Mathematics - 50 One-Mark Questions of Algebra and Calculus
What the section tests
Mathematics supplies 50 of the 200 one-mark questions on the AP ECET paper. With no negative marking, a fast, clean run through these 50 is where ranks are decided. The scope follows the diploma common core: matrices and determinants, partial fractions, trigonometric identities and equations, straight lines, differentiation with its applications (tangents, rates, maxima–minima), integration with definite integrals and area, and first-order differential equations.
Rules worth memorising
- A square system AX = B has a unique solution only when rank(A) = rank([A|B]) = n; infinitely many solutions when the ranks are equal but below n; no solution when they differ.
- Split into partial fractions only after the fraction is proper - if numerator degree ≥ denominator degree, divide first.
- Standard results: d/dx(sin x) = cos x; ∫dx/x = ln|x| + C; ∫₀^{π/2} sin²x dx = π/4.
- When you substitute in a definite integral, change the limits along with the variable.
Worked example
Evaluate ∫₀^{π/2} sin²x dx. Write sin²x = (1 − cos 2x)/2. The integral becomes ½[x − (sin 2x)/2] evaluated from 0 to π/2 = ½[(π/2 − 0) − (0 − 0)] = π/4. Cross-check by symmetry: cos²x over the same interval also gives π/4, and the two must add to π/2. They do.
Traps that cost marks
- Omitting +C in an indefinite integral; the keyed option usually carries it.
- Leaving the calculator in degree mode while the limits are in radians.
- Judging consistency of three-variable systems by the discriminant instead of by rank.
- Sign slips when the denominator has a repeated linear factor such as (x − 2)².
- Forgetting that a derivative test needs the second derivative to classify a stationary point.