🚨 Don't Lose Marks Here!

Common Mistakes Notebook 🙅‍♀️

These are the exact mistakes that cost students marks every year. Read this before the exam and Ishu (My Wife) will NOT repeat them! 🎯

01

Relations & Functions

Saves 1-2 Marks
❌ Wrong
Checking only Reflexivity and Symmetry — forgetting Transitivity
✅ Right
Check ALL THREE: Reflexive + Symmetric + Transitive for Equivalence
💡 Partial marks are NOT given if one condition is missing. Check all 3!
Saves 2 Marks
❌ Wrong
Saying inverse exists for any function $f(x)$
✅ Right
Inverse exists ONLY if $f$ is Bijective (One-One AND Onto both)
💡 Always prove bijection before writing inverse!
❌ Wrong
Writing $fog = gof$ directly
✅ Right
$fog \neq gof$ in general. Always compute separately!
💡 Apply the rightmost function FIRST in composition.
02

Inverse Trigonometric Functions

Saves 2 Marks
❌ Wrong
$\sin^{-1}(\sin(2\pi)) = 2\pi$
✅ Right
$\sin^{-1}(\sin(2\pi)) = 0$ — must reduce to principal value range $[-\pi/2, \pi/2]$
💡 $\sin^{-1}(\sin\theta) = \theta$ ONLY if $\theta \in [-\pi/2, \pi/2]$. Otherwise reduce first!
Saves 2 Marks
❌ Wrong
$\tan^{-1}2 + \tan^{-1}3 = \tan^{-1}\!\left(\frac{5}{-5}\right) = \tan^{-1}(-1)$
✅ Right
Since $xy = 6 > 1$, add $\pi$: result $= \pi + \tan^{-1}(-1) = \pi - \pi/4 = 3\pi/4$
💡 When $xy > 1$ in $\tan^{-1}x + \tan^{-1}y$, ALWAYS add $\pi$ if $x, y > 0$!
❌ Wrong
$\cos^{-1}(-x) = -\cos^{-1}(x)$
✅ Right
$\cos^{-1}(-x) = \pi - \cos^{-1}(x)$ — NOT negative!
💡 Only $\sin^{-1}$, $\tan^{-1}$, $\cot^{-1}$, $\text{cosec}^{-1}$ are odd. $\cos^{-1}$ and $\sec^{-1}$ are NOT.
03

Matrices

Saves 2 Marks
❌ Wrong
$(AB)^T = A^T B^T$
✅ Right
$(AB)^T = B^T A^T$ — ORDER REVERSES!
💡 Same for inverse: $(AB)^{-1} = B^{-1}A^{-1}$. Order always reverses!
❌ Wrong
Multiplying any two matrices $A \times B$ directly
✅ Right
Check compatibility first: columns of $A$ must equal rows of $B$. $m\times n$ × $n\times p$ = $m\times p$
💡 Alwasy note the order of the resulting matrix too!
04

Determinants

Saves 3 Marks
❌ Wrong
$|kA| = k|A|$ for a 3×3 matrix
✅ Right
$|kA| = k^n|A|$ where $n$ = order. For 3×3: $|kA| = k^3|A|$
💡 Scalar multiplies each row, so for $n$ rows: $k^n$ factor comes out!
Saves 2 Marks
❌ Wrong
Forgetting ± sign in minors/cofactors. Writing all cofactors as positive.
✅ Right
Cofactor sign: $(-1)^{i+j} \times M_{ij}$. Signs follow checkerboard: $+ - + / - + - / + - +$
💡 Draw the checkerboard sign pattern before writing cofactors!
❌ Wrong
Taking Area of triangle as negative when det is negative
✅ Right
Area = $\frac{1}{2}|\text{det}|$ — always take absolute value! Area is never negative.
💡 But for collinearity check: det = 0 (no absolute value needed)
05

Continuity & Differentiability

Saves 3 Marks
❌ Wrong
Checking only LHL = RHL for continuity and declaring function continuous
✅ Right
3 conditions: LHL = RHL = $f(c)$. Missing the third condition = wrong answer!
💡 Write all 3 steps explicitly: LHL, RHL, f(c), then conclude.
Saves 2 Marks
❌ Wrong
Differentiating $x^x$ as $x \cdot x^{x-1}$
✅ Right
Use log differentiation: $y = x^x \Rightarrow \ln y = x\ln x \Rightarrow \frac{y'}{y} = 1+\ln x$
💡 Power rule works only when base is variable & exponent is constant!
❌ Wrong
Concluding: if $f$ is continuous then it's differentiable
✅ Right
Diff ⟹ Continuous. But Continuous ⟹ NOT necessarily diff. $|x|$ is the classic example!
💡 Board often asks: "Is $f(x)=|x|$ differentiable at 0?" — Answer: NO!
06

Application of Derivatives

Saves 3 Marks
❌ Wrong
Forgetting to check endpoints for absolute maxima/minima in a closed interval
✅ Right
Evaluate $f$ at all critical points AND both endpoints [$a$, $b$]. Compare all values!
💡 Absolute max/min is often at endpoints, not critical points!
❌ Wrong
Writing tangent equation without finding slope: $y - y_1 = m(x - x_1)$ with wrong $m$
✅ Right
Always find $m = f'(x_1)$ first by substituting the given point in the derivative!
💡 Normal slope = $-1/m$. If slope is 0, tangent is horizontal: $y = y_1$
07

Integrals

Saves 3 Marks
❌ Wrong
Forgetting $+C$ in indefinite integrals every single time
✅ Right
ALWAYS write $+C$ at the end of every indefinite integral. 1 mark deducted if missing!
💡 Make it a habit. $+C$ = free marks. Never miss it!
Saves 2 Marks
❌ Wrong
Not substituting limits after substitution in definite integral: keeping old limits with new variable
✅ Right
When substituting $u = g(x)$, change limits too: new limits = $g(a)$ and $g(b)$
💡 OR: back-substitute to $x$ before applying original limits. Either way — don't mix!
❌ Wrong
$\int \frac{1}{x^2} dx = \ln x^2$
✅ Right
$\int x^{-2} dx = \frac{x^{-1}}{-1} = -\frac{1}{x} + C$. Only $\int \frac{1}{x} dx = \ln|x|+C$!
💡 Log rule applies ONLY to $1/x$, not $1/x^2$, $1/x^3$, etc.!
08

Application of Integrals

Saves 3 Marks
❌ Wrong
Getting negative area and writing it as the answer
✅ Right
Area is always positive! If integral gives negative, take $|$absolute value$|$. Split if curve crosses x-axis.
💡 Always sketch a rough graph — it shows if region is above or below x-axis!
❌ Wrong
Not finding intersection points before setting up area integral between two curves
✅ Right
Solve $f(x) = g(x)$ first for intersection points — these become the limits of integration!
💡 Without correct limits, full marks are impossible even if integration is right.
09

Differential Equations

Saves 2 Marks
❌ Wrong
Forgetting $+C$ in general solution of DE
✅ Right
General solution MUST have arbitrary constant $C$. Particular solution uses given condition to find $C$.
💡 Losing $C$ in general solution = losing marks guaranteed!
Saves 2 Marks
❌ Wrong
Using I.F. method when variables are separable — overcomplicating
✅ Right
First check if separable: can you write as $f(x)dx = g(y)dy$? If yes, separate — it's faster!
💡 I.F. method is for Linear DE $\frac{dy}{dx}+Py=Q$ only!
10

Vector Algebra

Saves 2 Marks
❌ Wrong
$\vec{a} \times \vec{b} = \vec{b} \times \vec{a}$
✅ Right
$\vec{a} \times \vec{b} = -(\vec{b} \times \vec{a})$ — cross product is anti-commutative!
💡 Dot product IS commutative: $\vec{a}\cdot\vec{b} = \vec{b}\cdot\vec{a}$ ✅
❌ Wrong
Saying $|\vec{a}+\vec{b}| = |\vec{a}| + |\vec{b}|$ always
✅ Right
$|\vec{a}+\vec{b}|^2 = |\vec{a}|^2 + 2\vec{a}\cdot\vec{b} + |\vec{b}|^2$. Equality holds only when parallel!
💡 Use the expansion formula when magnitude of sum is asked.
11

3D Geometry

Saves 3 Marks
❌ Wrong
Confusing direction ratios with direction cosines. Writing DR directly as DC.
✅ Right
DCs: $l = \frac{a}{\sqrt{a^2+b^2+c^2}}$. Always normalize! $l^2+m^2+n^2 = 1$ (DCs), not DRs!
💡 DRs are proportional to DCs. Don't use DRs where DCs are asked — check!
❌ Wrong
Forgetting absolute value in distance from plane: getting negative distance
✅ Right
$d = \frac{|ax_1+by_1+cz_1+d|}{\sqrt{a^2+b^2+c^2}}$ — always absolute value in numerator!
💡 Distance is always a positive scalar. Never negative!
12

Linear Programming

Saves 3 Marks
❌ Wrong
Evaluating objective function at only ONE or TWO corner points
✅ Right
Evaluate $Z$ at ALL corner points. Create a table showing all values. Then conclude!
💡 CBSE wants a proper table of corner points and $Z$ values. Skip = lose marks!
❌ Wrong
Shading the wrong side of constraint inequalities
✅ Right
Substitute $(0,0)$ into inequality. If satisfied → shade side containing origin. If not → shade opposite.
💡 The feasible region is the COMMON shaded area satisfying ALL constraints!
13

Probability

Saves 3 Marks
❌ Wrong
Not computing $P(A)$ using Total Probability before applying Bayes'
✅ Right
Step 1: Find $P(A)= \sum P(E_i)P(A|E_i)$. Step 2: Then apply Bayes' formula using this $P(A)$.
💡 Bayes' denominator IS the Total Probability. Compute it first, separately!
Saves 2 Marks
❌ Wrong
Confusing independent events with mutually exclusive events
✅ Right
Independent: $P(A\cap B)=P(A)P(B)$. Mutually exclusive: $P(A\cap B)=0$. These are DIFFERENT!
💡 Two mutually exclusive events with non-zero probability are NEVER independent!
❌ Wrong
In Binomial distribution, using $n$ and $p$ but forgetting to compute $q = 1-p$
✅ Right
$q = 1 - p$. Variance $= npq$. Always find $q$ first. Never use $p$ in place of $q$!
💡 Mean $=np$, Variance $=npq$. Variance < Mean always in binomial!