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11 multiple-choice questions and 10 flashcards on Systems of Equations and Matrices, about 7% of the High School Pre-Calculus bank. Every one carries a written rationale.
Systems of Equations and Matrices is one of 10 chapters in CoStudy's High School Pre-Calculus bank, and it holds 11 of the bank's 150 multiple-choice questions — roughly 7% of the total. That proportion is not arbitrary: chapters follow the certifying body's published exam outline, and the number of questions in each is set by that domain's published weight, so the share of your practice time this chapter takes matches the share of the real exam it accounts for.
Studying by chapter is worth doing once you have a diagnostic score. A single overall percentage tells you whether you are close; it does not tell you which domain is dragging. Working a weak chapter in isolation, and re-testing it in isolation, is the fastest way to move a score that has stalled — and it is why the mock exams in CoStudy report by domain rather than as one number.
8 questions drawn from this chapter, with the full rationale shown — the controlling principle behind the right answer, and why each wrong option tempts and fails.
End behavior: lim(x→∞) (2x³ - 5x + 1)/(x³ + 4) = ?
Answer: E — 2 (leading-term ratio: 2x³/x³ = 2; degrees equal, ratio of leading coefficients)
Rational function end behavior: if degrees equal, limit = ratio of leading coefficients (2/1 = 2). If num degree < denom: 0. If num degree > denom: ±∞. Horizontal asymptote y = 2.
Identify the conic: 4x² + 9y² − 36 = 0.
Answer: D — Ellipse
D) Rewrite as x²/9 + y²/4 = 1 — ellipse. A) Circle would need equal coefficients on x² and y². B) Hyperbola needs opposite signs. C) Parabola has only one squared term.
Given matrices A = [[1, 2], [3, 4]] and B = [[2, 0], [1, 1]], the product AB equals:
Answer: A — [[4, 2], [10, 4]]
A) AB row-by-column: (1·2+2·1, 1·0+2·1; 3·2+4·1, 3·0+4·1) = (4, 2; 10, 4). B) Entrywise product (wrong rule). C) Computed BA instead of AB — order matters! D) Transposed the result.
Identify the conic: x² − 4y² + 8y − 8 = 0.
Answer: B — Hyperbola
B) Complete square in y: x² − 4(y² − 2y) − 8 = 0 → x² − 4(y−1)² + 4 − 8 = 0 → x² − 4(y−1)² = 4 → x²/4 − (y−1)² = 1. Opposite signs → hyperbola. A) Same signs would make an ellipse. C) Only one squared term. D) Equal coefficients.
Sum of the geometric series 2 + 6 + 18 + 54 + 162:
Answer: C — 242
C) S = a(rⁿ − 1)/(r − 1) with a = 2, r = 3, n = 5 → 2(243 − 1)/(3 − 1) = 2(242)/2 = 242. A) Only the last term. B) The 6th term would be 486. D) Approximate but incorrect arithmetic.
Magnitude of vector ⟨3, 4⟩:
Answer: B — 5 (|v| = √(3² + 4²) = √25 = 5)
Vector magnitude: ||v|| = √(v₁² + v₂² + ...). ⟨3,4⟩: √(9+16) = √25 = 5. Direction = arctan(4/3). Vectors in pre-calc lead to physics applications and dot products.
Dot product of ⟨2, 3⟩ and ⟨4, -1⟩:
Answer: C — 5 (2·4 + 3·(-1) = 8 - 3 = 5)
Dot product: u·v = u₁v₁ + u₂v₂. (2)(4) + (3)(-1) = 8 - 3 = 5. Geometric meaning: ||u||·||v||·cos(θ). Zero dot product means perpendicular vectors.
Two vectors are PERPENDICULAR iff their dot product is:
Answer: E — 0 (zero dot product ⟺ orthogonal/perpendicular vectors)
u·v = ||u||·||v||·cos θ. If perpendicular, θ = 90°, cos = 0, so u·v = 0. Useful test for orthogonality and decomposition.
4 cards from the 10 in this chapter.
What is the eccentricity of an ellipse?
e = c/a, where c² = a² − b². 0 ≤ e < 1.
How is a conic section formed?
By intersecting a plane with a double cone — angle determines circle, ellipse, parabola, or hyperbola.
What does the value of a vs. b tell you about an ellipse?
If a > b, major axis is horizontal; if b > a, major axis is vertical.
What is the eccentricity of a hyperbola?
e > 1.
These are a sample. The full Systems of Equations and Matrices chapter runs 21 items with per-chapter progress tracking, on the web and in the iOS app.
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