Q.1) Prove that $\frac{\pi}{3} > \frac{1}{\sqrt{53}}\cos^{-1}\frac{3}{5} > \frac{\pi}{3} - \frac{1}{8}$ using Lagrange's mean value theorem. (Nov-2022)

Q.2) Find the minimum and maximum value of $f(x,y)=x^3+3xy^2-3x^2-3y^2+4$. (Nov-2022)

Q.3) Find C of Cauchy's Mean value theorem on [a, b] for the function $f(x)=e^x$ and $g(x)=e^{-x}, (a,b>0)$. (Nov-2022)

Q.4) State Rolle's theorem hence verify for $f(x)=x^2+2x$ defined in the interval [-2, 0]. (June-2023, Dec-2024)

Q.5) Find the first six terms of the expansions of the function $e^x\log(1+y)$ in a Taylor series in the neighbourhood of the point (0, 0). (June-2023)

Q.6) The temperature $u(x,y,z)$ at any point in space is $u=400xz^2$ find the highest temperature on surface of the sphere $x^2+y^2+z^2=1$. (June-2023)

Q.7) If $u=x^2\tan^{-1}\frac{y}{x}-y^2\tan^{-1}\frac{x}{y}$ find the value of $\frac{\partial^2u}{\partial x \partial y}$. (June-2023)

Q.8) Find shortest distance from the origin to the curve $x^2+4xy+6y^2=140$. (June-2023)

Q.9) Find $\frac{du}{dt}$ if $u=x^2+y^2, x=a\cos t, y=b\sin t$. (June-2023, Dec-2023)

Q.10) State Lagrange's theorem hence verify for $f(x)=x^2+2x$ defined in the interval [-2, 0]. (Dec-2023)

Q.11) Find the first six terms of the expansions of the function $e^x\cos y$ in a Taylor series in the neighbourhood of the point (0, 0). (Dec-2023)

Q.12) Estimate the extreme values of the function $x^3+y^3-63(x+y)+12xy$. (Dec-2023)

Q.13) If $u=\left(\frac{y-x}{xy}\right)\left(\frac{z-x}{xz}\right)$ find the value of $x^2u_x+y^2u_y+z^2u_z$. (Dec-2023)

Q.14) Show that the rectangular solid of maximum volume that can be inscribed in a given sphere is a cube. (Dec-2023, Dec-2024, June-2025)

Q.15) Expand $(1+x)^x$ by Maclaurin's Theorem. (June-2024)

Q.16) Find the Maximum value of $u=\sin x \sin y \sin(x+y)$. (June-2024)

Q.17) Find the Maximum and Minimum value of $u=a^2x^2+b^2y^2+c^2z^2$, where $x^2+y^2+z^2=1$ and $lx+my+nz=0$. (June-2024)

Q.18) Expand $\log x$ in power $(x-1)$ by Taylor's theorem and hence find the value $\log 1.1$. (June-2024)

Q.19) Find the points where the function $x^3+y^3-3axy$ has maximum or minimum value. (Dec-2024)

Q.20) Find the Taylor's expansion of $y=\sin x$ about point $x=\pi/2$. (Dec-2024)

Q.21) Verify Rolle's Theorem for the function $y=x^2+2, a=-2$ and $b=2$. (Dec-2024)

Q.22) If $u=\tan^{-1}\frac{x^3+y^3}{x-y}$ then prove that $x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}=\sin 2u$. (June-2025)

Q.23) Find the first 3 terms in the Maclaurin series for i) $\sin^{-1}x$ ii) $xe^{-x}$. (June-2025)

Q.24) Find the Taylor series for the function $x^4+x-2$ centered at a=1. (June-2025)

Q.25) If $x^x y^y z^z = C$ then show that $\frac{\partial^2 z}{\partial x \partial y}=-(x\log ex)^{-1}$. (June-2025)

Q.1) Find the directional derivative of $\phi = xy^2 + yz^3$ at the point (2, -1, 1) in the direction of the vector $\vec{i} + 2\vec{j} + 2\vec{k}$. (Predicted)

Q.2) Discuss the conditions for Maxima and Minima of functions of two variables using Lagrange's Multipliers method. (Predicted)

Q.3) Expand $f(x, y) = e^x \sin y$ in powers of $x$ and $y$ as far as terms of third degree. (Predicted)

Q.4) Verify Mean Value Theorem for $f(x) = \log x$ in $[1, e]$. (Predicted)

Q.5) If $u = f(r)$ where $r^2 = x^2 + y^2$, prove that $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = f''(r) + \frac{1}{r}f'(r)$. (Predicted)

Q.1) Prove that $\Gamma(n)\Gamma(1-n)=\frac{\pi}{\sin n\pi}$. (Nov-2022)

Q.2) By Changing the order of integration, evaluate $\int_{0}^{1}\int_{0}^{\sqrt{1-x^2}}y^2dydx$. (Nov-2022)

Q.3) Find the area of a plane in the form of a quadrant of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. (Nov-2022)

Q.4) Change the order of integration in $\int_{0}^{1}\int_{x^2}^{2-x}xy\ dy\ dx$ and hence evaluate. (June-2023, Dec-2023)

Q.5) Evaluate $\iint e^{2x+3y}dxdy$ over the triangle bounded by $x=0, y=0$ and $x+y=1$. (June-2023)

Q.6) Show that $\beta(l,m)=\frac{\Gamma(l)\Gamma(m)}{\Gamma(l+m)}$. (Dec-2023, Dec-2024)

Q.7) i) Find the value of $\Gamma(\frac{3}{2})$.
ii) Evaluate $\int_{0}^{1}x^3(1-\sqrt{x})^2dx$. (Dec-2023)

Q.8) Find the volume of the solid generated by the revolution of the Cardioid $r=a(1+\cos\theta)$ about the initial line. (June-2024)

Q.9) Prove that $\int_{0}^{\infty}\cos(x^2)dx=\frac{1}{2}\sqrt{\frac{\pi}{2}}$. (June-2024)

Q.10) Show that the surfaces area of the solid generated by revolution of the loop of the curve $x=t^2, y=t-\frac{1}{3}t^3$ about the x axis is $3\pi$. (June-2024, June-2025)

Q.11) Evaluate $\iint xy\ dx\ dy$ where the region of integration is $x+y<1$ in the positive quadrant. (June-2024, June-2025)

Q.12) The part of the parabola $y^2=4ax$ cut off by the latus rectum revolves about the tangent at the vertex. Find the volume of the reel thus generated. (Dec-2024)

Q.13) Prove that: $\int_{0}^{1}\frac{dx}{\sqrt{1-x^4}} = \frac{(\Gamma(1/4))^2}{6\sqrt{2\pi}}$. (Dec-2024)

Q.14) Evaluate: i) $\int_{0}^{2}x^2(1-x)^3dx$
ii) $\int_{0}^{1}\sqrt{x(1-x)}dx$. (Dec-2024)

Q.15) Prove that the surface area of the solid generated by the revolution of the ellipse $x^2/a^2+y^2/b^2=1$ about the major axis is : $2(\pi ab) \cdot \{\sqrt{1-e^2} + \frac{\sin^{-1}e}{e}\}$. (Dec-2024)

Q.16) Change the order of integration in the following integral and then evaluate $\int_{0}^{\infty}\int_{x}^{\infty}\frac{e^{-y}}{y}dydx$. (June-2025)

Q.17) Evaluate $\int_{0}^{\infty}\int_{x}^{\infty}\frac{e^{-y}}{y}dydx$. (June-2025)

Q.1) Evaluate $\int_{0}^{\infty} e^{-x^2} dx$ using Gamma function. (Predicted)

Q.2) Find the volume of the solid generated by revolving the area bounded by the curve $y^2 = x^3$ and the line $x = 4$ about the x-axis. (Predicted)

Q.3) Change the order of integration and evaluate $\int_{0}^{\infty}\int_{0}^{x} x e^{-x^2/y} dy dx$. (Predicted)

Q.4) Express $\int_{0}^{1} x^m (1-x^n)^p dx$ in terms of Beta function. (Predicted)

Q.5) Evaluate $\iiint xyz \, dx \, dy \, dz$ over the region bounded by $x=0, y=0, z=0$ and $x+y+z=1$. (Predicted)

Q.1) Obtain the Fourier series to represent $f(x)=x\sin x, 0(Nov-2022)

Q.2) Test the convergence of the series $\sum_{n=1}^{\infty}(\sqrt{n+1}-\sqrt{n-1})$. (Nov-2022)

Q.3) Test the convergence of the series $1+\frac{x}{2}+\frac{x^2}{5}+\frac{x^3}{10}+...+\frac{x^n}{n^2+1}+...$ (June-2023, Dec-2023)

Q.4) Expand as a half range $f(x)=x\sin x$ series and cosine series for the interval $0(June-2023, Dec-2023)

Q.5) Expand $f(x)=x\sin x, 0(June-2023)

Q.6) Find the $a_0$ and $a_n$ if the function $f(x)=x+x^2$ is expanded in Fourier series defined in (-1, 1). (June-2023)

Q.7) Show that Sequence ($x_n$) where $|x|<1$ converge to 0. (June-2024)

Q.8) Find the Fourier Series for the function $f(x)=x\sin x, (-\pi(June-2024)

Q.9) Find the Fourier Series for the function $f(x)=x+x^2, (-\pi(June-2024, June-2025)

Q.10) Show that the following series is Convergent. $\frac{1}{4} - \frac{1}{4^2} + \frac{1}{4^3} - \frac{1}{4^4} + \dots$ (Dec-2024)

Q.11) Obtain the Half-Range Sine Series for \( f(x) = e^x \) in \( 0 < x < 1 \). (Dec-2024)

Q.12) Show that the sequence $(n^{1/n})$ converge to 1. (Dec-2024, Dec-2024)

Q.13) Find the half range sine series for $f(x)=x(\pi-x)$ in $(0,\pi)$. Hence Deduce that $1-\frac{1}{3^3}+\frac{1}{5^3}-\cdots=\frac{\pi^3}{32}$. (June-2025)

Q.1) Test the convergence of the series $\sum_{n=1}^{\infty} \frac{n!}{n^n}$. (Predicted)

Q.2) Find the Fourier series expansion for $f(x) = |\sin x|$ in $(-\pi, \pi)$. (Predicted)

Q.3) Discuss the convergence of the geometric series $\sum_{n=0}^{\infty} ar^n$. (Predicted)

Q.4) Obtain the half-range cosine series for $f(x) = (x-1)^2$ in the interval $0 < x < 1$. (Predicted)

Q.5) Use Parseval’s identity to evaluate $\sum_{n=1}^{\infty} \frac{1}{n^4}$. (Predicted)

Q.1) Let W be a subspace of a finite dimensional vector space V(F). Then $\dim(V/W) = \dim V - \dim W$. (Nov-2022)

Q.2) Show that $T: V_2(R) \to V_3(R)$ is defined as $T(a, b) = (a-b, b-a, -a)$ is linear transformation. (Nov-2022)

Q.3) If $w_1$ and $w_2$ be two subspace of V(F) then Show that $w_1 \cap w_2$ also subspace of V(F). (June-2024)

Q.4) Show that the set $w=\{(a,b,0):a,b \in R\}$ is subspace of $R^3$. (Dec-2024)

Q.5) Are the following vectors LD? If so express one of these as a LC of other two.
$X_1=(1,3,4,2)$, $X_2=(3,-5,2,2)$, $X_3=(-2,1,-3,2)$. (Dec-2024)

Q.6) Let $W_1$ and $W_2$ be subspaces of a vector space V and assume that $W_1 \cap W_2 = \{0\}$. Let $w_1 \in W_1$ and $w_2 \in W_2$ be such that $w_1 \neq 0$ and $w_2 \neq 0$. Prove that $\{w_1, w_2\}$ is linearly independent. (June-2025)

Q.7) i) Prove that $W_1 \cap W_2$ is a subspace of V.
ii) Give an example to show that $W_1 \cup W_2$ need not be a subspace of V.
iii) Is $W_1 \cup W_2$ a subspace of V? (June-2025)

Q.1) Define Linear Dependence and Independence. Check if vectors (1, 2, 1), (2, 1, 4), (4, 5, 6) are linearly dependent. (Predicted)

Q.2) Define Basis and Dimension of a Vector Space. Show that the set $S = \{(1, 2, 1), (2, 1, 0), (1, -1, 2)\}$ forms a basis for $R^3$. (Predicted)

Q.3) State Rank-Nullity Theorem. Find the rank and nullity of the linear transformation $T: R^3 \to R^2$ defined by $T(x,y,z) = (x+y, y+z)$. (Predicted)

Q.4) Let $T: R^3 \to R^3$ be a linear transformation defined by $T(x,y,z) = (x+2y-z, y+z, x+y-2z)$. Find a basis for the range of T. (Predicted)

Q.5) Prove that the intersection of any collection of subspaces of a vector space V is a subspace of V. (Predicted)

Q.1) Verify Cayley-Hamilton theorem for the matrix $A = \begin{pmatrix} 1 & -2 & 2 \\ 1 & -2 & 3 \\ 0 & -1 & 2 \end{pmatrix}$. Hence find $A^{-1}$. (Nov-2022)

Q.2) Examine the consistency of the system of the following equations. If consistent, solve the equations. $x+y+z=3$, $x+2y+3z=4$, $x+4y+9z=6$. (Nov-2022)

Q.3) Diagonalize the matrix $A = \begin{pmatrix} 1 & 1 & 0 \\ 6 & 2 & 0 \\ 1 & 0 & 3 \end{pmatrix}$. (Nov-2022)

Q.4) i) If A is a skew symmetric matrix then show that $A^2$ is a symmetric matrix.
ii) Find eigen values of the matrix $\begin{pmatrix} 5 & 4 \\ 1 & 2 \end{pmatrix}$. (June-2023)

Q.5) Find the inverse of $\begin{pmatrix} 0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1 \end{pmatrix}$ by using elementary row transformations. (June-2023, Dec-2023)

Q.6) Verify Cayley Hamilton theorem for the matrix A and hence find $A^{-1}$ for $\begin{pmatrix} 1 & 2 & -2 \\ -1 & 3 & 0 \\ 0 & -2 & 1 \end{pmatrix}$. (June-2023)

Q.7) Test the consistency and hence, solve the following set of equations. $x+2y-z=3$, $3x-y+2z=1$, $2x-2y+3z=2$, $x-y+z=-1$. (June-2023, Dec-2023)

Q.8) Transform the following matrix into normal form and hence find its rank $\begin{pmatrix} 5 & 3 & 14 & 4 \\ 0 & 1 & 2 & 1 \\ 1 & -1 & 2 & 0 \end{pmatrix}$. (Dec-2023)

Q.9) Find the eigen values and eigen vectors of matrix $\begin{pmatrix} 2 & -2 & 2 \\ 1 & 1 & 1 \\ 1 & 3 & -1 \end{pmatrix}$. (Dec-2023, June-2025)

Q.10) Show that the following equations are consistent and solve them.
$x-y+2z=4$, $3x+y+4z=6$, $x+y+z=1$. (June-2024)

Q.11) Find the Characteristic equation of the matrix $\begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & -4 \\ 1 & 0 & -1 \end{pmatrix}$ and hence find the Eigen values and Eigen vectors. (June-2024)

Q.12) Show that the following matrix A is Diagonalizable. $A = \begin{pmatrix} 1 & 0 & -1 \\ 1 & 2 & 1 \\ 2 & 2 & 3 \end{pmatrix}$. (June-2024)

Q.13) Investigate for what values of $\lambda$ and $\mu$ the simultaneous equations. $X+Y+Z=6$, $X+2Y+3Z=10$, $X+2Y+\lambda Z=\mu$. (June-2024, June-2025)

Q.14) Find a similarity transformation that diagonalise the matrix. $A = \begin{pmatrix} -2 & 2 & -3 \\ 2 & 1 & -6 \\ -1 & -2 & 0 \end{pmatrix}$. (Dec-2024)

Q.15) Find the Eigen value and Corresponding Eigen Vectors of the following Matrix. $\begin{pmatrix} 8 & -6 & 2 \\ -6 & 7 & -4 \\ 2 & -4 & 3 \end{pmatrix}$. (Dec-2024)

Q.16) Diagonalizable the matrix. $\begin{pmatrix} 3 & -1 & 1 \\ -1 & 5 & -1 \\ 1 & -1 & 3 \end{pmatrix}$. (June-2025)

Q.17) Reduce the matrix $\begin{pmatrix} 3 & 2 & -1 \\ 4 & 2 & 6 \\ 7 & 4 & 5 \end{pmatrix}$ to the normal form, hence find its rank. (June-2025)

Q.1) State and prove Cayley-Hamilton Theorem. Verify it for the matrix $A = \begin{pmatrix} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{pmatrix}$ and find $A^{-1}$. (Predicted)

Q.2) Find the rank of the matrix $A = \begin{pmatrix} 1 & 2 & 3 & 4 \\ 2 & 1 & 4 & 3 \\ 3 & 0 & 5 & -10 \end{pmatrix}$ by reducing it to normal form. (Predicted)

Q.3) Investigate the values of $\lambda$ and $\mu$ so that the equations $2x+3y+5z=9$, $7x+3y-2z=8$, $2x+3y+\lambda z=\mu$ have (i) no solution, (ii) a unique solution, (iii) an infinite number of solutions. (Predicted)

Q.4) Determine the eigenvalues and eigenvectors of the matrix $A = \begin{pmatrix} 6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3 \end{pmatrix}$. (Predicted)

Q.5) Diagonalize the matrix $A = \begin{pmatrix} 8 & -6 & 2 \\ -6 & 7 & -4 \\ 2 & -4 & 3 \end{pmatrix}$ and hence find $A^4$. (Predicted)