Full Grade 10 test

Order the roots a < b. The product (x − a)(x − b) is positive outside the roots, negative between them and zero at the roots. Strict > and < exclude roots; ≤ and ≥ include them. Select the required intervals.

Order the roots a < b. The product (x − a)(x − b) is positive outside the roots, negative between them and zero at the roots. Strict > and < exclude roots; ≤ and ≥ include them. Select the required intervals.

If x + y = s and xy = p, both numbers are roots of t² − st + p = 0. Two distinct roots give two ordered pairs by swapping them. If x < y is required, put the smaller first; equal roots give one pair.

Dividing a total V in ratio a:b gives a + b equal units. First share = V × a/(a + b), second share = V × b/(a + b). Check that they sum to V.

In a triangle, a longer side faces a larger angle and a shorter side faces a smaller angle. Equal sides face equal angles.

With order important and no repetition, arranging n elements gives n! possibilities and assigning r distinct roles gives n!/(n − r)!. An unordered pair gives n(n − 1)/2. A code with r positions, n symbols per position and repetition allowed gives nʳ.

Equate the y expressions: x² = c. For c > 0 there are two pairs (−√c, c) and (√c, c); for c = 0 one pair (0, 0); for c < 0 no real solutions.

First require x ≠ a. Multiply 1/(x − a) = b by nonzero x − a: 1 = b(x − a). For b ≠ 0, x = a + 1/b. Check the original equation.

Cosine rule: c² = a² + b² − 2ab cos C. With three sides, cos C = (a² + b² − c²)/(2ab), then use arccos. For C = 90°, cos C = 0, giving Pythagoras.