Engineering Mathematics-I Unit 2 – Differential Calculus-I: important questions for AKTU
Unit 2 (Differential Calculus-I) questions that AKTU repeats most often. This unit carries about 34 marks per paper. Start with the repeated questions, then the most asked topics.
Most repeated Unit 2 questions
- If $z = x^4y^2 \sin^{-1}(\frac{x}{y}) + \log x - \log y$, show that $x \frac{\partial z}{\partial x} + y \frac{\partial z}{\partial y} = 6x^4y^2 \sin^{-1}(\frac{x}{y})$. (7 marks, 2026, Euler's Theorem for Homogeneous Functions) – also asked in 2019, 2020, 2021, 2023, 2024, 2025
- If $y = e^{\tan^{-1}x}$, prove that $(1 + x^2)y_{n+2} + [(2n+2)x - 1]y_{n+1} + n(n+1)y_n = 0$. (10 marks, 2021, Successive Differentiation (nth-Order Derivatives)) – also asked in 2019, 2020, 2022, 2023, 2024, 2025, 2026
- If $u=y^2e^{y/x}+x^2\tan^{-1}\left(\frac{x}{y}\right)$, show that (i) $x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}=2u$. (ii) $x^2\frac{\partial^2u}{\partial x^2}+2xy\frac{\partial^2u}{\partial x\partial y}+y^2\frac{\partial^2u}{\partial y^2}=2u$. (7 marks, 2024, Euler's Theorem for Homogeneous Functions) – also asked in 2019, 2020, 2021, 2023, 2025, 2026
Most important Unit 2 topic
- Successive Differentiation (nth-Order Derivatives) (Unit 2: Differential Calculus-I) – asked 12 times in 2019, 2020, 2021, 2022, 2023, 2024, 2025, 2026
Most asked Unit 2 topics
- Successive Differentiation (nth-Order Derivatives) – asked 12 times
- Euler's Theorem for Homogeneous Functions – asked 10 times
- Rolle's, Lagrange's and Cauchy's Mean Value Theorems – asked 7 times
More Unit 2 previous year questions
- If $y = x^3 \log_e x$, prove that $\frac{d^4y}{dx^4} = \frac{6}{x}$. (2 marks, 2026, Successive Differentiation (nth-Order Derivatives))
- If $u(x, y) = (\sqrt{x} + \sqrt{y})^5$, then find the degree of the function $u(x, y)$. (2 marks, 2026, Euler's Theorem for Homogeneous Functions)
- Find the $n^{th}$ derivative of $\frac{2x+1}{(2x-1)(2x+3)}$. (7 marks, 2026, Successive Differentiation (nth-Order Derivatives))
- Trace the curve: $y^2(a+x) = x^2(3a-x)$. (7 marks, 2026, Curve Tracing)
- If $u = \frac{x^2+y^2}{x+y}$, find the value of $x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y}$. (2 marks, 2025, Euler's Theorem for Homogeneous Functions)
Unit 2 syllabus topics
- Successive Differentiation (nth-Order Derivatives)
- Leibnitz Theorem
- Curve Tracing
- Partial Derivatives
- Euler's Theorem for Homogeneous Functions
- Total Derivative
- Change of Variables
- Rolle's, Lagrange's and Cauchy's Mean Value Theorems
- Envelopes of Families of Curves
- Continuity and Differentiability