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Quiz Chapter 3: Systems of Equations

10 questions · Form 4 Additional Mathematics Bab 3: Systems of Equations

Question 1 of 10Score: 0

Find the coordinates of the turning/intersection point where the line y = 4x - 4 touches the curve y = x² cleanly as a tangent.

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 coordinates of the turning/intersection point where the line y = 4x - 4 touches the curve y = x² cleanly as a tangent.

  1. (2, 4)
  2. (1, 0)
  3. (-2, 4)
  4. (4, 16)
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Answer: A

Equate y: x² = 4x - 4 => x² - 4x + 4 = 0 => (x - 2)² = 0 => x = 2. Then y = (2)² = 4. Touch point is (2, 4).

2. If a straight line y = mx + c is tangent to a curve x² + y² = 5, what must be the value of the discriminant B² - 4AC when the line is substituted into the curve?

  1. B² - 4AC > 0
  2. B² - 4AC < 0
  3. B² - 4AC = 0
  4. B² - 4AC ≥ 0
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Answer: C

A line being tangent to a curve means it touches the curve at exactly one point (two equal real roots), which corresponds to B² - 4AC = 0.

3. When solving simultaneous equations involving one linear and one non-linear equation, what is the best first step?

  1. Differentiate the non-linear equation
  2. Express one variable in terms of another using the linear equation
  3. Square both equations
  4. Set the discriminant of the non-linear equation to zero
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Answer: B

The standard substitution method requires expressing one variable as the subject from the linear equation first, then substituting it into the non-linear equation.

4. If x - y = 3 and x² + y² = 29, find the value of xy.

  1. 10
  2. 20
  3. 12
  4. 15
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Answer: A

Square (x - y = 3): (x - y)² = 9 => x² - 2xy + y² = 9. Substitute x² + y² = 29: 29 - 2xy = 9 => 2xy = 20 => xy = 10.

5. Find the points of intersection between the line y = 2x - 1 and the parabola y = x² - 4.

  1. (-1, -3) and (3, 5)
  2. (1, 1) and (-3, -7)
  3. (3, 5) and (-1, 3)
  4. (0, -1) and (2, 3)
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Answer: A

Equate y: x² - 4 = 2x - 1 => x² - 2x - 3 = 0 => (x - 3)(x + 1) = 0 => x = 3 or x = -1. When x = 3, y = 5. When x = -1, y = -3. Points are (3, 5) and (-1, -3).

6. Solve the system: 2x + y = 7 and x² + xy = 12. What is one of the possible values for x?

  1. x = 3
  2. x = 5
  3. x = 1
  4. x = 6
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Answer: A

Substitute y = 7 - 2x into x² + xy = 12: x² + x(7 - 2x) = 12 => x² + 7x - 2x² = 12 => -x² + 7x - 12 = 0 => x² - 7x + 12 = 0 => (x - 3)(x - 4) = 0 => x = 3 or x = 4.

7. Two numbers have a sum of 12 and a product of 35. Find the larger number.

  1. 7
  2. 5
  3. 8
  4. 9
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Answer: A

x + y = 12 and xy = 35 => x(12 - x) = 35 => x² - 12x + 35 = 0 => (x - 7)(x - 5) = 0. Numbers are 7 and 5. Larger is 7.

8. Which of the following equations is a linear equation in three variables?

  1. 2x + 3y - z² = 5
  2. xy + z = 4
  3. 3x - 4y + 2z = 9
  4. 1x + y - 2z = 0
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Answer: C

A linear equation in three variables must have each variable raised to the power of 1 without product terms like xy. Thus, 3x - 4y + 2z = 9 is linear.

9. Solve the non-linear system: x - 2y = 1 and x² - 4y² = 5. Find x + 2y.

  1. 5
  2. 1
  3. 25
  4. 3
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Answer: A

Use difference of squares: x² - 4y² = (x - 2y)(x + 2y) = 5. Given x - 2y = 1, we get 1(x + 2y) = 5 => x + 2y = 5.

10. If three planes in 3D space intersect along a common straight line, how many solutions does the system of three linear equations have?

  1. Exactly one solution
  2. No solution
  3. Infinitely many solutions
  4. Exactly three solutions
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Answer: C

When three planes intersect along a line, every point on that line is a common solution, resulting in infinitely many solutions.

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