Mathematics
The mathematics section tests FSc/HSSC-level manipulation under time pressure: algebra, sequences, trigonometry, coordinate geometry, and basic calculus. Most lost marks come from signs, domains, and formula slips rather than from unfamiliar theory.
Core rules
- For a quadratic ax²+bx+c=0, use b²−4ac to test the nature of roots, −b/a for the sum, and c/a for the product. After squaring an equation, check for extraneous roots.
- Arithmetic sequence: a_n=a+(n−1)d. Geometric sequence: a_n=ar^{n−1}, and S_n=a(1−r^n)/(1−r) for r≠1.
- In trigonometry, track quadrant signs and use sin²x+cos²x=1. Give all solutions in the stated interval.
- For lines, parallel slopes are equal and perpendicular slopes multiply to −1.
- Differentiate x^n as nx^{n−1}; use the chain rule for composite functions, and evaluate definite integrals by substituting the limits.
Worked example
Solve 2sin²x−sinx−1=0 for 0≤x<2π. Let y=sinx. Then 2y²−y−1=0=(2y+1)(y−1). So y=−1/2 or y=1. sinx=1 gives x=π/2. sinx=−1/2 gives x=7π/6 and 11π/6. All three lie in the interval. Answer: {π/2, 7π/6, 11π/6}.
Traps
Do not divide by sinx and lose solutions. Do not drop the ± when taking a square root. Do not mix degrees and radians in calculus. Reject roots outside the stated domain, and reread whether the question says not or except.