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Calculus I & II
Linear Algebra
Calculus I & II Quiz
Q1 [Limits and continuity] Evaluate lim (x → 3) of (x² − 9)/(x − 3).
0
3
Does not exist
6
Q2 [Derivatives and applications] What is the derivative of f(x) = 5x⁴ − 3x?
20x³ − 3
20x³ − 3x
5x³ − 3
20x⁴ − 3
Q3 [Integrals and techniques of integration] Evaluate ∫ from 0 to 2 of (2x + 1) dx.
4
5
8
6
Q4 [Sequences and series] What is the sum of the series 1 + 1/3 + 1/9 + 1/27 + ... (forever)?
4/3
3/2
3
The series diverges.
Q5 [Applications of integration] Find the area between y = x² and the x-axis from x = 0 to x = 3.
27
3
9
6
Q6 [Limits and continuity] Evaluate lim (x → ∞) of (3x² + 1)/(5x² − x).
3/5
0
∞
−1
Q7 [Derivatives and applications] What is the derivative of f(x) = x²eˣ?
2xeˣ
eˣ(x² + 2x)
x²eˣ
2x + eˣ
Q8 [Integrals and techniques of integration] Find ∫ x cos x dx.
x sin x + cos x + C
x sin x − cos x + C
(x²/2) sin x + C
−x sin x + cos x + C
Q9 [Sequences and series] Which of these series converges?
Σ 1/n²
Σ 1/n
Σ 1/√n
Σ n/(n + 1)
Q10 [Applications of integration] The region under y = √x from x = 0 to x = 4 is rotated about the x-axis. What is the volume?
16π
8π
4π
(16/3)π
Q11 [Limits and continuity] Evaluate lim (x → 0) of sin(5x)/x.
1
0
5
1/5
Q12 [Derivatives and applications] The radius of a sphere grows at 2 cm/s. How fast is the volume growing when r = 5 cm? (V = (4/3)πr³)
100π cm³/s
200π cm³/s
(1000/3)π cm³/s
40π cm³/s
Q13 [Integrals and techniques of integration] Evaluate ∫ from 0 to 1 of x e^(x²) dx.
(e − 1)/2
e − 1
e/2
(e − 1)/3
Q14 [Sequences and series] What is the radius of convergence of Σ (n = 1 to ∞) of xⁿ/(n · 2ⁿ)?
1/2
1
∞
2
Q15 [Applications of integration] What is the average value of f(x) = x² on the interval [0, 3]?
9
4.5
3
1
Q16 [Derivatives and applications] For the curve x² + xy + y² = 7, find dy/dx at the point (1, 2).
−5/4
−4/5
4/5
−2
Q17 [Integrals and techniques of integration] Find ∫ 1/(x² − 1) dx.
ln abs(x² − 1) + C
(1/2) ln abs((x + 1)/(x − 1)) + C
(1/2) ln abs((x − 1)/(x + 1)) + C
arctan x + C
Q18 [Limits and continuity] Evaluate lim (x → ∞) of (1 + 2/x)ˣ.
1
e
e²
∞
Q19 [Applications of integration] Find the arc length of y = (2/3)x^(3/2) from x = 0 to x = 3.
16/3
2√3
7/3
14/3
Q20 [Sequences and series] What is the coefficient of x³ in the Maclaurin series of sin(2x)?
−1/6
−8
4/3
−4/3
Linear Algebra Quiz
Q1 [Linear systems and row reduction] How many solutions does the system x + y = 2 and 2x + 2y = 5 have?
Exactly one
No solution
Infinitely many
Exactly two
Q2 [Matrix operations and inverses] Let A = [1 2; 3 4] and B = [0 1; 1 0]. What is AB?
[3 4; 1 2]
[0 2; 3 0]
[2 1; 4 3]
[1 3; 2 4]
Q3 [Determinants] What is the determinant of [3 5; 2 4]?
2
22
−2
7
Q4 [Span, independence and basis] Which vector is in the span of [1, 0, 1] and [0, 1, 1]?
[2, 3, 1]
[2, 3, 5]
[1, 1, 1]
[0, 0, 1]
Q5 [Eigenvalues and eigenvectors] What are the eigenvalues of A = [2 0; 0 5]?
7 and 10
2 and 5
0 and 5
1 and 1
Q6 [Matrix operations and inverses] If A is a 2×3 matrix and B is a 3×4 matrix, what is the size of AB?
2×4
3×3
4×2
AB is not defined
Q7 [Linear systems and row reduction] Solve the system with augmented matrix [1 2 | 5; 3 −1 | 1].
x = 2, y = 1
x = 3, y = 1
x = −1, y = 3
x = 1, y = 2
Q8 [Determinants] What is the determinant of [1 2 0; 0 3 1; 2 0 1]?
−1
3
11
7
Q9 [Matrix operations and inverses] What is the inverse of A = [2 1; 5 3]?
[3 1; 5 2]
[−3 1; 5 −2]
[3 −1; −5 2]
[1/2 1; 1/5 1/3]
Q10 [Span, independence and basis] Which set of vectors is linearly independent?
{[1, 2], [2, 4]}
{[1, 0, 2], [0, 1, 3], [1, 1, 5]}
{[1, 1, 0], [0, 1, 1], [1, 0, 1]}
{[1, 2, 3], [4, 5, 6], [7, 8, 10], [1, 0, 0]}
Q11 [Eigenvalues and eigenvectors] What are the eigenvalues of A = [4 1; 2 3]?
2 and 5
−2 and −5
4 and 3
1 and 6
Q12 [Determinants] A is a 3×3 matrix with det A = 4. What is det(2A)?
32
8
16
64
Q13 [Linear systems and row reduction] A system in x, y, z row reduces to [1 0 2 | 3; 0 1 −1 | 1; 0 0 0 | 0]. What can you conclude?
No solution
Infinitely many solutions, with 1 free variable
Exactly one solution
Infinitely many solutions, with 2 free variables
Q14 [Span, independence and basis] What is the dimension of the null space of A = [1 2 3; 2 4 6]?
1
3
0
2
Q15 [Eigenvalues and eigenvectors] v = [1, 1] is an eigenvector of A = [3 1; 2 2]. What is its eigenvalue?
1
5
4
2
Q16 [Matrix operations and inverses] A and B are invertible n×n matrices. Which statement is always true?
(AB)⁻¹ = A⁻¹B⁻¹
(A + B)⁻¹ = A⁻¹ + B⁻¹
AB = BA
(AB)⁻¹ = B⁻¹A⁻¹
Q17 [Determinants] For which value of k is [1 2 k; 0 1 1; 1 0 2] NOT invertible?
k = −4
k = 4
k = 0
k = 2
Q18 [Span, independence and basis] Which set is a basis for W = {(x, y, z) : x + y + z = 0}?
{[1, −1, 0], [1, 0, −1]}
{[1, −1, 0], [−1, 1, 0]}
{[1, 1, 1]}
{[1, −1, 0], [0, 1, −1], [1, 0, −1]}
Q19 [Eigenvalues and eigenvectors] Which vector is an eigenvector of A = [1 2; 2 1] for the eigenvalue λ = 3?
[1, −1]
[2, 1]
[1, 1]
[1, 2]
Q20 [Linear systems and row reduction] For what value of k does the system x + y + z = 1, x + 2y + 3z = 2, 2x + 3y + kz = 4 have no solution?
k = 2
k = −4
No value of k
k = 4
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