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KEAM Mathematics: Algebra and Trigonometry Flashcards
50 question-and-answer cards covering Mathematics: Algebra and Trigonometry as it is examined in KEAM. 24 of them are printed below, taken from across the deck — no signup, no paywall on the preview.
24 sample cards from the Mathematics: Algebra and Trigonometry deck
Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.
Give the nth term and the sum of the first n terms of a geometric progression (GP).
a_n = a·r^(n−1); S_n = a(r^n − 1)/(r − 1) for r ≠ 1.
Give the sum to infinity of a GP and its convergence condition.
S_∞ = a/(1 − r), valid only when |r| < 1.
State the relation between AM, GM, and HM of two positive numbers.
AM ≥ GM ≥ HM, and GM² = AM × HM (so GM is the geometric mean of AM and HM).
Give the standard summation formulas for Σn, Σn², and Σn³.
Σn = n(n+1)/2; Σn² = n(n+1)(2n+1)/6; Σn³ = [n(n+1)/2]².
Define a symmetric matrix and a skew-symmetric matrix.
Symmetric: A' = A (a_ij = a_ji). Skew-symmetric: A' = −A (a_ij = −a_ji), so all diagonal entries are 0.
State the conditions for matrix addition and matrix multiplication to be defined.
Addition: matrices must be of the same order. Multiplication AB: number of columns of A must equal number of rows of B.
Is matrix multiplication commutative? State key properties.
No, AB ≠ BA in general. Multiplication is associative (AB)C = A(BC) and distributive A(B+C) = AB + AC.
How can any square matrix be expressed in terms of symmetric and skew-symmetric matrices?
A = (1/2)(A + A') + (1/2)(A − A'), where (A+A')/2 is symmetric and (A−A')/2 is skew-symmetric.
Give the formula for the determinant of a 2×2 matrix [[a, b],[c, d]].
Determinant = ad − bc.
State the properties det(AB) and det(kA) for an n×n matrix.
det(AB) = det(A)·det(B); det(kA) = k^n · det(A) for an n×n matrix.
How does the determinant change when two rows (or columns) are interchanged, or when a row is a multiple of another?
Interchanging two rows/columns multiplies the determinant by −1; if two rows/columns are identical or proportional, the determinant is 0.
Give the formula for the area of a triangle with vertices (x₁,y₁), (x₂,y₂), (x₃,y₃) using a determinant.
Area = (1/2) |det[[x₁,y₁,1],[x₂,y₂,1],[x₃,y₃,1]]|.
Define the adjoint of a square matrix and give A·(adj A).
adj A = transpose of the cofactor matrix. A·(adj A) = (adj A)·A = |A|·I.
State the formula for the inverse of a matrix and the condition for its existence.
A⁻¹ = (1/|A|)·adj A, which exists iff |A| ≠ 0 (A is non-singular).
For the system AX = B, how do you classify solutions using |A|?
If |A| ≠ 0: unique solution X = A⁻¹B (consistent). If |A| = 0 and (adj A)B = 0: infinitely many solutions. If |A| = 0 and (adj A)B ≠ 0: no solution (inconsistent).
State Cramer's rule for a system of equations.
For AX = B with D = |A| ≠ 0, each variable x_i = D_i / D, where D_i is D with the i-th column replaced by B.
Give the values of sin, cos, and tan at 0°, 30°, 45°, 60°, and 90°.
sin: 0, 1/2, 1/√2, √3/2, 1. cos: 1, √3/2, 1/√2, 1/2, 0. tan: 0, 1/√3, 1, √3, undefined.
State the three Pythagorean trigonometric identities.
sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.
Write the compound angle formulas for sin(A±B) and cos(A±B).
sin(A±B) = sinA cosB ± cosA sinB; cos(A±B) = cosA cosB ∓ sinA sinB.
Give the double angle formulas for sin 2θ, cos 2θ, and tan 2θ.
sin 2θ = 2 sinθ cosθ; cos 2θ = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1; tan 2θ = 2tanθ/(1 − tan²θ).
Give the general solutions of sin θ = 0, cos θ = 0, and tan θ = 0.
sin θ = 0 ⇒ θ = nπ; cos θ = 0 ⇒ θ = (2n+1)π/2; tan θ = 0 ⇒ θ = nπ, for n ∈ ℤ.
State the principal value ranges (domains/ranges) of sin⁻¹x, cos⁻¹x, and tan⁻¹x.
sin⁻¹x: domain [−1,1], range [−π/2, π/2]; cos⁻¹x: domain [−1,1], range [0, π]; tan⁻¹x: domain ℝ, range (−π/2, π/2).
State the complementary identities sin⁻¹x + cos⁻¹x and tan⁻¹x + cot⁻¹x.
sin⁻¹x + cos⁻¹x = π/2 (for x ∈ [−1,1]); tan⁻¹x + cot⁻¹x = π/2 (for all real x).
State the Law of Sines and the Law of Cosines for a triangle.
Law of Sines: a/sinA = b/sinB = c/sinC = 2R. Law of Cosines: a² = b² + c² − 2bc·cosA (and cyclic forms).
What this deck covers
The Mathematics: Algebra and Trigonometry deck follows the KEAM Mathematics: Algebra and Trigonometry syllabus — 4 chapters and 12 topics — so questions land on material that is genuinely examinable rather than trivia around it. That works out to roughly 12.5 cards per chapter.
Answers are written to be recallable, not just readable — averaging about 84 characters, which is long enough to carry the reasoning and short enough to say out loud.
A deck like this earns its keep on the second and third pass. Read the syllabus first so you know the shape of the subject, then use the cards to find the specific facts that have not stuck.
Mathematics: Algebra and Trigonometry flashcards FAQ
How many Mathematics: Algebra and Trigonometry flashcards are in this KEAM deck?
50 cards. This page previews 24 of them, sampled evenly across the deck so you can judge the difficulty before installing anything.
Are these KEAM flashcards free?
Yes. The preview here is free to read with no signup, and the full 50-card deck is free inside the Examius app.
What do the Mathematics: Algebra and Trigonometry cards cover?
They follow the KEAM Mathematics: Algebra and Trigonometry syllabus — 4 chapters and 12 topics — so the questions track what is actually examinable.
How should I use these flashcards?
Read the syllabus first so you know the shape of the subject, then drill the deck. Examius schedules each card with spaced repetition, so cards you keep missing come back sooner and ones you know drift further apart.