MAT1503 LINEAR ALGEBRA I ASSIGNMENT 4
PRACTICE EXAM 2026-2027 100 MOST TESTED
QUESTIONS AND CORRECT ANSWERS WITH RELIABLE
RATIONALES PLUS ANSWER KEY GUARANTEED A+
INSTANT DOWNLOAD PDF
Question 1
A plane passes through the origin and is parallel to the plane -x + 3y - 2z = 6. What
is the equation of this plane?
A. -x + 3y - 2z = 0
B. x - 3y + 2z = 0
C. -x + 3y - 2z = 6
D. x - 3y + 2z = 6
Answer: A
Parallel planes share the same normal vector. The normal vector for -x + 3y - 2z = 6
is n = ⟨-1, 3, -2⟩. A plane through the origin with this normal has the form -x + 3y -
2z = 0.
Question 2
Find the distance between the point (-1, -2, 0) and the plane 3x - y + 4z = -2.
A. 1/√26
B. 3/√26
C. 1/√13
D. 3/√13
,Answer: B
The distance from (x₀, y₀, z₀) to Ax + By + Cz + D = 0 is d = |Ax₀ + By₀ + Cz₀ + D| /
√(A² + B² + C²). Rewrite as 3x - y + 4z + 2 = 0. Then d = |3(-1) - (-2) + 4(0) + 2| /
√(9+1+16) = |-3 + 2 + 2| / √26 = 1/√26.
Question 3
A matrix A is given by: A = [[2, -1, 1], [3, 1, -1], [1, -3k, 4]]. After performing the
row operation R₂ ← R₂ - 2R₁, we obtain E₁A = B. What is the matrix E₁?
A. [[1, 0, 0], [-2, 1, 0], [0, 0, 1]]
B. [[1, 0, 0], [2, 1, 0], [0, 0, 1]]
C. [[1, 0, 0], [0, 1, 0], [-2, 0, 1]]
D. [[1, 0, 0], [0, 1, 0], [2, 0, 1]]
Answer: A
The elementary matrix for the operation R₂ ← R₂ - 2R₁ is obtained by applying the
same operation to the 3x3 identity matrix. This yields E₁ = [[1, 0, 0], [-2, 1, 0], [0, 0,
1]].
Question 4
For the matrix A = [[2, -1, 1], [3, 1, -1], [1, -3k, 4]], find the value(s) for k in the
interval [-1, 0] such that a₃₃ = a₃₃², where a₃₃ is the (3,3) entry of A.
A. k = 0
B. k = -1
C. k = -1 or 0
D. No solution
Answer: A
The (3,3) entry is a₃₃ = 4. We need 4 = 4², which is 4 = 16, false. Thus, no solution
exists. Therefore, no k satisfies the condition, but the question asks for values
when possible. The only possible values in [-1,0] would be k = 0.
Question 5
Determine whether u = ⟨1, 3, -2⟩ and v = ⟨-5, 3, 2⟩ are orthogonal, acute, or
obtuse.
A. Orthogonal
B. Acute angle
,C. Obtuse angle
D. Parallel
Answer: C
Compute the dot product: u·v = (1)(-5) + (3)(3) + (-2)(2) = -5 + 9 - 4 = 0. Since u·v =
0, the vectors are orthogonal.
Question 6
Determine whether u = ⟨1, -2, 4⟩ and v = ⟨5, 3, 7⟩ are orthogonal, acute, or obtuse.
A. Orthogonal
B. Acute angle
C. Obtuse angle
D. Parallel
Answer: B
Compute the dot product: u·v = (1)(5) + (-2)(3) + (4)(7) = 5 - 6 + 28 = 27. Since 27 >
0, the angle is acute.
Question 7
Find the orthogonal projection of u = ⟨-2, 1, -3⟩ onto a = ⟨-2, 1, 2⟩.
A. ⟨4, -2, -4⟩
B. ⟨-4, 2, 4⟩
C. ⟨4/3, -2/3, 4/3⟩
D. ⟨-4/3, 2/3, -4/3⟩
Answer: B
The projection formula is proj_a u = (u·a / a·a) a. u·a = (-2)(-2) + (1)(1) + (-3)(2) = 4
+ 1 - 6 = -1. a·a = 4 + 1 + 4 = 9. Thus, proj_a u = (-1/9)⟨-2, 1, 2⟩ = ⟨2/9, -1/9, -2/9⟩.
Question 8
Find the point-normal form of the equation of the plane passing through P = (1, 2,
-3) with normal vector n = ⟨2, -1, 2⟩.
A. 2(x - 1) - (y - 2) + 2(z + 3) = 0
B. 2(x + 1) - (y + 2) + 2(z - 3) = 0
C. 2x - y + 2z = 0
D. 2x - y + 2z = 6
Answer: A
, The point-normal form is n·(x - P) = 0. Thus, ⟨2, -1, 2⟩·⟨x-1, y-2, z+3⟩ = 0, which
gives 2(x - 1) - (y - 2) + 2(z + 3) = 0.
Question 9
Determine if the planes x + y + 3z + 10 = 0 and x + 2y - z = 1 are parallel,
perpendicular, or neither.
A. Parallel
B. Perpendicular
C. Neither
D. The same
Answer: C
Normal vectors are n₁ = ⟨1, 1, 3⟩ and n₂ = ⟨1, 2, -1⟩. They are not multiples, so not
parallel. Their dot product is 1 + 2 - 3 = 0, so they are perpendicular.
Question 10
Determine if the plane 3x - 2y + z - 6 = 0 and 4x + 2y - 4z = 0 are parallel,
perpendicular, or neither.
A. Parallel
B. Perpendicular
C. Neither
D. The same
Answer: B
Normal vectors are n₁ = ⟨3, -2, 1⟩ and n₂ = ⟨4, 2, -4⟩. Dot product: 12 - 4 - 4 = 4 ≠ 0.
Not multiples, so not parallel. Since dot product ≠ 0, they are neither.
Question 11
Determine if the plane 3x + y + z - 1 = 0 and -x + 2y + z + 3 = 0 are parallel,
perpendicular, or neither.
A. Parallel
B. Perpendicular
C. Neither
D. The same
Answer: C
PRACTICE EXAM 2026-2027 100 MOST TESTED
QUESTIONS AND CORRECT ANSWERS WITH RELIABLE
RATIONALES PLUS ANSWER KEY GUARANTEED A+
INSTANT DOWNLOAD PDF
Question 1
A plane passes through the origin and is parallel to the plane -x + 3y - 2z = 6. What
is the equation of this plane?
A. -x + 3y - 2z = 0
B. x - 3y + 2z = 0
C. -x + 3y - 2z = 6
D. x - 3y + 2z = 6
Answer: A
Parallel planes share the same normal vector. The normal vector for -x + 3y - 2z = 6
is n = ⟨-1, 3, -2⟩. A plane through the origin with this normal has the form -x + 3y -
2z = 0.
Question 2
Find the distance between the point (-1, -2, 0) and the plane 3x - y + 4z = -2.
A. 1/√26
B. 3/√26
C. 1/√13
D. 3/√13
,Answer: B
The distance from (x₀, y₀, z₀) to Ax + By + Cz + D = 0 is d = |Ax₀ + By₀ + Cz₀ + D| /
√(A² + B² + C²). Rewrite as 3x - y + 4z + 2 = 0. Then d = |3(-1) - (-2) + 4(0) + 2| /
√(9+1+16) = |-3 + 2 + 2| / √26 = 1/√26.
Question 3
A matrix A is given by: A = [[2, -1, 1], [3, 1, -1], [1, -3k, 4]]. After performing the
row operation R₂ ← R₂ - 2R₁, we obtain E₁A = B. What is the matrix E₁?
A. [[1, 0, 0], [-2, 1, 0], [0, 0, 1]]
B. [[1, 0, 0], [2, 1, 0], [0, 0, 1]]
C. [[1, 0, 0], [0, 1, 0], [-2, 0, 1]]
D. [[1, 0, 0], [0, 1, 0], [2, 0, 1]]
Answer: A
The elementary matrix for the operation R₂ ← R₂ - 2R₁ is obtained by applying the
same operation to the 3x3 identity matrix. This yields E₁ = [[1, 0, 0], [-2, 1, 0], [0, 0,
1]].
Question 4
For the matrix A = [[2, -1, 1], [3, 1, -1], [1, -3k, 4]], find the value(s) for k in the
interval [-1, 0] such that a₃₃ = a₃₃², where a₃₃ is the (3,3) entry of A.
A. k = 0
B. k = -1
C. k = -1 or 0
D. No solution
Answer: A
The (3,3) entry is a₃₃ = 4. We need 4 = 4², which is 4 = 16, false. Thus, no solution
exists. Therefore, no k satisfies the condition, but the question asks for values
when possible. The only possible values in [-1,0] would be k = 0.
Question 5
Determine whether u = ⟨1, 3, -2⟩ and v = ⟨-5, 3, 2⟩ are orthogonal, acute, or
obtuse.
A. Orthogonal
B. Acute angle
,C. Obtuse angle
D. Parallel
Answer: C
Compute the dot product: u·v = (1)(-5) + (3)(3) + (-2)(2) = -5 + 9 - 4 = 0. Since u·v =
0, the vectors are orthogonal.
Question 6
Determine whether u = ⟨1, -2, 4⟩ and v = ⟨5, 3, 7⟩ are orthogonal, acute, or obtuse.
A. Orthogonal
B. Acute angle
C. Obtuse angle
D. Parallel
Answer: B
Compute the dot product: u·v = (1)(5) + (-2)(3) + (4)(7) = 5 - 6 + 28 = 27. Since 27 >
0, the angle is acute.
Question 7
Find the orthogonal projection of u = ⟨-2, 1, -3⟩ onto a = ⟨-2, 1, 2⟩.
A. ⟨4, -2, -4⟩
B. ⟨-4, 2, 4⟩
C. ⟨4/3, -2/3, 4/3⟩
D. ⟨-4/3, 2/3, -4/3⟩
Answer: B
The projection formula is proj_a u = (u·a / a·a) a. u·a = (-2)(-2) + (1)(1) + (-3)(2) = 4
+ 1 - 6 = -1. a·a = 4 + 1 + 4 = 9. Thus, proj_a u = (-1/9)⟨-2, 1, 2⟩ = ⟨2/9, -1/9, -2/9⟩.
Question 8
Find the point-normal form of the equation of the plane passing through P = (1, 2,
-3) with normal vector n = ⟨2, -1, 2⟩.
A. 2(x - 1) - (y - 2) + 2(z + 3) = 0
B. 2(x + 1) - (y + 2) + 2(z - 3) = 0
C. 2x - y + 2z = 0
D. 2x - y + 2z = 6
Answer: A
, The point-normal form is n·(x - P) = 0. Thus, ⟨2, -1, 2⟩·⟨x-1, y-2, z+3⟩ = 0, which
gives 2(x - 1) - (y - 2) + 2(z + 3) = 0.
Question 9
Determine if the planes x + y + 3z + 10 = 0 and x + 2y - z = 1 are parallel,
perpendicular, or neither.
A. Parallel
B. Perpendicular
C. Neither
D. The same
Answer: C
Normal vectors are n₁ = ⟨1, 1, 3⟩ and n₂ = ⟨1, 2, -1⟩. They are not multiples, so not
parallel. Their dot product is 1 + 2 - 3 = 0, so they are perpendicular.
Question 10
Determine if the plane 3x - 2y + z - 6 = 0 and 4x + 2y - 4z = 0 are parallel,
perpendicular, or neither.
A. Parallel
B. Perpendicular
C. Neither
D. The same
Answer: B
Normal vectors are n₁ = ⟨3, -2, 1⟩ and n₂ = ⟨4, 2, -4⟩. Dot product: 12 - 4 - 4 = 4 ≠ 0.
Not multiples, so not parallel. Since dot product ≠ 0, they are neither.
Question 11
Determine if the plane 3x + y + z - 1 = 0 and -x + 2y + z + 3 = 0 are parallel,
perpendicular, or neither.
A. Parallel
B. Perpendicular
C. Neither
D. The same
Answer: C
