Course Outline
Check off chapters as you study them, and take each module's test when you're ready. Everything is saved locally in your browser — nothing is uploaded, no account needed.
A note on these questions
These are practice questions written in JEE-style by the Bryn Flow team, covering the correct syllabus concepts for each chapter — they are not reproduced from official past-year papers. Concept notes are written for exam revision specifically, condensed to the formulas and techniques that actually get tested.
Module 1 — Algebra I
1 Sets, Relations and Functions
A set with n elements has 2ⁿ subsets. A relation is reflexive if every element relates to itself, symmetric if aRb implies bRa, and transitive if aRb and bRc imply aRc — a relation with all three properties is called an equivalence relation.
A function f: A → B is one-one (injective) if distinct inputs always give distinct outputs, and onto (surjective) if every element of B is hit by some input. To find an inverse function, swap x and y in y = f(x) and solve for the new y.
2 Complex Numbers and Quadratic Equations
Powers of i cycle with period 4: i¹=i, i²=−1, i³=−i, i⁴=1. The modulus of z = a+bi is |z| = √(a²+b²).
For a quadratic ax² + bx + c = 0: sum of roots = −b/a, product of roots = c/a. The discriminant b² − 4ac determines the nature of the roots — negative means the two roots are complex conjugates.
3 Matrices and Determinants
For a 2×2 matrix [[a,b],[c,d]], the determinant is ad − bc. A square matrix is called singular when its determinant is zero — and a matrix has an inverse only when it's non-singular.
Matrix multiplication is not commutative in general (AB ≠ BA), unlike ordinary number multiplication — a common source of errors if assumed otherwise.
4 Permutations and Combinations
ⁿPᵣ = n!/(n−r)! counts ordered arrangements; ⁿCᵣ = n!/(r!(n−r)!) counts unordered selections. A useful symmetry: ⁿCᵣ = ⁿC₍ₙ₋ᵣ₎.
The number of ways to arrange n distinct objects in a row is n!. The fundamental principle of counting: if one task can be done in m ways and a second, independent task in n ways, both together can be done in m × n ways.
Module 2 — Algebra II
5 Binomial Theorem
The general term in the expansion of (x+y)ⁿ is Tᵣ₊₁ = ⁿCᵣ xⁿ⁻ʳ yʳ. The expansion has exactly n+1 terms. Setting x = y = 1 gives the sum of all binomial coefficients: 2ⁿ.
A unique middle term exists only when n is even (the (n/2 + 1)th term); when n is odd, there are two middle terms instead.
6 Sequences and Series
Arithmetic progression (AP): nth term = a + (n−1)d, sum of n terms = n/2 [2a + (n−1)d]. Geometric progression (GP): nth term = arⁿ⁻¹, and for an infinite GP with |r| < 1, the sum converges to a/(1−r).
The AM-GM inequality states the arithmetic mean of positive numbers is always ≥ their geometric mean, with equality only when all the numbers are equal.
7 Mathematical Reasoning
De Morgan's laws: the negation of "p AND q" is "NOT p OR NOT q," and vice versa. The contrapositive of "if p then q" is "if not q then not p" — it's logically equivalent to the original statement, while the converse ("if q then p") generally is not.
A statement that's true under every possible truth assignment is a tautology. "p OR q" is false only in the single case where both p and q are false.
Module 3 — Differential Calculus
8 Limits and Continuity
A key standard limit: limₓ→₀ (sin x)/x = 1. Another: limₓ→∞ (1 + 1/x)ˣ = e. A function is continuous at a point a if the left-hand limit, right-hand limit, and the function's value at a all agree.
A removable discontinuity occurs when the limit exists at a point but doesn't match the function's actual value there (or the function is undefined there) — it can be "fixed" by simply redefining the function at that single point.
9 Differentiability and Applications of Derivatives
Differentiability at a point implies continuity there, but the converse is false — a function can be continuous yet have a sharp corner where no derivative exists (like |x| at x=0). The slope of the tangent to y = f(x) at a point equals f'(x) evaluated there.
A function is increasing where f'(x) > 0. At a local maximum, f'(x) = 0 and the second derivative test requires f''(x) < 0 to confirm it (a positive second derivative would indicate a local minimum instead).
10 Indefinite Integration
Integration reverses differentiation: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C (for n ≠ −1), and ∫(1/x) dx = ln|x| + C. The +C represents the arbitrary constant, since any constant's derivative is zero.
Integration by parts: ∫u dv = uv − ∫v du — useful when the integrand is a product of two functions of different types (e.g. a polynomial times an exponential).
11 Definite Integration and Applications
The fundamental theorem of calculus connects definite integrals to antiderivatives: ∫ₐᵇ f(x)dx = F(b) − F(a). Trivially, ∫ₐᵃ f(x)dx = 0. A useful symmetry property: ∫₀ᵃ f(x)dx = ∫₀ᵃ f(a−x)dx.
The area under a curve y = f(x) above the x-axis, between x=a and x=b, is given by ∫ₐᵇ f(x)dx — this extends naturally to finding the area between two curves by subtracting one integral from another.
Module 4 — Differential Equations & Coordinate Geometry I
12 Differential Equations
The order of a differential equation is the order of the highest derivative it contains. A variable-separable equation can be rearranged so all terms with one variable are on one side, all terms with the other on the opposite side, then integrated independently.
For a first-order linear equation dy/dx + Py = Q, the integrating factor is e^∫P dx. A "particular solution" satisfies given initial or boundary conditions, unlike the "general solution," which retains arbitrary constants.
13 Straight Lines
The slope between two points is (y₂−y₁)/(x₂−x₁). Two lines are parallel if their slopes are equal, and perpendicular if the product of their slopes is −1.
The perpendicular distance from a point (x₀,y₀) to the line ax + by + c = 0 is |ax₀ + by₀ + c| / √(a² + b²) — a formula that comes up constantly across coordinate geometry problems.
14 Circles
The general equation x² + y² + 2gx + 2fy + c = 0 represents a circle with center (−g, −f) and radius √(g² + f² − c). A circle centered at the origin with radius r has the simpler form x² + y² = r².
A line is tangent to a circle exactly when the perpendicular distance from the center to the line equals the radius — greater distance means no intersection, and smaller distance means the line is a secant crossing the circle at two points.
15 Conic Sections
Eccentricity classifies conics: e = 0 for a circle, e < 1 for an ellipse, e = 1 for a parabola, and e > 1 for a hyperbola. The standard parabola y² = 4ax has its focus at (a, 0).
An ellipse's defining property: the sum of distances from any point on it to the two foci is constant, equal to 2a. A hyperbola's analogous property uses the difference of those distances instead of the sum.
Module 5 — Geometry & Vectors
16 Three Dimensional Geometry
Direction cosines of a line satisfy l² + m² + n² = 1. Two lines are perpendicular in 3D exactly when the dot product of their direction ratios equals zero — the same test used for vectors in general.
A plane's general equation is ax + by + cz + d = 0, where (a,b,c) is the plane's normal vector direction. The distance from a point (x₁,y₁,z₁) to this plane is |ax₁+by₁+cz₁+d| / √(a²+b²+c²) — the natural 3D extension of the 2D point-to-line distance formula.
17 Vector Algebra
The dot product of two perpendicular vectors is always zero — a quick perpendicularity test. The cross product a × b produces a vector perpendicular to both a and b, with magnitude |a||b|sinθ — useful directly for computing the area of a parallelogram spanned by two vectors.
A unit vector has magnitude exactly 1; any nonzero vector can be converted to a unit vector by dividing by its own magnitude.
18 Trigonometric Ratios and Identities
The core identity: sin²θ + cos²θ = 1. Sum formulas: sin(A+B) = sinA cosB + cosA sinB, and the analogous formula for cosine. The double-angle identity sin(2θ) = 2 sinθ cosθ is derived directly by setting A = B = θ in the sum formula.
Standard angle values are worth memorizing solidly: sin(90°) = 1, cos(90°) = 0, and the familiar values at 0°, 30°, 45°, 60°, and 90° come up constantly.
Module 6 — Trigonometry & Statistics
19 Trigonometric Equations and Inverse Trigonometric Functions
The general solution of sinθ = sinα is θ = nπ + (−1)ⁿα; the general solution of cosθ = cosα is θ = 2nπ ± α, for integer n. These "general solution" forms capture every angle satisfying the equation, not just the one in the standard range.
Inverse trig functions have restricted ranges to stay single-valued: sin⁻¹x has principal range [−π/2, π/2], while cos⁻¹x has principal range [0, π].
20 Statistics and Probability
Variance measures spread around the mean; standard deviation is simply its square root. For two independent events A and B, P(A ∩ B) = P(A) × P(B) — a relationship that fails for dependent events.
Conditional probability: P(A|B) = P(A ∩ B) / P(B) — the probability of A given that B has already occurred, which underlies Bayes' theorem for "reversing" conditional probabilities.