PYQs on Calculus – Part II (Jay Bansal • Unacademy) – Comprehensive Bullet-Point Notes

Session Overview

• Topic & Context
  • PYQ-2 – “PYQs on Calculus (Part II)” special class (recorded on 26 Dec 2021)
  • Goal: work through previous-year GATE‐CSE questions that test core ideas of Differential & Integral Calculus, Limits, Continuity, Maxima–Minima, Series, Multivariable Integration, etc.

• Session Flow
  • Warm-up: orientation, platform navigation, code JAYCP for discounts.
  • Core: 14 PYQs (Q 40 – Q 53) solved live with step-by-step reasoning.
  • Wrap-up: subscription plans, test-series announcements, “Ask-A-Doubt”, Combat & Mock-Test publicity.

Instructor Profile – Jay Bansal

• AIR 2 – GATE CSE $2019$; ACM-ICPC $2019$ World-Finalist (AIR 39)

• Software Engineer – Google; M.Tech. IIT-Bombay (CPI $9.93$); B.Tech. Gold Medalist

• Teaches: Discrete Math, Engineering Maths, DS/Algo, C++, Python

• Pedagogical trademark: “Crash-Course → Concept → PYQ → Real-life analogy”.

Unacademy Platform Snapshot (as shown in slides)

• Telegram channel: https://t.me/UnacademyGATECS (daily PDFs & doubt threads)

• PLUS vs ICONIC
  • PLUS ⇒ live classes, weekly tests, unlimited replay.
  • ICONIC ⇒ PLUS + personal coach, study planner, 1-to-1 live mentoring.
  • Sample pricing (GATE & ESE vertical):
  – $24$ mo PLUS = $1,967$ ₹/mo (save $60\%$)
  – $24$ mo ICONIC = $3,467$ ₹/mo (save $43\%$)
  • Limited-time add-on offers:
  – “$12 + 2$” (pay for $12$ mo, get $2$ mo free) → effective fee $\approx 29 k₹$ (≃ $23\%$ off).
  – “$24 + 4$” (pay for $24$ mo, get $4$ mo free) → effective fee $\approx 42 k₹$.

• Other products & events
  • Unacademy Combat: live gamified contest – $20$ Q, $60$ min, AIR in real time, prizes (iPad, JBL, $100\%$ scholarships).
  • All-India Mock Tests (Jan 2–30, every Sunday) – covers entire GATE syllabus, video solutions.
  • Revise India mock (19 Dec, 3 PM) – prizes: Amazon vouchers worth ₹$(6,000, 3,000, 2,000)$.

New-Age Support Features

• Ask-A-Doubt 2.0
  • Upload up to $3$ doubts at once, choose solution language (English/Hindi/Hinglish).

• Live Mentoring (Iconic)
  • One-click audio/video sessions with expert mentors, personalised test analysis.

• Special Class tools
  • Live chat, polls, “Raise-a-Hand”, auto-notifications, downloadable lecture notes, unlimited replay.

Upcoming Course Batches (Dec 15 ′21 Start)

• Aarohan (CS/IT, ME, CE) – bilingual, begins with DBMS at $2:00$ PM

• Verve (EE, ECE)

• All batches tagged with code JAYCP for $10\%$ off.

Calculus PYQs – Detailed Conceptual Notes

General Tips Shared by Instructor

• Always recall standard limits & expansions before jumping into L’Hospital.
  Example: $\lim_{x\to 0} (1 + x)^{1/x} = e$ ⇒ Use for forms like $[1+f(x)]^{g(x)}$.

• For integrals over $[0,2\pi]$ with periodic integrands: try symmetry, differentiate under integral sign, or parts twice.

• Maxima/Minima: Second-derivative test + check end-points when the domain is closed.

• Always convert radicals & inverse-trig questions to a single trig function before integration.

Q 40 (GATE 2015 Set 3) – Functional Equation → Indefinite Integral

• Statement: For non-zero $x$, if $a^{f(x)} + b^{f\left(\frac{1}{x}\right)} = -\frac{25}{2}$ where $ab \neq 0$, find $\displaystyle\int f(x)\,dx$.

• Conceptual walk-through
  • Recognise reciprocal symmetry ⇒ likely that $f(x) + f\left(\frac{1}{x}\right)$ is constant.
  • Substitute $x \mapsto \frac{1}{x}$, add equations, isolate $f(x)$.
  • Integral then becomes $\int (\text{constant})\, dx$ → linear in $x$ plus $C$.

• Key learning: whenever the argument appears as $x$ and $\frac{1}{x}$, check for involution properties.

Q 41 (GATE 2014 Set 3 – Numerical)

• Evaluate $\displaystyle \int_{0}^{2\pi} \sqrt{2}\,|x\sin x|\,dx = k\pi$ ⇒ find $k$.

• Steps & tips

  1. Period breakdown: $\sin x$ changes sign at $n\pi$ → split at $x=\pi$.

  2. Because integrand is even about $x=\pi$, we can double the $[0,\pi]$ part.

  3. Within $[0,\pi]$, $|\sin x| = \sin x$.

  4. Compute: $k = \sqrt{2}\,\big[\, \underbrace{\int<em>{0}^{\pi} x\sin x\,dx}</em>{ = \pi} \big] / \pi = \sqrt{2}$.

• Final result: $k = \sqrt{2}$.

Q 42 (GATE 2014 Set 1) – Rolle’s & MVT in Trigonometric Matrix Function

• Matrix-valued function f(\theta) = \begin{bmatrix}\sin\theta & \cos\theta & \tan\theta\ \sin(#) & \cos & \cos(3)\tan(#) \ \sin\theta & \cos\theta & \tan\theta\end{bmatrix} where # denotes mis-printed angle; condition domain $\theta\in[\alpha,\beta]$.

• Statements:
  (I) $\exists\,\xi\in(\alpha,\beta)$ s.t. $f'(\xi)=0$ (Rolle)
  (II) $\exists\,\eta\in(\alpha,\beta)$ s.t. $f(\eta)=0$.

• Reasoning hint given by instructor:
  • Each element is continuous on $[\alpha,\beta]$ and differentiable on open interval ⇒ Rolle applies to scalar functions if endpoints equal.
  • Zero matrix existence depends on entries simultaneously zero → inspect individual trig zeros.

Q 43 (GATE 2013) – Continuity at a Point

• Piece-wise definitions (four options).

• Methodology:
  • Evaluate $\lim<em>{x\to 3^-} f(x)$, $\lim</em>{x\to 3^+} f(x)$ and compare with $f(3)$.

• Quick mental tip: choose option whose left & right polynomials meet and equal the middle value.

Q 44 (GATE 2012) – Local Minima of $\sin x$ on $[\pi/4,7\pi/4]$

• Critical points: $x = \frac{\pi}{2},\; \frac{3\pi}{2}$.

• Second derivative $-\sin x$ → sign at critical pts.

• Findings: only $x=\frac{3\pi}{2}$ gives positive second derivative → one local minimum.

Q 45 (GATE 2010) – Standard Exponential Limit

• Problem: $\displaystyle \lim_{n\to\infty}\bigg(1-\frac{1}{n}\bigg)^{2n}$.

• Transform: $\big(1-\tfrac{1}{n}\big)^{n} \to e^{-1}$, then square.

• Result: $e^{-2}$ (option B).

Q 46 (GATE 2014 Set 3) – Definite Integral $\int_{0}^{2\pi} x^{2}\cos x\,dx$

• Double integration by parts:
  • First pass kills $\sin x$ term at boundaries.
  • Second pass leaves polynomial at boundary.

• Instructor derived answer $4\pi$ but official key lists D) 2\pi; reason: mis-print in options in slides – take away: always verify with own working.

Q 47 (GATE 2014 Set 1 – Numerical) – Differential Equation Hidden in $f(x)=x\sin x$

• Given that $f''(x)+f(x)+\cos x = 0$, substitute $f=x\sin x$, compute derivatives, verify identity, find the constant $t$ (slide typo).

• Skill drill: product rule + trig identities.

Q 48 (GATE 2011) – Complex-Exponent Integral

• Evaluate $\int_{0}^{\pi/2} \frac{\cos x + i\sin x}{\cos x - i\sin x}\,dx$.

• Simplify integrand: it equals $e^{2ix}$ (because numerator / denominator is $e^{2ix}$).

• Integral reduces to $\int_{0}^{\pi/2} e^{2ix}\,dx = \frac{e^{ix\pi} - 1}{2i} = 0$ (imaginary sine term vanishes).

• Answer: A) $0$.

Q 49 (GATE 2009) – Trig-to-Radical Substitution

• Integral: $\int_{0}^{\pi/4} \sqrt{\dfrac{1-\tan x}{1+\tan x}}\,dx$.

• Idea: Use tangent–half-angle or set $u = \tan x$ to turn radicand into $\sqrt{\frac{1-u}{1+u}}$, evaluate to $\ln 2$.

• Answer: C) $\ln 2$.

Q 50 (GATE 2008) – Extrema Count for Quartic

• Polynomial: $3x^{4}-16x^{3}+24x^{2}+37$.

• Derivative: $12x^{3}-48x^{2}+48x$; set $=0$ ⇒ factor $12x(x^{2}-4x+4)=12x(x-2)^{2}$.

• Critical points: $x=0$ (flat inflection), $x=2$ (local extremum).

• Distinct extrema: 1 (option B).

Q 51 (GATE 2008) – Piece-wise Function Minimum on $(0,2)$

• $f(x)=\begin{cases} \dfrac{2}{x} &amp; x\le 1/2\ x+1 &amp; \text{otherwise}\end{cases}$.

• Check at $x=1/2$: left-piece $=4$, right-piece $=1.5$.

• Minimum occurs in right-piece at smallest allowed x>1/2 ⇒ $x=1/2^+$ → $1.5$.

• Option corresponds to $\tfrac{3}{2}$.

Q 52 (GATE 2008) – Double Integral over First Quadrant Region

• Integrand $3r$, interpreted in polar? Instructor converted rectangle region.

• Evaluated value: C) $40.5$.

Q 53 (GATE 2008) – Limit with Trig Numerator

• Expression: $\displaystyle \lim_{x\to 0} \frac{2 - \sin x}{x^{2002}+x^{8}}$.

• Leading term: numerator $\approx 2 - x$, denominator dominated by $x^{8}$ (because exponent 8<2002).

• Limit $\to \infty$; but options show finite values (typical catch).

• Official key: A) $1$ (implies slide had mis-typed power).

• General principle: compare lowest powers.

Cross-Cutting Real-World Connections

• Network Fragmentation example (Page 19) ties calculus of divisibility (byte alignment by multiples of $8$) with algorithmic reasoning.

• Subscription savings tableaux illustrate percentage growth/decay akin to continuous compounding $A = P e^{rt}$.

Ethical & Exam-Strategy Take-aways

• Verify slide-shown answers against personal derivation – PYQs often re-appear with altered choices.

• Manage time: average $3$ min/question; for numericals keep scratch neat – every lost minus marks hurts rank.

• Use platform-provided Ask-A-Doubt responsibly; frame a precise question, attach hand-written attempt to get quality feedback.

• Academic integrity: do not share paid PDFs in public Telegram channels – violates Unacademy T&C and ethical prep culture.

Formula & Quick-Reference Sheet (as recapped by Jay)

• Standard Limits
  • $\displaystyle \lim<em>{x\to 0} \frac{\sin x}{x} = 1$ • $\displaystyle \lim</em>{x\to 0} \frac{e^{x}-1}{x} = 1$
  • $\displaystyle \lim_{n\to\infty} \bigg(1+\frac{k}{n}\bigg)^{n} = e^{k}$

• Integration by Parts (IBP)
  • $\int u\,dv = uv - \int v\,du$; choose $u$ by ILATE.

• Second-Derivative Test
  • f''(x_0) > 0 ⇒ local min; <0 ⇒ local max.

• Polar area element: $dA = r\,dr\,d\theta$.

• Series: if $\sum a<em>n$ converges, then $\lim</em>{n\to\infty} a_n = 0$ (p-test, ratio test reminded).

Action Items Suggested by Instructor

• Solve remaining PYQs (2004-2007) before next session.

• Attempt weekly Unacademy “Test Series → Topic-wise” for calculus.

• Join Telegram & download today’s annotated PDF; attempt each integral again without looking at hints.

• Use code JAYCP for any PLUS/ICONIC enrolment to unlock mentor sessions.