Mathematics and Statistics practice questions

Answer: B) 1

• AIncorrect. The z-term is negative.
• BCorrect. \(4 + 0 - 3 = 1\).
• CIncorrect. The dot product is a scalar.
• DIncorrect. Recheck each product.

Answer: D) 2

With the antiderivative −cos x, the integral of sin x from 0 to π equals −cos π + cos 0 = 2.

• AIncorrect. π is the length of the interval, which would be the integral of the constant 1, not of sin x. Because sin x ≤ 1 and is below 1 on most of [0, π], the area is less than π.
• BIncorrect. The integral of sin x is zero over a full period (0 to 2π), not over the half period from 0 to π.
• CIncorrect. 1 is the value of the integral from 0 to π/2 only, which is half of the requested interval.
• DCorrect. The antiderivative is −cos x, so the integral is [−cos π] − [−cos 0] = 1 + 1 = 2.

Answer: D) 6

• AIncorrect. That's the coefficient.
• BIncorrect. That's \(y(3)\).
• CIncorrect. That isn't the derivative.
• DCorrect. \(dy/dx = 2x\).

Answer: A) 0.271

• ACorrect. One minus the chance of no failures, 1 − 0.9³.
• BIncorrect. That slightly overstates it.
• CIncorrect. That's all three.
• DIncorrect. That isn't the question.

Answer: A) 6 kN·m directed in the positive z-direction

M = r × F = (2i) × (3j) = 6(i × j) = 6k kN·m, which points in the positive z-direction by the right-hand rule. Reversing the order gives −6k.

• ACorrect. r × F = (2i) × (3j) = 6(i × j). By the right-hand rule, i × j = k, so M = +6k kN·m.
• BIncorrect. This is F × r. The cross product is anticommutative (j × i = −k), so reversing the order flips the sign.
• CIncorrect. Zero is the dot product r · F, because the vectors are perpendicular. For perpendicular vectors the cross product has its largest magnitude, |r||F| = 6.
• DIncorrect. The magnitude is right, but a cross product is perpendicular to both r (along x) and F (along y). It cannot point along the force's own direction.

Answer: D) 0.9025

For independent events, multiply the probabilities: the chance that both parts are good is \(0.95^2 = 0.9025\).

• AIncorrect. \(0.05 \times 0.05 = 0.0025\) is the probability that both parts are defective, not that both are good.
• BIncorrect. 0.95 is the probability that one part is good. For both to be good, the two probabilities must be multiplied.
• CIncorrect. This subtracts 5% twice from 1. Probabilities of independent events multiply; they don't add or subtract this way.
• DCorrect. The parts are independent, so P(both good) = \(0.95 \times 0.95 = 0.9025\).

Answer: C) 4

• AIncorrect. That's \(f(1, 2)\).
• BIncorrect. That's \(\partial f/\partial y\) plus extra terms.
• CCorrect. \(\partial f/\partial x = 2xy = 2(1)(2)\).
• DIncorrect. Include y in \(2xy\).

Answer: C) \(\begin{bmatrix} 4 \\ 6 \end{bmatrix}\)

Each entry of the product is the dot product of a matrix row with the vector: (2)(1) + (1)(2) = 4 and (0)(1) + (3)(2) = 6.

• AIncorrect. This multiplies the transpose of the matrix by the vector (2·1 + 0·2, 1·1 + 3·2).
• BIncorrect. This multiplies only the diagonal entries by the vector components and drops the off-diagonal term in row 1.
• CCorrect. Row 1: 2·1 + 1·2 = 4. Row 2: 0·1 + 3·2 = 6.
• DIncorrect. A 2 × 2 matrix times a 2 × 1 vector gives a 2 × 1 vector, not a single number.

Answer: A) 15/216

For exactly k successes in n independent trials, P = C(n,k)p^k(1 − p)^(n−k). Here that is 3(1/6)²(5/6) = 15/216.

• ACorrect. By the binomial formula, C(3,2)(1/6)²(5/6) = 3 × 5/216 = 15/216 ≈ 0.069.
• BIncorrect. This is the probability of one particular order, such as six, six, non-six. It leaves out the C(3,2) = 3 arrangements.
• CIncorrect. (1/6)² covers only the two sixes. It leaves out the (5/6) factor for the third roll and the number of arrangements.
• DIncorrect. This is the probability of at least two sixes, 15/216 + 1/216. The 1/216 is the chance of three sixes, which the question excludes.

Answer: D) 13

• AIncorrect. That isn't the calculation.
• BIncorrect. Interpolate between them.
• CIncorrect. That isn't halfway.
• DCorrect. Halfway between 10 and 16.