📌 How to use: Click 📘 Course to open the full textbook at the exact concept page. Click 📝 Exam to jump directly to the question in the question bank PDF. 🎯 Year pills: Dec 25 = most recent exams (bold & blue). 📊 Badges indicate question style & frequency.
| 📖 Question (with marks) | 📅 Exam References | 🔗 PDF Links |
|---|---|---|
| ⚡ TOPIC: Tests for Convergence (Comparison, Ratio, Root, Integral) | ||
| Check convergence: \(\sum_{n=1}^{\infty} \frac{n}{(n+1)(n+2)(n+3)}\) [06] 🔥 High Yield🧮 Numerical | Dec 25 Apr 24 Feb 24 | 📘 Course Pg 19 📝 Exam Pg 13 |
| Examine convergence: \(\sum_{n=1}^{\infty} \frac{n!}{(2n+1)!}\) [06] 🔥 High Yield📖 Theory | Apr 25 Dec 23 Jun 24 | 📘 Course Pg 27 📝 Exam Pg 5 |
| Test convergence of series: \(\sum_{n=1}^{\infty} \frac{2n^2+3n}{5+n^5}\) [06] 🧮 Numerical | Feb 24 Dec 25 | 📘 Course Pg 21 📝 Exam Pg 5 |
| 🎯 TOPIC: Maclaurin & Taylor Series Expansions | ||
| Find Maclaurin’s expansion of \(f(x) = \ln(1+4x)\) upto \(x^4\) terms. [06] 🔥 High Yield📖 Theory | Apr 25 Dec 24 Jun 23 | 📘 Course Pg 58 📝 Exam Pg 13 |
| Maclaurin’s series for \(e^{x}\sin x\) up to \(x^3\) term. [07] 📋 Detailed | Dec 25 Feb 23 | 📘 Course Pg 61 📝 Exam Pg 7 |
| Expand \(\tan^{-1}(y/x)\) in powers of \((x-1)\) and \((y-1)\) using Taylor’s theorem. [07] 📊 Algorithm | Apr 24 Dec 25 | 📘 Course Pg 151 📝 Exam Pg 7 |
| 📖 Question (with marks) | 📅 Exam References | 🔗 PDF Links |
|---|---|---|
| 📌 TOPIC: First & Second Order Partial Derivatives, Euler’s Theorem | ||
| If \(u = \ln\left(\frac{x^7+y^7+z^7}{x+y+z}\right)\), find \(x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}+z\frac{\partial u}{\partial z}\). [06] 🔥 High Yield📖 Euler | Apr 25 Dec 24 Jun 24 | 📘 Course Pg 115 📝 Exam Pg 5 |
| If \(u = \sec^{-1}\left(\frac{x^3-y^3}{x+y}\right)\), show \(x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}=2\cot u\) and evaluate \(x^2 u_{xx}+2xy u_{xy}+ y^2 u_{yy}\). [06]📋 Long | Dec 25 Feb 24 | 📘 Course Pg 113 📝 Exam Pg 3 |
| 🔗 TOPIC: Chain Rule & Total Differentiation | ||
| If \(z = xy^2 + x^2y\), \(x = at^2\), \(y=2at\), find \(\frac{dz}{dt}\). [06]🧮 Numerical | Apr 25 Dec 23 | 📘 Course Pg 127 📝 Exam Pg 3 |
| If \(u = f(y-z, z-x, x-y)\), prove \(\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}+\frac{\partial u}{\partial z}=0\). [07]📖 Theory | Feb 24 Dec 25 | 📘 Course Pg 141 📝 Exam Pg 9 |
| 📖 Question (with marks) | 📅 Exam References | 🔗 PDF Links |
|---|---|---|
| ✈️ TOPIC: Tangent Plane & Normal Line | ||
| Find equations of tangent plane & normal line to surface \(2x^2 + y^2 + 2z = 3\) at point \((2,1,-3)\). [06]📊 Diagram | Apr 25 Dec 24 | 📘 Course Pg 158 📝 Exam Pg 3 |
| 📊 TOPIC: Maxima & Minima of Two Variables | ||
| Find extreme values of \(f(x,y)=x^3+3xy^2-3x^2-3y^2+7\). [07]🔥 High Yield🧮 Numerical | Dec 25 Apr 24 Jun 24 | 📘 Course Pg 167 📝 Exam Pg 7 |
| Examine maxima/minima of \(f(x,y) = x^3 + y^3 - 3axy\). [07]📋 Detailed | Feb 24 Apr 25 | 📘 Course Pg 172 📝 Exam Pg 14 |
| ⚖️ TOPIC: Lagrange’s Multiplier Method | ||
| Find maximum value of \(x^2 y^3 z^4\) subject to \(2x+3y+4z = a\) using Lagrange’s multipliers. [07]🔥 High Yield📖 Optimization | Dec 25 Jun 24 Apr 24 | 📘 Course Pg 183 📝 Exam Pg 7 |
| A rectangular solid without lid from \(12 m^2\) cardboard: maximize volume. [06]🧮 Numerical | Feb 23 Apr 25 | 📘 Course Pg 185 📝 Exam Pg 13 |
| 📖 Question (with marks) | 📅 Exam References | 🔗 PDF Links |
|---|---|---|
| ∞ TOPIC: Improper Integrals & Beta-Gamma Functions | ||
| Evaluate \(\int_0^{\infty} x^5 e^{-3x} dx\) using Gamma function. [06]🔥 High Yield🧮 Numerical | Apr 25 Dec 23 Feb 24 | 📘 Course Pg 188 📝 Exam Pg 6 |
| Show that \(\int_0^{\pi/2} \tan^p x \, dx = \frac{\pi}{2}\sec\frac{p\pi}{2}\) and state restriction on p. [06]📖 Theory📊 Derivation | Dec 24 Apr 25 | 📘 Course Pg 196 📝 Exam Pg 2 |
| Evaluate \(\int_{0}^{1} x^4 (1-\sqrt{x})^5 dx\) using Beta function. [06]📋 Long | Jun 23 Dec 25 | 📘 Course Pg 198 📝 Exam Pg 6 |
| Evaluate \(\int_{0}^{\infty} \frac{x^{8}(1-x^{6})}{(1+x)^{24}}dx\). [06]🔹 Rare | Apr 24 | 📘 Course Pg 200 📝 Exam Pg 21 |
| 📖 Question (with marks) | 📅 Exam References | 🔗 PDF Links |
|---|---|---|
| 📐 TOPIC: Double Integrals (Cartesian & Change of Order) | ||
| Change order of integration: \(\int_0^1 \int_0^x e^{y^2} dy dx\) and evaluate. [06]🔥 High Yield🧮 Numerical | Dec 25 Apr 24 Feb 24 | 📘 Course Pg 246 📝 Exam Pg 4 |
| Evaluate \(\int_0^2\int_0^{\sqrt{4-x^2}} (x^2+y^2) dy dx\) by changing to polar coordinates. [06]📊 Polar | Apr 25 Dec 23 | 📘 Course Pg 255 📝 Exam Pg 4 |
| 📦 TOPIC: Triple Integrals & Volume Applications | ||
| Find volume of solid bounded by coordinate planes & plane \(x+y+z=1\). [06]🔥 High Yield🧮 Numerical | Dec 25 Feb 24 Jun 23 | 📘 Course Pg 266 📝 Exam Pg 22 |
| Use double integration to find area between parabolas \(y^2 = 36x\) and \(x^2 = 36y\). [06]📋 Long📊 Area | Apr 24 Dec 25 | 📘 Course Pg 262 📝 Exam Pg 14 |
| Volume of solid generated by revolving \(y = 2-x^2\) about y-axis from \(x=0\) to \(x=2\). [06]📖 Revolution | Apr 25 Dec 24 | 📘 Course Pg 217 📝 Exam Pg 2 |
| Evaluate \(\iiint_R (x^2+y^2+z^2) dxdydz\) where R is region \(x=0,y=0,z=0, x+y+z=a\). [07]🔥 High Yield🧮 Triple | Dec 24 Apr 25 Feb 23 | 📘 Course Pg 275 📝 Exam Pg 12 |
| 🌀 Surface Area by Integration | ||
| Find surface area generated by revolving curve \(y = e^x\), \(0\le x\le 1\) about x-axis. [07]📊 Diagram📋 Long | Dec 25 Apr 24 | 📘 Course Pg 231 📝 Exam Pg 12 |
| Arc of \(y = \sqrt{4-x^2}\) from \(-1\) to \(1\) revolved about x-axis: find surface area. [07]🧮 Numerical | Feb 24 Apr 25 | 📘 Course Pg 229 📝 Exam Pg 22 |