Maths CAE 1&2 PDF 2024-2025 Galgotias

## Galgotias College of Engineering and Technology, Greater Noida Continuous Assessment Exam (CAE) -1 : Odd Semester 2024 - 2025 | Course/Branch | B. Tech (ALL) | Semester | 1 | |---|---|---|---| | Subject Name | Engineering Mathematics-I | Max. Marks | 40 | | Subject Code | BAS103 | Time | 90 min...

## Galgotias College of Engineering and Technology, Greater Noida Continuous Assessment Exam (CAE) -1 : Odd Semester 2024 - 2025 | Course/Branch | B. Tech (ALL) | Semester | 1 | |---|---|---|---| | Subject Name | Engineering Mathematics-I | Max. Marks | 40 | | Subject Code | BAS103 | Time | 90 min | **CO-1**: Remember the basics of matrices and apply the concept of rank for solving linear simultaneous equations. **CO-2**: Apply the concept of differentiation to find nth derivative, Leibnitz theorem, partial derivatives, and total derivatives. ### Section - A (CO - 1) # Attempt both the questions # 20 Marks **Q.1**: Attempt any FIVE questions. Each question is of two marks. (5 x 2 = 10 Marks) a) Reduce the matrix $$ \begin{bmatrix} 3 & 1 \\ 11 & 1 \\ 13 & 1 \\ \end{bmatrix} $$ into normal form and find its rank. (K1) b) For the given matrix $$ A = \begin{bmatrix} -2 & -5 \\ -3 & 1 \\ \end{bmatrix} $$ and $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix}$ , Prove that $A^3 = 19A + 30I$. (Κ2) c) If the Eigen values of matrix $A$ are 1, 1, then find the Eigen values of $A^2 + 2A + 3I$. (K1) d) Verify whether the matrix $$ A = \begin{bmatrix} -7 & 0 & 3i \\ 0 & 5i & 1 \\ 3i & 1 & 0 \\ \end{bmatrix} $$ is a Hermitian matrix? (K1) e) Find the value of 'b' for which the rank of the matrix $$ A = \begin{bmatrix} 1 & 5 \\ 0 & 3 \\ 4 & 2 \\ b & 13 \\ 1 & 10 \\ \end{bmatrix} $$ is 2.(K1) f) Find the sum and product of the eigen values of the matrix $$ A = \begin{bmatrix} 1 & 6 & 1 \\ 2 & 0 & 0 \\ 0 & 0 & 3 \\ \end{bmatrix} $$ .(K1) **Q.2**: Attempt any TWO questions. Each question is of 5 marks. (2 x 5 = 10 Marks) a) Find nonsingular matrices $P$ and $Q$ such that $PAQ$ is normal form for the matrix $$ A = \begin{bmatrix} 1 & 1 & 2 \\ 1 & 2 & 3 \\ 1 & 0 & -1 \\ \end{bmatrix} $$ .(K1) b) For what values of $\lambda$ and $\mu$, the system of linear equations: $x + y + z = 6, x + 2y + 5z = 10, 2x + 3y + z = \lambda$ has (i) a unique solution (ii) no solution (iii) infinite solutions. Also find the solution for $\lambda = 2$ and $\mu = 8$.(K1) c) Find the eigen values and corresponding eigen vectors of the matrix $$ A = \begin{bmatrix} 1 & 1 & 1 \\ 2 & 1 & 1 \\ 3 & 3 & 2 \\ \end{bmatrix} $$ .(Κ1) d) Find the Characteristic equation of the matrix $$ A = \begin{bmatrix} 2 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \\ \end{bmatrix} $$ and hence, compute $A^{-1}$. Also find the matrix represented by $A^8 - 5A^7 + 7A^6 - 3A^5 + A^4 - 5A^3 + 8A^2 - 2A + 1$.(K1) ### Section – B (CO - 2) # Attempt both the questions # 20 Marks **Q.3**: Attempt any FIVE questions. Each question is of two marks. (5 x 2 = 10 Marks) a) Find the nth derivative of $log(ax + x^2)$. (K1) b) If $y = sin(asin^{-1}x)$, then show that $(1-x^2)y''-xy'+a^2y = 0$. (K1) c) Find nth derivative of $x^3 e^{3x}$. (Κ1) d) Find nth derivative of $\frac{1}{(x^2 - 4x + 3)}$. (Κ1) e) If $y = sin2x sin3x$, then find $y_n$. (K1) f) If $y = tan^{-1}(\frac{2x}{1-x^2})$, then find $y_n$ (K1) **Q.4**: Attempt any TWO questions. Each question is of 5 marks. (2 x 5 = 10 Marks) a) If $y = cos(msin^{-1} x)$, prove that $(1 − x^2)y_{n+2} − (2n + 1)xy_{n+1}+(m^2 − n^2)y_n=0$.Also find $(y_n)_o$. (K1) b) If $y^m + y^{-m}+\frac{1}{y^m + y^{-m}} = 2x$, then prove that $(x^2 - 1)y_{n+2}+(2n + 1)xy_{n+1} + (n^2 – m^2)y_n =0$. (K2) c) If $y = (sin^{-1}x)^2$, then show that $(1 - x^2)y_{n+2}- (2n+1)x y_{n+1} - n(n - 1)y_n = 0$ and calculate $y_n(0)$. (K2) d) fy = $e^{tan^{-1}x}$, then prove that $(1 + x^2)y_{n+2} + [(2n + 2)x - 1]y_{n+1} + n(n + 1)y_n = 0$. (K2) ## Galgotias College of Engineering and Technology, Greater Noida Continuous Assessment Exam (CAF) II: Odd Semester 2024-2025 | Course/Branch | B.Tech (ALL) | Semester | 1 | |---|---|---|---| | Subject Name | Engineering Mathematics-I | Max. Marks | 40 | | Subject Code | BAS103 | Time | 90 min | **CO-3**: Apply partial differentiation for evaluating extreme values, expansion of function and Jacobian. **CO-4**: Apply the methods of multiple integral find a volume, and Dirichlet's theorem in multiple integral. ### Section - A (CO-3) # Attempt both the questions # 20 Marks **Q.1**: Attempt any FIVE questions. Each question is of two marks. (5 x 2 = 10 Marks) a) Find the critical/stationary points of the function $f(x,y) = x^2 + y^2 + 6x + 12$. (K1) b) If $u = x^2 - y^2$ and $v = 2xy$, find $\frac{d(u,v)}{d(x,y)}$ . (K1) c) Show that $u = x^2 + y^2$ and $v = y^2 - x^2$ are functionally dependent.(K2) d) If $f(x,y)= tan(\frac{x}{y})$, find $\frac{d(u,v)}{d(x,y)}$ and $\frac{d(u,v)}{d(x,y)}$ . (K1) e) For given function $f(x,y)= x^2 + y^3-3axy$, find $r$, $s$ and $t$. (K1) f) If $F_1 (u, v, x, y)=0$ and $F_2(u, v, x, y)=0$ are two implicit functions, then the value of $\frac{d(x,y)}{d(u,v)}$? (K1) **Q.2**: Attempt any TWO questions. Each question is of 5 marks. (2 10 Marks) a) Find Taylor's series expansion for the function $x^3 + y^3 -2x + 2y$ about (1,1) upto second degree terms. (K1) b) If $u$, $v$, $w$ are the roots of the equation $(x-x)^3 + (x-y)^3 + (x-z)^3 = 0$, in A, then find $\frac{d(u,v,w)}{d(x,y,z)}$? (K2) c) A rectangular box, which is open at the top has a capacity of 256 cubic feet. Determine the dimensions of the box such that the least material is required for the construction of the box. (K5) d) Use Lagrange's method of multipliers to find the shortest distance from the point (1,2,-1) to the sphere $x^2 + y^2 + z^2 = 24$.(Κ1) ### Section – B (CO - 4 ) # Attempt both the questions # 20 Marks **Q.3**: Attempt any FIVE questions. Each question is of two marks. (5 x 2 = 10 Marks) a) Find the value of the integral $\int_{0}^{1} \frac{x}{x^2+4}dx$ . (K1) b) Find the value of the integral $\int_{0}^{8} \int_{0}^{x} x dxdy$. (K1) c) Change the order of integration $\int_{0}^{1} \int_{x^2}^{x} f(x, y) dy dx..$ (K1) d) Find the value of the integral $\int_{0}^{1} \int_{0}^{y} xy dydx$. (K1) e) Find the value of the integral $\int_{0}^{1} \frac{1}{\sqrt{1-x^2}} dx$. (K1) f) Scratch the region for $\int_{0}^{1} \int_{x}^{1} xe^{y} dx dy$ and find the value of the integral. (K1) **Q.4**: Attempt any TWO questions. Each question is of 5 marks. (2 x 5 = 10 Marks) a) Change the order of integration $\int_{0}^{2} \int_{x/2}^{x} xy dydx$ and evaluate the same. (K5) b) Evaluate $\int\int y dxdy$ over the region enclosed by the parabola $x^2 = y$ and the line $y = x+2$. (K5) c) Find the value of the integral $\int_{0}^{1} \int_{0}^{1} \frac{\sqrt{(1-x)(1-y)}}{4} dxdy$. (K1) d) Prove that $\int_{0}^{1} \int_{x}^{1} (x^2 + 3y^2) dydx = \int_{0}^{1} \int_{0}^{y} (x^2 + 3y^2)dxdy$. (K2)