Math With Amar · Guided practice
10,000 ways to practice.
One idea at a time.
Each chapter gives you a distinct mathematical scenario: one question to attempt, a hint, a worked solution, and an animated example you can control.
10,000 questions · 10,000 animated scenarios · 14 study levels
Choose a starting point
Practice at your level.
Grade labels are starting points; school sequences vary. Advanced chapters assume the prerequisites listed on their pages.
Attempt · Explain · Animate
Grade 12 practice chapters
401–420 of 600 matching chapters · Page 21 of 30
Clear filtersGrade 12 · Chapter 401
Cubic derivatives · Cubic coefficient a=1; Linear coefficient b=2.5
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=2.5.
Question · Worked solution · Animated example
Open practice chapterGrade 12 · Chapter 402
The product rule · Linear shift a=0.5; 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=0.5; Exponential rate b=-0.75.
Question · Worked solution · Animated example
Open practice chapterGrade 12 · Chapter 403
Implicit circle tangents · Circle radius r=1; Final angle (degrees)=330
Use angle θ=198°. Find both coordinates and the tangent slope; identify a vertical tangent instead of dividing by zero when necessary. Givens: Circle radius r=1; Final angle (degrees)=330.
Question · Worked solution · Animated example
Open practice chapterGrade 12 · Chapter 404
Signed accumulation · Slope a=1.5; Subtracted intercept b=0.5
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=0.5.
Question · Worked solution · Animated example
Open practice chapterGrade 12 · Chapter 405
Substitution and bounds · Exponent factor k=0.7; 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=0.7; Final upper bound B=0.5.
Question · Worked solution · Animated example
Open practice chapterGrade 12 · Chapter 406
Differential growth models · Initial amount a=1.5; Rate k=0.5
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=0.5.
Question · Worked solution · Animated example
Open practice chapterGrade 12 · Chapter 407
Binomial probability · Trial count n=7; Success probability p=0.45
Use success count k=4. Calculate P(X=k), P(X≤k), and the expected count. Givens: Trial count n=7; Success probability p=0.45.
Question · Worked solution · Animated example
Open practice chapterGrade 12 · Chapter 408
Vector components and angles · Vector length r=2; Final angle (degrees)=165
Evaluate at the signed angle θ=99° (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)=165.
Question · Worked solution · Animated example
Open practice chapterGrade 12 · Chapter 409
Cubic derivatives · Cubic coefficient a=1.5; Linear coefficient b=3
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=3.
Question · Worked solution · Animated example
Open practice chapterGrade 12 · Chapter 410
The product rule · Linear shift a=1; Exponential rate b=-0.5
Evaluate at x=0.2. Find the product value, its derivative, and the contribution from differentiating the exponential factor. Givens: Linear shift a=1; Exponential rate b=-0.5.
Question · Worked solution · Animated example
Open practice chapterGrade 12 · Chapter 411
Implicit circle tangents · Circle radius r=1.5; Final angle (degrees)=30
Use angle θ=18°. Find both coordinates and the tangent slope; identify a vertical tangent instead of dividing by zero when necessary. Givens: Circle radius r=1.5; Final angle (degrees)=30.
Question · Worked solution · Animated example
Open practice chapterGrade 12 · Chapter 412
Signed accumulation · Slope a=2; Subtracted intercept b=1
Integrate on 0≤x≤2.4. Find the signed integral, total geometric area, and zero crossing of the line. Givens: Slope a=2; Subtracted intercept b=1.
Question · Worked solution · Animated example
Open practice chapterGrade 12 · Chapter 413
Substitution and bounds · Exponent factor k=0.75; Final upper bound B=0.75
Integrate from x=0 to x=0.45. 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=0.75.
Question · Worked solution · Animated example
Open practice chapterGrade 12 · Chapter 414
Differential growth models · Initial amount a=2; Rate k=-0.5
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=2; Rate k=-0.5.
Question · Worked solution · Animated example
Open practice chapterGrade 12 · Chapter 415
Vector components and angles · Vector length r=2.5; Final angle (degrees)=180
Evaluate at the signed angle θ=108° (three fifths of that final angle). Find its dot product with (1,0), its vertical component, and its length. Givens: Vector length r=2.5; Final angle (degrees)=180.
Question · Worked solution · Animated example
Open practice chapterGrade 12 · Chapter 416
Cubic derivatives · Cubic coefficient a=2; Linear coefficient b=-3
Evaluate at x=0.4. Find the function value and first and second derivatives at that input. Givens: Cubic coefficient a=2; Linear coefficient b=-3.
Question · Worked solution · Animated example
Open practice chapterGrade 12 · Chapter 417
The product rule · Linear shift a=1.5; Exponential rate b=-0.25
Evaluate at x=0.2. Find the product value, its derivative, and the contribution from differentiating the exponential factor. Givens: Linear shift a=1.5; Exponential rate b=-0.25.
Question · Worked solution · Animated example
Open practice chapterGrade 12 · Chapter 418
Signed accumulation · Slope a=2.5; Subtracted intercept b=1.5
Integrate on 0≤x≤2.4. Find the signed integral, total geometric area, and zero crossing of the line. Givens: Slope a=2.5; Subtracted intercept b=1.5.
Question · Worked solution · Animated example
Open practice chapterGrade 12 · Chapter 419
Substitution and bounds · Exponent factor k=0.8; Final upper bound B=1
Integrate from x=0 to x=0.6. Find the transformed upper bound under u=kx² and the exact integral, then give a decimal approximation. Givens: Exponent factor k=0.8; Final upper bound B=1.
Question · Worked solution · Animated example
Open practice chapterGrade 12 · Chapter 420
Differential growth models · Initial amount a=2.5; Rate k=-0.4
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=2.5; Rate k=-0.4.
Question · Worked solution · Animated example
Open practice chapter