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Grade 12 practice chapters

1. Grade 12 · Chapter 261

   Washer volumes · Outer radius factor a=1; Inner radius factor b=0.6

   Use 0≤x≤1.2 and inspect the cross-section at x=1.2. Calculate the outer radius, washer area, and accumulated volume. Givens: Outer radius factor a=1; Inner radius factor b=0.6.

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2. Grade 12 · Chapter 262

   Differential growth models · Initial amount a=0.5; Rate k=0

   Evaluate at t=2.4 time units. Find the amount, instantaneous rate, and multiplier relative to the initial amount; check the rate against the given differential equation. Givens: Initial amount a=0.5; Rate k=0.

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3. Grade 12 · Chapter 263

   Binomial probability · Trial count n=2; Success probability p=0.15

   Use success count k=1. Calculate P(X=k), P(X≤k), and the expected count. Givens: Trial count n=2; Success probability p=0.15.

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4. Grade 12 · Chapter 264

   Cubic derivatives · Cubic coefficient a=0.5; Linear coefficient b=1

   Evaluate at x=0.4. Find the function value and first and second derivatives at that input. Givens: Cubic coefficient a=0.5; Linear coefficient b=1.

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5. Grade 12 · Chapter 265

   The product rule · Linear shift a=0; Exponential rate b=0.75

   Evaluate at x=0.2. Find the product value, its derivative, and the contribution from differentiating the exponential factor. Givens: Linear shift a=0; Exponential rate b=0.75.

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6. Grade 12 · Chapter 266

   Signed accumulation · Slope a=1; Subtracted intercept b=3

   Integrate on 0≤x≤2.4. Find the signed integral, total geometric area, and zero crossing of the line. Givens: Slope a=1; Subtracted intercept b=3.

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7. Grade 12 · Chapter 267

   Substitution and bounds · Exponent factor k=0.7; Final upper bound B=2

   Integrate from x=0 to x=1.2. Find the transformed upper bound under u=kx² and the exact integral, then give a decimal approximation. Givens: Exponent factor k=0.7; Final upper bound B=2.

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8. Grade 12 · Chapter 268

   Washer volumes · Outer radius factor a=1.5; Inner radius factor b=0.7

   Use 0≤x≤1.2 and inspect the cross-section at x=1.2. Calculate the outer radius, washer area, and accumulated volume. Givens: Outer radius factor a=1.5; Inner radius factor b=0.7.

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9. Grade 12 · Chapter 269

   Differential growth models · Initial amount a=1; Rate k=0.1

   Evaluate at t=2.4 time units. Find the amount, instantaneous rate, and multiplier relative to the initial amount; check the rate against the given differential equation. Givens: Initial amount a=1; Rate k=0.1.

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10. Grade 12 · Chapter 270

    Vector components and angles · Vector length r=1.5; Final angle (degrees)=105

    Evaluate at the signed angle θ=63° (three fifths of that final angle). Find its dot product with (1,0), its vertical component, and its length. Givens: Vector length r=1.5; Final angle (degrees)=105.

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11. Grade 12 · Chapter 271

    Cubic derivatives · Cubic coefficient a=1; Linear coefficient b=1.5

    Evaluate at x=0.4. Find the function value and first and second derivatives at that input. Givens: Cubic coefficient a=1; Linear coefficient b=1.5.

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12. Grade 12 · Chapter 272

    The product rule · Linear shift a=0.5; Exponential rate b=1

    Evaluate at x=0.2. Find the product value, its derivative, and the contribution from differentiating the exponential factor. Givens: Linear shift a=0.5; Exponential rate b=1.

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13. Grade 12 · Chapter 273

    Implicit circle tangents · Circle radius r=2; Final angle (degrees)=270

    Use angle θ=162°. Find both coordinates and the tangent slope; identify a vertical tangent instead of dividing by zero when necessary. Givens: Circle radius r=2; Final angle (degrees)=270.

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14. Grade 12 · Chapter 274

    Signed accumulation · Slope a=1.5; Subtracted intercept b=3.5

    Integrate on 0≤x≤2.4. Find the signed integral, total geometric area, and zero crossing of the line. Givens: Slope a=1.5; Subtracted intercept b=3.5.

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15. Grade 12 · Chapter 275

    Substitution and bounds · Exponent factor k=0.75; Final upper bound B=0.5

    Integrate from x=0 to x=0.3. Find the transformed upper bound under u=kx² and the exact integral, then give a decimal approximation. Givens: Exponent factor k=0.75; Final upper bound B=0.5.

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16. Grade 12 · Chapter 276

    Washer volumes · Outer radius factor a=2; Inner radius factor b=0.8

    Use 0≤x≤1.2 and inspect the cross-section at x=1.2. Calculate the outer radius, washer area, and accumulated volume. Givens: Outer radius factor a=2; Inner radius factor b=0.8.

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17. Grade 12 · Chapter 277

    Differential growth models · Initial amount a=1.5; Rate k=0.2

    Evaluate at t=2.4 time units. Find the amount, instantaneous rate, and multiplier relative to the initial amount; check the rate against the given differential equation. Givens: Initial amount a=1.5; Rate k=0.2.

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18. Grade 12 · Chapter 278

    Binomial probability · Trial count n=4; Success probability p=0.25

    Use success count k=3. Calculate P(X=k), P(X≤k), and the expected count. Givens: Trial count n=4; Success probability p=0.25.

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19. Grade 12 · Chapter 279

    Vector components and angles · Vector length r=2; Final angle (degrees)=120

    Evaluate at the signed angle θ=72° (three fifths of that final angle). Find its dot product with (1,0), its vertical component, and its length. Givens: Vector length r=2; Final angle (degrees)=120.

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20. Grade 12 · Chapter 280

    Cubic derivatives · Cubic coefficient a=1.5; Linear coefficient b=2

    Evaluate at x=0.4. Find the function value and first and second derivatives at that input. Givens: Cubic coefficient a=1.5; Linear coefficient b=2.

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