Full Grade 12 test

Use tan α = sin α / cos α for cos α ≠ 0. Rearrange a sin x + b = 0 to sin x = −b/a for a ≠ 0. If this is outside [−1, 1], there is no solution. Otherwise use sine solutions and retain all in the specified interval.

Divide the same distance from the mean by each standard deviation. Smaller spread gives greater probability within the same symmetric interval. Outside the interval the comparison reverses.

Treat the adjacent pair as one block. Arrange that block together with the remaining books. Check separately whether the two internal orders are both allowed.

The oblique segment, its orthogonal projection and the perpendicular to the plane form a right triangle. First find the perpendicular length using the Pythagorean theorem. The sine of the angle between the segment and the plane is the perpendicular length divided by the oblique length.

Evaluate the polynomial at x = 1 and x = −1. Adding these values cancels odd-power contributions and doubles even-power contributions. Account for that doubling.

Standardize both interval bounds and subtract the lower cumulative probability from the upper one. Use Φ(−t) = 1 − Φ(t). The answer is approximate.

The diagonal section of a right prism is a rectangle. Its width is the base diagonal √(a² + b²) and height h. Section area = h√(a² + b²).

Integrate to obtain F(x) with an unknown constant. Substitute the given x and F value to determine the constant. Then evaluate the requested function value.

Apply the product and chain rules. Factor out the positive exponential, examine the signs of the remaining factors and include the boundary t = 0.

Count all teams of the required size, then subtract teams with no certificate holders. This includes teams with one or more certificate holders without counting them multiple times.

Calculate the probability of one sequence with the specified numbers of successes and failures. Count the ways to choose the success positions and multiply. Failure probability is 1 − p.

Direct substitution gives the indeterminate form 0/0. Factor the difference of squares and cancel the common factor away from the approach point. Then calculate the limit of the simplified expression.

Let t = sin(x) and solve the quadratic. Reject t values outside [−1; 1]. For each remaining value, find all valid x; the number of t values is not the number of x solutions.

Use the binomial theorem to select the term with the requested power of x. Its coefficient includes a combination count and powers of both binomial factors. Account for the sign of a negative constant.

Use (axⁿ)′ = anxⁿ⁻¹, the derivative of a constant is 0 and derivatives add across sums. Thus (ax² + bx + c)′ = 2ax + b. Substitute the point x for a derivative value; for f′(x) = r, solve 2ax + b = r.

In a probability tree, multiply successive probabilities along a path, using the appropriate conditional probabilities. Add probabilities of mutually exclusive favourable paths. For disjoint A and B with equally likely outcomes, P(A or B) = (a + b)/n.

Tangent and cosine are related by 1 + tan²α = 1 / cos²α.

1. Call the given tangent value t. First calculate t², then add 1.
2. Rearrange the identity to obtain cos²α = 1 / (1 + t²).
3. Take the square root. Since α is acute, its cosine is positive, so cos α = 1 / √(1 + t²).
4. Evaluate the expression and round only the final result to the nearest hundredth.

Do not confuse t² with t or omit the square root. The cosine of an acute angle lies between 0 and 1.