RTU Kota B.Tech 1st Year Engineering Mathematics-I Question Paper 2026

Paper DetailsSubject Code: 1FY2-01Examination: B.Tech I - Sem (Main/Back/Re-back)Exam Month: January 2026Total Marks: 70Time Duration: 3 HoursRTU Engineering Mathematics-I 2026 Paper ContentPART – A (Short Answer Questions)10 Questions x 2 Marks = 20 MarksDefine the Rank of a Matrix.State Cayley-Hamilton Theorem.Evaluate $\lim_{x \to 0} \frac{\sin x - x}{x^3}$ using L'Hospital's Rule.State the condition for the convergence of a p-series.What is the Asymptote of a curve?Write the formula for the Radius of Curvature in Cartesian form.Define Beta and Gamma functions.State Euler's Theorem for homogeneous functions.Find the first-order partial derivatives of $u = x^y$.What are Taylor’s Series expansions for $\sin x$?PART – B (Analytical Questions)Attempt any five (5 x 4 Marks = 20 Marks)Find the Eigenvalues and Eigenvectors of a $3 \times 3$ matrix.Test the convergence of the series: $\sum \frac{\sqrt{n}}{n^2 + 1}$.Expand $e^x \cos y$ in powers of $x$ and $y$ up to second-degree terms using Taylor’s Theorem.If $u = f(y-z, z-x, x-y)$, prove that $\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} + \frac{\partial u}{\partial z} = 0$.Find the Maxima and Minima of the function $f(x, y) = x^3 + y^3 - 3axy$.Evaluate $\int_0^\infty \sqrt{x} e^{-x^2} dx$ using Gamma functions.Trace the curve $y^2(a-x) = x^3$ (Cissoid of Diocles).PART – C (Descriptive Questions)Attempt any three (3 x 10 Marks = 30 Marks)Matrices: Reduce the following matrix to Normal Form and find its rank. Solve the system of linear equations using Gauss-Elimination method.Differential Calculus: If $y = (x^2 - 1)^n$, prove that $(x^2 - 1)y_{n+2} + 2xy_{n+1} - n(n+1)y_n = 0$ using Leibniz’s Theorem.Partial Differentiation: Discuss the method of Lagrange’s Multipliers to find the dimensions of a rectangular box of maximum volume with a given surface area.Multiple Integrals: Change the order of integration in $\int_0^1 \int_x^{\sqrt{x}} f(x, y) dy dx$ and evaluate.Applications: Find the area between the parabolas $y^2 = 4ax$ and $x^2 = 4ay$ using double integration.Download Section[Download RTU Maths-I Jan 2026 Question Paper PDF]