to2\pi ]$ and express $i$ as a Fourier series. (xx)	2016	None
	Find the harmonics and amplitude spectrum of $f(t)=\pi ,-\pi \le t<0;\pi -t,0\le t\le \pi$ considering the logical period of $f(t)$. (xx)	2016	None
	Find the half range sine series for the function $f(x)=x,for0<x<\pi /2;\pi -x,for\pi /2<x<\pi$. (11)	2016	None
	Find the Fourier series of sines and cosines of multiples of x which represents $f(x)$ in the interval $-\pi <x<\pi$ where, $f(x)=0,when-\pi <x<0;\frac{\pi x}{4},when0<x<\pi$, such that $f(2\pi +x)=f(x)$. (10)	2015	None
	Develop Fourier series of a periodic function $f(x)$ in the interval $(-2,2)$ defined by $f(x)=0$ for $-2<x<0$ and $f(x)=1$ for $0<x<2$. (15)	2015	None
Fourier Transform & Integral	Find the Fourier integral of the function $f(x)=e^{-kx}$ when $x>0$ and $f(-x)=-f(x)$ for $k>0$ and hence show that ${\int }_0^{\infty }\frac{usinux}{k^2+u^2}du=\frac{\pi }{2}{e}^{-kx}$. (06)	2025	None
	Solve $\frac{\partial u}{\partial t}=\frac{{\partial }^{2}u}{\partial x^2}$; $x>0,t>0$ subject to the condition $u(0,t)=0$, $u(x,0)=1;0<x<1,0;x>1$ and $u(x,t)$ odd. (18)	2025	None
	Find the Fourier integral of the function $f(x)=e^{-kx}$, $x>0$ and $f(-x)=f(x)$ for $k>0$. (06)	2024	None
	Find the Fourier transform of the function $f(x) = {1-x^2, |x|<1;\ 0, |
	Find the Fourier integral representation of the function $p(t)=0;t<0,0.5;t=0,e^{-t};t>0$. (11)	2023	None
	Find the Fourier cosine transform of $f(x)=1/(1+x^2)$ and hence derive the Fourier sine transform of $\varphi (x)=x/(1+x^2)$. (12)	2023	None
	Obtain inverse Fourier transform of the following signal: (i) $X(j\omega )=\frac{3j\omega +10}{(j\omega {)}^2+6j\omega +8}$ (ii) $X(j\omega )=\frac{1+3j\omega }{(j\omega +3{)}^2}$. (12)	2023	None
	By applying the Fourier series integral formula to the function $f(x)=1,0<x<k;1/2,x=k;0,x>k$ Prove that $f(x)=\frac{2}{\pi }{\int }_0^{\infty }\frac{1-cosku}{u}sinuxdu$. (10)	2022	None
	Evaluate the Fourier transform of $(x) = {1-x^2, |x|<1;\ 0, |
	Given that $f(x) = {1; |x|< 1,\ 0; |
	Find the Fourier integral representation of the function, $g(x)=0,x<0;sinx,0\le x\le \pi ;0,x>\pi$. (14)	2020	None
	Find the Fourier transform of $F(x)=1,when-a<x<a;0,elsewhere$. Hence show that $\frac{\pi }{2}={\int }_0^{\infty }\frac{sin(\lambda a)}{\lambda }d\lambda$. (13)	2018	None
	Find the Fourier transform of $F(x) = {1-x^2, |x|<1;\ 0, |
	Find the Fourier sine transform of $g(x)=x,0<x<1;2-x,1<x<2;0,x>2$. (10)	2017	None
	Find the finite Fourier cosine transform of the function $F(x)=2x$; $0<x<4$. (10)	2017	None
	Find the Fourier transform of $f(t) = {1, |t|\le b;\ 0, |
	Find the Fourier cosine transform of $f(x)=e^{-2x}+4e^{-3x}$. (10)	2016	None
	Find the finite Fourier sine transform of $F(x)=x$ such that $0<x<2$. (08)	2015	None

(Note: “xx” marks in 2016 Fourier questions mean the marks were physically cut off from the provided scan margin. The questions are maintained verbatim with mathematical notations adjusted using universally accepted text equivalents for markdown readability.)