Unit notes and practice questions
Everything here is for unit 5 of the Grade 11 Mathematics textbook: what the chapter covers, the short notes, practice questions with their answers, and the lesson video where one exists.
The cards filed under this chapter on the Grade 11 Mathematics page. Open one at a time.
When you use it
Use to calculate the dot product, find the angle between vectors, or check if two vectors are perpendicular.
Watch out
Two vectors are perpendicular if and only if their scalar product is zero. The dot product is a scalar value, not a vector.
Drafted from Grade 11 Mathematics, pages 251-312, then checked twice before it went up
When you use it
Use to find the equation of the tangent line when the circle is centered at the origin (0, 0).
Watch out
This equation only applies when the circle center is at (0, 0). For shifted centers, use the general formula.
Drafted from Grade 11 Mathematics, pages 251-312, then checked twice before it went up
When you use it
Use to find the equation of the tangent line to a circle with center (x_0, y_0) at a given point of tangency (x_1, y_1).
Watch out
Do not confuse the point of tangency with the variable coordinates of the line. The right hand side must be r^2 and not r.
Drafted from Grade 11 Mathematics, pages 251-312, then checked twice before it went up
10 multiple-choice questions written against this chapter, each with its answer and the reason. Try one before you open the answer.
Question 1
When using Cramer's rule to solve a system of n linear equations in n variables, how many determinants must be evaluated?
Answer: B n + 1
Why
The textbook states that solving a system of n linear equations in n variables using Cramer's rule requires evaluating n + 1 determinants.
Question 2
According to the textbook classification, which of the following is a vector quantity?
Answer: C Velocity
Why
Velocity requires both magnitude and direction, so it is classified as a vector quantity. Mass, time, and temperature are scalar quantities.
Question 3
Three points (x1, y1), (x2, y2), and (x3, y3) in the xy-plane are collinear if and only if the determinant of the matrix with rows [x1 y1 1], [x2 y2 1], and [x3 y3 1] is equal to what value?
Answer: A 0
Why
Three points in the xy-plane lie on the same straight line if and only if the determinant of the 3 by 3 matrix formed by their coordinates and ones is 0.
Question 4
If A is an invertible matrix of order n, what is the determinant of its inverse, det(A-1)?
Answer: B 1 / det(A)
Why
By the properties of determinants in the textbook, if A is invertible, then det(A-1) = 1 / det(A).
Question 5
What is the area of a triangle with vertices at (0, 5), (0, 0), and (5, 0) in the xy-plane?
Answer: C 12.5 square units
Why
The determinant evaluating the vertices gives 25. The area is (1/2) * 25 = 12.5 square units.
Question 6
A student in Hawassa solves a linear system AX = B of n equations in n variables and finds |A| = 0. What can be concluded about the system?
Answer: B It has either no solution or infinitely many solutions
Why
When |A| = 0, Cramer's rule cannot be used. The textbook states there are two possibilities: the system has no solution or it has infinitely many solutions.
Question 7
Using determinants, what is the general equation of the line passing through the points P(2, 0) and R(0, 3)?
Answer: A 3x + 2y - 6 = 0
Why
Setting the determinant of [x y 1; 2 0 1; 0 3 1] to 0 yields -2(y - 3) - 3x = 0, which simplifies to 3x + 2y - 6 = 0.
Question 8
What is the area of the triangle with vertices P(1, 1), Q(1, 3), and R(4, 5)?
Answer: B 3 square units
Why
The determinant of the coordinate matrix is -6. Since area is positive, Area = -(1/2)(-6) = 3 square units.
Question 9
Let A and B be square matrices of order 3 with det(A) = 2 and det(B) = -3. What is the value of det(2AB)?
Answer: C -48
Why
For a 3 by 3 matrix, det(2M) = 23 det(M) = 8 det(M). Thus, det(2AB) = 8 * det(A) * det(B) = 8 * 2 * (-3) = -48.
Question 10
An interpolating polynomial p(x) = a + bx + cx2 fits the data points (0, 1), (1, 2), and (2, 22). What is the estimated value of p(3)?
Answer: D 61
Why
Solving the system gives p(x) = 1 - 8.5x + 9.5x2. Evaluating at x = 3 gives p(3) = 1 - 8.5(3) + 9.5(9) = 1 - 25.5 + 85.5 = 61.
The app holds 18 questions on this unit, marked as you go. These are the first 10.
Unit 5 runs from page 251 to page 312 of the Ministry of Education edition.
Ministry of Education edition, English, 494 pages
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