CBSE Applied Mathematics · Class XII · 241
Unit Notes, Formulas & Exam Tips
Select a unit above · Notes · Exam Tips · Common Mistakes
🎯 Practice
🏆 Perform
📖 Notes & Formulas
📊 Syllabus & Weightage
Syllabus & marks weightage — CBSE Applied Mathematics 2026–27
I
Numbers, Quantification & Numerical Applications
Modular arithmetic · Congruence modulo · Alligation & mixture · Boats & streams · Pipes & cisterns · Races & games · Numerical inequalities
11M II
Algebra
Matrices & types · Equality & transpose · Symmetric & skew-symmetric · Algebra of matrices · Determinants · Inverse of matrix · Solving simultaneous equations using Cramer's Rule & Matrix Method
10M III
Calculus
Parametric & Higher Order Differentiation · Rate of Change · Increasing/Decreasing Functions · Maxima & Minima · Marginal Cost/Revenue · Integration & Applications: Cost-Revenue, Consumer/Producer Surplus, Area under Curve · Differential Equations
15M IV
Probability Distributions
Probability distribution table · Mean & variance · Binomial distribution · Poisson distribution · Normal distribution · Z-scores
10M V
Inferential Statistics
Population & sample · Sampling methods · Hypothesis testing · Null & alternate hypothesis · One-sample t-test · Type I & Type II errors
5M VI
Time-based Data
Index numbers · Time series · Trend analysis · Moving averages · Method of least squares
6M VII
Financial Mathematics
Perpetuity (Type I & II) · Sinking Fund · EMI & Amortization · Evaluation of Bonds · CAGR · Depreciation
15M VIII
Linear Programming
Formulation of LPP · Graphical method · Feasible region · Corner point method · Optimal solution
8M
Exam Paper Pattern — CBSE 2026–27
📋 CBSE Class 12 Applied Mathematics — Question Paper Pattern (2026–27)
Section
Question Type
No. of Qs
Marks each
Total
A MCQ & Assertion-Reason 20 1 20 B Short Answer – I (VSA) 5 2 10 C Short Answer – II 6 3 18 D Long Answer 4 5 20 E Case Study 3 4 10 Total 80
Internal choice in Sections B, C, D and E · Duration: 3 hours · Maximum marks: 80
Ready to test your knowledge?
🎯 Practice Questions
🏆 Check Your Performance
Unit I — Numbers, Quantification & Numerical Applications
📖 Notes
💡 Exam Tips
⚠️ Common Mistakes
Unit Notes & Formulas
Unit I — Numbers, Quantification & Numerical Applications
I
Numbers, Quantification & Numerical Applications
11 marks in board exam · CBSE Applied Mathematics Class 12
📐 Formula Cards
💡 Exam Tips
⚠️ Common Mistakes
In alligation, always draw the cross diagram — it's faster than algebra in the exam.
For boat/stream problems, always define $u$ (boat speed) and $v$ (stream speed) at the start.
In pipe problems, convert "fills in $x$ hours" to rate $= 1/x$ per hour before adding.
Races: "A beats B by $x$ m in a $d$-m race" means when A runs $d$, B runs $d-x$.
Modulo: always reduce to $[0, m)$ range. For negative numbers add $m$ repeatedly.
In inequalities: mark critical points, test one value from each interval.
Forgetting to flip inequality when multiplying/dividing by a negative number.
Confusing upstream (against current) with downstream (with current).
In alligation, using absolute prices instead of differences from mean price.
In pipes, adding rates when one pipe empties — it should be subtracted.
Not converting mixed numbers ($1\frac{1}{2}$ hours = 90 min) before substituting.
II
Algebra
10 marks in board exam · CBSE Applied Mathematics Class 12
📐 Formula Cards
💡 Exam Tips
⚠️ Common Mistakes
For 2×2 inverse: swap diagonal elements, negate off-diagonal, divide by determinant.
Cofactor sign pattern: $+\,-\,+$ / $-\,+\,-$ / $+\,-\,+$ — memorise this grid.
If two rows/columns are proportional, determinant is 0 — no need to expand.
For Assertion-Reason on matrices: check the statement first with a 2×2 example.
In system of equations, write in matrix form $AX = B$ and solve $X = A^{-1}B$.
Confusing adjoint with cofactor matrix — adj is the TRANSPOSE of the cofactor matrix.
Not applying the sign pattern when computing cofactors.
Forgetting to divide by $|A|$ when computing $A^{-1}$.
Confusing $(AB)^T = B^T A^T$ with $(AB)^T = A^T B^T$ (wrong order).
In Cramer's rule, replacing the wrong column when forming $D_x$, $D_y$, $D_z$.
III
Calculus
15 marks in board exam · CBSE Applied Mathematics Class 12
📐 Formula Cards
💡 Exam Tips
⚠️ Common Mistakes
For maxima/minima: find $f'(x)=0$, then check $f''(x)$ — negative means maximum.
Consumer surplus: equilibrium is where demand curve = supply curve ($D(x) = S(x)$).
For parametric differentiation: $\frac{d^2y}{dx^2} = \frac{d(dy/dx)/dt}{dx/dt}$ — don't forget to divide by $dx/dt$ again.
Integration by substitution: let $u$ = the inner function, find $du$, substitute.
In differential equations, always add constant of integration $C$ and apply initial condition.
Forgetting $+C$ in indefinite integration — costs marks every time.
For $\int \frac{dx}{ax+b}$: answer is $\frac{1}{a}\ln|ax+b|+C$, not $\ln|ax+b|+C$.
Confusing $MR$ and $AR$ — marginal revenue is derivative of total revenue, not $R/x$.
In $d^2y/dx^2$ for parametric: students differentiate w.r.t. $t$ instead of $x$.
Wrong limits in definite integration for consumer/producer surplus — always from 0 to $x_0$.
IV
Probability Distributions
10 marks in board exam · CBSE Applied Mathematics Class 12
📐 Formula Cards
💡 Exam Tips
⚠️ Common Mistakes
For Binomial: identify $n$ (trials), $p$ (success prob), $r$ (required successes) clearly.
Use Poisson when the question mentions "rare events", "defects per unit", or large $n$ small $p$.
For Normal: always convert to Z-score first, then use given table values.
$P(X > a) = 1 - P(X \leq a)$: use complement rule when easier.
For finding $k$ in distribution: use $\sum P(X) = 1$ to get one equation, $E(X)$ for the second.
Confusing Mean ($np$) with Variance ($npq$) — Mean is always ≥ Variance for Binomial.
Using $P(X=r) = e^{-\lambda}\lambda^r$ without dividing by $r!$ in Poisson.
In Normal, students forget to standardise: must compute $Z = (X-\mu)/\sigma$ first.
Not verifying $\sum P(X) = 1$ when constructing a distribution table — partial marks lost.
Confusing ${}^nC_r$ (combinations) with ${}^nP_r$ (permutations) in Binomial.
V
Inferential Statistics
5 marks in board exam · CBSE Applied Mathematics Class 12
📐 Formula Cards
💡 Exam Tips
⚠️ Common Mistakes
Always state $H_0$ and $H_1$ explicitly — 1 mark each in many mark schemes.
One-tailed test: "greater than", "increased", "improved" — use one-tail critical value.
Two-tailed test: "different from", "changed" — use $\alpha/2$ in each tail.
The t-statistic formula is given in Section A (16): choose the right one from the options.
Using $df = n$ instead of $n-1$ for one-sample t-test.
Forgetting to take square root when computing $s/\sqrt{n}$.
Confusing one-tailed and two-tailed critical values from the t-table.
Not writing the conclusion in plain English: "We reject/fail to reject $H_0$. Hence..."
VI
Time-based Data
6 marks in board exam · CBSE Applied Mathematics Class 12
📐 Formula Cards
💡 Exam Tips
⚠️ Common Mistakes
Always set up the X, Y, XY, X² table — examiner expects it, and it earns step marks.
For odd number of years: choose middle year as origin. For even: choose midpoint (use ±½, ±3/2…).
3-year MA: first value centred at 2nd year, last at (n-1)th year — 2 values lost.
4-year centred MA: always show both the 4-pt MA AND the centred value columns.
"Average change per year" in trend = value of $b$ (slope).
In 4-year MA, skipping the centring step — just averaging 4 values is NOT the centred MA.
Using actual year numbers as X instead of deviation from origin — $\Sigma X$ won't be 0.
Confusing $b = \Sigma XY / \Sigma X^2$ with $b = \Sigma XY / \Sigma X$ (wrong denominator).
Predicting for a future year: compute $X = \text{future year} - \text{origin}$, then substitute.
VII
Financial Mathematics
15 marks in board exam · CBSE Applied Mathematics Class 12
📐 Formula Cards
💡 Exam Tips
⚠️ Common Mistakes
Flat rate EMI: simple — total interest = P × rate × years; total ÷ months.
Reducing balance EMI: use the $(1+i)^{-n}$ value given in the question directly.
Perpetuity: $PV = R/i$ — nothing else. Don't over-complicate.
Sinking fund: you need to ACCUMULATE $S$; solve for $R$ (the annual deposit).
CAGR: always express as percentage at the end — multiply by 100.
Using annual interest rate directly in EMI formula instead of monthly rate $i = r/1200$.
Confusing $n$ (months) with years in EMI — always convert years × 12.
For sinking fund, forgetting to subtract scrap value from cost to find $S$.
In bond price, using annual coupon rate directly — must compute $R = $ face value × rate.
CAGR: computing $(EV/SV)^{1/n}$ but forgetting to subtract 1.
VIII
Linear Programming
8 marks in board exam · CBSE Applied Mathematics Class 12
📐 Formula Cards
💡 Exam Tips
⚠️ Common Mistakes
Draw the graph neatly — all lines labelled, feasible region shaded, corner points marked.
Find corner points by solving pairs of boundary lines simultaneously.
Check ALL corner points — the maximum might not be at the "obvious" vertex.
For formulation questions: define variables clearly, write objective function, then list ALL constraints including non-negativity.
Redundant constraint check: substitute the corner point of the feasible region into the constraint — if satisfied strictly, it may be redundant.
Shading the WRONG region — shade the region that satisfies the inequality (test a point like origin).
Missing the non-negativity constraints ($x \geq 0$, $y \geq 0$) in formulation.
Computing only 2–3 corner points instead of ALL vertices of the feasible region.
Forgetting to write the maximum/minimum value of $Z$ and where it occurs.
🎥 Free Video Tutorials
Need a formula reference while learning?
Step-by-step video explanations for all 8 units — free on YouTube. Watch a concept video, then explore the formula cards here.
📋
Applied Maths Formula Deck — All 8 Units
One organised PDF · Instant download · Board exam pattern
Every formula for all 8 units in one printable PDF
Chapter-wise layout for quick reference during revision
Perfect for daily revision and last-minute exam prep
Keep this handy on exam day. Works brilliantly alongside the formula cards — a perfect companion for board exam prep.
Ready to test your knowledge?
🎯 Practice Questions
🏆 Check Your Performance
📚 Board PYQs
★ Practice Set
Source:
CBSE Sample Paper
Board Papers
Sample + Board
Unit
All Units
Unit I — Numbers
Unit II — Algebra
Unit III — Calculus
Unit IV — Probability
Unit V — Statistics
Unit VI — Time-based Data
Unit VII — Financial Maths
Unit VIII — LPP
Year
All Years
2026
2025
2024
2023
Paper Type
CBSE Sample Paper
Board Papers
Sample + Board
Question Type
All Types
MCQ · 1M
Assertion-Reason · 1M
VSA · 2M
SA · 3M
LA · 5M
Case Study
👆 MCQ attempt mode: Click an option first, then click Show solution to see if you're correct.
Browse by unit
Unit I
Numbers, Quantification & Numerical Applications
11 marks
Modular arithmetic Alligation Boats & streams Pipes & cisterns Inequalities
Practice Questions
PYQs
Unit II
Algebra
10 marks
Matrices Determinants Inverse Cramer's Rule
Practice Questions
PYQs
Unit III
Calculus
15 marks
Differentiation Applications Integration Area under curve Differential eq.
Practice Questions
PYQs
Unit IV
Probability Distributions
10 marks
PDT Binomial Poisson Normal distribution Z-scores
Practice Questions
PYQs
Unit V
Inferential Statistics
5 marks
Sampling Hypothesis testing t-test Type I & II errors
Practice Questions
PYQs
Unit VI
Time-based Data
6 marks
Index numbers Time series Moving averages Least squares
Practice Questions
PYQs
Unit VII
Financial Mathematics
15 marks
EMI Sinking fund CAGR Depreciation Bonds Perpetuity
Practice Questions
PYQs
Unit VIII
Linear Programming
8 marks
Formulation Graphical method Feasible region Corner point
Practice Questions
PYQs
📄 Question Papers Available — CBSE Applied Mathematics Class 12
📋 CBSE Sample Paper2025–26
📄 Board Paper2026
📄 Board Paper2025
📝 Mock Test I
📝 Mock Test II
📚 Understand
★
AI Practice Set
Powered by Claude · Personalised for you
Unit
— Select Unit —
Unit I — Number System
Unit II — Algebra
Unit III — Calculus
Unit IV — Probability
Unit V — Statistics
Unit VI — Time-based Data
Unit VII — Financial Maths
Unit VIII — Linear Programming
Question Type
Mixed Questions
MCQ Only (1 mark)
Assertion-Reason (1 mark)
Short Answer (2–3M)
Long Answer (5M)
Case Study (4M)
How Many
5 questions
10 questions
15 questions
20 questions
★
Generate Set
★
Configure and generate your practice set above
Questions will appear here · Practice sources only (not CBSE papers)
🎥 Free Video Tutorials
Need conceptual clarity before practising?
Step-by-step video explanations for Probability Distributions, Financial Mathematics, Calculus, and more — free on YouTube.
📋
Applied Maths Formula Deck — All 8 Units
One organised PDF · Instant download · Board exam pattern
Every formula for all 8 units in one printable PDF
Chapter-wise layout for quick reference during revision
Perfect for daily revision and last-minute exam prep
Keep this handy on exam day. Works brilliantly alongside these practice questions — a perfect companion for board exam prep.
Ready to go further?
📖 Revise Formulas
🏆 Check Your Performance
📋 Exam format:
Sec A: 20×1M · Sec B: 5×2M · Sec C: 6×3M · Sec D: 4×5M · Sec E: 3×4M · Total: 80M · 3 hrs
How to use these papers: Always attempt the question paper first under timed conditions — 3 hours, no breaks. Only open the solution after you've attempted every question. This is the single most effective exam preparation technique.
📌
Continue where you left off
You have an in-progress paper
Resume →
or
↺ Reset
×
CBSE Sample Paper 2025-26
Board Paper 2026
Board Paper 2025
Board Paper 2024
Mock Test I
Mock Test II
⏱
00:00:00
Section A
⏸ Paused
⏸ Pause
⏹ Stop
Want to attempt a timed test?
▶ Start Timer
⏱ How the timer works
Click Start Timer when you begin
Each section is timed independently
Switch sections freely — time follows you
Use ⏸ Pause for breaks (water, breaks etc.)
Click View Solutions to lock a section's time
Timer auto-stops on Section E solutions
Use Stop Timer at the bottom when done
Closing the tab pauses — resumes on return
Read instructions on hover
Applied Mathematics (Code 241) · Class XII · 80 marks · 3 hours
↺ Start Over
Section:
A · MCQ (1M)
B · VSA (2M)
C · SA (3M)
D · LA (5M)
E · Case Study (4M)
Section A
Multiple Choice Questions — 1 mark each (Q.1 to Q.18) · Assertion-Reason (Q.19 to Q.20)
Section A of 5
View section solutions ↺ Start Over
✨ My Report
← Previous
Next section →
⏹ Stop Timer & See Time Summary
⏱ Time Summary
Section Questions Time Spent Status
Review Answers
📖 Not happy with a paper attempt? Go back to unit-wise practice.
After attempting any paper, identify which units need more work. Head to the Unit Notes & Question Bank tab → filter by that unit → practise Board PYQs, CBSE Sample Paper questions, and extra practice problems. Once you feel confident, come back here and attempt another paper under timed conditions.
Browse by unit
The question bank has 500+ solved questions — MCQs, Assertion-Reason, Short Answer, Long Answer and Case Studies — covering all 8 units from CBSE Board Papers 2022–2025 and CBSE Sample Papers. Every question includes a step-by-step solution and common error hint.
Unit I
Numbers, Quantification & Numerical Applications
11 marks
Practice Questions
PYQs
Unit II
Algebra
10 marks
Practice Questions
PYQs
Unit III
Calculus
15 marks
Practice Questions
PYQs
Unit IV
Probability Distributions
10 marks
Practice Questions
PYQs
Unit V
Inferential Statistics
5 marks
Practice Questions
PYQs
Unit VI
Time-based Data
6 marks
Practice Questions
PYQs
Unit VII
Financial Mathematics
15 marks
Practice Questions
PYQs
Unit VIII
Linear Programming
8 marks
Practice Questions
PYQs
🎥 Free Video Tutorials
Need conceptual clarity before practising?
Step-by-step video explanations for Probability Distributions, Financial Mathematics, Calculus, and more — free on YouTube. Watch a unit video, then attempt the questions here.
📋
Applied Maths Formula Deck — All 8 Units
One organised PDF · Instant download · Board exam pattern
Every formula for all 8 units in one printable PDF
Chapter-wise layout for quick reference during revision
Perfect for daily revision and last-minute exam prep
Attempting papers without formulas memorised? Keep this PDF handy — a quick scan before each section saves marks.
Want to practise before attempting a full paper?
📖 Revise Formulas First
🎯 Practice Questions