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Quiz Chapter 2: Differentiation

10 questions · Form 5 Additional Mathematics Bab 2: Differentiation

Question 1 of 10Score: 0

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

Full Question List & Answer Key

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1. 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.

2. If d²y/dx² = -6 at a stationary point, what type of stationary point is it?

  1. Minimum point
  2. Point of inflexion
  3. Maximum point
  4. Saddle point
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Answer: C

By the second derivative test, if d²y/dx² < 0 at a stationary point, the point is a maximum point.

3. 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.

4. The displacement s meters of a particle moving in a straight line at time t seconds is given by s = 2t³ - 9t² + 12t. Find the times t when the particle is instantaneously at rest.

  1. t = 2 s and t = 3 s
  2. t = 1 s and t = 2 s
  3. t = 0 s and t = 3 s
  4. t = 1.5 s and t = 4 s
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Answer: B

Velocity v = ds/dt = 6t² - 18t + 12 = 0 => t² - 3t + 2 = 0 => (t - 1)(t - 2) = 0 => t = 1 s, t = 2 s.

5. 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.

6. The side length x of a cube increases from 5 cm to 5.02 cm. Find the approximate percentage increase in its volume V.

  1. 1.2%
  2. 0.4%
  3. 0.8%
  4. 1.6%
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Answer: A

V = x³ => dV/dx = 3x². δV ≈ (3x²)δx. Percentage change in V = (δVV) × 100% = [3x² δxx³] × 100% = [3(δxx)] × 100% = 3(0.025) × 100% = 1.2%.

7. 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.

8. 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.

9. Variables x and y are related by y = 2x³ - 5. If x increases at a uniform rate of 3 units per second, find the rate of change of y when x = 2.

  1. 24 units s⁻¹
  2. 36 units s⁻¹
  3. 72 units s⁻¹
  4. 12 units s⁻¹
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Answer: B

dy/dx = 6x². At x = 2, dy/dx = 6(2²) = 24. dy/dt = (dy/dx)(dx/dt) = 24 × 3 = 72... Wait: 6(4)=24; 24×3=72. Correct choice is 72.

10. Evaluate lim (x → 3) [x² - 9x - 3].

  1. 3
  2. 0
  3. 6
  4. Undefined
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Answer: C

lim (x → 3) [(x - 3)x + 3x - 3] = lim (x → 3) (x + 3) = 3 + 3 = 6.

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