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🎯 Part A
Class 10 Mathematics
01
Real Numbers
1.1 Euclid's Division Lemma
\begin{gather*}
a = bq + r, \text{where} 0 \le r < b \\
\\
\Rightarrow b = (a - r) / q \\
\Rightarrow q = (a - r) / b \\
\Rightarrow r = a - bq \\
\Rightarrow a - r = bq \\
\Rightarrow a mod b = r
\end{gather*}
1.2 HCF & LCM Relationship
\begin{gather*}
HCF(a, b) \times LCM(a, b) = a \times b \\
\\
\Rightarrow LCM(a, b) = (a \times b) / HCF(a, b) \\
\Rightarrow HCF(a, b) = (a \times b) / LCM(a, b) \\
\Rightarrow a = HCF \times (a/HCF) \\
\Rightarrow b = HCF \times (b/HCF) \\
\Rightarrow LCM = HCF \times (a/HCF) \times (b/HCF) \\
\\
\text{For} three numbers: \\
HCF(a,b,c) \times LCM(a,b,c) \neq a\times b\times c [\text{NOT} directly] \\
\\
LCM(a,b,c) = (a\times b\times c \times HCF(a,b,c)) / [HCF(a,b) \times HCF(b,c) \times HCF(a,c)]
\end{gather*}
1.3 Fundamental Theorem of Arithmetic
\begin{gather*}
Every composite number = product of primes (unique, order aside) \\
n = p_1^a_1 \times p_2^a_2 \times p_3^a_3 \times ... \times p_k^a_k \\
\\
HCF = product of SMALLEST powers of common prime factors \\
LCM = product of GREATEST powers of all prime factors
\end{gather*}
1.4 Rational & Irrational Numbers
\begin{gather*}
Rational number p/q terminates iff q = 2^m \times 5^n \\
\\
Number of decimal places = max(m, n) \\
\\
\text{If} q has prime factor other than 2 \text{or} 5 \Rightarrow non-terminating repeating \\
\\
\sqrt{p} \text{is} irrational \text{if} p \text{is} a prime number \\
\sqrt{a/b} \text{is} irrational \text{if} a/b \text{is} \text{not} a perfect square of rationals
\end{gather*}
1.5 Divisibility Conditions
\begin{gather*}
n^2 - 1 = (n-1)(n+1) \\
n(n+1) = always even \\
n(n+1)(n+2) = always divisible by 6 \\
n(n+1)(2n+1)/6 = sum of squares of first n natural numbers
\end{gather*}
1.6 Key Number Theory Identities
\begin{gather*}
(a + b)^2 = a^2 + 2ab + b^2 \\
(a - b)^2 = a^2 - 2ab + b^2 \\
a^2 - b^2 = (a+b)(a-b) \\
(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \\
(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 \\
a^3 + b^3 = (a+b)(a^2 - ab + b^2) \\
a^3 - b^3 = (a-b)(a^2 + ab + b^2) \\
\\
Twisted Forms: \\
ab = [(a+b)^2 - a^2 - b^2] / 2 \\
ab = [(a+b)^2 - (a-b)^2] / 4 \\
a^2 + b^2 = (a+b)^2 - 2ab \\
a^2 + b^2 = (a-b)^2 + 2ab \\
(a+b)^2 - (a-b)^2 = 4ab \\
(a+b)^2 + (a-b)^2 = 2(a^2+b^2)
\end{gather*}
02
Polynomials
2.1 Degree and Types
\begin{gather*}
Linear: ax + b = 0 (degree 1) \Rightarrow 1 zero \\
Quadratic: ax^2 + bx + c = 0 (degree 2) \Rightarrow 2 zeros \\
Cubic: ax^3+bx^2+cx+d = 0 (degree 3) \Rightarrow 3 zeros
\end{gather*}
2.2 Quadratic Polynomial — Relationship Between Zeros and Coefficients
\begin{gather*}
p(x) = ax^2 + bx + c, zeros: \alpha , \beta \\
\\
Sum of zeros: \alpha + \beta = -b/a \\
Product of zeros: \alpha \times \beta = c/a \\
\\
\Rightarrow b = -a(\alpha + \beta ) \\
\Rightarrow c = a(\alpha \times \beta ) \\
\Rightarrow a/b = -1/(\alpha +\beta ) \\
\Rightarrow b/c = -(\alpha +\beta )/(\alpha \beta ) \\
\\
Twisted Derived Forms: \\
\alpha ^2 + \beta ^2 = (\alpha +\beta )^2 - 2\alpha \beta = b^2/a^2 - 2c/a = (b^2 - 2ac)/a^2 \\
(\alpha - \beta )^2 = (\alpha +\beta )^2 - 4\alpha \beta = b^2/a^2 - 4c/a = (b^2 - 4ac)/a^2 \\
|\alpha - \beta | = √[(\alpha +\beta )^2 - 4\alpha \beta ] = \sqrt{b^2-4ac} / |a| \\
\alpha ^2 - \beta ^2 = (\alpha +\beta )(\alpha -\beta ) \\
1/\alpha + 1/\beta = (\alpha +\beta )/(\alpha \beta ) = -b/c \\
1/(\alpha \beta ) = a/c \\
\alpha /\beta + \beta /\alpha = (\alpha ^2+\beta ^2)/(\alpha \beta ) = (b^2-2ac)/(ac) \\
\alpha ^2 \times \beta ^2 = (\alpha \beta )^2 = c^2/a^2 \\
\alpha ^3 + \beta ^3 = (\alpha +\beta )^3 - 3\alpha \beta (\alpha +\beta ) = (-b/a)^3 - 3(c/a)(-b/a) \\
= (-b^3 + 3abc) / a^3 \\
\alpha ^3 - \beta ^3 = (\alpha -\beta )[(\alpha +\beta )^2-\alpha \beta ] \\
1/\alpha ^2 + 1/\beta ^2 = (\alpha ^2+\beta ^2)/(\alpha \beta )^2 = (b^2-2ac)/c^2 \\
(\alpha +1)(\beta +1) = \alpha \beta + \alpha + \beta + 1 = c/a - b/a + 1 = (c - b + a)/a \\
(\alpha -1)(\beta -1) = \alpha \beta - (\alpha +\beta ) + 1 = c/a + b/a + 1 = (c + b + a)/a \\
\alpha ^2\beta + \alpha \beta ^2 = \alpha \beta (\alpha +\beta ) = (c/a)(-b/a) = -bc/a^2
\end{gather*}
2.3 Formation of Quadratic from Given Zeros
\begin{gather*}
p(x) = x^2 - (\alpha +\beta )x + \alpha \beta [\text{when} a=1] \\
p(x) = k[x^2 - (\alpha +\beta )x + \alpha \beta ] [general, k \neq 0] \\
p(x) = k[x^2 - (sum of zeros)x + (product of zeros)]
\end{gather*}
2.4 Cubic Polynomial — Relationship with Zeros
\begin{gather*}
p(x) = ax^3 + bx^2 + cx + d, zeros: \alpha , \beta , \gamma \\
\\
\alpha + \beta + \gamma = -b/a \\
\alpha \beta + \beta \gamma + \gamma \alpha = c/a \\
\alpha \beta \gamma = -d/a \\
\\
Twisted Forms: \\
\alpha ^2 + \beta ^2 + \gamma ^2 = (\alpha +\beta +\gamma )^2 - 2(\alpha \beta +\beta \gamma +\gamma \alpha ) \\
= b^2/a^2 - 2c/a = (b^2-2ac)/a^2 \\
\\
\alpha ^3+\beta ^3+\gamma ^3-3\alpha \beta \gamma = (\alpha +\beta +\gamma )(\alpha ^2+\beta ^2+\gamma ^2-\alpha \beta -\beta \gamma -\gamma \alpha ) \\
\\
1/\alpha + 1/\beta + 1/\gamma = (\alpha \beta +\beta \gamma +\gamma \alpha )/(\alpha \beta \gamma ) = -c/d \\
\\
Formation: \\
p(x) = x^3 - (\alpha +\beta +\gamma )x^2 + (\alpha \beta +\beta \gamma +\gamma \alpha )x - \alpha \beta \gamma
\end{gather*}
2.5 Division Algorithm
\begin{gather*}
Dividend = Divisor \times Quotient + Remainder \\
p(x) = g(x) \times q(x) + r(x) \\
\\
\text{where} deg[r(x)] < deg[g(x)] \\
\\
\Rightarrow p(x) - r(x) = g(x) \times q(x) [p-r \text{is} divisible by g] \\
\Rightarrow q(x) = [p(x) - r(x)] / g(x)
\end{gather*}
2.6 Discriminant (Δ)
\begin{gather*}
\Delta = b^2 - 4ac \\
\\
\Delta > 0 \Rightarrow two distinct real zeros \\
\Delta = 0 \Rightarrow two equal real zeros (coincident) \\
\Delta < 0 \Rightarrow no real zeros (complex) \\
\\
Zeros: \alpha , \beta = [-b ± \sqrt{b^2-4ac}] / 2a [Quadratic Formula] \\
\\
\Rightarrow \alpha + \beta = -b/a (verify) \\
\Rightarrow \alpha - \beta = √\Delta / a (difference of roots) \\
\Rightarrow 2\alpha = (-b/a) + √\Delta /a = (-b + √\Delta )/a \\
\Rightarrow 2\beta = (-b/a) - √\Delta /a = (-b - √\Delta )/a
\end{gather*}
03
Pair of Linear Equations in Two Variables
3.1 Standard Form
\begin{gather*}
a_1x + b_1y + c_1 = 0 \\
a_2x + b_2y + c_2 = 0
\end{gather*}
3.2 Consistency Conditions
\begin{gather*}
Unique solution (consistent): a_1/a_2 \neq b_1/b_2 \\
Infinitely many (consistent): a_1/a_2 = b_1/b_2 = c_1/c_2 \\
No solution (inconsistent): a_1/a_2 = b_1/b_2 \neq c_1/c_2
\end{gather*}
3.3 Cross-Multiplication Method
\begin{gather*}
x y 1 \\
------- = ------- = -------- \\
b_1c_2-b_2c_1 c_1a_2-c_2a_1 a_1b_2-a_2b_1 \\
\\
\Rightarrow x = (b_1c_2 - b_2c_1) / (a_1b_2 - a_2b_1) \\
\Rightarrow y = (c_1a_2 - c_2a_1) / (a_1b_2 - a_2b_1) \\
\\
Let D = a_1b_2 - a_2b_1 \\
Dx = b_1c_2 - b_2c_1 [replace a-column with c values, negated] \\
Dy = c_1a_2 - c_2a_1 \\
x = Dx/D, y = Dy/D
\end{gather*}
3.4 Substitution Method (Steps)
\begin{gather*}
From eq1: y = (c_1 - a_1x) / b_1 \\
Substitute in eq2: a_2x + b_2[(c_1-a_1x)/b_1] + c_2 = 0 \\
Solve \text{for} x, then back-substitute \text{for} y
\end{gather*}
3.5 Elimination Method
\begin{gather*}
Multiply eq1 by a_2, eq2 by a_1: \\
a_1a_2x + b_1a_2y = -c_1a_2 \\
a_1a_2x + b_2a_1y = -c_2a_1 \\
Subtract: y(b_1a_2 - b_2a_1) = c_2a_1 - c_1a_2 \\
\Rightarrow y = (c_2a_1 - c_1a_2)/(b_1a_2 - b_2a_1)
\end{gather*}
3.6 Graphical Interpretation
\begin{gather*}
Lines intersect \Rightarrow unique solution \Rightarrow (x,y) = point of intersection \\
Lines parallel \Rightarrow no solution \Rightarrow slopes equal, y-intercepts differ \\
Lines coincident \Rightarrow infinite solutions \Rightarrow same line \\
\\
Slope of a_1x + b_1y + c_1 = 0 \text{is} m = -a_1/b_1
\end{gather*}
04
Quadratic Equations
4.1 Standard Form & Quadratic Formula
\begin{gather*}
ax^2 + bx + c = 0 (a \neq 0) \\
\\
x = [-b ± \sqrt{b^2 - 4ac}] / 2a \\
\\
x_1 = (-b + \sqrt{D}) / 2a \\
x_2 = (-b - \sqrt{D}) / 2a \\
\\
\text{where} D = b^2 - 4ac (Discriminant)
\end{gather*}
4.2 Nature of Roots
\begin{gather*}
D > 0 \Rightarrow two distinct real roots \\
D = 0 \Rightarrow two equal real roots: x = -b/2a \\
D < 0 \Rightarrow no real roots (imaginary) \\
D \text{is} perfect square \Rightarrow rational roots \\
D > 0 but \text{not} perfect square \Rightarrow irrational roots
\end{gather*}
4.3 Sum and Product of Roots
\begin{gather*}
x_1 + x_2 = -b/a \\
x_1 \times x_2 = c/a \\
x_1 - x_2 = \sqrt{D}/a = \sqrt{b^2-4ac}/a \\
\\
Equation from roots: x^2 - (x_1+x_2)x + x_1x_2 = 0
\end{gather*}
4.4 Completing the Square
\begin{gather*}
ax^2 + bx + c = 0 \\
x^2 + (b/a)x + c/a = 0 \\
(x + b/2a)^2 = b^2/4a^2 - c/a = (b^2-4ac)/4a^2 \\
x + b/2a = ± \sqrt{b^2-4ac} / 2a \\
x = -b/2a ± \sqrt{b^2-4ac}/2a
\end{gather*}
4.5 Special Cases & Factoring Tricks
\begin{gather*}
\text{If} sum of coefficients = 0: x=1 \text{is} always a root \\
i.e., a + b + c = 0 \Rightarrow x=1 \text{and} x=c/a \\
\\
\text{If} a + c = b: x = -1 \text{is} always a root \\
\\
\text{For} x^2 - Sx + P = 0: roots are S±\sqrt{S^2-4P}/2 \\
\\
Factoring: ax^2+bx+c = a(x-x_1)(x-x_2) \\
= a[x^2 - (x_1+x_2)x + x_1x_2]
\end{gather*}
4.6 Common Root Condition
\begin{gather*}
\text{If} ax^2+bx+c=0 \text{and} dx^2+ex+f=0 share a common root r: \\
ar^2+br+c=0 \text{and} dr^2+er+f=0 \\
\Rightarrow r = (bf-ce)/(cd-af) = (ae-bd)/(bf-ce) [cross multiply] \\
\\
Condition: (bf-ce)(ae-bd) = (cd-af)^2
\end{gather*}
05
Arithmetic Progressions (AP)
5.1 General Term (nth Term)
\begin{gather*}
aₙ = a + (n-1)d \\
\\
\text{where}: a = first term \\
d = common difference = aₙ - aₙ₋_1 \\
n = number of terms \\
\\
\Rightarrow n = (aₙ - a)/d + 1 \\
\Rightarrow d = (aₙ - a)/(n-1) \\
\Rightarrow a = aₙ - (n-1)d \\
\Rightarrow aₙ = l = last term \\
\\
Last term: l = a + (n-1)d \\
Number of terms: n = (l-a)/d + 1
\end{gather*}
5.2 Sum of n Terms
\begin{gather*}
Sₙ = n/2 [2a + (n-1)d] \\
Sₙ = n/2 [a + l] (\text{when} last term l \text{is} known) \\
Sₙ = n/2 [a + aₙ] \\
\\
\Rightarrow aₙ = Sₙ - Sₙ₋_1 (nth term from sum) \\
\Rightarrow 2a + (n-1)d = 2Sₙ/n \\
\Rightarrow d = 2(Sₙ - na) / [n(n-1)] \\
\Rightarrow a = [2Sₙ - n(n-1)d] / 2n \\
\\
\text{If} Sₙ = An^2 + Bn: \\
aₙ = A(2n-1) + B (nth term) \\
a_1 = A + B \\
d = 2A
\end{gather*}
5.3 Important Sum Formulas
\begin{gather*}
Sum of first n natural numbers: \\
\Sigma n = n(n+1)/2 \\
\\
Sum of first n odd numbers: \\
1+3+5+...+(2n-1) = n^2 \\
\\
Sum of first n even numbers: \\
2+4+6+...+2n = n(n+1) \\
\\
Sum of squares of first n naturals: \\
\Sigma n^2 = n(n+1)(2n+1)/6 \\
\\
Sum of cubes of first n naturals: \\
\Sigma n^3 = [n(n+1)/2]^2
\end{gather*}
5.4 Arithmetic Mean
\begin{gather*}
AM between a \text{and} b: A = (a+b)/2 \\
\Rightarrow a, A, b are in AP ↔ A-a = b-A ↔ 2A = a+b \\
\\
n AMs between a \text{and} b: \\
A_1 = a + d, A_2 = a + 2d, ..., Aₙ = a + nd \\
\text{where} d = (b-a)/(n+1) \\
\\
Sum of n AMs between a \text{and} b = n(a+b)/2
\end{gather*}
5.5 Properties of AP
\begin{gather*}
\text{If} a, b, c are in AP: 2b = a + c \Rightarrow b-a = c-b \\
\\
\text{If} each term multiplied by k \Rightarrow still AP with same ratio \\
\text{If} constant added \Rightarrow still AP with same d \\
\text{If} reversed \Rightarrow still AP with common difference -d \\
\\
aₘ + aₙ = aₚ + aₚ \text{where} m+n = p+q (same sum of indices \Rightarrow same sum of terms) \\
\\
Middle term of AP with odd n = Sₙ/n = average \\
\\
3 numbers in AP: a-d, a, a+d [sum = 3a] \\
4 numbers in AP: a-3d, a-d, a+d, a+3d [sum = 4a] \\
5 numbers in AP: a-2d, a-d, a, a+d, a+2d [sum = 5a]
\end{gather*}
5.6 Twisted AP Identities
\begin{gather*}
aₘ - aₙ = (m-n)d \\
aₘ/aₙ = [a+(m-1)d] / [a+(n-1)d] \\
\\
\text{If} aₘ = n \text{and} aₙ = m: \\
d = (n-m)/(m-n) = -1 \\
aₘ₊ₙ = 0 \\
\\
\text{If} Sₘ/Sₙ = (am+b)/(cm+d) [ratio of sums given]: \\
aₘ/aₙ = [a(2m-1)+b] / [c(2m-1)+d] [replace n by 2n-1] \\
\\
Sₘ = Sₙ (m\neq n) \Rightarrow Sₘ₊ₙ = 0 \Rightarrow aₘ₊ₙ₊_1 = 0
\end{gather*}
06
Triangles
6.1 Basic Proportionality Theorem (Thales' Theorem)
\begin{gather*}
\text{If} DE ∥ BC in △ABC, D on AB, E on AC: \\
AD/DB = AE/EC \\
\\
Equivalent Forms: \\
AD/AB = AE/AC (part/whole ratio) \\
DB/AB = EC/AC \\
AB/AD = AC/AE \\
AB/DB = AC/EC \\
DB/AD = EC/AE
\end{gather*}
6.2 Converse of BPT
\begin{gather*}
\text{If} AD/DB = AE/EC, then DE ∥ BC
\end{gather*}
6.3 Angle Bisector Theorem
\begin{gather*}
\text{If} AD bisects ∠A in △ABC: \\
BD/DC = AB/AC \\
\\
\Rightarrow BD = (AB \times BC)/(AB + AC) \\
\Rightarrow DC = (AC \times BC)/(AB + AC)
\end{gather*}
6.4 Criteria for Similarity
\begin{gather*}
AAA / AA: two pairs of equal angles \Rightarrow similar \\
SSS: all three sides proportional \Rightarrow similar \\
SAS: two sides proportional \text{and} included angle equal \Rightarrow similar \\
\\
\text{If} △ABC ~ △DEF: \\
AB/DE = BC/EF = CA/FD = k (scale factor) \\
∠A = ∠D, ∠B = ∠E, ∠C = ∠F
\end{gather*}
6.5 Properties of Similar Triangles
\begin{gather*}
\text{If} △ABC ~ △DEF with ratio k: \\
\\
Ratio of perimeters = k = AB/DE \\
Ratio of areas = k^2 = (AB/DE)^2 \\
Ratio of altitudes = k \\
Ratio of medians = k \\
Ratio of angle bisectors = k \\
Ratio of circumradii = k \\
Ratio of inradii = k \\
\\
Area(△ABC)/Area(△DEF) = (AB/DE)^2 = (BC/EF)^2 = (CA/FD)^2
\end{gather*}
6.6 Pythagoras Theorem & Converse
\begin{gather*}
In right △ABC (right angle at C): \\
AB^2 = BC^2 + AC^2 \\
(Hypotenuse)^2 = (Base)^2 + (Perpendicular)^2 \\
\\
\Rightarrow BC^2 = AB^2 - AC^2 \\
\Rightarrow AC^2 = AB^2 - BC^2 \\
\Rightarrow BC = \sqrt{AB^2 - AC^2} \\
\\
Converse: \text{If} AB^2 = BC^2 + AC^2, then ∠C = 90° \\
\\
Pythagorean Triplets (a,b,c) \text{where} a^2+b^2=c^2: \\
(3,4,5), (5,12,13), (8,15,17), (7,24,25), (9,40,41) \\
(6,8,10), (10,24,26) [multiples of above] \\
General: (m^2-n^2, 2mn, m^2+n^2) \text{for} m>n>0
\end{gather*}
6.7 Important Results from Pythagoras
\begin{gather*}
In △ABC, \text{if} altitude CD ⊥ AB: \\
CD^2 = AD \times DB [geometric mean relation] \\
BC^2 = BD \times BA [altitude on hypotenuse] \\
AC^2 = AD \times AB \\
CD = (AC \times BC)/AB \\
\\
Median formula (in △ABC, m_a = median from A): \\
m_a^2 = (2b^2 + 2c^2 - a^2)/4 \\
\\
Stewart's Theorem: \\
b^2m + c^2n - a(mn + d^2) = 0 \\
[cevian d divides BC into m \text{and} n]
\end{gather*}
6.8 Congruence Criteria
\begin{gather*}
SSS: three sides equal \\
SAS: two sides \text{and} included angle equal \\
ASA: two angles \text{and} included side equal \\
AAS: two angles \text{and} non-included side equal \\
RHS: right angle, hypotenuse, one side equal
\end{gather*}
07
Coordinate Geometry
7.1 Distance Formula
\begin{gather*}
d = √[(x_2-x_1)^2 + (y_2-y_1)^2] \\
\\
\Rightarrow d^2 = (x_2-x_1)^2 + (y_2-y_1)^2 \\
\Rightarrow (x_2-x_1)^2 = d^2 - (y_2-y_1)^2 \\
\\
Distance from origin O(0,0) to P(x,y): \\
d = \sqrt{x^2 + y^2} \\
\\
Distance between P(x_1,y_1) \text{and} Q(x_2,y_2): \\
PQ = √[(x_1-x_2)^2 + (y_1-y_2)^2]
\end{gather*}
7.2 Section Formula
\begin{gather*}
Point P dividing A(x_1,y_1) \text{and} B(x_2,y_2) in ratio m:n \\
\\
INTERNAL division: \\
P = [(mx_2 + nx_1)/(m+n), (my_2 + ny_1)/(m+n)] \\
\\
EXTERNAL division: \\
P = [(mx_2 - nx_1)/(m-n), (my_2 - ny_1)/(m-n)] \\
\\
Finding ratio m:n given P(x,y) on AB: \\
m/n = (x - x_1)/(x_2 - x) [using x-coordinate] \\
m/n = (y - y_1)/(y_2 - y) [using y-coordinate]
\end{gather*}
7.3 Midpoint Formula
\begin{gather*}
Midpoint M of A(x_1,y_1) \text{and} B(x_2,y_2): \\
M = [(x_1+x_2)/2, (y_1+y_2)/2] \\
\\
\Rightarrow x_1 + x_2 = 2xₘ \\
\Rightarrow x_1 = 2xₘ - x_2 [find other endpoint \text{if} midpoint known]
\end{gather*}
7.4 Centroid of Triangle
\begin{gather*}
G = [(x_1+x_2+x_3)/3, (y_1+y_2+y_3)/3] \\
\\
\Rightarrow x_1+x_2+x_3 = 3xG \\
\Rightarrow \text{If} G = (0,0): x_1+x_2+x_3 = 0, y_1+y_2+y_3 = 0
\end{gather*}
7.5 Area of Triangle
\begin{gather*}
Area = ½ |x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)| \\
\\
Expanded: \\
= ½ |x_1y_2 - x_1y_3 + x_2y_3 - x_2y_1 + x_3y_1 - x_3y_2| \\
\\
Shoelace form: \\
= ½ |(x_1y_2 - x_2y_1) + (x_2y_3 - x_3y_2) + (x_3y_1 - x_1y_3)| \\
\\
Collinearity condition (Area = 0): \\
x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) = 0
\end{gather*}
7.6 Slope of a Line
\begin{gather*}
m = (y_2-y_1)/(x_2-x_1) = tan \theta \\
\\
\Rightarrow \theta = arctan(m) (angle with positive x-axis) \\
\\
Parallel lines: m_1 = m_2 \\
Perpendicular: m_1 \times m_2 = -1 \Rightarrow m_2 = -1/m_1 \\
\\
Slope-intercept form: y = mx + c \\
Point-slope form: y - y_1 = m(x - x_1) \\
Two-point form: (y-y_1)/(y_2-y_1) = (x-x_1)/(x_2-x_1) \\
Intercept form: x/a + y/b = 1
\end{gather*}
7.7 Area of Quadrilateral & Polygon
\begin{gather*}
Area of quadrilateral ABCD: \\
= ½ |d_1 \times d_2 \times sin \theta | (diagonals d_1, d_2 at angle \theta ) \\
\\
\text{Or} using vertices A,B,C,D: \\
= ½ |(x_1-x_3)(y_2-y₄) - (x_2-x₄)(y_1-y_3)| [diagonals AC, BD] \\
\\
Using Shoelace: \\
= ½ |(x_1y_2-x_2y_1) + (x_2y_3-x_3y_2) + (x_3y₄-x₄y_3) + (x₄y_1-x_1y₄)|
\end{gather*}
7.8 Key Distance Facts
\begin{gather*}
Collinear: AB + BC = AC (\text{if} B \text{is} between A \text{and} C) \\
Isosceles: two distances equal \\
Equilateral: all three distances equal, each = same \\
Right angle: (longest)^2 = (other)^2 + (other)^2 \\
Rectangle: diagonals equal \text{and} bisect each other \\
Rhombus: all sides equal, diagonals bisect perpendicularly \\
Square: all sides equal + diagonals equal \\
Parallelogram: diagonals bisect each other (midpoints same)
\end{gather*}
08–09
Trigonometry & Applications
8.1 Trigonometric Ratios (Right Triangle)
\begin{gather*}
sin \theta = Opposite/Hypotenuse = P/H \\
cos \theta = Adjacent/Hypotenuse = B/H \\
tan \theta = Opposite/Adjacent = P/B \\
\\
cosec \theta = 1/sin \theta = H/P \\
sec \theta = 1/cos \theta = H/B \\
cot \theta = 1/tan \theta = B/P = cos \theta /sin \theta \\
\\
tan \theta = sin \theta /cos \theta \\
cot \theta = cos \theta /sin \theta
\end{gather*}
8.2 Reciprocal Relations
\begin{gather*}
sin \theta \times cosec \theta = 1 \Rightarrow cosec \theta = 1/sin \theta \\
cos \theta \times sec \theta = 1 \Rightarrow sec \theta = 1/cos \theta \\
tan \theta \times cot \theta = 1 \Rightarrow cot \theta = 1/tan \theta
\end{gather*}
8.3 Pythagorean Identities
\begin{gather*}
sin^2\theta + cos^2\theta = 1 ← PRIMARY \\
\\
\Rightarrow sin^2\theta = 1 - cos^2\theta \\
\Rightarrow cos^2\theta = 1 - sin^2\theta \\
\Rightarrow sin\theta = \sqrt{1-cos^2\theta } \\
\Rightarrow cos\theta = \sqrt{1-sin^2\theta } \\
\\
1 + tan^2\theta = sec^2\theta ← SECONDARY \\
\\
\Rightarrow sec^2\theta - tan^2\theta = 1 \\
\Rightarrow sec^2\theta - 1 = tan^2\theta \\
\Rightarrow tan^2\theta = sec^2\theta - 1 \\
\Rightarrow (sec\theta + tan\theta )(sec\theta - tan\theta ) = 1 \\
\Rightarrow sec\theta + tan\theta = 1/(sec\theta - tan\theta ) \\
\\
1 + cot^2\theta = cosec^2\theta ← TERTIARY \\
\\
\Rightarrow cosec^2\theta - cot^2\theta = 1 \\
\Rightarrow cosec^2\theta - 1 = cot^2\theta \\
\Rightarrow (cosec\theta + cot\theta )(cosec\theta - cot\theta ) = 1 \\
\Rightarrow cosec\theta + cot\theta = 1/(cosec\theta - cot\theta )
\end{gather*}
8.4 Standard Angle Values Table
\begin{gather*}
Angle \theta : 0° 30° 45° 60° 90° \\
───────────────────────────────────────────────────── \\
sin \theta : 0 1/2 1/\sqrt{2} \sqrt{3}/2 1 \\
cos \theta : 1 \sqrt{3}/2 1/\sqrt{2} 1/2 0 \\
tan \theta : 0 1/\sqrt{3} 1 \sqrt{3} ∞ \\
cosec \theta : ∞ 2 \sqrt{2} 2/\sqrt{3} 1 \\
sec \theta : 1 2/\sqrt{3} \sqrt{2} 2 ∞ \\
cot \theta : ∞ \sqrt{3} 1 1/\sqrt{3} 0 \\
\\
Memory trick \text{for} sin: \sqrt{0}/2, \sqrt{1}/2, \sqrt{2}/2, \sqrt{3}/2, \sqrt{4}/2 \\
= 0, 1/2, 1/\sqrt{2}, \sqrt{3}/2, 1
\end{gather*}
8.5 Complementary Angle Relations
\begin{gather*}
sin(90°-\theta ) = cos\theta \Rightarrow cos\theta = sin(90°-\theta ) \\
cos(90°-\theta ) = sin\theta \Rightarrow sin\theta = cos(90°-\theta ) \\
tan(90°-\theta ) = cot\theta \Rightarrow cot\theta = tan(90°-\theta ) \\
cot(90°-\theta ) = tan\theta \Rightarrow tan\theta = cot(90°-\theta ) \\
sec(90°-\theta ) = cosec\theta \Rightarrow cosec\theta = sec(90°-\theta ) \\
cosec(90°-\theta ) = sec\theta \Rightarrow sec\theta = cosec(90°-\theta )
\end{gather*}
8.6 Useful Identity Proofs (Templates)
\begin{gather*}
(sin\theta + cos\theta )^2 = 1 + 2sin\theta cos\theta \\
(sin\theta - cos\theta )^2 = 1 - 2sin\theta cos\theta \\
(sin\theta + cos\theta )^2 + (sin\theta - cos\theta )^2 = 2 \\
sin⁴\theta + cos⁴\theta = 1 - 2sin^2\theta cos^2\theta = 1 - 2sin^2\theta (1-sin^2\theta ) \\
sin⁶\theta + cos⁶\theta = 1 - 3sin^2\theta cos^2\theta \\
\\
tan\theta + cot\theta = 1/(sin\theta cos\theta ) = sec\theta cosec\theta \\
tan\theta - cot\theta = (sin^2\theta -cos^2\theta )/(sin\theta cos\theta ) = -cos2\theta /(sin\theta cos\theta ) \\
sec\theta + tan\theta = (1+sin\theta )/cos\theta \\
sec\theta - tan\theta = (1-sin\theta )/cos\theta = 1/(sec\theta +tan\theta ) \\
cosec\theta + cot\theta = (1+cos\theta )/sin\theta \\
cosec\theta - cot\theta = (1-cos\theta )/sin\theta = 1/(cosec\theta +cot\theta )
\end{gather*}
8.7 Heights and Distances (Chapter 9)
\begin{gather*}
Angle of Elevation: angle from horizontal UP to object \\
Angle of Depression: angle from horizontal DOWN to object \\
\\
tan(angle of elevation) = Height / Horizontal Distance \\
tan(angle of depression) = Height / Horizontal Distance \\
\\
Height h = d \times tan \theta \\
Distance d = h / tan \theta = h \times cot \theta \\
\\
\text{If} angle of elevation from A \text{is} \alpha , from B (further) \text{is} \beta : \\
(\alpha > \beta since A \text{is} closer) \\
Let AB = x, object height = h, base of object foot = C \\
h/BC = tan \alpha \Rightarrow BC = h/tan\alpha = h cot\alpha \\
h/AC = tan \beta \Rightarrow AC = h cot\beta \\
AB = AC - BC = h(cot\beta - cot\alpha ) \\
\Rightarrow h = AB/(cot\beta - cot\alpha ) = AB tan\alpha tan\beta /(tan\alpha - tan\beta ) \\
\\
Two buildings problem: \\
Height of taller - shorter = d \times (tan\alpha - tan\beta ) [careful about geometry]
\end{gather*}
10
Circles
10.1 Tangent Properties
\begin{gather*}
Tangent ⊥ radius at point of contact \\
\\
Length of tangent from external point P to circle (center O, radius r): \\
PQ = PT = \sqrt{PO^2 - r^2} \\
\\
\Rightarrow PO^2 = PQ^2 + r^2 (Pythagoras) \\
\Rightarrow r^2 = PO^2 - PQ^2 \\
\\
Tangents from same external point are equal: \\
PA = PB (A,B are points of tangency)
\end{gather*}
10.2 Angle Properties of Tangent
\begin{gather*}
∠OAP = ∠OBP = 90° (tangent-radius angle) \\
\\
In quadrilateral OAPB (P external, A,B tangent points): \\
∠APB + ∠AOB = 180° [opposite angles supplementary] \\
∠APO = ∠BPO [OP bisects angle between tangents] \\
∠AOT = ∠BOT [OP bisects angle between radii] \\
\\
OA = OB = r \\
PA = PB \\
OP = OP (common) \\
\Rightarrow △OAP ≅ △OBP (RHS congruence)
\end{gather*}
10.3 Circle Theorems
\begin{gather*}
Angle in semicircle = 90° \\
Angles in same segment are equal \\
Angle at center = 2 \times angle at circumference (same arc) \\
Opposite angles of cyclic quadrilateral sum to 180° \\
Exterior angle of cyclic quadrilateral = interior opposite angle \\
Equal chords subtend equal angles at center \\
Perpendicular from center to chord bisects the chord
\end{gather*}
10.4 Chord-Tangent Angle
\begin{gather*}
Angle between tangent \text{and} chord = angle in alternate segment \\
(Tangent-chord angle = inscribed angle on opposite side)
\end{gather*}
11
Areas Related to Circles
11.1 Basic Circle Formulas
\begin{gather*}
Area of circle = \pi r^2 \\
Circumference = 2\pi r = \pi d \\
Diameter d = 2r \\
Area = \pi (d/2)^2 = \pi d^2/4 \\
r = \sqrt{Area/\pi } \\
r = Circumference/(2\pi )
\end{gather*}
11.2 Sector Formulas
\begin{gather*}
Area of sector = (\theta /360°) \times \pi r^2 [\theta in degrees] \\
= (1/2)r^2\theta [\theta in radians] \\
= (1/2) \times l \times r [l = arc length] \\
\\
Arc length = (\theta /360°) \times 2\pi r = (\theta /180°) \times \pi r \\
= r\theta [\theta in radians] \\
\\
Perimeter of sector = 2r + l = 2r + (\theta /360°)\times 2\pi r \\
= 2r + (\pi r\theta /180°) \\
\\
\Rightarrow r = 2 \times Area / l [from A = ½lr] \\
\Rightarrow \theta = (l/r) radians = (l/r) \times (180°/\pi ) degrees
\end{gather*}
11.3 Segment Formulas
\begin{gather*}
Area of minor segment = Area of sector - Area of triangle \\
= (\theta /360°)\pi r^2 - (1/2)r^2 sin \theta \\
= r^2[(\pi \theta /360°) - (sin\theta /2)] \\
= (r^2/2)[\theta _rad - sin\theta ] [\theta in radians] \\
\\
Area of major segment = \pi r^2 - Area of minor segment \\
\\
Chord length = 2r sin(\theta /2) \\
Height of segment = r(1 - cos\theta /2) = r - r cos(\theta /2)
\end{gather*}
11.4 Combination Formulas
\begin{gather*}
Area of ring (annulus) = \pi (R^2-r^2) = \pi (R+r)(R-r) \\
\text{where} R=outer radius, r=inner radius \\
\\
Area of semi-circle = \pi r^2/2 \\
Perimeter of semi-circle = \pi r + 2r = r(\pi +2) \\
\\
Area of quadrant = \pi r^2/4 \\
Perimeter of quadrant = \pi r/2 + 2r = r(\pi /2 + 2) \\
\\
Shaded region problems: \\
Area of figure = Area_1 ± Area_2 ± Area_3...
\end{gather*}
12
Surface Areas and Volumes
12.1 Cuboid
\begin{gather*}
LSA (Lateral Surface Area) = 2h(l+b) \\
TSA (Total Surface Area) = 2(lb + bh + hl) \\
Volume = l \times b \times h \\
Diagonal = \sqrt{l^2+b^2+h^2} \\
Face diagonal (on lb face) = \sqrt{l^2+b^2} \\
\\
\Rightarrow l = V/(bh) \\
\Rightarrow LSA = TSA - 2lb [remove top \text{and} bottom]
\end{gather*}
12.2 Cube (side = a)
\begin{gather*}
LSA = 4a^2 \\
TSA = 6a^2 \\
Volume = a^3 \\
Diagonal = a\sqrt{3} \\
Face diagonal = a\sqrt{2} \\
\\
\Rightarrow a = (V)^(1/3) \\
\Rightarrow a = \sqrt{TSA/6}
\end{gather*}
12.3 Cylinder (radius r, height h)
\begin{gather*}
CSA (Curved SA) = 2\pi rh \\
TSA = 2\pi r(r+h) = 2\pi r^2 + 2\pi rh \\
Volume = \pi r^2h \\
\\
\Rightarrow r = CSA/(2\pi h) \\
\Rightarrow h = CSA/(2\pi r) \\
\Rightarrow h = V/(\pi r^2) \\
\Rightarrow r = \sqrt{V/\pi h} \\
\Rightarrow r+h = TSA/(2\pi r) [\text{if} r known]
\end{gather*}
12.4 Cone (radius r, height h, slant l)
\begin{gather*}
Slant height l = \sqrt{r^2+h^2} \\
CSA = \pi rl = \pi r\sqrt{r^2+h^2} \\
TSA = \pi rl + \pi r^2 = \pi r(l+r) \\
Volume = (1/3)\pi r^2h \\
\\
\Rightarrow h = \sqrt{l^2-r^2} \\
\Rightarrow r = \sqrt{l^2-h^2} \\
\Rightarrow l = \sqrt{r^2+h^2} \\
\Rightarrow r = CSA/(\pi l) \\
\Rightarrow h = 3V/(\pi r^2) \\
\Rightarrow r = \sqrt{3V/(\pi h})
\end{gather*}
12.5 Sphere (radius r)
\begin{gather*}
SA (Surface Area) = 4\pi r^2 \\
Volume = (4/3)\pi r^3 \\
\\
\Rightarrow r = \sqrt{SA/4\pi } \\
\Rightarrow r = (3V/4\pi )^(1/3) \\
\Rightarrow V = (SA)^(3/2) / (6√\pi ) [from eliminating r] \\
\Rightarrow SA^3 = 36\pi V^2 [classic relation]
\end{gather*}
12.6 Hemisphere (radius r)
\begin{gather*}
CSA = 2\pi r^2 \\
TSA = 3\pi r^2 [CSA + base circle \pi r^2] \\
Volume = (2/3)\pi r^3 \\
\\
\Rightarrow r = \sqrt{CSA/2\pi } \\
\Rightarrow r = \sqrt{TSA/3\pi } \\
\Rightarrow r = (3V/2\pi )^(1/3)
\end{gather*}
12.7 Frustum of Cone (R=bigger radius, r=smaller, h=height, l=slant)
\begin{gather*}
Slant height l = √[h^2 + (R-r)^2] \\
CSA = \pi (R+r)l \\
TSA = \pi (R+r)l + \pi R^2 + \pi r^2 = \pi [(R+r)l + R^2 + r^2] \\
Volume = (\pi h/3)(R^2 + r^2 + Rr) \\
\\
\Rightarrow h = 3V / [\pi (R^2+r^2+Rr)] \\
\Rightarrow l = √[h^2+(R-r)^2] \\
\\
\text{When} r=0 \Rightarrow cone: V = \pi R^2h/3 ✓ \\
\text{When} r=R \Rightarrow cylinder: V = \pi R^2h ✓
\end{gather*}
12.8 Combination of Solids
\begin{gather*}
Total Volume = V_1 + V_2 + ... \\
Total SA = SA of outer surface only (remove hidden parts) \\
\\
Hemisphere on cylinder: \\
TSA = CSA of cylinder + CSA of hemisphere + base circle \\
= 2\pi rh + 2\pi r^2 + \pi r^2 \\
= \pi r(2h + 3r) \\
Volume = \pi r^2h + (2/3)\pi r^3 = \pi r^2(h + 2r/3) \\
\\
Cone on cylinder: \\
TSA = base of cylinder + CSA cylinder + CSA cone \\
= \pi r^2 + 2\pi rh + \pi rl \\
Volume = \pi r^2H + (1/3)\pi r^2h_cone
\end{gather*}
12.9 Volume Conversion Formulas
\begin{gather*}
\text{If} solid A melted \text{and} recast into solid B: \\
Volume of A = Volume of B (\text{if} no wastage) \\
n \times Volume of B = Volume of A (n pieces of B from A) \\
\\
\text{If} melted cube of side a \Rightarrow n spheres of radius r: \\
a^3 = n \times (4/3)\pi r^3 \\
\Rightarrow n = 3a^3/(4\pi r^3) \\
\Rightarrow r = a \times (3/4\pi n)^(1/3)
\end{gather*}
13
Statistics
13.1 Mean (Ungrouped)
\begin{gather*}
x̄ = (x_1+x_2+...+xₙ)/n = \Sigma xᵢ/n = \Sigma fᵢxᵢ/\Sigma fᵢ [frequency dist] \\
\\
\Rightarrow \Sigma xᵢ = n \times x̄ \\
\Rightarrow n = \Sigma xᵢ/x̄
\end{gather*}
13.2 Mean (Grouped Data — Direct Method)
\begin{gather*}
x̄ = \Sigma fᵢxᵢ / \Sigma fᵢ = \Sigma fᵢxᵢ / N \\
\\
\text{where} xᵢ = class mark = (lower limit + upper limit)/2 \\
fᵢ = frequency of ith class \\
N = \Sigma fᵢ = total frequency
\end{gather*}
13.3 Assumed Mean (Shortcut) Method
\begin{gather*}
x̄ = A + (\Sigma fᵢdᵢ/N) \\
\\
\text{where} A = assumed mean (usually central class mark) \\
dᵢ = xᵢ - A (deviation from assumed mean) \\
N = \Sigma fᵢ
\end{gather*}
13.4 Step Deviation Method
\begin{gather*}
x̄ = A + [(\Sigma fᵢuᵢ/N) \times h] \\
\\
\text{where} uᵢ = (xᵢ - A)/h \\
h = class width (size) \\
A = assumed mean \\
\\
\Rightarrow dᵢ = uᵢ \times h \\
\Rightarrow xᵢ = A + uᵢh
\end{gather*}
13.5 Median (Grouped Data)
\begin{gather*}
Median = L + [(N/2 - cf)/f] \times h \\
\\
\text{where} L = lower limit of median class \\
N = \Sigma fᵢ (total frequency) \\
cf = cumulative frequency of class before median class \\
f = frequency of median class \\
h = class width \\
\\
Steps: \\
1. Find N/2 \\
2. Find class with cf \ge N/2 \Rightarrow median class \\
3. Apply formula \\
\\
Twisted: \\
\Rightarrow L = Median - [(N/2 - cf)/f] \times h \\
\Rightarrow cf = N/2 - f(Median - L)/h
\end{gather*}
13.6 Mode (Grouped Data)
\begin{gather*}
Mode = L + [f_1-f₀ / (2f_1-f₀-f_2)] \times h \\
\\
\text{where} L = lower limit of modal class \\
f_1 = frequency of modal class (highest frequency) \\
f₀ = frequency of class before modal class \\
f_2 = frequency of class after modal class \\
h = class width \\
\\
Twisted: \\
\Rightarrow L = Mode - [(f_1-f₀)/(2f_1-f₀-f_2)] \times h \\
\Rightarrow \text{If} f₀ = f_2: Mode = L + h/2 [symmetric case]
\end{gather*}
13.7 Empirical Relationship
\begin{gather*}
Mode = 3 \times Median - 2 \times Mean \\
\\
\Rightarrow Mean = (3 \times Median - Mode)/2 \\
\Rightarrow Median = (Mode + 2 \times Mean)/3 \\
\Rightarrow Mean - Mode = 3(Mean - Median) \\
\Rightarrow Mean - Median = (1/3)(Mean - Mode) \\
\\
[This holds approximately \text{for} moderately skewed distributions]
\end{gather*}
13.8 Ogive and Cumulative Frequency
\begin{gather*}
Less-than ogive: plot (upper limit, cf) \\
More-than ogive: plot (lower limit, N - cf) \\
\\
Median = x-coordinate \text{where} two ogives intersect
\end{gather*}
14
Probability
14.1 Classical (Theoretical) Probability
\begin{gather*}
P(E) = n(E)/n(S) = Number of favorable outcomes / Total outcomes \\
\\
0 \le P(E) \le 1 \\
P(certain event) = 1 \\
P(impossible event) = 0
\end{gather*}
14.2 Complementary Probability
\begin{gather*}
P(Ē) = 1 - P(E) (E-bar = complement of E) \\
P(E) + P(Ē) = 1 \\
\Rightarrow P(E) = 1 - P(Ē) \\
\Rightarrow P(Ē) = 1 - P(E) \\
\\
P(\text{not} E) = 1 - P(E) \\
P(at least one) = 1 - P(none)
\end{gather*}
14.3 Standard Sample Spaces
\begin{gather*}
Coin toss: S = {H, T}, n(S) = 2 \\
Two coins: S = {HH,HT,TH,TT}, n(S) = 4 \\
Three coins: n(S) = 8 \\
n coins: n(S) = 2ⁿ \\
\\
One die: n(S) = 6 \\
Two dice: n(S) = 36 \\
Cards (full deck): n(S) = 52 \\
- Hearts: 13, Diamonds: 13, Clubs: 13, Spades: 13 \\
- Red cards: 26, Black cards: 26 \\
- Face cards (J,Q,K): 12 (3 per suit) \\
- Aces: 4, Kings: 4, Queens: 4, Jacks: 4 \\
- Non-face number cards: 36
\end{gather*}
14.4 Two Dice Probability Reference
\begin{gather*}
Sum = 2: {(1,1)} \Rightarrow P = 1/36 \\
Sum = 3: {(1,2),(2,1)} \Rightarrow P = 2/36 = 1/18 \\
Sum = 4: 3 ways \Rightarrow P = 3/36 = 1/12 \\
Sum = 5: 4 ways \Rightarrow P = 4/36 = 1/9 \\
Sum = 6: 5 ways \Rightarrow P = 5/36 \\
Sum = 7: 6 ways (MAXIMUM) \Rightarrow P = 6/36 = 1/6 \\
Sum = 8: 5 ways \Rightarrow P = 5/36 \\
Sum = 9: 4 ways \Rightarrow P = 4/36 \\
Sum = 10: 3 ways \Rightarrow P = 3/36 = 1/12 \\
Sum = 11: 2 ways \Rightarrow P = 2/36 = 1/18 \\
Sum = 12: {(6,6)} \Rightarrow P = 1/36 \\
\\
Doublets: (1,1),(2,2),(3,3),(4,4),(5,5),(6,6) \Rightarrow P = 6/36 = 1/6
\end{gather*}
14.5 Addition Rule
\begin{gather*}
P(A∪B) = P(A) + P(B) - P(A∩B) \\
\\
\text{If} A \text{and} B are mutually exclusive: P(A∩B) = 0 \\
P(A∪B) = P(A) + P(B) \\
\\
P(A∩B) = P(A) + P(B) - P(A∪B)
\end{gather*}
14.6 Odds
\begin{gather*}
Odds in favor of E = P(E) : P(Ē) = n(E) : n(Ē) \\
Odds against E = P(Ē) : P(E) = n(Ē) : n(E) \\
\\
\text{If} odds in favor = m:n: \\
P(E) = m/(m+n) \\
P(Ē) = n/(m+n)
\end{gather*}
🎓 Part B
Class 12 Mathematics
B1
Relations and Functions
1.1 Types of Relations
\begin{gather*}
Empty Relation: R = ∅ \\
Universal Relation: R = A \times A \\
Reflexive: (a,a) ∈ R \text{for} all a ∈ A \\
Symmetric: (a,b) ∈ R \Rightarrow (b,a) ∈ R \\
Transitive: (a,b)∈R \text{and} (b,c)∈R \Rightarrow (a,c)∈R \\
Equivalence: Reflexive + Symmetric + Transitive
\end{gather*}
1.2 Types of Functions
\begin{gather*}
One-one (injective): f(a)=f(b) ⟹ a=b \\
Onto (surjective): Range = Codomain \\
Bijective: One-one + Onto \\
\\
Number of functions from A to B: |B|^|A| = n(B)^n(A) \\
Number of one-one functions: ⁿ⁽ᴮ⁾Pₙ₍ₐ₎ = P(n(B), n(A)) [\text{if} n(B)\ge n(A)] \\
Number of onto functions (|A|=m, |B|=n): \Sigma (-1)^k C(n,k)(n-k)^m [inclusion-exclusion]
\end{gather*}
1.3 Composition of Functions
\begin{gather*}
(g∘f)(x) = g(f(x)) \\
(f∘g)(x) = f(g(x)) \\
\\
(g∘f) \neq (f∘g) in general [\text{not} commutative] \\
(h∘g)∘f = h∘(g∘f) [associative] \\
\\
\text{If} f \text{and} g both one-one \Rightarrow g∘f one-one \\
\text{If} f \text{and} g both onto \Rightarrow g∘f onto \\
\text{If} g∘f one-one \Rightarrow f \text{is} one-one \\
\text{If} g∘f onto \Rightarrow g \text{is} onto
\end{gather*}
1.4 Inverse Function
\begin{gather*}
\text{If} f: A\Rightarrow B bijective, then f⁻¹: B\Rightarrow A exists \\
f⁻¹(f(x)) = x \\
f(f⁻¹(y)) = y \\
f∘f⁻¹ = f⁻¹∘f = I (identity) \\
\\
To find f⁻¹: let y = f(x), solve \text{for} x in terms of y
\end{gather*}
1.5 Binary Operations
\begin{gather*}
Closure: a*b ∈ A \text{for} all a,b ∈ A \\
Commutative: a*b = b*a \\
Associative: (a*b)*c = a*(b*c) \\
Identity element e: a*e = e*a = a \\
Inverse of a: a*a⁻¹ = e (\text{if} identity exists) \\
\\
Number of binary operations on set with n elements = n^(n^2) \\
Number of commutative binary operations = n^(n(n+1)/2)
\end{gather*}
B2
Inverse Trigonometric Functions
2.1 Domain and Range
\begin{gather*}
Function Domain Range (Principal) \\
sin⁻¹x [-1, 1] [-\pi /2, \pi /2] \\
cos⁻¹x [-1, 1] [0, \pi ] \\
tan⁻¹x (-∞, ∞) = ℝ (-\pi /2, \pi /2) \\
cosec⁻¹x (-∞,-1]∪[1,∞) [-\pi /2,\pi /2]\{0} \\
sec⁻¹x (-∞,-1]∪[1,∞) [0,\pi ]\{\pi /2} \\
cot⁻¹x (-∞, ∞) = ℝ (0, \pi )
\end{gather*}
2.2 Basic Identities
\begin{gather*}
sin⁻¹x + cos⁻¹x = \pi /2 (|x| \le 1) \\
tan⁻¹x + cot⁻¹x = \pi /2 (x ∈ ℝ) \\
sec⁻¹x + cosec⁻¹x = \pi /2 (|x| \ge 1) \\
\\
\Rightarrow cos⁻¹x = \pi /2 - sin⁻¹x \\
\Rightarrow sin⁻¹x = \pi /2 - cos⁻¹x \\
\Rightarrow cot⁻¹x = \pi /2 - tan⁻¹x \\
\Rightarrow tan⁻¹x = \pi /2 - cot⁻¹x
\end{gather*}
2.3 Negative Argument
\begin{gather*}
sin⁻¹(-x) = -sin⁻¹x (odd function) \\
tan⁻¹(-x) = -tan⁻¹x (odd function) \\
cosec⁻¹(-x) = -cosec⁻¹x (odd function) \\
\\
cos⁻¹(-x) = \pi - cos⁻¹x (\text{not} odd!) \\
sec⁻¹(-x) = \pi - sec⁻¹x \\
cot⁻¹(-x) = \pi - cot⁻¹x
\end{gather*}
2.4 Reciprocal Relations
\begin{gather*}
sin⁻¹(1/x) = cosec⁻¹x (x \ge 1 \text{or} x \le -1) \\
cos⁻¹(1/x) = sec⁻¹x (x \ge 1 \text{or} x \le -1) \\
tan⁻¹(1/x) = cot⁻¹x (x > 0) \\
tan⁻¹(1/x) = cot⁻¹x - \pi (x < 0) \\
\\
\Rightarrow cosec⁻¹x = sin⁻¹(1/x) \\
\Rightarrow sec⁻¹x = cos⁻¹(1/x) \\
\Rightarrow cot⁻¹x = tan⁻¹(1/x) \text{for} x>0, cot⁻¹x = \pi + tan⁻¹(1/x) \text{for} x<0
\end{gather*}
2.5 Double Angle Formulas
\begin{gather*}
2sin⁻¹x = sin⁻¹(2x\sqrt{1-x^2}) (|x| \le 1/\sqrt{2}) \\
2cos⁻¹x = cos⁻¹(2x^2-1) (0 \le x \le 1) \\
2tan⁻¹x = tan⁻¹(2x/(1-x^2)) (|x| < 1) \\
2tan⁻¹x = sin⁻¹(2x/(1+x^2)) (|x| \le 1) \\
2tan⁻¹x = cos⁻¹((1-x^2)/(1+x^2)) (x \ge 0) \\
3tan⁻¹x = tan⁻¹(3x-x^3)/(1-3x^2) (|x|<1/\sqrt{3})
\end{gather*}
2.6 Sum and Difference Formulas
\begin{gather*}
sin⁻¹x ± sin⁻¹y = sin⁻¹[x\sqrt{1-y^2} ± y\sqrt{1-x^2}] \\
cos⁻¹x ± cos⁻¹y = cos⁻¹[xy ∓ \sqrt{1-x^2}\sqrt{1-y^2}] \\
\\
tan⁻¹x + tan⁻¹y = tan⁻¹[(x+y)/(1-xy)] (xy < 1) \\
= \pi + tan⁻¹[(x+y)/(1-xy)] (xy > 1, x>0, y>0) \\
= -\pi + tan⁻¹[(x+y)/(1-xy)] (xy > 1, x<0, y<0) \\
\\
tan⁻¹x - tan⁻¹y = tan⁻¹[(x-y)/(1+xy)] (xy > -1) \\
= \pi + tan⁻¹[(x-y)/(1+xy)] (x>0, xy<-1) \\
= -\pi + tan⁻¹[(x-y)/(1+xy)] (x<0, xy<-1) \\
\\
Derived: \\
tan⁻¹(1/2) + tan⁻¹(1/3) = \pi /4 \\
tan⁻¹1 + tan⁻¹2 + tan⁻¹3 = \pi
\end{gather*}
2.7 Conversion Formulas
\begin{gather*}
sin⁻¹x = cos⁻¹(\sqrt{1-x^2}) (x \ge 0) \\
= tan⁻¹(x/\sqrt{1-x^2}) (|x|<1) \\
\\
cos⁻¹x = sin⁻¹(\sqrt{1-x^2}) (x \ge 0) \\
= tan⁻¹(\sqrt{1-x^2}/x) \\
\\
tan⁻¹x = sin⁻¹(x/\sqrt{1+x^2}) \\
= cos⁻¹(1/\sqrt{1+x^2}) \\
\\
\text{For} \theta = sin⁻¹x: \\
sin\theta =x, cos\theta =\sqrt{1-x^2}, tan\theta =x/\sqrt{1-x^2} \\
\\
\text{For} \theta = cos⁻¹x: \\
cos\theta =x, sin\theta =\sqrt{1-x^2}, tan\theta =\sqrt{1-x^2}/x \\
\\
\text{For} \theta = tan⁻¹x: \\
tan\theta =x, sin\theta =x/\sqrt{1+x^2}, cos\theta =1/\sqrt{1+x^2}
\end{gather*}
B3
Matrices
3.1 Matrix Notation
\begin{gather*}
A = [aᵢⱼ]ₘₓₙ (m rows, n columns) \\
Order = m \times n, Total elements = mn \\
aᵢⱼ = element in ith row, jth column
\end{gather*}
3.2 Types of Matrices
\begin{gather*}
Row matrix: 1 \times n \\
Column matrix: m \times 1 \\
Square matrix: m = n \\
Diagonal matrix: aᵢⱼ = 0 \text{if} i \neq j \\
Scalar matrix: Diagonal with all diagonal elements equal \\
Identity matrix: I, diagonal with all 1s \\
Zero matrix: O, all elements 0 \\
Upper triangular: aᵢⱼ = 0 \text{if} i > j \\
Lower triangular: aᵢⱼ = 0 \text{if} i < j
\end{gather*}
3.3 Transpose
\begin{gather*}
\text{If} A = [aᵢⱼ]ₘₓₙ, then Aᵀ = [aⱼᵢ]ₙₓₘ \\
\\
Properties: \\
(Aᵀ)ᵀ = A \\
(A+B)ᵀ = Aᵀ + Bᵀ \\
(kA)ᵀ = kAᵀ \\
(AB)ᵀ = BᵀAᵀ ← REVERSED ORDER! \\
\\
Symmetric matrix: A = Aᵀ \Rightarrow aᵢⱼ = aⱼᵢ \\
Skew-symmetric: A = -Aᵀ \Rightarrow aᵢⱼ = -aⱼᵢ, diagonal = 0 \\
\\
Any square matrix A: \\
A = ½(A+Aᵀ) + ½(A-Aᵀ) [symmetric + skew-symmetric]
\end{gather*}
3.4 Matrix Operations
\begin{gather*}
Addition: (A+B)ᵢⱼ = aᵢⱼ + bᵢⱼ [same order] \\
Scalar: (kA)ᵢⱼ = kaᵢⱼ \\
Multiplication: (AB)ᵢⱼ = \Sigma _k aᵢ_kb_kⱼ [A: m\times n, B: n\times p \Rightarrow AB: m\times p] \\
\\
AB \neq BA in general [\text{not} commutative] \\
A(BC) = (AB)C [associative] \\
A(B+C) = AB+AC [distributive] \\
AO = OA = O \\
AI = IA = A
\end{gather*}
3.5 Elementary Row/Column Operations
\begin{gather*}
Rᵢ ↔ Rⱼ [swap rows i \text{and} j] \\
Rᵢ \Rightarrow kRᵢ [multiply row i by k \neq 0] \\
Rᵢ \Rightarrow Rᵢ + kRⱼ [add k times row j to row i]
\end{gather*}
3.6 Invertible Matrix
\begin{gather*}
AA⁻¹ = A⁻¹A = I \\
A⁻¹ = adjA / |A| (\text{if} |A| \neq 0) \\
\\
(A⁻¹)⁻¹ = A \\
(AB)⁻¹ = B⁻¹A⁻¹ ← REVERSED! \\
(Aᵀ)⁻¹ = (A⁻¹)ᵀ \\
(kA)⁻¹ = (1/k)A⁻¹ \\
|A⁻¹| = 1/|A|
\end{gather*}
3.7 2×2 Matrix — Quick Formulas
\begin{gather*}
A = [a b; c d] \\
\\
|A| = ad - bc \\
A⁻¹ = (1/(ad-bc)) \times [d -b; -c a] \\
\\
adj A = [d -b; -c a] (transpose of cofactor matrix) \\
\\
\text{For} 2\times 2: adj A \text{is} obtained by: \\
swapping diagonal elements \\
negating off-diagonal elements
\end{gather*}
B4
Determinants
4.1 Determinant of 2×2
\begin{gather*}
|A| = |a b| = ad - bc \\
|c d|
\end{gather*}
4.2 Determinant of 3×3 (Expansion along R₁)
\begin{gather*}
|a_1 b_1 c_1| \\
|a_2 b_2 c_2| = a_1(b_2c_3-b_3c_2) - b_1(a_2c_3-a_3c_2) + c_1(a_2b_3-a_3b_2) \\
|a_3 b_3 c_3| \\
\\
= a_1M_1_1 - b_1M_1_2 + c_1M_1_3 \\
(Mᵢⱼ = minor of element aᵢⱼ) \\
\\
Cofactor Cᵢⱼ = (-1)^(i+j) Mᵢⱼ \\
Sign pattern \text{for} cofactors: \\
+ - + \\
- + - \\
+ - +
\end{gather*}
4.3 Properties of Determinants
\begin{gather*}
|Aᵀ| = |A| \\
|kA|ₙₓₙ = kⁿ|A| \\
|AB| = |A||B| \\
|A⁻¹| = 1/|A| \\
|A^2| = |A|^2 \\
|Aⁿ| = |A|ⁿ \\
|adj A| = |A|^(n-1) [\text{for} n\times n matrix] \\
\\
\text{If} any row/column = 0 \Rightarrow |A| = 0 \\
\text{If} two rows/columns identical \Rightarrow |A| = 0 \\
\text{If} one row \text{is} k\times another row \Rightarrow |A| = 0 \\
Swapping two rows/columns \Rightarrow |A| changes sign
\end{gather*}
4.4 Adjoint and Inverse
\begin{gather*}
adj A = transpose of cofactor matrix \\
A(adj A) = (adj A)A = |A| \times I \\
\\
A⁻¹ = adj A / |A| (|A| \neq 0) \\
\\
|adj A| = |A|^(n-1) \\
adj(AB) = (adj B)(adj A) \\
adj(adj A) = |A|^(n-2) \times A [n\times n] \\
adj(Aᵀ) = (adj A)ᵀ \\
adj(kA) = k^(n-1) adj A
\end{gather*}
4.5 Cramer's Rule
\begin{gather*}
\text{For} AX = B: \\
x = D_1/D, y = D_2/D, z = D_3/D \\
\\
D = |A| (coefficient determinant) \\
D_1 = replace column 1 of A by B \\
D_2 = replace column 2 of A by B \\
D_3 = replace column 3 of A by B \\
\\
\text{If} D \neq 0 \Rightarrow unique solution \\
\text{If} D = 0 \text{and} D_1 = D_2 = D_3 = 0 \Rightarrow infinitely many / no solution \\
\text{If} D = 0 \text{and} any Dᵢ \neq 0 \Rightarrow no solution (inconsistent)
\end{gather*}
4.6 Area Using Determinants
\begin{gather*}
Area of △ with vertices (x_1,y_1), (x_2,y_2), (x_3,y_3): \\
\\
Area = ½ |x_1 y_1 1| \\
|x_2 y_2 1| \\
|x_3 y_3 1| \\
\\
Collinear \text{if} Area = 0: \\
|x_1 y_1 1| \\
|x_2 y_2 1| = 0 \\
|x_3 y_3 1| \\
\\
Equation of line through (x_1,y_1) \text{and} (x_2,y_2): \\
|x y 1| \\
|x_1 y_1 1| = 0 \\
|x_2 y_2 1| \\
\Rightarrow (y-y_1)(x_2-x_1) = (x-x_1)(y_2-y_1)
\end{gather*}
B5
Continuity and Differentiability
5.1 Continuity Condition
\begin{gather*}
f \text{is} continuous at x=a \text{if}: \\
1. f(a) \text{is} defined \\
2. lim[x\Rightarrow a] f(x) exists \\
3. lim[x\Rightarrow a] f(x) = f(a) \\
\\
LHL = lim[x\Rightarrow a⁻] f(x) = lim[h\Rightarrow 0] f(a-h) \\
RHL = lim[x\Rightarrow a⁺] f(x) = lim[h\Rightarrow 0] f(a+h) \\
Continuous ↔ LHL = RHL = f(a)
\end{gather*}
5.2 Standard Derivatives
\begin{gather*}
d/dx (constant) = 0 \\
d/dx (x) = 1 \\
d/dx (xⁿ) = nxⁿ⁻¹ \\
d/dx (\sqrt{x}) = 1/(2\sqrt{x}) \\
d/dx (1/x) = -1/x^2 \\
d/dx (1/xⁿ) = -n/x^(n+1) \\
\\
Exponential & Log: \\
d/dx (eˣ) = eˣ \\
d/dx (aˣ) = aˣ ln a \\
d/dx (ln x) = 1/x \\
d/dx (log_a x) = 1/(x ln a) \\
\\
Trigonometric: \\
d/dx (sin x) = cos x \\
d/dx (cos x) = -sin x \\
d/dx (tan x) = sec^2x \\
d/dx (cot x) = -cosec^2x \\
d/dx (sec x) = sec x tan x \\
d/dx (cosec x) = -cosec x cot x \\
\\
Inverse Trig: \\
d/dx (sin⁻¹x) = 1/\sqrt{1-x^2} \\
d/dx (cos⁻¹x) = -1/\sqrt{1-x^2} \\
d/dx (tan⁻¹x) = 1/(1+x^2) \\
d/dx (cot⁻¹x) = -1/(1+x^2) \\
d/dx (sec⁻¹x) = 1/(|x|\sqrt{x^2-1}) \\
d/dx (cosec⁻¹x) = -1/(|x|\sqrt{x^2-1})
\end{gather*}
5.3 Rules of Differentiation
\begin{gather*}
Sum/Difference: (u ± v)' = u' ± v' \\
Scalar: (cu)' = cu' \\
Product: (uv)' = u'v + uv' ← Product Rule (Leibniz) \\
Quotient: (u/v)' = (u'v - uv')/v^2 ← Quotient Rule (v\neq 0) \\
Chain: d/dx[f(g(x))] = f'(g(x))·g'(x) ← Chain Rule \\
\\
Extended Product Rule: \\
(uvw)' = u'vw + uv'w + uvw' \\
\\
Extended Chain Rule: \\
d/dx[f(g(h(x)))] = f'(g(h(x)))·g'(h(x))·h'(x)
\end{gather*}
5.4 Implicit Differentiation
\begin{gather*}
\text{For} F(x,y) = 0: \\
dy/dx = -(∂F/∂x)/(∂F/∂y) [partial derivatives method] \\
\\
\text{Or} differentiate both sides w.r.t. x, treating y as function of x \\
\\
Example: x^2 + y^2 = r^2 \\
\Rightarrow 2x + 2y(dy/dx) = 0 \\
\Rightarrow dy/dx = -x/y
\end{gather*}
5.5 Parametric Differentiation
\begin{gather*}
x = f(t), y = g(t) \\
dy/dx = (dy/dt)/(dx/dt) = g'(t)/f'(t) \\
\\
d^2y/dx^2 = (d/dt[dy/dx]) / (dx/dt)
\end{gather*}
5.6 Logarithmic Differentiation
\begin{gather*}
\text{For} y = [f(x)]^g(x): \\
ln y = g(x) ln[f(x)] \\
(1/y)(dy/dx) = g'(x)ln[f(x)] + g(x)·f'(x)/f(x) \\
dy/dx = y \times [g'(x)ln f(x) + g(x)f'(x)/f(x)] \\
\\
\text{For} y = (f_1·f_2·...)/(g_1·g_2·...): \\
ln y = ln f_1 + ln f_2 + ... - ln g_1 - ln g_2 - ... \\
(1/y)y' = f_1'/f_1 + f_2'/f_2 + ... - g_1'/g_1 - ...
\end{gather*}
5.7 Higher Order Derivatives
\begin{gather*}
y = f(x) \\
y' = f'(x) = dy/dx [first derivative] \\
y'' = f''(x) = d^2y/dx^2 [second derivative] \\
y''' = f'''(x) = d^3y/dx^3 [third derivative] \\
yⁿ = f⁽ⁿ⁾(x) = dⁿy/dxⁿ [nth derivative] \\
\\
Leibniz Formula (nth derivative of product): \\
(uv)ⁿ = \Sigma _k₌₀ⁿ C(n,k) u⁽ᵏ⁾ v⁽ⁿ⁻ᵏ⁾
\end{gather*}
5.8 Rolle's Theorem
\begin{gather*}
\text{If} f: [a,b]\Rightarrow ℝ \text{is}: \\
1. Continuous on [a,b] \\
2. Differentiable on (a,b) \\
3. f(a) = f(b) \\
Then ∃ c ∈ (a,b) such that f'(c) = 0
\end{gather*}
5.9 Mean Value Theorem (Lagrange's MVT)
\begin{gather*}
\text{If} f: [a,b]\Rightarrow ℝ \text{is}: \\
1. Continuous on [a,b] \\
2. Differentiable on (a,b) \\
Then ∃ c ∈ (a,b) such that: \\
f'(c) = [f(b)-f(a)] / (b-a) \\
\\
\Rightarrow f(b) - f(a) = f'(c)(b-a) \\
\Rightarrow c = value(s) satisfying the above \\
\\
[Geometric meaning: slope of chord = slope of tangent at c]
\end{gather*}
B6
Application of Derivatives
6.1 Rate of Change
\begin{gather*}
Rate of change of y w.r.t. x = dy/dx \\
Rate of change of y w.r.t. t = dy/dt \\
\\
\text{If} y = f(x) \text{and} x = g(t): \\
dy/dt = (dy/dx)(dx/dt) \\
\\
Rate of change of area of circle w.r.t. radius: \\
dA/dr = 2\pi r \\
\\
Rate of change of volume of sphere w.r.t. radius: \\
dV/dr = 4\pi r^2 \\
\\
Rate of change of volume of cube w.r.t. side: \\
dV/da = 3a^2
\end{gather*}
6.2 Tangent and Normal
\begin{gather*}
Equation of tangent at (x_1,y_1): \\
y - y_1 = m(x - x_1), \text{where} m = (dy/dx) at (x_1,y_1) \\
\\
Equation of normal at (x_1,y_1): \\
y - y_1 = -1/m (x - x_1), \text{where} m = (dy/dx) at (x_1,y_1) \\
\\
Slope of tangent: m_t = f'(x_1) \\
Slope of normal: m_n = -1/f'(x_1) \\
\\
m_t \times m_n = -1 (tangent ⊥ normal) \\
\\
Length of tangent = y\sqrt{1+m^2}/m \\
Length of normal = y\sqrt{1+m^2} \\
Length of subtangent = y/m = y/(dy/dx) \\
Length of subnormal = y \times m = y(dy/dx) \\
\\
Tangent parallel to x-axis: dy/dx = 0 \\
Tangent parallel to y-axis: dy/dx \Rightarrow ∞ (dx/dy = 0) \\
Tangent passes through origin: y/x = dy/dx (i.e., y=mx, m=dy/dx)
\end{gather*}
6.3 Increasing and Decreasing Functions
\begin{gather*}
f \text{is} strictly increasing on (a,b) \text{if} f'(x) > 0 ∀ x∈(a,b) \\
f \text{is} strictly decreasing on (a,b) \text{if} f'(x) < 0 ∀ x∈(a,b) \\
f \text{is} constant on (a,b) \text{if} f'(x) = 0 ∀ x∈(a,b) \\
\\
At a critical point: f'(c) = 0 \\
\\
Monotonically increasing: f(x_1) < f(x_2) whenever x_1 < x_2 \\
Monotonically decreasing: f(x_1) > f(x_2) whenever x_1 < x_2
\end{gather*}
6.4 Maxima and Minima
\begin{gather*}
First Derivative Test: \\
f'(c) = 0 \text{and} \\
f' changes + to - at c \Rightarrow local maximum \\
f' changes - to + at c \Rightarrow local minimum \\
f' doesn't change sign \Rightarrow neither (inflection point) \\
\\
Second Derivative Test: \\
f'(c) = 0 \text{and} \\
f''(c) < 0 \Rightarrow local maximum at x=c; f(c) = local max value \\
f''(c) > 0 \Rightarrow local minimum at x=c; f(c) = local min value \\
f''(c) = 0 \Rightarrow test inconclusive (use first derivative test) \\
\\
Global/Absolute maximum/minimum on [a,b]: \\
Compare f(a), f(b), \text{and} all f(c) \text{where} f'(c)=0
\end{gather*}
6.5 Optimization Formulas
\begin{gather*}
Rectangle of maximum area \text{for} given perimeter P: \\
Square: side = P/4, Area = P^2/16 \\
\\
Rectangle of minimum perimeter \text{for} given area A: \\
Square: side = \sqrt{A}, Perimeter = 4\sqrt{A} \\
\\
Cylinder of maximum volume: r = h \\
\\
Cone of maximum volume inscribed in sphere of radius R: \\
h = 4R/3, r = 2R\sqrt{2}/3 \\
\\
Rectangle inscribed in circle of radius R: \\
Maximum area square: side = R\sqrt{2}, Area = 2R^2 \\
\\
Cylinder inscribed in sphere of radius R: \\
Maximum volume: h = 2R/\sqrt{3}, r = R\sqrt{2/3} \\
V_max = 4\pi R^3/(3\sqrt{3}) \\
\\
Wire of length L bent to maximize area: \\
\text{If} circle: r = L/2\pi , A = L^2/4\pi \\
\text{If} square: side = L/4, A = L^2/16 \\
Circle has more area than square \text{for} same perimeter \\
\\
Box (open top) from square sheet of side a, corners cut x: \\
V = x(a-2x)^2 \\
dV/dx = 0 \Rightarrow x = a/6 \text{for} max volume \\
V_max = 2a^3/27
\end{gather*}
6.6 Approximations (Linear Approximation)
\begin{gather*}
\Delta y ≈ dy = f'(x)·\Delta x [\text{for} small \Delta x] \\
\\
f(x + \Delta x) ≈ f(x) + f'(x)·\Delta x \\
\\
Relative error: dy/y \\
Percentage error: (dy/y)\times 100 \\
\\
\text{If} y = xⁿ: \Delta y/y ≈ n(\Delta x/x) [percentage error multiplied by n]
\end{gather*}
B7
Integrals
7.1 Standard Integration Formulas
\begin{gather*}
∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n \neq -1) \\
∫ 1/x dx = ln|x| + C \\
∫ eˣ dx = eˣ + C \\
∫ aˣ dx = aˣ/ln a + C \\
∫ 1 dx = x + C \\
\\
Trigonometric: \\
∫ sin x dx = -cos x + C \\
∫ cos x dx = sin x + C \\
∫ tan x dx = ln|sec x| + C = -ln|cos x| + C \\
∫ cot x dx = ln|sin x| + C \\
∫ sec x dx = ln|sec x + tan x| + C \\
∫ cosec x dx = ln|cosec x - cot x| + C \\
\\
∫ sec^2x dx = tan x + C \\
∫ cosec^2x dx = -cot x + C \\
∫ sec x tan x dx = sec x + C \\
∫ cosec x cot x dx = -cosec x + C \\
\\
Special: \\
∫ 1/\sqrt{1-x^2} dx = sin⁻¹x + C = -cos⁻¹x + C \\
∫ -1/\sqrt{1-x^2} dx = cos⁻¹x + C \\
∫ 1/(1+x^2) dx = tan⁻¹x + C = -cot⁻¹x + C \\
∫ 1/(x\sqrt{x^2-1}) dx = sec⁻¹x + C = -cosec⁻¹x + C
\end{gather*}
7.2 Standard Forms (Key Templates)
\begin{gather*}
∫ 1/(x^2+a^2) dx = (1/a)tan⁻¹(x/a) + C \\
\\
∫ 1/(x^2-a^2) dx = (1/2a)ln|(x-a)/(x+a)| + C (|x|>a) \\
\\
∫ 1/(a^2-x^2) dx = (1/2a)ln|(a+x)/(a-x)| + C (|x|
7.3 (a) Substitution Method (u-substitution)
\begin{gather*}
∫ f(g(x))·g'(x) dx = ∫ f(u) du [let u = g(x)] \\
\\
Common substitutions: \\
\sqrt{a^2-x^2} \Rightarrow x = a sin\theta \\
\sqrt{a^2+x^2} \Rightarrow x = a tan\theta \\
\sqrt{x^2-a^2} \Rightarrow x = a sec\theta \\
(a-x)/(a+x) \text{or} similar \Rightarrow x = a cos2\theta
\end{gather*}
7.3 (b) Integration by Parts (IBP)
\begin{gather*}
∫ u dv = uv - ∫ v du \\
\\
\text{or} ∫ u·v dx = u(∫v dx) - ∫[u'(∫v dx)] dx \\
\\
ILATE priority rule (choose u in this order): \\
I - Inverse trig (sin⁻¹x, tan⁻¹x...) \\
L - Logarithmic (ln x, log x) \\
A - Algebraic (xⁿ, polynomials) \\
T - Trigonometric (sin x, cos x...) \\
E - Exponential (eˣ, aˣ) \\
\\
Standard IBP results: \\
∫ xeˣ dx = eˣ(x-1) + C \\
∫ x^2eˣ dx = eˣ(x^2-2x+2) + C \\
∫ xⁿeˣ dx = eˣ[xⁿ - nxⁿ⁻¹ + n(n-1)xⁿ⁻^2 - ...] + C \\
∫ x sin x dx = -x cos x + sin x + C \\
∫ x cos x dx = x sin x + cos x + C \\
∫ x^2 sin x dx = -x^2cos x + 2x sin x + 2cos x + C \\
∫ ln x dx = x ln x - x + C \\
∫ x ln x dx = x^2/2 ln x - x^2/4 + C \\
∫ sin⁻¹x dx = x sin⁻¹x + \sqrt{1-x^2} + C \\
∫ cos⁻¹x dx = x cos⁻¹x - \sqrt{1-x^2} + C \\
∫ tan⁻¹x dx = x tan⁻¹x - (1/2)ln(1+x^2) + C \\
\\
Special formula: \\
∫ eˣ[f(x) + f'(x)] dx = eˣ f(x) + C
\end{gather*}
7.3 (c) Partial Fractions
\begin{gather*}
Proper fraction: degree(numerator) < degree(denominator) \\
\\
Case 1: Distinct linear factors \\
P(x)/[(ax+b)(cx+d)] = A/(ax+b) + B/(cx+d) \\
\\
Case 2: Repeated linear factors \\
P(x)/(ax+b)^2 = A/(ax+b) + B/(ax+b)^2 \\
P(x)/(ax+b)^3 = A/(ax+b) + B/(ax+b)^2 + C/(ax+b)^3 \\
\\
Case 3: Irreducible quadratic \\
P(x)/[(ax+b)(x^2+bx+c)] = A/(ax+b) + (Bx+C)/(x^2+bx+c) \\
\\
Case 4: Improper \Rightarrow divide first \\
\text{If} degree(P) \ge degree(Q): P/Q = quotient + remainder/Q
\end{gather*}
7.3 (d) Integration of Rational Trig Functions
\begin{gather*}
∫ 1/(a + b sinx) dx: let t = tan(x/2) \\
sinx = 2t/(1+t^2), cosx = (1-t^2)/(1+t^2), dx = 2dt/(1+t^2) \\
\\
∫ 1/(a sinx + b cosx) dx = (1/\sqrt{a^2+b^2}) ln|tan(x/2 + \alpha )| + C \\
\text{where} tan \alpha = a/b \\
\\
R sin(x+\alpha ) form: a sinx + b cosx = R sin(x+\alpha ) \\
R = \sqrt{a^2+b^2}, tan \alpha = b/a \\
\\
R cos(x-\alpha ) form: a cosx + b sinx = R cos(x-\alpha ) \\
R = \sqrt{a^2+b^2}, tan \alpha = b/a
\end{gather*}
7.4 Definite Integral Properties
\begin{gather*}
∫ₐᵇ f(x) dx = F(b) - F(a) [\text{where} F'(x)=f(x)] \\
\\
P1: ∫ₐᵇ f(x) dx = -∫ᵦₐ f(x) dx \\
P2: ∫ₐᵃ f(x) dx = 0 \\
P3: ∫ₐᵇ f(x) dx = ∫ₐᶜ f(x) dx + ∫ᶜᵇ f(x) dx \\
P4: ∫ₐᵇ f(x) dx = ∫ₐᵇ f(a+b-x) dx ← KING Property \\
P5: ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a-x) dx ← Special King \\
P6: ∫₀^2ᵃ f(x) dx = 2∫₀ᵃ f(x) dx [\text{if} f(2a-x) = f(x)] \\
= 0 [\text{if} f(2a-x) = -f(x)] \\
P7: ∫₋ₐᵃ f(x) dx = 2∫₀ᵃ f(x) dx [\text{if} f even: f(-x)=f(x)] \\
= 0 [\text{if} f odd: f(-x)=-f(x)] \\
P8: ∫₀ⁿᵀ f(x) dx = n∫₀ᵀ f(x) dx [\text{if} f periodic with period T]
\end{gather*}
7.5 Definite Integral as Limit of Sum
\begin{gather*}
∫ₐᵇ f(x) dx = lim[n\Rightarrow ∞] h \Sigma _k₌₀ⁿ⁻¹ f(a+kh) \\
\\
\text{where} h = (b-a)/n \\
\\
\text{Or}: ∫₀¹ f(x) dx = lim[n\Rightarrow ∞] (1/n) \Sigma _k₌_1ⁿ f(k/n)
\end{gather*}
7.6 Important Definite Integrals
\begin{gather*}
∫₀^(\pi /2) sin x dx = ∫₀^(\pi /2) cos x dx = 1 \\
\\
∫₀^\pi sin x dx = 2, ∫₀^\pi cos x dx = 0 \\
\\
∫₀^(\pi /2) sin^2x dx = ∫₀^(\pi /2) cos^2x dx = \pi /4 \\
\\
∫₀^(\pi /2) sinⁿx dx: \\
n=2: \pi /4 \\
n=3: 2/3 \\
n=4: 3\pi /16 \\
\\
∫₀^(\pi /2) ln(sinx) dx = ∫₀^(\pi /2) ln(cosx) dx = -(\pi /2)ln2 \\
\\
∫₀^\pi x f(sinx) dx = (\pi /2)∫₀^\pi f(sinx) dx [using King property]
\end{gather*}
7.7 Walli's Formula
\begin{gather*}
∫₀^(\pi /2) sinⁿx dx = ∫₀^(\pi /2) cosⁿx dx = \\
[(n-1)(n-3)...3·1]/[n(n-2)...4·2] \times \pi /2 (n even) \\
[(n-1)(n-3)...4·2]/[n(n-2)...3·1] (n odd)
\end{gather*}
B8
Application of Integrals
8.1 Area Under a Curve
\begin{gather*}
Area between y=f(x) \text{and} x-axis, from x=a to x=b: \\
A = ∫ₐᵇ |f(x)| dx \\
\\
\text{If} f(x) \ge 0: A = ∫ₐᵇ f(x) dx \\
\text{If} f(x) \le 0: A = -∫ₐᵇ f(x) dx = ∫ₐᵇ |f(x)| dx \\
\\
Area between x=g(y) \text{and} y-axis, from y=c to y=d: \\
A = ∫ᶜᵈ |g(y)| dy
\end{gather*}
8.2 Area Between Two Curves
\begin{gather*}
Area between y=f(x) \text{and} y=g(x), a to b [f(x) \ge g(x)]: \\
A = ∫ₐᵇ [f(x) - g(x)] dx \\
\\
Area between x=f(y) \text{and} x=g(y) [f(y) \ge g(y)]: \\
A = ∫ᶜᵈ [f(y) - g(y)] dy \\
\\
Finding intersection points: solve f(x) = g(x)
\end{gather*}
8.3 Standard Areas
\begin{gather*}
Circle x^2+y^2=r^2: Area = \pi r^2 (full) \\
Area = \pi r^2/2 (semicircle) \\
\\
Ellipse x^2/a^2+y^2/b^2=1: Area = \pi ab \\
\\
Area of parabola y=x^2 \text{and} y=x: \\
Intersect at x=0,1 \\
A = ∫₀¹(x-x^2)dx = [x^2/2 - x^3/3]₀¹ = 1/2-1/3 = 1/6 \\
\\
Area of ellipse arc with x-axis: ∫₀ᵃ b\sqrt{1-x^2/a^2}dx = \pi ab/4 [first quadrant]
\end{gather*}
B9
Differential Equations
9.1 Order, Degree, and Type
\begin{gather*}
Order = highest derivative present \\
Degree = power of highest order derivative (\text{when} polynomial in derivatives) \\
Degree undefined \text{if} sin(y'), eʸ' etc. \\
\\
Linear DE: highest power of y \text{and} its derivatives \text{is} 1 \\
Non-linear: otherwise
\end{gather*}
9.2 Variable Separable Method
\begin{gather*}
dy/dx = f(x)·g(y) \\
\\
\Rightarrow dy/g(y) = f(x)dx \\
\Rightarrow ∫ dy/g(y) = ∫ f(x)dx + C \\
\\
Particular solution: use initial condition to find C
\end{gather*}
9.3 Homogeneous Differential Equation
\begin{gather*}
dy/dx = f(x,y)/g(x,y) \text{where} f,g are homogeneous of same degree \\
\\
Substitution: y = vx \Rightarrow dy/dx = v + x(dv/dx) \\
\\
v + x(dv/dx) = F(v) \\
x(dv/dx) = F(v) - v \\
∫ dv/[F(v)-v] = ∫ dx/x + C \\
\\
[After integrating, back-substitute v = y/x]
\end{gather*}
9.4 Linear Differential Equation (First Order)
\begin{gather*}
dy/dx + P(x)y = Q(x) \\
\\
Integrating Factor (\text{IF}) = e^(∫P dx) \\
\\
Solution: y \times \text{IF} = ∫[Q \times \text{IF}] dx + C \\
\\
\Rightarrow ye^(∫Pdx) = ∫[Q·e^(∫Pdx)] dx + C \\
\\
\text{For} dx/dy + P(y)x = Q(y) [x as dependent]: \\
\text{IF} = e^(∫P(y)dy) \\
x \times \text{IF} = ∫[Q(y)·\text{IF}] dy + C
\end{gather*}
9.5 Bernoulli's Equation
\begin{gather*}
dy/dx + P(x)y = Q(x)yⁿ (n \neq 0, 1) \\
\\
Substitution: z = y^(1-n) \\
dz/dx = (1-n)y^(-n)(dy/dx) \\
\\
Becomes linear: dz/dx + (1-n)P(x)z = (1-n)Q(x)
\end{gather*}
9.6 General Solutions
\begin{gather*}
∫ dy/y = ln|y| \\
∫ dy/y^2 = -1/y \\
∫ dy/(1+y^2) = tan⁻¹y \\
\\
\text{For} xdy + ydx = d(xy) [exact differential] \\
\text{For} (xdy-ydx)/x^2 = d(y/x) \\
\text{For} (ydx-xdy)/y^2 = d(x/y) \\
\text{For} (xdx+ydy) = (1/2)d(x^2+y^2)
\end{gather*}
9.7 Important Differential Equations and Solutions
\begin{gather*}
dy/dx = ky: y = Ce^(kx) [growth/decay] \\
d^2y/dx^2 + n^2y = 0: y = A cos(nx) + B sin(nx) \\
d^2y/dx^2 - n^2y = 0: y = Ae^(nx) + Be^(-nx) \\
d^2y/dx^2 = f(x): integrate twice \\
\\
Population: dP/dt = kP \Rightarrow P = P₀e^(kt) \\
Radioactive: dN/dt = -λN \Rightarrow N = N₀e^(-λt) \\
Newton's cooling: dT/dt = -k(T-T₀) \Rightarrow T-T₀ = (T_1-T₀)e^(-kt)
\end{gather*}
B10
Vector Algebra
10.1 Vector Notation
\begin{gather*}
Position vector of A(x,y,z): a⃗ = xî + yĵ + zk̂ \\
\\
Magnitude: |a⃗| = \sqrt{x^2+y^2+z^2} \\
\\
Unit vector: â = a⃗/|a⃗| \\
\\
Zero vector: 0⃗ = 0î + 0ĵ + 0k̂ \\
\\
Direction cosines: l = x/|a⃗|, m = y/|a⃗|, n = z/|a⃗| \\
l^2 + m^2 + n^2 = 1 \\
\\
Direction ratios: proportional to (x,y,z)
\end{gather*}
10.2 Vector Operations
\begin{gather*}
Addition: a⃗ + b⃗ = (a_1+b_1)î + (a_2+b_2)ĵ + (a_3+b_3)k̂ \\
Subtraction: a⃗ - b⃗ = (a_1-b_1)î + (a_2-b_2)ĵ + (a_3-b_3)k̂ \\
Scalar multiple: ka⃗ = ka_1î + ka_2ĵ + ka_3k̂ \\
\\
|a⃗ + b⃗|^2 = |a⃗|^2 + 2a⃗·b⃗ + |b⃗|^2 \\
|a⃗ - b⃗|^2 = |a⃗|^2 - 2a⃗·b⃗ + |b⃗|^2 \\
|a⃗ + b⃗|^2 + |a⃗ - b⃗|^2 = 2(|a⃗|^2 + |b⃗|^2) [Parallelogram law] \\
|a⃗ + b⃗|^2 - |a⃗ - b⃗|^2 = 4(a⃗·b⃗)
\end{gather*}
10.3 Dot Product (Scalar Product)
\begin{gather*}
a⃗·b⃗ = |a⃗||b⃗|cos\theta = a_1b_1 + a_2b_2 + a_3b_3 \\
\\
\Rightarrow cos\theta = a⃗·b⃗ / (|a⃗||b⃗|) \\
\Rightarrow \theta = cos⁻¹[a⃗·b⃗ / (|a⃗||b⃗|)] \\
\\
Perpendicular ↔ a⃗·b⃗ = 0 (a⃗ \neq 0⃗, b⃗ \neq 0⃗) \\
Parallel ↔ a⃗\times b⃗ = 0⃗ \\
\\
î·î = ĵ·ĵ = k̂·k̂ = 1 \\
î·ĵ = ĵ·k̂ = k̂·î = 0 \\
\\
a⃗·a⃗ = |a⃗|^2 \\
a⃗·b⃗ = b⃗·a⃗ [commutative] \\
\\
Projection of a⃗ on b⃗ = (a⃗·b⃗)/|b⃗| \\
Component of a⃗ along b⃗ = (a⃗·b⃗)/|b⃗| [scalar] \\
Vector projection = [(a⃗·b⃗)/|b⃗|^2] b⃗
\end{gather*}
10.4 Cross Product (Vector Product)
\begin{gather*}
a⃗\times b⃗ = |a⃗||b⃗|sin\theta n̂ \\
\\
|a⃗\times b⃗| = |a⃗||b⃗|sin\theta \\
\\
\Rightarrow sin\theta = |a⃗\times b⃗| / (|a⃗||b⃗|) \\
\\
a⃗\times b⃗ = |î ĵ k̂ | \\
|a_1 a_2 a_3| \\
|b_1 b_2 b_3| \\
\\
= î(a_2b_3-a_3b_2) - ĵ(a_1b_3-a_3b_1) + k̂(a_1b_2-a_2b_1) \\
\\
î\times î = ĵ\times ĵ = k̂\times k̂ = 0⃗ \\
î\times ĵ = k̂, ĵ\times k̂ = î, k̂\times î = ĵ [cyclic] \\
ĵ\times î = -k̂, k̂\times ĵ = -î, î\times k̂ = -ĵ [anti-cyclic] \\
\\
a⃗\times b⃗ = -(b⃗\times a⃗) [anti-commutative] \\
\\
Area of parallelogram = |a⃗\times b⃗| \\
Area of triangle = ½|a⃗\times b⃗| \\
Area of △ABC = ½|AB⃗\times AC⃗|
\end{gather*}
10.5 Scalar Triple Product
\begin{gather*}
[a⃗ b⃗ c⃗] = a⃗·(b⃗\times c⃗) = scalar \\
\\
= |a_1 a_2 a_3| \\
|b_1 b_2 b_3| \\
|c_1 c_2 c_3| \\
\\
Volume of parallelepiped = |[a⃗ b⃗ c⃗]| \\
Volume of tetrahedron = (1/6)|[a⃗ b⃗ c⃗]| \\
\\
Coplanar vectors ↔ [a⃗ b⃗ c⃗] = 0 \\
\\
[a⃗ b⃗ c⃗] = [b⃗ c⃗ a⃗] = [c⃗ a⃗ b⃗] [cyclic] \\
[a⃗ b⃗ c⃗] = -[b⃗ a⃗ c⃗] [swap any two \Rightarrow sign changes]
\end{gather*}
10.6 Vector Triple Product
\begin{gather*}
a⃗\times (b⃗\times c⃗) = (a⃗·c⃗)b⃗ - (a⃗·b⃗)c⃗ [BAC-CAB rule] \\
(a⃗\times b⃗)\times c⃗ = (a⃗·c⃗)b⃗ - (b⃗·c⃗)a⃗
\end{gather*}
10.7 Section Formula in Vectors
\begin{gather*}
\text{If} P divides AB in ratio m:n internally: \\
p⃗ = (mb⃗ + na⃗)/(m+n) \\
\\
Midpoint: p⃗ = (a⃗ + b⃗)/2 \\
\\
Externally: p⃗ = (mb⃗ - na⃗)/(m-n)
\end{gather*}
B11
Three Dimensional Geometry
11.1 Direction Cosines and Ratios
\begin{gather*}
\text{For} line with direction ratios (a,b,c): \\
l = a/\sqrt{a^2+b^2+c^2} \\
m = b/\sqrt{a^2+b^2+c^2} \\
n = c/\sqrt{a^2+b^2+c^2} \\
\\
l^2+m^2+n^2 = 1 always \\
\\
cos \alpha = l, cos \beta = m, cos \gamma = n \\
(\alpha , \beta , \gamma = angles with x,y,z axes) \\
\\
cos^2\alpha + cos^2\beta + cos^2\gamma = 1 \\
sin^2\alpha + sin^2\beta + sin^2\gamma = 2 [derived]
\end{gather*}
11.2 Angle Between Two Lines
\begin{gather*}
Lines with DRs (a_1,b_1,c_1) \text{and} (a_2,b_2,c_2): \\
cos \theta = |a_1a_2+b_1b_2+c_1c_2| / [\sqrt{a_1^2+b_1^2+c_1^2}·\sqrt{a_2^2+b_2^2+c_2^2}] \\
\\
Parallel: a_1/a_2 = b_1/b_2 = c_1/c_2 \\
Perpendicular: a_1a_2 + b_1b_2 + c_1c_2 = 0 \\
\\
With DCs (l_1,m_1,n_1) \text{and} (l_2,m_2,n_2): \\
cos \theta = |l_1l_2 + m_1m_2 + n_1n_2|
\end{gather*}
11.3 Equation of a Line
\begin{gather*}
Vector Form: \\
r⃗ = a⃗ + λb⃗ [passing through point a⃗, direction b⃗] \\
r⃗ = a⃗ + λ(b⃗-a⃗) [passing through a⃗ \text{and} b⃗] \\
\\
Cartesian/Symmetric Form: \\
(x-x_1)/a = (y-y_1)/b = (z-z_1)/c = λ \\
x = x_1+aλ, y = y_1+bλ, z = z_1+cλ
\end{gather*}
11.4 Distance Between Two Lines
\begin{gather*}
Skew lines (r⃗=a⃗_1+λb⃗_1 \text{and} r⃗=a⃗_2+μb⃗_2): \\
d = |(a⃗_2-a⃗_1)·(b⃗_1\times b⃗_2)| / |b⃗_1\times b⃗_2| \\
\\
Parallel lines (b⃗_1 ∥ b⃗_2=b⃗): \\
d = |(a⃗_2-a⃗_1)\times b⃗| / |b⃗| \\
\\
Intersecting lines: d = 0 \\
Coplanar lines: (a⃗_2-a⃗_1)·(b⃗_1\times b⃗_2) = 0
\end{gather*}
11.5 Equation of a Plane
\begin{gather*}
Vector Form: \\
r⃗·n̂ = d [n̂ = unit normal, d = distance from origin] \\
r⃗·n⃗ = D [n⃗ = normal vector (\text{not} unit)] \\
\\
Cartesian Form: \\
ax + by + cz = d [a,b,c = normal direction] \\
ax + by + cz + d = 0 \\
\\
Normal form: lx + my + nz = p \\
\\
Intercept Form: \\
x/\alpha + y/\beta + z/\gamma = 1 [\alpha ,\beta ,\gamma = x,y,z intercepts] \\
\\
Plane through 3 points A(x_1,y_1,z_1), B, C: \\
|x-x_1 y-y_1 z-z_1| \\
|x_2-x_1 y_2-y_1 z_2-z_1| = 0 \\
|x_3-x_1 y_3-y_1 z_3-z_1|
\end{gather*}
11.6 Angle Between Two Planes
\begin{gather*}
Planes a_1x+b_1y+c_1z+d_1=0 \text{and} a_2x+b_2y+c_2z+d_2=0: \\
cos \theta = |a_1a_2+b_1b_2+c_1c_2| / [\sqrt{a_1^2+b_1^2+c_1^2}·\sqrt{a_2^2+b_2^2+c_2^2}] \\
\\
Parallel: a_1/a_2 = b_1/b_2 = c_1/c_2 \\
Perpendicular: a_1a_2 + b_1b_2 + c_1c_2 = 0
\end{gather*}
11.7 Distance Formulas in 3D
\begin{gather*}
Distance between P(x_1,y_1,z_1) \text{and} Q(x_2,y_2,z_2): \\
d = √[(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2] \\
\\
Distance from point P(x_1,y_1,z_1) to plane ax+by+cz+d=0: \\
d = |ax_1+by_1+cz_1+d| / \sqrt{a^2+b^2+c^2} \\
\\
Distance between parallel planes ax+by+cz=d_1 \text{and} ax+by+cz=d_2: \\
d = |d_1-d_2| / \sqrt{a^2+b^2+c^2} \\
\\
Foot of perpendicular from P(x_1,y_1,z_1) to plane ax+by+cz+d=0: \\
(x-x_1)/a = (y-y_1)/b = (z-z_1)/c = -(ax_1+by_1+cz_1+d)/(a^2+b^2+c^2)
\end{gather*}
11.8 Angle Between Line and Plane
\begin{gather*}
Line: r⃗ = a⃗ + λb⃗, Plane: r⃗·n⃗ = d \\
\\
sin φ = |b⃗·n⃗| / (|b⃗||n⃗|) [φ = angle with plane] \\
cos \theta = |b⃗·n⃗| / (|b⃗||n⃗|) [\theta = angle with normal] \\
\\
Line ∥ plane ↔ b⃗·n⃗ = 0 \\
Line ⊥ plane ↔ b⃗ ∥ n⃗ ↔ b⃗\times n⃗ = 0⃗ \\
Line in plane ↔ b⃗·n⃗=0 \text{AND} a⃗·n⃗=d
\end{gather*}
11.9 Family of Planes
\begin{gather*}
Plane through intersection of P_1=0 \text{and} P_2=0: \\
P_1 + λP_2 = 0 (\text{for} varying λ) \\
\\
(a_1x+b_1y+c_1z+d_1) + λ(a_2x+b_2y+c_2z+d_2) = 0
\end{gather*}
B12
Linear Programming
12.1 Standard Formulation
\begin{gather*}
Objective Function: Z = ax + by (maximize \text{or} minimize) \\
Subject to constraints: a_1x+b_1y \le /\ge /= c_1 etc. \\
Non-negativity: x \ge 0, y \ge 0 \\
\\
Feasible Region: set of all points satisfying all constraints \\
Optimal Solution: max/min value of Z at a corner point \\
\\
Corner Point Method: \\
1. Graph all constraints \\
2. Find feasible region \\
3. Find corner points (vertices) \\
4. Evaluate Z at each corner point \\
5. Max/min Z \text{is} the answer
\end{gather*}
12.2 Fundamental Theorem
\begin{gather*}
\text{If} optimal solution exists \Rightarrow it \text{is} at a corner point of feasible region \\
\\
\text{If} Z = c (constant) \text{is} parallel to a boundary \Rightarrow Z takes same value \\
along that boundary (infinite solutions) \\
\\
Unbounded feasible region \Rightarrow Z may \text{or} may \text{not} have optimum
\end{gather*}
12.3 Key Results
\begin{gather*}
\text{For} maximize Z = ax+by: \\
\Rightarrow Look \text{for} corner with largest x \text{if} a>b, etc. \\
\\
\text{For} minimize Z = ax+by: \\
\Rightarrow Look \text{for} corner nearest to origin (usually) \\
\\
\text{If} feasible region \text{is} empty \Rightarrow no solution (infeasible problem) \\
\text{If} feasible region \text{is} bounded \Rightarrow both max \text{and} min exist \\
\text{If} unbounded \Rightarrow only one of max/min may exist
\end{gather*}
B13
Probability (Class 12)
13.1 Conditional Probability
\begin{gather*}
P(A|B) = P(A∩B)/P(B) [probability of A given B, P(B)\neq 0] \\
\\
\Rightarrow P(A∩B) = P(B)·P(A|B) = P(A)·P(B|A) \\
\Rightarrow P(B|A) = P(A∩B)/P(A) \\
\Rightarrow P(A|B) · P(B) = P(B|A) · P(A) \\
\\
Properties: \\
0 \le P(A|B) \le 1 \\
P(S|B) = 1 \\
P(A'|B) = 1 - P(A|B) \\
P(A∪C|B) = P(A|B) + P(C|B) - P(A∩C|B)
\end{gather*}
13.2 Multiplication Theorem
\begin{gather*}
P(A∩B) = P(A)·P(B|A) = P(B)·P(A|B) \\
\\
P(A∩B∩C) = P(A)·P(B|A)·P(C|A∩B) \\
\\
Independent events A \text{and} B: \\
P(A|B) = P(A) \text{and} P(B|A) = P(B) \\
P(A∩B) = P(A)·P(B) \\
\\
Pairwise independent \neq mutually independent \\
Mutually independent ↔ every subset satisfies multiplication rule
\end{gather*}
13.3 Total Probability Theorem
\begin{gather*}
\text{If} B_1, B_2, ..., Bₙ are mutually exclusive \text{and} exhaustive events: \\
P(A) = \Sigma ᵢ P(Bᵢ)·P(A|Bᵢ) \\
= P(B_1)P(A|B_1) + P(B_2)P(A|B_2) + ... + P(Bₙ)P(A|Bₙ)
\end{gather*}
13.4 Bayes' Theorem
\begin{gather*}
P(Bᵢ|A) = P(Bᵢ)·P(A|Bᵢ) / \Sigma ⱼ P(Bⱼ)·P(A|Bⱼ) \\
\\
P(Bᵢ)·P(A|Bᵢ) \\
= ───────────────────────────────────────────── \\
P(B_1)P(A|B_1) + P(B_2)P(A|B_2) + ... + P(Bₙ)P(A|Bₙ) \\
\\
Two hypotheses B_1, B_2 (prior probs p_1, p_2 = 1-p_1): \\
P(B_1|A) = p_1·P(A|B_1) / [p_1·P(A|B_1) + p_2·P(A|B_2)]
\end{gather*}
13.5 Random Variables and Probability Distribution
\begin{gather*}
\text{If} X \text{is} a random variable taking values x_1, x_2, ..., xₙ with \\
probabilities p_1, p_2, ..., pₙ: \\
\\
Conditions: pᵢ \ge 0 \text{for} all i, \Sigma pᵢ = 1 \\
\\
Mean (Expected Value): \\
E(X) = μ = \Sigma xᵢpᵢ = x_1p_1 + x_2p_2 + ... + xₙpₙ \\
\\
Variance: \\
Var(X) = \sigma ^2 = E(X^2) - [E(X)]^2 \\
= \Sigma xᵢ^2pᵢ - (\Sigma xᵢpᵢ)^2 \\
= E[(X-μ)^2] = \Sigma (xᵢ-μ)^2pᵢ \\
\\
Standard Deviation: \\
\sigma = \sqrt{Var}(X) = √[E(X^2) - (E(X))^2] \\
\\
Properties: \\
E(aX+b) = aE(X) + b \\
Var(aX+b) = a^2Var(X) \\
Var(aX) = a^2Var(X) \\
Var(X+b) = Var(X)
\end{gather*}
13.6 Binomial Distribution
\begin{gather*}
X ~ B(n, p) [n = trials, p = success prob, q = 1-p] \\
\\
P(X = r) = C(n,r) · pʳ · qⁿ⁻ʳ (r = 0,1,2,...,n) \\
\\
\text{where} C(n,r) = n! / [r!(n-r)!] = ⁿCᵣ \\
\\
Mean: E(X) = np \\
Variance: Var(X) = npq = np(1-p) \\
SD: \sigma = \sqrt{npq} \\
\\
Mode: (n+1)p [\text{if} integer, two modes: (n+1)p \text{and} (n+1)p-1] \\
⌊(n+1)p⌋ [\text{if} \text{not} integer] \\
\\
P(X=0) = qⁿ \\
P(X=n) = pⁿ \\
P(X\ge 1) = 1 - P(X=0) = 1 - qⁿ \\
\\
Recurrence: P(X=r+1)/P(X=r) = [(n-r)/(r+1)] \times (p/q) \\
\\
\text{For} large n, small p (np=λ fixed) \Rightarrow Poisson approximation: \\
P(X=r) ≈ e^(-λ)λʳ/r!
\end{gather*}
13.7 Important Probability Results
\begin{gather*}
P(A only) = P(A∩B') = P(A) - P(A∩B) \\
P(B only) = P(A'∩B) = P(B) - P(A∩B) \\
P(exactly one of A,B) = P(A) + P(B) - 2P(A∩B) \\
P(neither A nor B) = 1 - P(A∪B) = 1 - P(A) - P(B) + P(A∩B) \\
\\
\text{For} three events: \\
P(A∪B∪C) = P(A)+P(B)+P(C)-P(A∩B)-P(B∩C)-P(C∩A)+P(A∩B∩C) \\
\\
\text{If} A \text{and} B independent: \\
A \text{and} B' independent \\
A' \text{and} B independent \\
A' \text{and} B' independent \\
\\
P(A∩B) = P(A)P(B) [independent] \\
P(A∩B) = 0 [mutually exclusive] \\
Independent \neq mutually exclusive (unless P(A)=0 \text{or} P(B)=0)
\end{gather*}
🔥 Part C
Additional / Advanced Formulas
C1
Advanced Algebra Identities
Algebraic Expansions
\begin{gather*}
(a+b+c)^2 = a^2+b^2+c^2+2ab+2bc+2ca \\
(a-b-c)^2 = a^2+b^2+c^2-2ab+2bc-2ca \\
(a+b-c)^2 = a^2+b^2+c^2+2ab-2bc-2ca \\
\\
a^2+b^2+c^2 = (a+b+c)^2 - 2(ab+bc+ca) \\
ab+bc+ca = [(a+b+c)^2-(a^2+b^2+c^2)]/2 \\
\\
(a+b+c)^3 = a^3+b^3+c^3+3(a+b)(b+c)(c+a) \\
a^3+b^3+c^3-3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ca) \\
= ½(a+b+c)[(a-b)^2+(b-c)^2+(c-a)^2] \\
\\
\text{If} a+b+c=0: a^3+b^3+c^3 = 3abc \\
\\
(a+b)⁴ = a⁴+4a^3b+6a^2b^2+4ab^3+b⁴ \\
(a-b)⁴ = a⁴-4a^3b+6a^2b^2-4ab^3+b⁴ \\
a⁴-b⁴ = (a+b)(a-b)(a^2+b^2) \\
a⁴+b⁴ = (a^2+b^2)^2-2a^2b^2 \\
\\
Binomial: (a+b)ⁿ = \Sigma _k₌₀ⁿ C(n,k)aⁿ⁻ᵏbᵏ \\
General term: T(r+1) = C(n,r)aⁿ⁻ʳbʳ \\
Middle term: T((n/2)+1) \text{for} even n \\
\\
Sophie Germain Identity: \\
a⁴+4b⁴ = (a^2+2b^2+2ab)(a^2+2b^2-2ab)
\end{gather*}
C2
Complete Trigonometry Identities
Compound Angle Formulas
\begin{gather*}
sin(A+B) = sinA cosB + cosA sinB \\
sin(A-B) = sinA cosB - cosA sinB \\
cos(A+B) = cosA cosB - sinA sinB \\
cos(A-B) = cosA cosB + sinA sinB \\
\\
tan(A+B) = (tanA + tanB)/(1 - tanA tanB) \\
tan(A-B) = (tanA - tanB)/(1 + tanA tanB) \\
\\
sin(A+B)·sin(A-B) = sin^2A - sin^2B = cos^2B - cos^2A \\
cos(A+B)·cos(A-B) = cos^2A - sin^2B = cos^2B - sin^2A
\end{gather*}
Double Angle Formulas
\begin{gather*}
sin 2A = 2 sinA cosA = 2tanA/(1+tan^2A) \\
cos 2A = cos^2A - sin^2A = 1-2sin^2A = 2cos^2A-1 = (1-tan^2A)/(1+tan^2A) \\
tan 2A = 2tanA/(1-tan^2A) \\
\\
sin^2A = (1-cos2A)/2 \\
cos^2A = (1+cos2A)/2 \\
tan^2A = (1-cos2A)/(1+cos2A) \\
\\
sin A = 2sin(A/2)cos(A/2) \\
cos A = 1-2sin^2(A/2) = 2cos^2(A/2)-1
\end{gather*}
Triple Angle Formulas
\begin{gather*}
sin 3A = 3sinA - 4sin^3A \\
cos 3A = 4cos^3A - 3cosA \\
tan 3A = (3tanA - tan^3A)/(1-3tan^2A) \\
\\
sin^3A = (3sinA - sin3A)/4 \\
cos^3A = (3cosA + cos3A)/4
\end{gather*}
Product to Sum Formulas
\begin{gather*}
2sinA cosB = sin(A+B) + sin(A-B) \\
2cosA sinB = sin(A+B) - sin(A-B) \\
2cosA cosB = cos(A+B) + cos(A-B) \\
2sinA sinB = cos(A-B) - cos(A+B)
\end{gather*}
Sum to Product Formulas
\begin{gather*}
sinC + sinD = 2sin((C+D)/2)cos((C-D)/2) \\
sinC - sinD = 2cos((C+D)/2)sin((C-D)/2) \\
cosC + cosD = 2cos((C+D)/2)cos((C-D)/2) \\
cosC - cosD = -2sin((C+D)/2)sin((C-D)/2)
\end{gather*}
Important Trig Values
\begin{gather*}
sin 15° = (\sqrt{6}-\sqrt{2})/4 cos 15° = (\sqrt{6}+\sqrt{2})/4 \\
sin 18° = (\sqrt{5}-1)/4 cos 36° = (\sqrt{5}+1)/4 \\
sin 36° = \sqrt{10-2\sqrt{5}}/4 cos 18° = \sqrt{10+2\sqrt{5}}/4 \\
sin 22.5° = \sqrt{(\sqrt{2}-1}/(2\sqrt{2})) \\
tan 15° = 2-\sqrt{3} tan 75° = 2+\sqrt{3}
\end{gather*}
C3
Mensuration Advanced Formulas
\begin{gather*}
Regular polygon (n sides, side a): \\
Perimeter = na \\
Interior angle = (n-2)\times 180°/n \\
Exterior angle = 360°/n \\
Area = (na^2/4)cot(\pi /n) \\
Sum of interior angles = (n-2)\times 180° \\
\\
Heron's Formula (triangle with sides a,b,c): \\
s = (a+b+c)/2 [semi-perimeter] \\
Area = √[s(s-a)(s-b)(s-c)] \\
\\
Area using angles (triangle): \\
Area = (1/2)ab sinC = (1/2)bc sinA = (1/2)ca sinB \\
\\
Circumradius R of triangle: \\
R = abc/(4 \times Area) = a/(2sinA) \\
\\
Inradius r of triangle: \\
r = Area/s = (s-a)tan(A/2) \\
\\
Cosine Rule: \\
a^2 = b^2+c^2-2bc cosA \\
b^2 = a^2+c^2-2ac cosB \\
c^2 = a^2+b^2-2ab cosC \\
\Rightarrow cosA = (b^2+c^2-a^2)/(2bc) \\
\\
Sine Rule: a/sinA = b/sinB = c/sinC = 2R
\end{gather*}
C4
Coordinate Geometry Advanced (Class 12 Level)
\begin{gather*}
Standard curves: \\
Circle: x^2+y^2+2gx+2fy+c=0 \\
Center: (-g,-f), Radius: \sqrt{g^2+f^2-c} \\
\\
Parabola: y^2=4ax (opens right, vertex origin) \\
Directrix: x=-a, Focus: (a,0), Axis: y=0 \\
y^2=-4ax (opens left) \\
x^2=4ay (opens up), x^2=-4ay (opens down) \\
\\
Ellipse: x^2/a^2+y^2/b^2=1 (a>b>0) \\
c^2=a^2-b^2 (c=focal distance) \\
e=c/a<1 (eccentricity) \\
Foci: (±c,0), Vertices: (±a,0) \\
Directrices: x=±a/e=±a^2/c \\
\\
Hyperbola: x^2/a^2-y^2/b^2=1 \\
b^2=c^2-a^2, e=c/a>1 \\
Asymptotes: y=±(b/a)x \\
Conjugate: y^2/a^2-x^2/b^2=1 \\
\\
Tangent to circle x^2+y^2=r^2 at (x_1,y_1): xx_1+yy_1=r^2 \\
\\
Tangent to parabola y^2=4ax at (x_1,y_1): yy_1=2a(x+x_1) \\
\\
Tangent to ellipse x^2/a^2+y^2/b^2=1 at (x_1,y_1): xx_1/a^2+yy_1/b^2=1 \\
\\
Tangent of slope m to parabola y^2=4ax: y=mx+a/m \\
Tangent of slope m to ellipse: y=mx±\sqrt{a^2m^2+b^2}
\end{gather*}
C5
Limits — Key Formulas
\begin{gather*}
Standard Limits: \\
lim[x\Rightarrow 0] sinx/x = 1 \\
lim[x\Rightarrow 0] tanx/x = 1 \\
lim[x\Rightarrow 0] (1-cosx)/x = 0 \\
lim[x\Rightarrow 0] (1-cosx)/x^2 = 1/2 \\
lim[x\Rightarrow 0] sin⁻¹x/x = 1 \\
lim[x\Rightarrow 0] tan⁻¹x/x = 1 \\
lim[x\Rightarrow 0] (eˣ-1)/x = 1 \\
lim[x\Rightarrow 0] (aˣ-1)/x = ln a \\
lim[x\Rightarrow 0] (ln(1+x))/x = 1 \\
lim[x\Rightarrow ∞] (1+1/x)ˣ = e \\
lim[x\Rightarrow 0] (1+x)^(1/x) = e \\
lim[x\Rightarrow a] (xⁿ-aⁿ)/(x-a) = naⁿ⁻¹ \\
\\
L'Hôpital's Rule (0/0 \text{or} ∞/∞ form): \\
lim f(x)/g(x) = lim f'(x)/g'(x) \\
\\
Special: lim[x\Rightarrow ∞] xⁿ/eˣ = 0 (exponential grows faster) \\
lim[x\Rightarrow ∞] ln x/xⁿ = 0 (power grows faster than log)
\end{gather*}
C6
Sequences and Series (Additional)
Geometric Progression (GP)
\begin{gather*}
aₙ = ar^(n-1) [nth term] \\
r = common ratio = aₙ₊_1/aₙ \\
\\
Sum of n terms: \\
Sₙ = a(rⁿ-1)/(r-1) (r \neq 1, r > 1 preferred) \\
Sₙ = a(1-rⁿ)/(1-r) (r \neq 1, r < 1 preferred) \\
Sₙ = na (r = 1) \\
\\
Sum to infinity (|r|<1): \\
S∞ = a/(1-r) \\
\\
Product of first n terms of GP: \\
P = aⁿ \times r^(n(n-1)/2) \\
\\
GM between a \text{and} b: G = \sqrt{ab} \\
AM \ge GM \text{for} positive reals: (a+b)/2 \ge \sqrt{ab} \\
HM between a \text{and} b: H = 2ab/(a+b) \\
AM \ge GM \ge HM \\
GM^2 = AM \times HM \\
\\
3 terms in GP: a/r, a, ar [product = a^3] \\
4 terms in GP: a/r^3, a/r, ar, ar^3 [product = a⁴]
\end{gather*}
Harmonic Progression (HP)
\begin{gather*}
\text{If} a,b,c in HP: 1/a, 1/b, 1/c in AP \\
\Rightarrow 2/b = 1/a + 1/c \\
\Rightarrow b = 2ac/(a+c) \\
\\
nth term of HP: 1/(A + (n-1)D)
\end{gather*}
AGP (Arithmetico-Geometric Progression)
\begin{gather*}
a, (a+d)r, (a+2d)r^2, ... \\
S∞ = a/(1-r) + dr/(1-r)^2 [|r|<1]
\end{gather*}
C7
Combinatorics (for Probability)
\begin{gather*}
Permutation: ⁿPᵣ = n!/(n-r)! [order matters] \\
Combination: ⁿCᵣ = n!/[r!(n-r)!] [order doesn't matter] \\
\\
ⁿCᵣ = ⁿCₙ₋ᵣ \\
ⁿCᵣ + ⁿCᵣ₋_1 = ⁿ⁺¹Cᵣ [Pascal's identity] \\
ⁿCₒ = ⁿCₙ = 1 \\
ⁿC_1 = n \\
ⁿC_2 = n(n-1)/2 \\
ⁿCᵣ = (n/r)ⁿ⁻¹Cᵣ₋_1 \\
\\
n! = n \times (n-1)! \\
0! = 1 \\
\\
Circular permutation: (n-1)! \\
Necklace/bracelet: (n-1)!/2 \\
\\
Multinomial: n!/(n_1!n_2!...n_k!) \\
\\
Stars \text{and} bars: distributing n identical into k distinct: \\
C(n+k-1, k-1)
\end{gather*}
C8
Complex Numbers
\begin{gather*}
z = a + ib [a = real part, b = imaginary part] \\
i = \sqrt{-1}, i^2 = -1, i^3 = -i, i⁴ = 1 \\
\\
Modulus: |z| = \sqrt{a^2+b^2} \\
Argument: arg(z) = \theta = tan⁻¹(b/a) [in correct quadrant] \\
\\
Conjugate: z̄ = a - ib \\
z + z̄ = 2a = 2Re(z) \\
z - z̄ = 2ib = 2i·Im(z) \\
z·z̄ = |z|^2 = a^2+b^2 \\
(z̄_1+z_2) = z̄_1+z̄_2 \\
(z_1z_2) = z̄_1·z̄_2 \\
\\
Polar form: z = r(cos\theta + i sin\theta ) = reⁱ\theta \\
r = |z|, \theta = arg(z) \\
\\
De Moivre's theorem: (cos\theta +i sin\theta )ⁿ = cos(n\theta )+i sin(n\theta ) \\
|z_1z_2| = |z_1||z_2| \\
arg(z_1z_2) = arg(z_1)+arg(z_2) \\
\\
Triangle inequality: \\
|z_1+z_2| \le |z_1|+|z_2| \\
|z_1-z_2| \ge ||z_1|-|z_2|| \\
\\
nth roots of unity: zⁿ = 1 \\
z_k = cos(2\pi k/n)+i sin(2\pi k/n) \text{for} k=0,1,...,n-1 \\
Sum of nth roots = 0 \\
Product of nth roots = (-1)^(n-1) \\
\\
Cube roots of unity: 1, ω, ω^2 \\
ω = (-1+i\sqrt{3})/2 = e^(2\pi i/3) \\
1+ω+ω^2 = 0 \\
ω^3 = 1 \\
ω^2 = (-1-i\sqrt{3})/2
\end{gather*}
C9
Complete Derivative Table (Quick Reference)
\begin{gather*}
f(x) f'(x) \\
───────────────────────────────────────── \\
c (constant) 0 \\
x 1 \\
xⁿ nxⁿ⁻¹ \\
\sqrt{x} 1/(2\sqrt{x}) \\
1/x -1/x^2 \\
1/xⁿ -n/x^(n+1) \\
eˣ eˣ \\
eᵃˣ aeᵃˣ \\
aˣ aˣ ln a \\
ln x 1/x \\
log_a x 1/(x ln a) \\
sin x cos x \\
cos x -sin x \\
tan x sec^2x \\
cot x -cosec^2x \\
sec x sec x tan x \\
cosec x -cosec x cot x \\
sin(ax+b) a cos(ax+b) \\
cos(ax+b) -a sin(ax+b) \\
tan(ax+b) a sec^2(ax+b) \\
sin⁻¹x 1/\sqrt{1-x^2} \\
cos⁻¹x -1/\sqrt{1-x^2} \\
tan⁻¹x 1/(1+x^2) \\
cot⁻¹x -1/(1+x^2) \\
sec⁻¹x 1/(|x|\sqrt{x^2-1}) \\
cosec⁻¹x -1/(|x|\sqrt{x^2-1}) \\
sin⁻¹(x/a) 1/\sqrt{a^2-x^2} \\
cos⁻¹(x/a) -1/\sqrt{a^2-x^2} \\
tan⁻¹(x/a) a/(a^2+x^2) \\
|x| x/|x| = sgn(x) (x\neq 0) \\
xˣ xˣ(1+ln x) \\
f(g(x)) f'(g(x))·g'(x)
\end{gather*}
C10
Complete Integral Table (Quick Reference)
\begin{gather*}
f(x) ∫f(x)dx \\
──────────────────────────────────────────────────────────── \\
xⁿ (n\neq -1) xⁿ⁺¹/(n+1) + C \\
1/x ln|x| + C \\
eˣ eˣ + C \\
eᵃˣ eᵃˣ/a + C \\
aˣ aˣ/ln a + C \\
sin x -cos x + C \\
cos x sin x + C \\
tan x ln|sec x| + C \\
cot x ln|sin x| + C \\
sec x ln|sec x + tan x| + C \\
cosec x ln|cosec x - cot x| + C \\
sec^2x tan x + C \\
cosec^2x -cot x + C \\
sec x tan x sec x + C \\
cosec x cot x -cosec x + C \\
1/\sqrt{1-x^2} sin⁻¹x + C \\
-1/\sqrt{1-x^2} cos⁻¹x + C \\
1/(1+x^2) tan⁻¹x + C \\
-1/(1+x^2) cot⁻¹x + C \\
1/(x^2+a^2) (1/a)tan⁻¹(x/a) + C \\
1/(x^2-a^2) (1/2a)ln|(x-a)/(x+a)| + C \\
1/(a^2-x^2) (1/2a)ln|(a+x)/(a-x)| + C \\
1/\sqrt{a^2-x^2} sin⁻¹(x/a) + C \\
1/\sqrt{x^2+a^2} ln|x+\sqrt{x^2+a^2}| + C \\
1/\sqrt{x^2-a^2} ln|x+\sqrt{x^2-a^2}| + C \\
\sqrt{a^2-x^2} (x/2)\sqrt{a^2-x^2}+(a^2/2)sin⁻¹(x/a)+C \\
\sqrt{x^2+a^2} (x/2)\sqrt{x^2+a^2}+(a^2/2)ln|x+\sqrt{x^2+a^2}|+C \\
\sqrt{x^2-a^2} (x/2)\sqrt{x^2-a^2}-(a^2/2)ln|x+\sqrt{x^2-a^2}|+C \\
eˣ[f(x)+f'(x)] eˣf(x) + C \\
x eˣ eˣ(x-1) + C \\
x^2 eˣ eˣ(x^2-2x+2) + C \\
ln x x ln x - x + C \\
x ln x (x^2/2)ln x - x^2/4 + C \\
sin⁻¹x x sin⁻¹x + \sqrt{1-x^2} + C \\
tan⁻¹x x tan⁻¹x - (1/2)ln(1+x^2) + C \\
cos⁻¹x x cos⁻¹x - \sqrt{1-x^2} + C \\
sin^2x x/2 - sin(2x)/4 + C \\
cos^2x x/2 + sin(2x)/4 + C \\
tan^2x tan x - x + C \\
cot^2x -cot x - x + C \\
sin^3x -cos x + cos^3x/3 + C \\
cos^3x sin x - sin^3x/3 + C
\end{gather*}
C11
Matrices — Additional Important Results
\begin{gather*}
Trace(A) = sum of diagonal elements \\
Trace(AB) = Trace(BA) \\
Rank of matrix = number of non-zero rows in row echelon form \\
\\
\text{For} A (m\times n): rank(A) \le min(m,n) \\
A \text{is} invertible ↔ rank(A) = n (full rank) \\
A \text{is} singular ↔ |A| = 0 ↔ rank(A) < n \\
\\
Cayley-Hamilton Theorem: \\
Every square matrix satisfies its own characteristic equation. \\
\text{For} A 2\times 2 with char. eq λ^2-(trA)λ+|A|=0: \\
A^2 - (trA)A + |A|I = 0 \\
\Rightarrow A⁻¹ = (A - (trA)I) / |A| [\text{for} 2\times 2 non-singular] \\
\\
\text{For} 2\times 2 A = [a b; c d]: \\
Characteristic equation: λ^2 - (a+d)λ + (ad-bc) = 0 \\
λ_1+λ_2 = a+d = Trace(A) \\
λ_1·λ_2 = ad-bc = |A| \\
\\
Eigenvalues of diagonal matrix = diagonal elements \\
Eigenvalues of triangular matrix = diagonal elements \\
|A| = product of eigenvalues \\
Trace = sum of eigenvalues \\
\\
\text{If} A \text{is} orthogonal: AAᵀ = AᵀA = I \Rightarrow A⁻¹ = Aᵀ \\
\text{If} A \text{is} idempotent: A^2 = A \\
\text{If} A \text{is} nilpotent: Aⁿ = O \text{for} some n \\
\text{If} A \text{is} involutory: A^2 = I \Rightarrow A⁻¹ = A
\end{gather*}
C12
Special Technique Formulas (Exam Tricks)
Rationalization Tricks
\begin{gather*}
1/(a+b\sqrt{c}) = (a-b\sqrt{c})/(a^2-b^2c) \\
\\
1/(\sqrt{a}+\sqrt{b}) = (\sqrt{a}-\sqrt{b})/(a-b) \\
\\
Nested radicals: \sqrt{a+\sqrt{b}} = \sqrt{(a+√(a^2-b})/2) + \sqrt{(a-√(a^2-b})/2)
\end{gather*}
Telescoping Sums
\begin{gather*}
\Sigma [f(k)-f(k-1)] = f(n) - f(0) [collapses to endpoints] \\
\\
1/(k(k+1)) = 1/k - 1/(k+1) \\
1/(k(k+1)(k+2)) = (1/2)[1/(k(k+1)) - 1/((k+1)(k+2))] \\
\\
\Sigma 1/(k(k+1)) from k=1 to n = 1 - 1/(n+1) = n/(n+1)
\end{gather*}
Inequality Formulas
\begin{gather*}
AM-GM: (a+b)/2 \ge \sqrt{ab} [equality \text{when} a=b] \\
(a_1+a_2+...+aₙ)/n \ge (a_1a_2...aₙ)^(1/n) \\
\\
Cauchy-Schwarz: (a_1b_1+a_2b_2+...+aₙbₙ)^2 \le (a_1^2+...+aₙ^2)(b_1^2+...+bₙ^2) \\
\\
Triangle inequality: |a+b| \le |a| + |b| \\
||a|-|b|| \le |a-b| \\
\\
\text{For} positive x: x + 1/x \ge 2 [min value 2 at x=1] \\
x^2 + 1/x^2 \ge 2
\end{gather*}
Useful Number Patterns
\begin{gather*}
1+2+3+...+n = n(n+1)/2 \\
1+3+5+...+(2n-1) = n^2 \\
2+4+6+...+2n = n(n+1) \\
1^2+2^2+...+n^2 = n(n+1)(2n+1)/6 \\
1^3+2^3+...+n^3 = [n(n+1)/2]^2 \\
1\times 2+2\times 3+...+n(n+1) = n(n+1)(n+2)/3 \\
1\times 1!+2\times 2!+...+n\times n! = (n+1)!-1
\end{gather*}
Important Constants and Values
\begin{gather*}
\pi ≈ 3.14159265... \\
e ≈ 2.71828182... \\
\sqrt{2} ≈ 1.41421356... \\
\sqrt{3} ≈ 1.73205080... \\
\sqrt{5} ≈ 2.23606797... \\
1/\sqrt{2} = \sqrt{2}/2 ≈ 0.7071 \\
\sqrt{3}/2 ≈ 0.8660 \\
ln 2 ≈ 0.6931 \\
ln 10 ≈ 2.3026 \\
log_1₀ e ≈ 0.4343 \\
\\
Radian to degree: 1 rad = 180°/\pi ≈ 57.296° \\
Degree to radian: 1° = \pi /180 rad ≈ 0.01745 rad \\
\\
30° = \pi /6 rad 45° = \pi /4 rad \\
60° = \pi /3 rad 90° = \pi /2 rad \\
120° = 2\pi /3 rad 135° = 3\pi /4 rad \\
150° = 5\pi /6 rad 180° = \pi rad \\
270° = 3\pi /2 rad 360° = 2\pi rad
\end{gather*}
Differential Calculus — Twisted Forms
\begin{gather*}
\text{If} y = sin(ax+b): y' = a cos(ax+b), y'' = -a^2 sin(ax+b), yⁿ = aⁿ sin(ax+b+n\pi /2) \\
\text{If} y = cos(ax+b): yⁿ = aⁿ cos(ax+b+n\pi /2) \\
\text{If} y = eᵃˣ: yⁿ = aⁿeᵃˣ \\
\text{If} y = ln x: y' = 1/x, y'' = -1/x^2, yⁿ = (-1)^(n-1)(n-1)!/xⁿ \\
\\
Partial derivatives (\text{for} Lagrange multipliers): \\
∂/∂x[f(x,y)] - treat y as constant \\
\\
Jacobian: J = ∂(u,v)/∂(x,y) = |∂u/∂x ∂u/∂y| \\
|∂v/∂x ∂v/∂y| \\
\\
Euler's theorem \text{for} homogeneous f of degree n: \\
x(∂f/∂x) + y(∂f/∂y) = nf \\
\\
Total differential: df = (∂f/∂x)dx + (∂f/∂y)dy
\end{gather*}
3D Geometry — Extra Formulas
\begin{gather*}
Sphere with center (h,k,l) \text{and} radius r: \\
(x-h)^2+(y-k)^2+(z-l)^2 = r^2 \\
\\
General sphere: x^2+y^2+z^2+2ux+2vy+2wz+d=0 \\
Center: (-u,-v,-w), Radius: \sqrt{u^2+v^2+w^2-d} \\
\\
Image of point P(\alpha ,\beta ,\gamma ) in plane ax+by+cz+d=0: \\
(x-\alpha )/a = (y-\beta )/b = (z-\gamma )/c = -2(a\alpha +b\beta +c\gamma +d)/(a^2+b^2+c^2) \\
\\
Centroid of tetrahedron (4 vertices): \\
G = [(x_1+x_2+x_3+x₄)/4, (y_1+y_2+y_3+y₄)/4, (z_1+z_2+z_3+z₄)/4] \\
\\
Volume of tetrahedron with one vertex at origin, others A,B,C: \\
V = (1/6)|[a⃗ b⃗ c⃗]| \\
\\
Distance from line to line (skew): \\
d = |[(b⃗_1\times b⃗_2)·(a⃗_2-a⃗_1)]| / |b⃗_1\times b⃗_2|
\end{gather*}
Vector — Additional Identities
\begin{gather*}
|a⃗\times b⃗|^2 = |a⃗|^2|b⃗|^2 - (a⃗·b⃗)^2 [Lagrange's identity] \\
\\
(a⃗\times b⃗)·(c⃗\times d⃗) = (a⃗·c⃗)(b⃗·d⃗) - (a⃗·d⃗)(b⃗·c⃗) \\
\\
(a⃗\times b⃗)\times (c⃗\times d⃗) = [a⃗ b⃗ d⃗]c⃗ - [a⃗ b⃗ c⃗]d⃗ = [a⃗ c⃗ d⃗]b⃗ - [b⃗ c⃗ d⃗]a⃗ \\
\\
Area of parallelogram with diagonals d⃗_1, d⃗_2: \\
Area = ½|d⃗_1\times d⃗_2| \\
\\
Normal to plane through a⃗, b⃗, c⃗: \\
n⃗ = (b⃗-a⃗)\times (c⃗-a⃗) \\
\\
Angle bisector of a⃗ \text{and} b⃗ (unit vectors â,b̂): \\
bisector direction = â + b̂ [internal] \\
= â - b̂ [external]
\end{gather*}
Exam Strategy — Key Identity Shortcuts
\begin{gather*}
a^2 + b^2 + c^2 - ab - bc - ca = ½[(a-b)^2 + (b-c)^2 + (c-a)^2] \\
\ge 0 always \\
\\
(a+b+c)^2 = a^2+b^2+c^2 + 2(ab+bc+ca) \\
\\
\text{If} a+b+c=0: a^2+b^2+c^2 = -2(ab+bc+ca) \\
ab+bc+ca = -(a^2+b^2+c^2)/2 \\
a^3+b^3+c^3 = 3abc \\
\\
Determinant tricks: \\
R_1\Rightarrow R_1+R_2+R_3 (sum of rows): factor out the sum \\
C_1\Rightarrow C_1-C_2, C_2\Rightarrow C_2-C_3: create zeros \\
\text{If} rows in AP: middle row = average of outer rows \Rightarrow use R_2\Rightarrow 2R_2-R_1-R_3 \\
\\
Common factor: \\
|ka kb kc| = k \times |a b c| \\
|d e f | |d e f| \\
|g h i | |g h i|
\end{gather*}
CBSE Board Frequent Formula Combos
\begin{gather*}
Volume related conversion: \\
\text{If} a sphere of radius R \text{is} melted into n small spheres of radius r: \\
(4/3)\pi R^3 = n \times (4/3)\pi r^3 \\
\Rightarrow n = R^3/r^3 = (R/r)^3 \\
\\
Cone from circle (sector folded): \\
\text{If} a sector of radius L \text{and} angle \theta \text{is} folded into a cone: \\
Slant height of cone = L (radius of sector) \\
2\pi r = (\theta /360°) \times 2\pi L \\
\Rightarrow r = \theta L/360° \\
\Rightarrow h = \sqrt{L^2-r^2} \\
\Rightarrow V = (1/3)\pi r^2h \\
\\
Ratio of volumes of cone:cylinder:sphere (same r, same h=2r): \\
V_cone : V_cyl : V_sph = 1 : 3 : 2 [FAMOUS RATIO] \\
\\
Sphere inscribed in cylinder: \\
r_sphere = r_cyl = h_cyl/2 \\
SA_sphere/CSA_cyl = 1 [Equal surface areas!] \\
\\
Largest cone inscribed in cylinder: \\
same base \text{and} height \\
V_cone = (1/3)V_cyl \\
\\
Largest sphere inscribed in cube of side a: \\
r = a/2 \\
\\
Largest sphere inscribed in cone of radius R, height h: \\
r = Rh/(\sqrt{R^2+h^2}+R) = Rh/(l+R)
\end{gather*}
C13
Final Master Summary — The Golden Rules
\begin{gather*}
1. a^2 - b^2 = (a+b)(a-b) - always factorize first! \\
2. D = b^2-4ac - discriminant rules everything \\
3. sin^2+cos^2=1 - most common identity \\
4. AM\ge GM\ge HM - inequalities \\
5. P(A)+P(Ā)=1 - complement rule \\
6. aₙ=a+(n-1)d, Sₙ=n/2[2a+(n-1)d] - AP core \\
7. LCM\times HCF = a\times b - always \\
8. Area = ½|base\times height| - universally \\
9. Chain rule: d/dx[f(g(x))]=f'(g(x))g'(x) \\
10. ∫eˣ[f+f']=eˣf+C - IBP shortcut \\
\\
Key Values to Remember: \\
sin30=1/2, cos60=1/2, tan45=1, sin90=1, cos0=1 \\
\sqrt{2}≈1.414, \sqrt{3}≈1.732, \pi ≈3.14159, e≈2.71828 \\
\\
Key Theorems: \\
- Pythagoras: H^2=B^2+P^2 \\
- Thales (BPT): AD/DB=AE/EC \\
- MVT: f'(c)=(f(b)-f(a))/(b-a) \\
- Bayes: P(H|E)=P(H)P(E|H)/P(E) \\
- Euclid: a=bq+r \\
- FTA: Every composite number has unique prime factorization
\end{gather*}
© CBSE Mathematics Formula Compendium 2026–27 · Class 10 & Class 12 · All Chapters · Complete Reference
Use with NCERT textbooks and previous year papers for best results.
Made by Akshit Tyagi — CBSE Mathematics Reference 2026–27