Full Grade 10 test

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.

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.

Sine rule: a/sin A = b/sin B. Thus b = a sin B / sin A, and sin B = b sin A / a. Pair each side with its opposite angle.

Pure substance mass = total mass × its percentage / 100. If p is the water percentage, the dry matter percentage is 100 − p, so its mass = total mass × (100 − p)/100.

Similar figures have equal corresponding angles and proportional sides. Lengths and perimeters scale by k; areas by k². Find k as target length divided by original length. Multiply area by k² to enlarge and divide to shrink. Sides 4 and 12 give k = 3 and an area factor of 9.

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.

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.

Choose the mandatory symbol position. Each remaining position may contain any allowed symbol except the mandatory one. Apply the product rule; the mandatory symbol cannot occur again.

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.

Mean μ = sum of values / count. Population variance = sum of squared deviations (x − μ)² / count. Standard deviation is the square root of variance. It is zero only when all values are equal.

Calculate red then blue and blue then red separately. The total number of available balls decreases for the second draw. The two cases are disjoint, so add their probabilities.

Supplementary-angle identities: sin(180° − α) = sin α, cos(180° − α) = −cos α and tan(180° − α) = −tan α. In quadrant II, sine is positive while cosine and tangent are negative.

At an intersection, both graphs have the same y-value.

1. Set a(x − m)² + n equal to k.
2. Subtract n and divide by a.
3. Take the square root to obtain two x-values symmetric about x = m.
4. The y-coordinate of both points is k.

The workers complete 1/a and 1/b jobs per hour. Add their rates. Time for t jobs = t/(1/a + 1/b) = tab/(a + b).

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.