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Grade 12 Quiz: Geometry — Easy Version 1
Grade 12 · MathSheets Name: ____________________ Date: __________
Answer every question. The lesson name before each question tells you which skill it checks — open that lesson if you need a reminder.
How to solve
This quiz mixes questions from different lessons, like a real test. Read each question carefully: the lesson name in front tells you what kind of problem it is. Check your answers, then practise the lessons where you made mistakes.
Example
Transformations of Graphs: P(-1, -5) is on the graph of y = f(x). Which point must be on the graph of y = f(x + 1)?
Answer: (-2, -5)
Not quite. The answer is
💡 The − 4 outside f moves the graph down 4, so subtract 4 from y: -3 − 4 = -7. x stays -3. So P moves to (-3, -7).
✓ Correct!
Not quite. The answer is
💡 The − 4 outside f moves the graph down 4, so subtract 4 from y: -3 − 4 = -7. x stays -3. So P moves to (-3, -7).
💡 How to solve: Transformations of Graphs
Changes outside f act on y: +k moves up, -f flips over the x-axis, a·f stretches y by a. Changes inside act on x the opposite way: f(x - h) moves right h, f(-x) flips over the y-axis, f(bx) divides x by b.
Try a similar one
P(-3, -5) is on the graph of y = f(x). Which point must be on the graph of y = f(x - 3)?
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 This matches the double-angle formula cos 2θ = cos2 θ − sin2 θ with θ = 30°, so it equals cos(2 × 30°) = cos(60°). From the 30°-60°-90° triangle, cos(60°) = 12.
✓ Correct!
Not quite. The answer is
💡 This matches the double-angle formula cos 2θ = cos2 θ − sin2 θ with θ = 30°, so it equals cos(2 × 30°) = cos(60°). From the 30°-60°-90° triangle, cos(60°) = 12.
💡 How to solve: Double-Angle Formulas
sin 2θ = 2 sin θ cos θ, cos 2θ = cos2 θ - sin2 θ = 1 - 2sin2 θ = 2cos2 θ - 1, and tan 2θ = 2tan θ / (1 - tan2 θ). First find the missing sin or cos with a right triangle (watch the quadrant signs), then substitute.
Try a similar one
sin θ = 2425 and θ is acute. Find the exact value of cos 2θ.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 Inside f, x + 4 moves the graph left 4 (the opposite of the sign you see), so subtract 4 from x: 4 − 4 = 0. y stays 1. So P moves to (0, 1).
✓ Correct!
Not quite. The answer is
💡 Inside f, x + 4 moves the graph left 4 (the opposite of the sign you see), so subtract 4 from x: 4 − 4 = 0. y stays 1. So P moves to (0, 1).
💡 How to solve: Transformations of Graphs
Changes outside f act on y: +k moves up, -f flips over the x-axis, a·f stretches y by a. Changes inside act on x the opposite way: f(x - h) moves right h, f(-x) flips over the y-axis, f(bx) divides x by b.
Try a similar one
P(-1, -2) is on the graph of y = f(x). Which point must be on the graph of y = f(x - 3)?
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 Use sin2 θ + cos2 θ = 1: cos2 θ = 1 − (817)2 = 1 − 64289 = 225289, so cos θ = ±1517. θ is acute, so cos θ = 1517. Double-angle formula: sin 2θ = 2 sin θ cos θ = 2 × (817) × (1517) = 240289. So sin 2θ = 240289.
✓ Correct!
Not quite. The answer is
💡 Use sin2 θ + cos2 θ = 1: cos2 θ = 1 − (817)2 = 1 − 64289 = 225289, so cos θ = ±1517. θ is acute, so cos θ = 1517. Double-angle formula: sin 2θ = 2 sin θ cos θ = 2 × (817) × (1517) = 240289. So sin 2θ = 240289.
💡 How to solve: Double-Angle Formulas
sin 2θ = 2 sin θ cos θ, cos 2θ = cos2 θ - sin2 θ = 1 - 2sin2 θ = 2cos2 θ - 1, and tan 2θ = 2tan θ / (1 - tan2 θ). First find the missing sin or cos with a right triangle (watch the quadrant signs), then substitute.
Try a similar one
sin θ = 725 and θ is acute. Find the exact value of cos 2θ.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 Inside f, x − 1 moves the graph right 1 (the opposite of the sign you see), so add 1 to x: 5 + 1 = 6. y stays 1. So P moves to (6, 1).
✓ Correct!
Not quite. The answer is
💡 Inside f, x − 1 moves the graph right 1 (the opposite of the sign you see), so add 1 to x: 5 + 1 = 6. y stays 1. So P moves to (6, 1).
💡 How to solve: Transformations of Graphs
Changes outside f act on y: +k moves up, -f flips over the x-axis, a·f stretches y by a. Changes inside act on x the opposite way: f(x - h) moves right h, f(-x) flips over the y-axis, f(bx) divides x by b.
Try a similar one
P(1, -2) is on the graph of y = f(x). Which point must be on the graph of y = f(x - 5)?
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 This matches the double-angle formula sin 2θ = 2 sin θ cos θ with θ = 30°, so it equals sin(2 × 30°) = sin(60°). From the 30°-60°-90° triangle, sin(60°) = √32.
✓ Correct!
Not quite. The answer is
💡 This matches the double-angle formula sin 2θ = 2 sin θ cos θ with θ = 30°, so it equals sin(2 × 30°) = sin(60°). From the 30°-60°-90° triangle, sin(60°) = √32.
💡 How to solve: Double-Angle Formulas
sin 2θ = 2 sin θ cos θ, cos 2θ = cos2 θ - sin2 θ = 1 - 2sin2 θ = 2cos2 θ - 1, and tan 2θ = 2tan θ / (1 - tan2 θ). First find the missing sin or cos with a right triangle (watch the quadrant signs), then substitute.
Try a similar one
sin θ = 817 and θ is acute. Find the exact value of cos 2θ.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 Inside f, x − 4 moves the graph right 4 (the opposite of the sign you see), so add 4 to x: 2 + 4 = 6. y stays -3. So P moves to (6, -3).
✓ Correct!
Not quite. The answer is
💡 Inside f, x − 4 moves the graph right 4 (the opposite of the sign you see), so add 4 to x: 2 + 4 = 6. y stays -3. So P moves to (6, -3).
💡 How to solve: Transformations of Graphs
Changes outside f act on y: +k moves up, -f flips over the x-axis, a·f stretches y by a. Changes inside act on x the opposite way: f(x - h) moves right h, f(-x) flips over the y-axis, f(bx) divides x by b.
Try a similar one
P(-5, 2) is on the graph of y = f(x). Which point must be on the graph of y = f(x) - 1?
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 Use sin2 θ + cos2 θ = 1: cos2 θ = 1 − (513)2 = 1 − 25169 = 144169, so cos θ = ±1213. θ is acute, so cos θ = 1213. Double-angle formula: sin 2θ = 2 sin θ cos θ = 2 × (513) × (1213) = 120169. So sin 2θ = 120169.
✓ Correct!
Not quite. The answer is
💡 Use sin2 θ + cos2 θ = 1: cos2 θ = 1 − (513)2 = 1 − 25169 = 144169, so cos θ = ±1213. θ is acute, so cos θ = 1213. Double-angle formula: sin 2θ = 2 sin θ cos θ = 2 × (513) × (1213) = 120169. So sin 2θ = 120169.
💡 How to solve: Double-Angle Formulas
sin 2θ = 2 sin θ cos θ, cos 2θ = cos2 θ - sin2 θ = 1 - 2sin2 θ = 2cos2 θ - 1, and tan 2θ = 2tan θ / (1 - tan2 θ). First find the missing sin or cos with a right triangle (watch the quadrant signs), then substitute.
Try a similar one
sin θ = 1517 and θ is acute. Find the exact value of cos 2θ.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 The − 5 outside f moves the graph down 5, so subtract 5 from y: -2 − 5 = -7. x stays -2. So P moves to (-2, -7).
✓ Correct!
Not quite. The answer is
💡 The − 5 outside f moves the graph down 5, so subtract 5 from y: -2 − 5 = -7. x stays -2. So P moves to (-2, -7).
💡 How to solve: Transformations of Graphs
Changes outside f act on y: +k moves up, -f flips over the x-axis, a·f stretches y by a. Changes inside act on x the opposite way: f(x - h) moves right h, f(-x) flips over the y-axis, f(bx) divides x by b.
Try a similar one
P(-4, -4) is on the graph of y = f(x). Which point must be on the graph of y = f(x) - 3?
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 This matches the double-angle formula sin 2θ = 2 sin θ cos θ with θ = 22.5°, so it equals sin(2 × 22.5°) = sin(45°). From the 45°-45°-90° triangle, sin(45°) = √22.
✓ Correct!
Not quite. The answer is
💡 This matches the double-angle formula sin 2θ = 2 sin θ cos θ with θ = 22.5°, so it equals sin(2 × 22.5°) = sin(45°). From the 45°-45°-90° triangle, sin(45°) = √22.
💡 How to solve: Double-Angle Formulas
sin 2θ = 2 sin θ cos θ, cos 2θ = cos2 θ - sin2 θ = 1 - 2sin2 θ = 2cos2 θ - 1, and tan 2θ = 2tan θ / (1 - tan2 θ). First find the missing sin or cos with a right triangle (watch the quadrant signs), then substitute.
Try a similar one
cos θ = 1213 and θ is acute. Find the exact value of sin 2θ.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
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Grade 12 Quiz: Geometry — Easy Version 1
Answer every question. The lesson name before each question tells you which skill it checks — open that lesson if you need a reminder.