The traps that lose marks

You may divide both sides by a quantity only if it cannot be zero.

Squaring can create solutions that do not satisfy the original equation.

a⁰ = 1 (for a ≠ 0) and a⁻ⁿ = 1/aⁿ.

√(x²) = |x|, not x.

For ax² + bx + c = 0: b²−4ac > 0 two real roots, = 0 one repeated root, < 0 no real roots.

For ax² + bx + c = 0: sum of roots = −b/a, product = c/a.

Solve by finding the roots and testing the sign of the parabola in each region.

log(x) is defined only for x > 0.

√(f(x)) needs f(x) ≥ 0.

A function has an inverse only if it is one-to-one (passes the horizontal line test).

(f∘g)(x) = f(g(x)) — apply g first, then f.

Trig equations have infinitely many solutions; a bounded question needs ALL solutions in the range.

−1 ≤ sin x ≤ 1 and −1 ≤ cos x ≤ 1, always.

sin²x + cos²x = 1.

Calculus and general solutions use radians; π radians = 180°.

d/dx f(g(x)) = f'(g(x)) · g'(x) — multiply by the derivative of the inside.

Product rule for f·g; chain rule for f(g(x)). They are not interchangeable.

Every indefinite integral needs + C.

A definite integral is F(upper) − F(lower).

f'(x) = 0 finds stationary points; the second derivative decides max/min.

Distance = √[(x₂−x₁)² + (y₂−y₁)²].

Perpendicular lines have gradients whose product is −1 (m₁m₂ = −1).

(x−a)² + (y−b)² = r² has centre (a, b) and radius r.

AP adds a common difference d; GP multiplies by a common ratio r.

A GP has a finite sum to infinity only if |r| < 1, and it is a/(1−r).

The number of terms from k = m to k = n is n − m + 1.

Without replacement, the denominator shrinks each draw; with replacement it stays the same.

Permutations (nPr) count ORDER; combinations (nCr) ignore it.

P(A or B) = P(A) + P(B) − P(A and B). The overlap is subtracted.

i² = −1, i³ = −i, i⁴ = 1, then it cycles.

|(a, b, c)| = √(a² + b² + c²).

Two non-zero vectors are perpendicular exactly when their dot product is 0.

'If P then Q' does not imply 'If Q then P'.

A single counterexample is enough to disprove a universal claim.

Sufficient guarantees it; necessary is required for it. They are not the same.