Algebra Formulas

Algebra1,001 Qs

Probability & Distributions

Conditional probability, Bayes theorem, independent events, total probability, and binomial distribution.

7 Formulas

Algebra658 Qs

Complex Numbers

Argand plane, polar & Euler forms, triangle inequalities, cube roots of unity, De Moivre's theorem, and geometric loci.

Algebra599 Qs

Permutations & Combinations (P&C)

Fundamental principle of counting, nPr, nCr, circular permutations, distribution into groups, and de-arrangement.

7 Formulas

Algebra551 Qs

Sequences & Series

Arithmetic, geometric, arithmetico-geometric (AGP), harmonic progressions, AM-GM-HM inequality, and telescoping series.

Algebra443 Qs

Binomial Theorem

General term T_{r+1}, middle term, greatest term, binomial coefficients identities, and remainder theorem.

Algebra315 Qs

Quadratic Equations

Roots of quadratic, discriminant nature, Vieta relations, condition for common roots, and location of roots.

7 Formulas

Algebra332 Qs

Statistics

Mean, variance, standard deviation, mean deviation, and effect of linear change of origin and scale.

Formulas Reference (Algebra)

8 Chapters match

AlgebraTop Priority (3-4 Qs in JEE)1,015 PYQs

Matrices & Determinants

Matrix algebra, determinant expansion, adjoint properties, inverse, Cramer's rule, and systems of linear equations.

Determinant Properties

|A^T| = |A|, \quad |AB| = |A||B|, \quad |kA| = k^n |A| \quad (n \times n \text{ matrix})

Scalar multiplication pulls out factor k raised to the dimension power n.

|A⁻¹| = 1 / |A|.

Adjoint Properties

A \cdot \text{adj}(A) = |A| I_n, \quad |\text{adj}(A)| = |A|^{n-1}, \quad |\text{adj}(\text{adj}(A))| = |A|^{(n-1)^2}

Crucial adjoint determinant power formulas tested almost every year in JEE Main.

For 3x3 matrix (n = 3): |adj(A)| = |A|² and |adj(adj(A))| = |A|⁴.

Matrix Inverse Formula

A^{-1} = \frac{1}{|A|} \text{adj}(A) \quad (|A| \ne 0)

Inverse exists if and only if matrix A is non-singular (|A| ≠ 0).

(AB)⁻¹ = B⁻¹ A⁻¹ (reversal law).

Cramer's Rule for Linear Systems

x = \frac{\Delta_x}{\Delta}, \quad y = \frac{\Delta_y}{\Delta}, \quad z = \frac{\Delta_z}{\Delta}

Determinant method for solving system of linear equations.

Δ ≠ 0: Unique solution. Δ = 0 with any Δ_i ≠ 0: Inconsistent (No solution).

Symmetric & Skew-Symmetric Matrices

A = \frac{A + A^T}{2} + \frac{A - A^T}{2}, \quad \text{Skew-Symmetric: } A^T = -A \implies a_{ii} = 0

Every square matrix is uniquely decomposable into symmetric and skew-symmetric parts.

Determinant of odd-order skew-symmetric matrix is ALWAYS zero.

Orthogonal Matrix Property

A \cdot A^T = I \implies |A| = \pm 1, \quad A^{-1} = A^T

Orthogonal matrices preserve lengths and angles.

Rows and columns form orthonormal vectors.

Cayley-Hamilton Theorem & Matrix Powers

\det(A - \lambda I) = 0 \implies P(A) = O, \quad A^2 - \text{tr}(A)A + |A|I = O \; (2 \times 2)

Every square matrix satisfies its own characteristic polynomial equation. Enables computing high powers and inverses.

To find A⁻¹, multiply the characteristic equation throughout by A⁻¹.

Special Matrix Types (Idempotent, Involutory, Nilpotent)

A^2 = A \; (\text{Idempotent}), \quad A^2 = I \; (\text{Involutory}), \quad A^k = O \; (\text{Nilpotent}), \quad \text{tr}(AB) = \text{tr}(BA)

Key algebraic matrix classes frequently tested in eigenvalue and matrix polynomial questions.

Eigenvalues of idempotent matrices can only be 0 or 1. Determinant of involutory matrix is ±1.

Homogeneous System of Linear Equations (AX = 0)

|A| \ne 0 \implies X = O \; (\text{Trivial Unique Solution}), \quad |A| = 0 \implies \text{Infinitely Many Non-Trivial Solutions}

Criterion for existence of non-zero solutions to homogeneous linear equations.

If a homogeneous system has a non-trivial solution, its determinant MUST equal 0.

AlgebraHigh Yield (2-3 Qs in JEE)1,001 PYQs

Probability & Distributions

Conditional probability, Bayes theorem, independent events, total probability, and binomial distribution.

Conditional Probability

P(A|B) = \frac{P(A \cap B)}{P(B)} \quad (P(B) > 0)

Probability of event A occurring given that B has already occurred.

If events are independent, P(A|B) = P(A).

Bayes' Theorem

P(E_i|A) = \frac{P(E_i) P(A|E_i)}{\sum_{j=1}^n P(E_j) P(A|E_j)}

Calculates reverse (posterior) probability using total probability denominator.

Write partition events E₁, E₂... clearly before evaluating branch likelihoods.

Binomial Distribution (Mean & Variance)

P(X = r) = {}^nC_r p^r q^{n-r}, \quad \text{Mean } \mu = np, \quad \text{Variance } \sigma^2 = npq

Probability distribution for r successes in n independent Bernoulli trials (p + q = 1).

Variance is ALWAYS strictly less than Mean for Binomial distributions (npq < np since q < 1).

Total Probability Theorem

P(A) = \sum_{i=1}^n P(E_i) \cdot P(A|E_i)

Law of total probability over a partition of sample space.

Forms the denominator of Bayes theorem.

Addition Theorem & 3-Event Inclusion-Exclusion

P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(AB) - P(BC) - P(CA) + P(ABC)

Probability of occurrence of at least one of three arbitrary events.

Probability of exactly two events occurring = P(AB) + P(BC) + P(CA) - 3 P(ABC).

Odds in Favor & Odds Against

\text{Odds in favor of } A = \frac{P(A)}{P(A')} = \frac{m}{n} \implies P(A) = \frac{m}{m + n}, \quad \text{Odds against } = \frac{n}{m}

Ratio of favorable outcomes to unfavorable outcomes.

If odds in favor of an event are 3:5, its probability is 3/(3 + 5) = 3/8.

Expectation & Variance of Discrete Random Variable

E(X) = \sum x_i p_i, \quad \text{Var}(X) = E(X^2) - [E(X)]^2 = \sum x_i^2 p_i - \mu^2, \quad \text{Var}(aX + b) = a^2 \text{Var}(X)

Mathematical expectation (weighted mean) and spread of a discrete probability distribution.

E(aX + b) = a E(X) + b. For independent random variables X and Y: Var(X ± Y) = Var(X) + Var(Y).

AlgebraVery High Yield (2-3 Qs in JEE)658 PYQs

Complex Numbers

Argand plane, polar & Euler forms, triangle inequalities, cube roots of unity, De Moivre's theorem, and geometric loci.

Polar & Euler Form

z = x + iy = r(\cos\theta + i\sin\theta) = r e^{i\theta} \quad (r = |z| = \sqrt{x^2 + y^2}, \; \theta = \arg(z))

Exponential Euler representation for complex numbers.

e^(iπ) + 1 = 0 (Euler's identity).

Triangle Inequalities

||z_1| - |z_2|| \le |z_1 \pm z_2| \le |z_1| + |z_2|

Bounds for modulus of sum and difference of complex numbers.

Equality holds when origin, z₁, and z₂ are collinear.

Cube Roots of Unity (1, ω, ω²)

1 + \omega + \omega^2 = 0, \quad \omega^3 = 1, \quad \omega = \frac{-1 + i\sqrt{3}}{2} = e^{i 2\pi/3}

Roots of equation z³ - 1 = 0.

1 + ω^n + ω^(2n) = 3 if n is multiple of 3, else 0.

De Moivre's Theorem

(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta) = e^{in\theta}

Powers and roots of trigonometric complex forms for integer and rational n.

Always convert to polar form before taking high powers like z²⁰²⁴.

Geometric Locus in Complex Plane

|z - z_0| = r \; (\text{Circle}), \quad |z - z_1| = |z - z_2| \; (\text{Perpendicular Bisector}), \quad \arg\left(\frac{z - z_1}{z - z_2}\right) = \theta \; (\text{Circular Arc})

Translating complex equations into 2D Cartesian geometric figures.

If θ = ±π/2, the locus is a circle with z₁ and z₂ as diametric endpoints.

Rotation Theorem (Coni's Theorem)

\frac{z_3 - z_1}{z_2 - z_1} = \frac{|z_3 - z_1|}{|z_2 - z_1|} e^{i\alpha}

Rotates directed line segment (z₂ - z₁) by an angle α counter-clockwise around pivot point z₁ to obtain direction of (z₃ - z₁).

For an equilateral triangle with vertices z₁, z₂, z₃: z₁² + z₂² + z₃² = z₁z₂ + z₂z₃ + z₃z₁.

Apollonius Circle & Conic Loci in Argand Plane

\frac{|z - z_1|}{|z - z_2|} = k \; (k \ne 1 \implies \text{Circle}), \quad |z - z_1| + |z - z_2| = 2a \; (2a > |z_1 - z_2| \implies \text{Ellipse})

Modulus ratio represents an Apollonius circle, while sum and difference of distances represent ellipses and hyperbolas.

If |z - z₁| + |z - z₂| = |z₁ - z₂|, locus is the line segment joining z₁ and z₂.

n-th Roots of Unity & Factorization

z^n = 1 \implies z_k = e^{i \frac{2k\pi}{n}} \quad (k = 0, 1, \dots, n-1), \quad \sum_{k=0}^{n-1} z_k = 0, \quad \prod_{k=0}^{n-1} z_k = (-1)^{n-1}

The n-th roots of unity form vertices of a regular n-gon inscribed in unit circle centered at origin.

Factorization identity: x^n - 1 = (x - 1)(x - α)(x - α²) ... (x - α^(n-1)).

AlgebraHigh Yield (2 Qs in JEE)599 PYQs

Permutations & Combinations (P&C)

Fundamental principle of counting, nPr, nCr, circular permutations, distribution into groups, and de-arrangement.

Basic Permutation & Combination Formulas

{}^nP_r = \frac{n!}{(n-r)!}, \quad {}^nC_r = \frac{n!}{r!(n-r)!}, \quad {}^nC_r + {}^nC_{r-1} = {}^{n+1}C_r

Fundamental counting formulas (Pascal's rule).

^nC_r = ^nC_{n-r}.

Number of Non-Negative Integral Solutions

x_1 + x_2 + \dots + x_r = n \implies \text{Solutions} = {}^{n + r - 1}C_{r - 1}

Stars and bars method: distributing n identical items into r distinct boxes.

For positive integral solutions (x_i ≥ 1), solutions = ^{n - 1}C_{r - 1}.

De-arrangement Formula (D_n)

D_n = n! \left[ 1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \dots + \frac{(-1)^n}{n!} \right]

Number of permutations such that none of the n objects appear in their original natural position.

D₁ = 0, D₂ = 1, D₃ = 2, D₄ = 9, D₅ = 44.

Division into Groups

\text{Equal Groups: } \frac{(mn)!}{(m!)^n n!}, \quad \text{Unequal Groups: } \frac{(m+n+p)!}{m! n! p!}

Formulas for partition into distinct or identical sets.

Divide by k! whenever k groups have the same size and are undistinguished.

Circular Permutations & Necklaces

P_{\text{circle}} = (n - 1)!, \quad P_{\text{necklace}} = \frac{(n - 1)!}{2}

Arrangements around a round table versus a necklace/garland where flip-over (clockwise = anticlockwise) makes halves identical.

If arranging n men and n women alternatively around a round table: (n - 1)! × n!.

Grid / Lattice Path Counting

\text{Paths from } (0, 0) \to (m, n) = \frac{(m + n)!}{m! n!} = {}^{m+n}C_m

Number of shortest monotonic paths on an m × n grid moving only rightward (R) and upward (U).

If passing through an intermediate point (p, q): Total Paths = {}^{p+q}C_p × {}^{(m-p)+(n-q)}C_{m-p}.

Distribution of Distinct Objects into Boxes

r^n \; (\text{Distinct items, Distinct boxes}), \quad \sum_{k=0}^r (-1)^{r-k} {}^rC_k k^n \; (\text{No empty box})

Fundamental distribution theorem with restrictions on empty containers using inclusion-exclusion.

Total number of functions from set of n elements to set of r elements is r^n; total onto functions is r! · S(n, r).

AlgebraHigh Yield (2-3 Qs in JEE)551 PYQs

Sequences & Series

Arithmetic, geometric, arithmetico-geometric (AGP), harmonic progressions, AM-GM-HM inequality, and telescoping series.

Arithmetic Progression (AP)

T_n = a + (n - 1)d, \quad S_n = \frac{n}{2}[2a + (n - 1)d] = \frac{n}{2}(a + l)

Standard formulas for linear progressions.

Three terms in AP are best assumed as (a - d, a, a + d).

Geometric Progression (GP)

T_n = a r^{n-1}, \quad S_n = \frac{a(1 - r^n)}{1 - r}, \quad S_\infty = \frac{a}{1 - r} \quad (|r| < 1)

Exponential series with common ratio r.

Three terms in GP are best assumed as (a/r, a, ar).

AM-GM-HM Inequality

\frac{a + b}{2} \ge \sqrt{ab} \ge \frac{2ab}{a + b} \iff \text{AM} \ge \text{GM} \ge \text{HM}

Fundamental algebraic inequality for positive real numbers. Equality holds iff all terms are equal.

Premier tool for finding minimum values of expressions like x + 1/x ≥ 2.

Sum of First n Powers of Natural Numbers

\sum n = \frac{n(n+1)}{2}, \quad \sum n^2 = \frac{n(n+1)(2n+1)}{6}, \quad \sum n^3 = \left[\frac{n(n+1)}{2}\right]^2

Standard power sum formulas for polynomial series evaluation.

Notice that ∑ n³ = (∑ n)².

Arithmetico-Geometric Progression (AGP) Sums

S_\infty = \frac{a}{1 - r} + \frac{d \cdot r}{(1 - r)^2}, \quad S_n = \frac{a}{1 - r} + \frac{d r(1 - r^{n-1})}{(1 - r)^2} - \frac{[a + (n-1)d]r^n}{1 - r}

Sum of finite and infinite series whose terms are products of an AP and a GP.

To evaluate manually: multiply by r and subtract (1 - r)S = a + d·r + d·r² + ... - [a + (n-1)d]r^n.

Method of Differences & Telescoping Series

S_n = \sum_{r=1}^n [f(r) - f(r-1)] = f(n) - f(0), \quad \frac{1}{r(r+1)(r+2)} = \frac{1}{2}\left[\frac{1}{r(r+1)} - \frac{1}{(r+1)(r+2)}\right]

Splitting general term T_r into difference of consecutive functions cancels all intermediate terms.

For products of factors in AP in the denominator: multiply and divide by (last factor - first factor).

Cauchy-Schwarz & Weighted AM-GM Inequalities

(\sum a_i^2)(\sum b_i^2) \ge (\sum a_i b_i)^2, \quad \frac{\sum m_i x_i}{\sum m_i} \ge \left(\prod x_i^{m_i}\right)^{\frac{1}{\sum m_i}}

Powerful analytical inequalities for finding minimum values of multivariable constrained expressions in JEE Advanced.

Equality holds in Cauchy-Schwarz iff a₁/b₁ = a₂/b₂ = ... = a_n/b_n.

Harmonic Progression (HP) & Relation Between Means

H = \frac{2ab}{a+b}, \quad G^2 = A \cdot H, \quad A \ge G \ge H \quad (a, b > 0)

Sequence whose reciprocals form an AP. Geometric mean is the geometric mean between AM and HM.

There is no general summation formula for HP series; solve HP problems by converting terms to AP reciprocals.

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Binomial Theorem

General term T_{r+1}, middle term, greatest term, binomial coefficients identities, and remainder theorem.

General Term in Binomial Expansion

(x + y)^n = \sum_{r=0}^n {}^nC_r x^{n-r} y^r, \quad T_{r+1} = {}^nC_r x^{n-r} y^r

Expansion has (n + 1) terms.

Independent of x term occurs when power of x in T_{r+1} equals 0.

Binomial Coefficient Sum Identities

\sum_{r=0}^n {}^nC_r = 2^n, \quad \sum_{r=0}^n (-1)^r {}^nC_r = 0, \quad \sum_{r=1}^n r \cdot {}^nC_r = n \cdot 2^{n-1}

Classic combinatorial summation identities.

Differentiate (1 + x)^n to find sums with r in the coefficient.

Middle Term in Expansion

n \text{ even} \implies T_{\frac{n}{2} + 1}; \quad n \text{ odd} \implies T_{\frac{n+1}{2}} \text{ and } T_{\frac{n+3}{2}}

Has the greatest binomial coefficient ^nC_{max}.

For even n, single middle term has maximum coefficient ^nC_{n/2}.

Fractional / Negative Index Binomial Series

(1 + x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \cdots \quad (|x| < 1), \quad T_{r+1} = \frac{n(n-1)\cdots(n-r+1)}{r!}x^r

Infinite convergent series for any rational n (negative or fractional). General term uses falling factorial in numerator.

Common cases: (1+x)^{-1} = 1 - x + x² - x³ + ⋯, (1-x)^{-1} = 1 + x + x² + x³ + ⋯, (1+x)^{-2} = 1 - 2x + 3x² - 4x³ + ⋯. All valid for |x| < 1.

Weighted Binomial Coefficient Sums

\sum_{r=1}^n r \cdot {}^nC_r = n \cdot 2^{n-1}, \quad \sum_{r=1}^n r^2 \cdot {}^nC_r = n(n+1) \cdot 2^{n-2}, \quad \sum_{r=0}^n \frac{1}{r+1}{}^nC_r = \frac{2^{n+1} - 1}{n + 1}

Weighted sums obtained by differentiating or integrating (1+x)^n and substituting x = 1.

Differentiate (1+x)^n once → Σ r·C(n,r)·x^(r-1); differentiate again → Σ r(r-1)·C(n,r)·x^(r-2). Add to get r²·C(n,r) sum.

Multinomial Theorem & Number of Terms

(x_1 + x_2 + \dots + x_k)^n = \sum \frac{n!}{r_1! r_2! \dots r_k!} x_1^{r_1} x_2^{r_2} \dots x_k^{r_k}, \quad \text{Total Terms} = {}^{n+k-1}C_{k-1}

Expansion of sum of k variables raised to integer power n where r₁ + r₂ + ... + r_k = n.

Coefficient of x₁^{r₁} x₂^{r₂}...x_k^{r_k} is the multinomial coefficient n! / (r₁! r₂!...r_k!).

Numerically Greatest Term (NGT)

\left| \frac{T_{r+1}}{T_r} \right| = \frac{n - r + 1}{r} \left| \frac{y}{x} \right| \ge 1 \implies r \le \frac{n + 1}{1 + |x/y|}

Determines the term having the largest magnitude in the expansion of (x + y)^n.

If m = (n+1)/(1 + |x/y|) is an integer, both T_m and T_{m+1} are equal and numerically greatest.

Divisibility & Remainder Finding

(1 + k)^n = 1 + nk + {^nC_2} k^2 + \dots \equiv 1 + nk \pmod{k^2}

Express base as (multiple of divisor ± remainder) to find remainders of large powers.

For finding remainder of 7¹⁰³ divided by 25: rewrite as 7 · (49)⁵¹ = 7 · (50 - 1)⁵¹.

AlgebraHigh Yield (1-2 Qs in JEE)315 PYQs

Quadratic Equations

Roots of quadratic, discriminant nature, Vieta relations, condition for common roots, and location of roots.

Quadratic Formula & Vieta's Relations

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \quad \alpha + \beta = -\frac{b}{a}, \quad \alpha\beta = \frac{c}{a}

Roots and coefficients of ax² + bx + c = 0.

Difference of roots: |α - β| = √(D) / |a|.

Nature of Roots & Discriminant D

D = b^2 - 4ac: \quad D > 0 \; (\text{Real, Distinct}), \quad D = 0 \; (\text{Real, Equal}), \quad D < 0 \; (\text{Complex Conjugates})

Discriminant determines the geometric intersection with the x-axis.

If a, b, c are rational and D is not a perfect square, roots occur in irrational conjugate pairs (p ± √q).

Location of Roots (Both Roots > k)

D \ge 0, \quad -\frac{b}{2a} > k, \quad a \cdot f(k) > 0

Simultaneous necessary and sufficient conditions for both roots to lie to the right of value k.

If roots lie on opposite sides of k, the ONLY condition needed is a * f(k) < 0.

Condition for Both Roots Common

\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}

Condition for two quadratic equations to have identical solutions.

If one equation has imaginary roots (D < 0), having ONE common root implies BOTH roots are common.

Condition for Exactly One Common Root

(c_1 a_2 - c_2 a_1)^2 = (a_1 b_2 - a_2 b_1)(b_1 c_2 - b_2 c_1), \quad x = \frac{c_1 a_2 - c_2 a_1}{a_1 b_2 - a_2 b_1}

Cross-multiplication determinant condition for a single shared root between two quadratics.

The common root can also be found directly by subtracting equations to eliminate the x² term.

Newton's Sum Theorem for Polynomial Roots

S_n = \alpha^n + \beta^n \implies a S_n + b S_{n-1} + c S_{n-2} = 0

Powerful recurrence relation connecting powers of roots for ax² + bx + c = 0.

Very frequent in JEE Main: e.g. finding (a₁₀ - 2a₈) / 2a₉ without solving for roots explicitly.

Transformation of Quadratic Equations

\text{Roots } k\alpha, k\beta \implies a(x/k)^2 + b(x/k) + c = 0; \quad \text{Roots } 1/\alpha, 1/\beta \implies c x^2 + b x + a = 0; \quad \text{Roots } \alpha \pm h \implies a(x \mp h)^2 + b(x \mp h) + c = 0

Symmetric transformations of equations without recalculating individual roots.

For roots α² and β², substitute x = √y and square to eliminate radicals.

AlgebraHigh Yield (1-2 Qs in JEE)332 PYQs

Statistics

Mean, variance, standard deviation, mean deviation, and effect of linear change of origin and scale.