10 questions · Form 4 Additional Mathematics Bab 3: Systems of Equations
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Prefer reading to quizzing? All 10 questions are listed below with the answer and explanation under each one.
1. If elimination yields the identity 0 = 0 when solving a system of three linear equations, what can be concluded?
Answer: B
A tautology such as 0 = 0 signifies dependent equations, meaning the planes intersect along a line or coincide, giving infinitely many solutions.
2. The perimeter of a rectangular garden is 28 m and its area is 40 m². Find the length and width of the garden.
Answer: A
Let length = x, width = y. 2(x + y) = 28 => x + y = 14 => y = 14 - x. Area: xy = 40 => x(14 - x) = 40 => x² - 14x + 40 = 0 => (x - 10)(x - 4) = 0. Dimensions are 10 m and 4 m.
3. Solve the system: 2x + y = 7 and x² + xy = 12. What is one of the possible values for x?
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.
4. How many intersection points exist between the straight line x + y = 5 and the curve x² - y = 1?
Answer: C
Substitute y = 5 - x into x² - y = 1: x² - (5 - x) = 1 => x² + x - 6 = 0 => (x + 3)(x - 2) = 0 => 2 real distinct roots (x = 2, x = -3), giving 2 intersection points.
5. Geometrically, what does a single linear equation in three variables ax + by + cz = d represent?
Answer: B
In 3D Cartesian coordinates, a first-degree linear equation in three variables represents a flat 2D plane.
6. When solving simultaneous equations involving one linear and one non-linear equation, what is the best first step?
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.
7. Solve the simultaneous equations x - y = 2 and x² + y² = 10 for positive values of x and y.
Answer: A
Substitute y = x - 2 into x² + y² = 10: x² + (x - 2)² = 10 => x² + x² - 4x + 4 = 10 => 2x² - 4x - 6 = 0 => x² - 2x - 3 = 0 => (x - 3)(x + 1) = 0. Positive x = 3, so y = 3 - 2 = 1.
8. If three planes in 3D space intersect along a common straight line, how many solutions does the system of three linear equations have?
Answer: C
When three planes intersect along a line, every point on that line is a common solution, resulting in infinitely many solutions.
9. Find the coordinates of the turning/intersection point where the line y = 4x - 4 touches the curve y = x² cleanly as a tangent.
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).
10. Solve the linear system: x + y + z = 6, 2x - y + z = 3, and x + 2y - z = 2. Find the value of x.
Answer: A
From Eq 1 + Eq 3: (x+y+z) + (x+2y-z) = 6 + 2 => 2x + 3y = 8. From Eq 1 + Eq 2: 3x + 2z = 9... Solving the system algebraically yields x = 1, y = 2, z = 3.