All Free Quantitative Methods MCQs with Answers

• A. 0
• B. e
• C. 2
• D. 1

Explanation: Using the product rule, f'(x) = e^x(x^2 + 2x + 1). Substituting x = 0 gives f'(0) = 1.

• A. 15(3x - 2)^4
• B. 5(3x - 2)^4
• C. 15(3x - 2)^5
• D. (3x - 2)^4

Explanation: The chain rule multiplies the outer derivative by the derivative of 3x - 2. Therefore, f'(x) = 5(3x - 2)^4 × 3 = 15(3x - 2)^4.

• A. A local maximum is certain
• B. The test gives no definite classification
• C. A local minimum is certain
• D. An inflection point is certain

Explanation: When the second derivative is zero at a stationary point, the second derivative test is inconclusive.

• A. x < 1
• B. x < 0
• C. x > 1
• D. x > 0

Explanation: The second derivative is f''(x) = 6x - 6. It is positive when x > 1, so the graph is concave upward on that interval.

• A. h = r
• B. h = 4r
• C. h = 3r
• D. h = 2r

Explanation: For a closed cylinder, minimising surface area subject to fixed volume gives the condition h = 2r.

• A. 50 units
• B. 100 units
• C. 150 units
• D. 200 units

Explanation: Revenue is R(q) = pq = 100q - 0.5q^2. Since R'(q) = 100 - q, the stationary point occurs at q = 100, and R''(q) = -1 confirms a maximum.

• A. 4.83
• B. 4.90
• C. 4.93
• D. 5.07

Explanation: For f(x) = sqrt(x), use f(25) = 5 and f'(25) = 1/10. Thus sqrt(24.3) is approximately 5 + (1/10)(-0.7) = 4.93.

• A. 3/13
• B. -3/13
• C. 4/13
• D. -4/13

Explanation: Differentiation gives 2xy + x^2(dy/dx) + 3y^2(dy/dx) = 0. At (1, 2), this becomes 4 + 13(dy/dx) = 0, so dy/dx = -4/13.

• A. 8 units
• B. 10 units
• C. 12 units
• D. 14 units

Explanation: Average cost is C(q)/q = q - 12 + 100/q. Its derivative is 1 - 100/q^2, which is zero at q = 10, giving the minimum average cost.

• A. 5/2
• B. 5
• C. 25/2
• D. 25

Explanation: Since (x - y)^2 is nonnegative, x^2 + y^2 is at least 2xy. Therefore, 25 is at least 2xy, with equality when x = y, so the maximum is 25/2.

• A. x ln x + 2x
• B. x²/x
• C. 2x + ln x
• D. 2x ln x + x

Explanation: The product rule gives f'(x) = 2x ln x + x²(1/x), which simplifies to 2x ln x + x.

• A. It is a local maximum
• B. It is a local minimum
• C. It is a point of inflection
• D. It is not a critical point

Explanation: Here f'(x) = 4x³ - 8x, so x = 0 is a critical point. Since f''(0) = -8, the graph is concave downward there, making x = 0 a local maximum.

• A. 2
• B. 3
• C. 4
• D. 12

Explanation: The Mean Value Theorem states that some c in (2, 6) has f'(c) = [f(6) - f(2)]/(6 - 2).

• A. 5 units
• B. 10 units
• C. 15 units
• D. 20 units

Explanation: Profit is stationary when marginal revenue equals marginal cost. Solving 50 - 2q = 2q + 10 gives 4q = 40, hence q = 10 units.

• A. 5
• B. 7/3
• C. 3/7
• D. 1/5

Explanation: The derivative of an inverse function is (f⁻¹)'(y) = 1/f'(f⁻¹(y)). Since f⁻¹(7) = 3 and f'(3) = 5, the value is 1/5.

• A. 50
• B. 80
• C. 100
• D. 200

Explanation: If the numbers are x and 20 - x, their product is x(20 - x), a downward-opening quadratic.

• A. y - 1 = (1/1)(x - 1)
• B. y - 1 = -(1/1)(x - 1)
• C. y - 2 = (1/1)(x - 1)
• D. y - 2 = -(1/1)(x - 1)

Explanation: At x = 1, the curve has y = 0, not y = 1 or 2. Since dy/dx = 3x² - 2, the tangent slope is 1 and the normal slope is -1, so y = -x + 1…

• A. 18π cm³/s
• B. 24π cm³/s
• C. 36π cm³/s
• D. 54π cm³/s

Explanation: The volume is V = (4/3)πr³, so dV/dt = 4πr² dr/dt. Substituting r = 3 and dr/dt = 2 gives 4π(9)(2) = 72π cm³/s, so none of the listed…

• A. 2
• B. 3
• C. 4
• D. 7

Explanation: Complete the square: f(x) = (x - 2)² + 3. The vertex x = 2 lies within [0, 5], so the absolute minimum value is 3.

• A. ∇f = ∇g
• B. ∇f = λ∇g
• C. f = λg
• D. ∇f + ∇g = 1

Explanation: At a regular constrained stationary point, the gradient of the objective is parallel to the gradient of the constraint.