10 questions · Form 4 Additional Mathematics Bab 3: Systems of Equations
Solve the linear system: x + y + z = 6, 2x - y + z = 3, and x + 2y - z = 2. Find the value of x.
Prefer reading to quizzing? All 10 questions are listed below with the answer and explanation under each one.
1. 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.
2. If the line y = kx - 2 does not intersect the curve y = x² + 2, find the range of values of k.
Answer: A
Equate y: x² + 2 = kx - 2 => x² - kx + 4 = 0. No intersection => B² - 4AC < 0 => (-k)² - 4(1)(4) < 0 => k² - 16 < 0 => (k - 4)(k + 4) < 0 => -4 < k < 4.
3. Given x + 2y - z = 4, 2x - y + z = 3, and 3x + y + 2z = 11, calculate the value of z.
Answer: B
Solving this system gives x = 1, y = 2, z = 2.
4. 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.
5. 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.
6. 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.
7. Two numbers have a sum of 12 and a product of 35. Find the larger number.
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. 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. Solve for y in the simultaneous equations: 3x + y = 1 and 2x² - y² = -7 for a positive value of y.
Answer: A
Substitute y = 1 - 3x into 2x² - y² = -7: 2x² - (1 - 3x)² = -7 => 2x² - (1 - 6x + 9x²) = -7 => -7x² + 6x - 1 = -7 => 7x² - 6x - 6 = 0... Alternate root check: for x = -1, y = 1 - 3(-1) = 4.
10. 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.