Unit 3: Calculus | Class 12 Applied Maths | Boundless Maths
Unit 3 of 8 15 Marks CBSE 2026–27 Second-Highest Weightage

Calculus
Unit 3 — Free Study Resources

CBSE Class 12 Applied Mathematics · Free MCQs, Solved Examples & Case Studies

Unit 3 carries 15 marks in the CBSE board exam — the second-highest weightage unit. Everything you need to master it: 6 interactive MCQs with instant feedback, 6 short-answer and 5 long-answer solved examples, and 3 board-pattern case studies. Covering Differentiation, Applications of Derivatives, Integrals, Applications of Integrals, and Differential Equations. All content is aligned to the CBSE 2026–27 syllabus and board exam pattern. Once you've attempted the questions and self-assessed your answers, tap ✨ My Report to see your personalised performance breakdown.

0
Board Exam Marks
0
Topics Covered
0
Practice MCQs
0
Solved Examples
0
Case Studies

Class 12 Applied Maths Unit 3 — Calculus Study Material

This page covers all topics in Unit 3 of CBSE Class 12 Applied Mathematics — carrying 15 marks in the board exam, making it the second-highest weightage unit. You'll find 6 MCQs with solutions, 6 short-answer and 5 long-answer questions with complete step-by-step workings, and 3 case studies based on Manufacturing, Water Tanks, and E-commerce. Topics include Differentiation, Applications of Derivatives (maxima, minima, marginal cost), Definite and Indefinite Integrals, Applications of Integrals (consumer surplus, area), and Differential Equations. All content is aligned to the CBSE 2026–27 syllabus.

Differentiation Maxima & Minima Marginal Cost & Revenue Definite Integrals Consumer Surplus Differential Equations CBSE 2026-27
Unit 3 · 15 Marks

Topics & Key Formulas

Five topics examined across MCQs, short answers, long answers, and case studies. Memorise the formulas below — they appear in multiple question types every year.

1. Differentiation

Higher-order derivatives and their applications

Power rule: \(\dfrac{d}{dx}(x^n) = nx^{n-1}\) Chain rule: \(\dfrac{dy}{dx} = \dfrac{dy}{du}\cdot\dfrac{du}{dx}\)

2. Applications of Derivatives

Rate of change, maxima–minima, marginal cost & revenue

\(MC = \dfrac{dC}{dx},\quad MR = \dfrac{dR}{dx}\) Max/min: \(f'(x)=0\); check sign of \(f''(x)\)

3. Integrals

Indefinite, definite, substitution, partial fractions, by parts

\(\displaystyle\int x^n\,dx = \dfrac{x^{n+1}}{n+1} + C\) \(\displaystyle\int_a^b f(x)\,dx = F(b)-F(a)\)

4. Applications of Integrals

Consumer & producer surplus, area under curves

\(CS = \displaystyle\int_0^{x_0} D(x)\,dx - p_0 x_0\) \(PS = p_0 x_0 - \displaystyle\int_0^{x_0} S(x)\,dx\)

5. Differential Equations

Order, degree, formation and solutions

Order = highest derivative present Degree = power of that derivative (if polynomial)
📐
All the formulas — not just for this unit, but all 8 units — in one print-ready PDF

Organised by topic · CBSE 2026–27 · Perfect for daily revision & last-minute exam prep

Get Formula Deck — ₹199 →
3
Calculus 15 Marks · Series of 7 Videos

Integration & Its Applications — Series (7 Videos)

▶ Watch Series on YouTube →
✓ Indefinite Integrals ✓ Definite Integrals ✓ Area Under the Curve ✓ Consumer & Producer Surplus

🕒 Coming Soon

Differentiation & Applications Differential Equations

Subscribe for free video lessons on all 8 units

New videos every week. Get notified so you never miss a lesson.

  • Unit 1 — Numbers & Quantification
  • Unit 2 — Algebra (Matrices)
  • Unit 3 — Calculus (current)
  • Unit 4 — Probability Distributions
  • Units 5–8 — coming soon
Subscribe Free
Interactive Practice

Practice MCQs — Unit 3 Calculus

Select your answer, then click Show Answer to check and reveal the full explanation. All questions are based on CBSE past papers and board exam pattern.

Question 1
What is the derivative of \(x^3 + 3x^2 + 4x + 5\)?
\(3x^2 + 6x + 4\)
\(x^3 + 3x^2 + 4\)
\(3x^2 + 6x + 5\)
\(x^2 + 6x + 4\)
Using the power rule \(\dfrac{d}{dx}(x^n)=nx^{n-1}\):

\(\dfrac{d}{dx}(x^3)=3x^2\),   \(\dfrac{d}{dx}(3x^2)=6x\),   \(\dfrac{d}{dx}(4x)=4\),   \(\dfrac{d}{dx}(5)=0\)

Adding all terms: derivative \(= 3x^2+6x+4\)
Question 2
What does \(\displaystyle\int(2x+3)\,dx\) equal?
\(x^2+3x+C\)
\(2x^2+3x+C\)
\(x^2+3+C\)
\(2x+3x+C\)
\(\displaystyle\int 2x\,dx = 2\cdot\dfrac{x^2}{2} = x^2\)   and   \(\displaystyle\int 3\,dx = 3x\)

Adding constant of integration: \(x^2+3x+C\)
Question 3
What does \(\displaystyle\int_0^2 x^2\,dx\) equal?
\(\dfrac{8}{3}\)
\(\dfrac{4}{3}\)
\(8\)
\(\dfrac{2}{3}\)
Antiderivative: \(\displaystyle\int x^2\,dx = \dfrac{x^3}{3}\)

Apply limits: \(\left[\dfrac{x^3}{3}\right]_0^2 = \dfrac{8}{3} - 0 = \dfrac{8}{3}\)
Question 4
What is the derivative of \(e^{3x}\)?
\(3e^{3x}\)
\(e^{3x}\)
\(\dfrac{e^{3x}}{3}\)
\(3e^x\)
Chain rule: \(\dfrac{d}{dx}[e^{ax}] = a\cdot e^{ax}\)

Here \(a=3\), so \(\dfrac{d}{dx}[e^{3x}] = 3e^{3x}\)
Question 5
What is the product rule for differentiation of \(uv\)?
\(u\dfrac{dv}{dx} + v\dfrac{du}{dx}\)
\(u\dfrac{dv}{dx} - v\dfrac{du}{dx}\)
\(\dfrac{du}{dx}\cdot\dfrac{dv}{dx}\)
\(u + v\)
The product rule: \(\dfrac{d}{dx}(uv) = u\dfrac{dv}{dx} + v\dfrac{du}{dx}\)

Always define \(u\) and \(v\) clearly at the start of your solution — examiners award method marks for this.
Question 6
What is the second derivative of \(x^4\)?
\(12x^2\)
\(4x^3\)
\(12x\)
\(24x\)
First derivative: \(\dfrac{d}{dx}(x^4) = 4x^3\)

Second derivative: \(\dfrac{d}{dx}(4x^3) = 12x^2\)
📐 Formula Deck
₹199

All 8 units in one print-ready PDF — formulas, tips & common mistakes

📐 Get Formula Deck →
🤖 AI-Powered
🤖 AI Q-Bank

Adaptive practice, instant feedback & timed board exam simulation —

🤖 Get AI Q-Bank →
Step-by-Step Solutions

Short Answer Questions

2-mark and 3-mark questions with complete working. Click Show Solution to reveal each answer.

Question 1
Find the derivative of \(f(x) = (2x+1)(3x^2-4)\).
Product rule: \(\dfrac{d}{dx}(uv) = u\dfrac{dv}{dx} + v\dfrac{du}{dx}\). Let \(u = 2x+1,\; v = 3x^2-4\).
\(u' = 2,\quad v' = 6x\)
\(f'(x) = (2x+1)(6x) + (3x^2-4)(2) = 12x^2+6x+6x^2-8\)
\(f'(x) = 18x^2 + 6x - 8\)
Self-assess: ✓ Saved
Question 2
Evaluate \(\displaystyle\int_1^3(4x^3-2x)\,dx\).
Integrate each term: \(\displaystyle\int 4x^3\,dx = x^4\),   \(\displaystyle\int (-2x)\,dx = -x^2\)
Antiderivative: \(F(x) = x^4 - x^2\)
At \(x=3\): \(F(3) = 81 - 9 = 72\)  |  At \(x=1\): \(F(1) = 1 - 1 = 0\)
\(\displaystyle\int_1^3(4x^3-2x)\,dx = 72 - 0 = \mathbf{72}\)
Self-assess: ✓ Saved
Question 3
A company's profit function is \(P(x) = -2x^2 + 40x - 50\), where \(x\) is units sold (hundreds). Find the number of units that maximises profit.
Differentiate: \(\dfrac{dP}{dx} = -4x + 40\)
Set to zero: \(-4x + 40 = 0 \Rightarrow x = 10\)
Second derivative test: \(\dfrac{d^2P}{dx^2} = -4 < 0\) → confirmed maximum ✓
\(P(10) = -2(100) + 40(10) - 50 = -200 + 400 - 50\) = ₹150 (hundreds)
Profit is maximised at \(x = 10\) hundred = 1,000 units  ·  Maximum profit = ₹150 thousand
Self-assess: ✓ Saved
Question 4
Find the derivative of \(y = \ln(x^2+5x+1)\).
Use chain rule for logarithms: \(\dfrac{d}{dx}[\ln f(x)] = \dfrac{f'(x)}{f(x)}\)
Here \(f(x) = x^2 + 5x + 1\), so \(f'(x) = 2x + 5\)
\(\dfrac{dy}{dx} = \dfrac{2x+5}{x^2+5x+1}\)
Self-assess: ✓ Saved
Question 5
Find the area bounded by \(y = x^2\), the \(x\)-axis, and the lines \(x=1\) and \(x=3\).
Area \(= \displaystyle\int_1^3 x^2\,dx = \left[\dfrac{x^3}{3}\right]_1^3\)
\(= \dfrac{27}{3} - \dfrac{1}{3} = 9 - \dfrac{1}{3} = \dfrac{26}{3}\)
Area \(= \dfrac{26}{3} \approx 8\tfrac{2}{3}\) square units
Self-assess: ✓ Saved
Question 6
The revenue function is \(R(x) = 100x - 0.5x^2\). Find the marginal revenue at \(x = 40\) units.
Marginal Revenue: \(MR = \dfrac{dR}{dx} = 100 - x\)
At \(x=40\): \(MR = 100 - 40 = 60\)
Marginal Revenue at 40 units = ₹60 per unit
Self-assess: ✓ Saved

📐 Never Forget a Formula Again

All Unit 3 formulas — and all 8 units — in one organised, printable PDF. Chapter-wise layout, exam tips, common mistakes noted alongside every formula.

  • All 8 units in one PDF — organised topic-wise
  • Pro tips & common mistakes alongside each formula
  • Perfect for daily revision and last-minute exam prep
  • Instant download · Print & pin up
₹199
One-time · Instant PDF
📐 Order Formula Deck →

🔒 Secured by Razorpay

📐 Formula Deck
₹199

All 8 units in one print-ready PDF — formulas, tips & common mistakes

📐 Get Formula Deck →
🤖 AI-Powered
🤖 AI Q-Bank

Adaptive practice that targets weak areas, instant feedback & timed board exam simulation —

🤖 Get AI Q-Bank →
4–5 Mark Questions

Long Answer Questions

4-mark and 5-mark practice questions with complete working. Click Show Solution to reveal each answer.

Question 1
A wire of length 28 cm is cut into two pieces. One piece is bent into a square, the other into a circle. Find the length of each piece so that the total enclosed area is minimum.
Let wire for square \(= x\) cm; wire for circle \(= (28-x)\) cm.
Area of square: \(\left(\dfrac{x}{4}\right)^2 = \dfrac{x^2}{16}\). Radius of circle: \(r = \dfrac{28-x}{2\pi}\).
Total area: \(A = \dfrac{x^2}{16} + \dfrac{(28-x)^2}{4\pi}\)
Differentiate and set to zero: \(\dfrac{dA}{dx} = \dfrac{x}{8} - \dfrac{28-x}{2\pi} = 0\)
Solving: \(\pi x = 4(28-x) \Rightarrow x(\pi+4) = 112 \Rightarrow x = \dfrac{112}{\pi+4}\approx 15.68\) cm.
\(\dfrac{d^2A}{dx^2} = \dfrac{1}{8}+\dfrac{1}{2\pi} > 0\) → confirmed minimum ✓
Length for square \(= \dfrac{112}{\pi+4} \approx 15.68\) cm  |  Length for circle \(= \dfrac{28\pi}{\pi+4} \approx 12.32\) cm
Self-assess: ✓ Saved
Question 2
Evaluate: \(\displaystyle\int \dfrac{x^2+1}{x^4+x^2+1}\,dx\)
Divide numerator and denominator by \(x^2\): \(\displaystyle\int \dfrac{1+\frac{1}{x^2}}{\left(x-\frac{1}{x}\right)^2+3}\,dx\)
Let \(t = x - \dfrac{1}{x}\), so \(dt = \left(1+\dfrac{1}{x^2}\right)dx\).
Integral becomes: \(\displaystyle\int \dfrac{dt}{t^2+(\sqrt{3})^2} = \dfrac{1}{\sqrt{3}}\tan^{-1}\!\left(\dfrac{t}{\sqrt{3}}\right)+C\)
\(\dfrac{1}{\sqrt{3}}\tan^{-1}\!\left(\dfrac{x^2-1}{x\sqrt{3}}\right)+C\)
Self-assess: ✓ Saved
Question 3
The marginal cost function is \(MC = 3x^2 - 10x + 3\). The total cost of producing 3 units is ₹49. Find the total cost function and the cost of producing 5 units.
Integrate \(MC\): \(\displaystyle\int(3x^2-10x+3)\,dx = x^3-5x^2+3x+k\)
Use \(C(3)=49\): \(27-45+9+k=49 \Rightarrow k=58\)
\(C(5) = 125-125+15+58 = 73\)
Total cost function: \(C(x) = x^3 - 5x^2 + 3x + 58\)
Cost of producing 5 units = ₹73
Self-assess: ✓ Saved
Question 4
Find the area of the region bounded by the parabola \(y^2 = 4x\) and the line \(y = 2x - 4\).
Find intersections. Substitute \(x = \dfrac{y^2}{4}\) into \(y = 2x-4\): \(y = \dfrac{y^2}{2}-4\).
\(y^2 - 2y - 8 = 0 \Rightarrow (y-4)(y+2)=0 \Rightarrow y=4\) or \(y=-2\)
Area \(= \dfrac{1}{4}\displaystyle\int_{-2}^{4}(8+2y-y^2)\,dy = \dfrac{1}{4}\left[8y+y^2-\dfrac{y^3}{3}\right]_{-2}^{4}\)
\(= \dfrac{1}{4}\times 60 = 15\)
Area = 15 square units
Self-assess: ✓ Saved
Question 5
Solve the differential equation: \((x+y)\,\dfrac{dy}{dx} = 1\), given \(y=0\) when \(x=1\).
Rewrite as \(\dfrac{dx}{dy} = x+y\), i.e. \(\dfrac{dx}{dy}-x=y\) — this is linear in \(x\).
Integrating factor: \(e^{-y}\). Multiply through: \(\dfrac{d}{dy}(xe^{-y}) = ye^{-y}\)
Integrate: \(xe^{-y} = \displaystyle\int ye^{-y}\,dy = -ye^{-y}-e^{-y}+C\)
So \(x = -y-1+Ce^y\). Apply \(y=0, x=1\): \(1=-1+C \Rightarrow C=2\)
\(x = -y - 1 + 2e^y\)
Self-assess: ✓ Saved
📐 Formula Deck
₹199

All 8 units in one print-ready PDF — formulas, tips & common mistakes

📐 Get Formula Deck →
🤖 AI-Powered
🤖 AI Q-Bank

Adaptive practice, instant feedback & timed board exam simulation —

🤖 Get AI Q-Bank →
4-Mark Questions

Case Study Questions

Board-pattern 4-mark case studies. Read the context carefully, then click Show Solution under each part.

🏭

Case Study 1: Manufacturing Cost Optimisation

A company manufactures electronic components. The total cost function (in thousands of rupees) for producing \(x\) units (in hundreds) is: \[C(x) = 2x^3 - 15x^2 + 36x + 10\] The selling price per unit is fixed at ₹200. The company wants to optimise production.
(i)

Find the marginal cost function \(dC/dx\).

\(C(x) = 2x^3-15x^2+36x+10\)
\(MC = \dfrac{dC}{dx} = 6x^2-30x+36\)
Self-assess: ✓ Saved
(ii)

Find the value of \(x\) for which marginal cost is minimum.

\(\dfrac{d(MC)}{dx} = 12x-30 = 0 \Rightarrow x = 2.5\)
\(\dfrac{d^2(MC)}{dx^2} = 12 > 0\) → confirms minimum ✓
Marginal cost is minimum at \(x = 2.5\) hundred = 250 units
Self-assess: ✓ Saved
(iii)(a)

Find the minimum marginal cost.

\(MC(2.5) = 6(6.25)-30(2.5)+36 = 37.5-75+36 = -1.5\)
Minimum Marginal Cost = ₹−1,500 per hundred units
Self-assess: ✓ Saved
(iii)(b) OR

If \(R(x) = 200x-2x^2\), find the production level that maximises profit.

\(P(x) = R(x)-C(x) = -2x^3+13x^2+164x-10\)
\(\dfrac{dP}{dx} = -6x^2+26x+164 = 0 \Rightarrow x \approx 6.73\)
Optimal production ≈ 673 units for maximum profit
Self-assess: ✓ Saved
🚰

Case Study 2: Water Tank Filling Problem

A cylindrical water tank has radius 5 metres. The height \(h\) (in metres) at time \(t\) (in minutes) is \(h(t) = 0.1t^2+0.5t\). (Use \(\pi = 3.14\))
(i)

Find the rate of change of height at \(t = 10\) minutes.

\(\dfrac{dh}{dt} = 0.2t+0.5\)
At \(t=10\): \(\dfrac{dh}{dt} = 2+0.5 = 2.5\) m/min
Rate of height increase at \(t=10\) min = 2.5 m/min
Self-assess: ✓ Saved
(ii)

Calculate the height after 20 minutes.

\(h(20) = 0.1(400)+0.5(20) = 40+10 = 50\) m
Height after 20 minutes = 50 metres
Self-assess: ✓ Saved
(iii)(a)

Find the volume of water filled in the first 10 minutes.

\(h(10) = 0.1(100)+5 = 15\) m
\(V = \pi r^2 h = 25\times 3.14\times 15\)
\(V = \mathbf{1{,}177.5}\) m³
Self-assess: ✓ Saved
(iii)(b) OR

Find the rate of change of volume at \(t = 15\) minutes.

\(\dfrac{dV}{dt} = 25\pi\cdot\dfrac{dh}{dt} = 25\times 3.14\times(0.2t+0.5)\)
At \(t=15\): \(\dfrac{dV}{dt} = 78.5\times(3+0.5) = 78.5\times 3.5\)
Rate of volume change = 274.75 m³/min
Self-assess: ✓ Saved
🛒

Case Study 3: E-Commerce Sales Analysis

An e-commerce company models daily sales revenue (₹ thousands) as a function of hours \(x\) after business start: \[S(x) = -x^3+9x^2+15x+20, \quad 0\le x\le 8\]
(i)

Find the rate of change of sales with respect to time.

\(\dfrac{dS}{dx} = -3x^2+18x+15\) (₹ thousand per hour)
Self-assess: ✓ Saved
(ii)

Find the time at which the sales rate is maximum.

\(\dfrac{d^2S}{dx^2} = -6x+18 = 0 \Rightarrow x = 3\) hours
\(\dfrac{d^3S}{dx^3} = -6 < 0\) → confirms maximum ✓
Sales rate is maximum at 3 hours after opening
Self-assess: ✓ Saved
(iii)(a)

Calculate the total sales revenue from hour 2 to hour 6.

\(S(6) = -216+324+90+20 = 218\)
\(S(2) = -8+36+30+20 = 78\)
Revenue increase \(= 218-78 = 140\)
Revenue from hour 2 to 6 = ₹140 thousand
Self-assess: ✓ Saved
(iii)(b) OR

Find when the sales revenue is exactly ₹100 thousand.

Set \(-x^3+9x^2+15x+20 = 100\), i.e. \(x^3-9x^2-15x+80 = 0\).
By numerical methods: \(x\approx 4.37\) hours.
Sales reach ₹100 thousand at approximately 4 hours 22 minutes
Self-assess: ✓ Saved
📐 Formula Deck
₹199

All formulas for all 8 units — organised, printable PDF for quick revision

📐 Get Formula Deck →
🤖 AI-Powered
🤖 AI Q-Bank

Adaptive case studies & MCQs that target your exact weak areas automatically —

🤖 Get AI Q-Bank →
Score Full Marks

Exam Tips — Unit 3

Mistakes students make in the board exam — and how to avoid them.

✓ Tip 1 — Always Verify with the Second Derivative Test

Finding where \(f'(x)=0\) is only half the answer. You must then compute \(f''(x)\) at that point and write explicitly: "Since \(f''(x) < 0\), this is a maximum" or "Since \(f''(x) > 0\), this is a minimum." Skipping this verification step costs 1 mark in almost every long-answer question on maxima and minima.

🔒  More exam tips for Unit 3 — integration step-writing, consumer surplus setup, differential equations conclusion format, and MCQ shortcuts — are all in the AI Question Bank.

🤖 Get All Tips in AI Q-Bank →

❌ Common Mistakes to Avoid in Unit 3

  • Not verifying maxima/minima using the second derivative test — always write the check explicitly
  • Forgetting the constant of integration \(+C\) in indefinite integrals
  • Not using the given boundary condition to find \(C\) when integrating MC to get TC
  • Confusing order and degree — degree is undefined for non-polynomial equations in derivatives
  • Applying limits in the wrong order in definite integrals (upper − lower, not lower − upper)
  • Not setting up the consumer/producer surplus formula before substituting values
Common Questions

Frequently Asked Questions

Questions students frequently ask about Unit 3 — topics, formulas, and exam strategy.

Unit 3 — Calculus carries 15 marks in the CBSE Class 12 Applied Mathematics board exam, making it the second-highest weightage unit. Questions can appear as 1-mark MCQs, 3-mark short answers, 4-mark long answers, and 4-mark case studies.
Five topics: (1) Higher-order Differentiation, (2) Applications of Derivatives (rate of change, maxima and minima, marginal cost and revenue), (3) Indefinite and Definite Integrals (substitution, partial fractions, integration by parts), (4) Applications of Integrals (consumer and producer surplus, area under curves), and (5) Differential Equations (order, degree, formation and solutions).
Marginal Cost \((MC) = \dfrac{dC}{dx}\) — the derivative of the total cost function.
Marginal Revenue \((MR) = \dfrac{dR}{dx}\).
To recover total cost from MC, integrate it and use the given boundary condition to find the constant of integration.
Consumer Surplus \(= \displaystyle\int_0^{x_0} D(x)\,dx - p_0 x_0\), where \(D(x)\) is the demand function, \(x_0\) is the equilibrium quantity, and \(p_0\) is the equilibrium price. It represents the total benefit consumers gain by paying the market price rather than what they were individually willing to pay.
Order = the highest derivative present in the equation.
Degree = the power of that highest derivative, provided the equation is polynomial in derivatives.
Important: if the equation contains \(\sin\!\left(\dfrac{dy}{dx}\right)\) or \(e^{dy/dx}\), the degree is not defined — state this explicitly in the exam for full marks.
The product rule: \(\dfrac{d}{dx}(uv) = u\dfrac{dv}{dx} + v\dfrac{du}{dx}\).
Always define \(u\) and \(v\) clearly at the start of your solution — examiners award partial marks for correct method even if arithmetic slips.
Based on CBSE past papers (2023–26): Maxima and Minima — every year as a long answer. Marginal Cost and Revenue — case study or short answer. Definite Integrals — MCQ and long answer. Area under curves — long answer. Differential Equations — long answer most years.
Complete Course

Explore All 8 Units

Free study material for every unit of the CBSE Class 12 Applied Maths syllabus.

🔢
Unit 1
Numbers & Quantification
 📐
Unit 2
Algebra — Matrices & Determinants
∫
Unit 3 · Current
Calculus
🎲
Unit 4
Probability Distributions
 📊
Unit 5
Inferential Statistics
 📈
Unit 6
Index Numbers & Time-based Data
 💰
Unit 7
Financial Mathematics
 📉
Unit 8
Linear Programming
Premium Study Aids

Upgrade Your Preparation

Two focused products to complement the free resources on this page — each designed for a different stage of preparation.

📐 Formula Deck
₹199
One-time · Instant PDF

All 8 Units in One Printable PDF

  • ✓Every formula organised chapter-wise
  • ✓Pro tips & common mistakes noted
  • ✓Perfect for daily revision & last-minute prep
📐 Order Formula Deck →

🔒 Razorpay · PDF to your email

🤖 AI-Powered
🤖 AI Question Bank
One-time · Instant access  

Adaptive Practice & Board Exam Simulation

  • ✓Questions adapt to your level in real time
  • ✓Instant feedback with detailed explanations
  • ✓Targets your weak areas automatically
  • ✓Timed 3-hour board exam simulation
  • ✓CBSE Sample Paper questions (2023–26)
🤖 Get AI Question Bank →

🔒 Razorpay · Instant access  

💬 Questions? WhatsApp us directly and we'll reply promptly.

💬 WhatsApp: +91-85952 36539
✨ AI Performance Report
Unit 3 — Calculus
Class XII Applied Maths · 29 questions · 53 marks
—/53
By Question Type
Sub-topic Breakdown
Focus Areas
✨ AI Study Insight
Self-assess a few questions above, then come back here.

🚀 Ready for the Full 8-Unit, 80-Mark Report?

The AI Question Bank gives you:

  • Full 8-Unit, 80-Mark Performance Report — always know exactly what to revise before your boards
  • Exam-focused AI-generated worksheets that target your weakest topics automatically
  • Daily Challenge to keep your practice streak alive every day

That's just a glimpse — there's a lot more inside.

Experience the Magical Dashboard → View All Features
Expert CBSE Coaching · Class 9–12