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Quiz Chapter 2: Differentiation

10 questions · Form 5 Additional Mathematics Bab 2: Differentiation

10 questions · plays in fullscreen

Full Question List & Answer Key

Prefer reading to quizzing? All 10 questions are listed below with the answer and explanation under each one.

1. Find the equation of the normal to the curve y = 3x² - 5 at x = 1.

  1. x - 6y - 13 = 0
  2. x + 6y + 11 = 0
  3. 6x + y - 4 = 0
  4. 6x - y - 8 = 0
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Answer: B

At x = 1, y = 3(1)² - 5 = -2. dy/dx = 6x => m_t = 6(1) = 6. Gradient of normal m_n = -16. Equation: y - (-2) = -16(x - 1) => 6y + 12 = -x + 1 => x + 6y + 11 = 0.

2. Given y = x³ - 6x² + 9x + 2, find the x-coordinates of the stationary points.

  1. x = -1 and x = -3
  2. x = 0 and x = 2
  3. x = 1 and x = 3
  4. x = 2 and x = 4
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Answer: C

dy/dx = 3x² - 12x + 9 = 0 => x² - 4x + 3 = 0 => (x - 1)(x - 3) = 0 => x = 1, x = 3.

3. Differentiate y = x²(2x + 1)³ with respect to x.

  1. 6x(2x + 1)²
  2. 2x(2x + 1)³ + 6x²(2x + 1)²
  3. 2x(2x + 1)³ + 3x²(2x + 1)²
  4. 6x²(2x + 1)²
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Answer: B

Using Product Rule (u = x², v = (2x+1)³): dy/dx = x³ derivative... = x²[3(2x+1)²(2)] + (2x+1)³[2x] = 6x²(2x+1)² + 2x(2x+1)³.

4. Find the derivative of y = 2x + 1x - 3 with respect to x.

  1. -7x - 3²
  2. 7x - 3²
  3. -5x - 3²
  4. 2x - 3²
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Answer: A

Using Quotient Rule (u = 2x+1, v = x-3): dy/dx = (x-3)(2) - (2x+1)(1)x-3² = 2x - 6 - 2x - 1x-3² = -7x-3².

5. Find the second derivative d²y/dx² for y = 2x⁵ - 4x³ + x.

  1. 40x³ - 24x
  2. 10x⁴ - 12x² + 1
  3. 40x⁴ - 24x²
  4. 20x³ - 12x
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Answer: A

dy/dx = 10x⁴ - 12x² + 1. d²y/dx² = d/dx(10x⁴ - 12x² + 1) = 40x³ - 24x.

6. Differentiate y = 4x³ - 5x² + 7x - 9 with respect to x.

  1. 12x² - 10x
  2. 12x² - 10x + 7
  3. 4x² - 5x + 7
  4. 12x³ - 10x² + 7
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Answer: B

dy/dx = d/dx(4x³) - d/dx(5x²) + d/dx(7x) - d/dx(9) = 12x² - 10x + 7.

7. Find the equation of the tangent to the curve y = x² + 3 at the point (1, 4).

  1. y = 2x + 2
  2. y = 2x + 4
  3. y = x + 3
  4. y = 4x
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Answer: A

dy/dx = 2x. At x = 1, m = 2(1) = 2. Equation: y - 4 = 2(x - 1) => y = 2x + 2.

8. The radius r of a circle increases at a rate of 0.2 cm s⁻¹. Find the rate of change of its area A when r = 5 cm.

  1. 1.0π cm² s⁻¹
  2. 4.0π cm² s⁻¹
  3. 0.5π cm² s⁻¹
  4. 2.0π cm² s⁻¹
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Answer: D

A = π r² => dA/dr = 2π r. Using Chain Rule: dA/dt = (dA/dr)(dr/dt) = (2π × 5)(0.2) = 2.0π cm² s⁻¹.

9. Find the gradient of the normal to the curve y = x² - 4x + 5 at point (3, 2).

  1. 2
  2. -12
  3. -2
  4. 12
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Answer: B

dy/dx = 2x - 4. At x = 3, m_t = 2(3) - 4 = 2. Gradient of normal m_n = -1m_t = -12.

10. Given y = 3x², evaluate dy/dx when x = 2.

  1. -34
  2. 34
  3. -38
  4. -32
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Answer: A

y = 3x⁻² => dy/dx = -6x⁻³ = -6x³. At x = 2, dy/dx = -62³ = -68 = -34.

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