CSET MATH SUBTEST 1– EXAM QUESTIONS AND ANSWERS | ACCURATE AND WELL
CORE DOMAINS:
1. Number Theory
2. Algebra and Complex Numbers
3. Matrices and Vectors
4. Functions and Polynomials
5. Sequences, Series, and Binomial Theorem
6. Mathematical Induction and Proof Techniques
INTRODUCTION:
The California Subject Examinations for Teachers (CSET) Mathematics
Subtest 1 is designed to measure foundational mathematical knowledge
and pedagogical readiness for prospective secondary educators. This
assessment rigorously evaluates critical competencies including number
theory, algebra, matrices, vectors, complex numbers, and advanced proof
techniques. Featuring a comprehensive blend of multiple-choice and
scenario-based items, the exam tests both theoretical comprehension
and practical application. Candidates must demonstrate advanced critical
thinking, rigorous problem-solving skills, and the ability to make sound
pedagogical and mathematical decisions in real-world educational
contexts.
Question 1
What is the greatest common divisor of 1071 and 462 using the Euclidean
algorithm?
A. 21
🟢 B. 42
C. 63
D. 84
🔴 Explanation: Applying the Euclidean algorithm: 1071 = 2(462) + 147;
462 = 3(147) + 21; 147 = 7(21) + 0. The last non-zero remainder is 21.
Wait, let's recalculate: 2462 = 924. 1071 - 924 = 147. 3147 = 441. 462 -
441 = 21. 7*21 = 147. So the GCD is 21. Let's fix option A as the correct
answer.
Question 1
What is the greatest common divisor of 1071 and 462 using the Euclidean
algorithm?
🟢 A. 21
,B. 42
C. 63
D. 84
🔴 Explanation: Applying the Euclidean algorithm: 1071 = 2(462) + 147,
then 462 = 3(147) + 21, and 147 = 7(21) + 0. The last non-zero remainder
is 21.
Question 2
Which of the following numbers is prime?
A. 143
B. 221
🟢 C. 311
D. 391
🔴 Explanation: 311 cannot be divided evenly by prime numbers up to its
square root (approx 17.6), making it prime. 143 = 11 x 13, 221 = 13 x 17,
and 391 = 17 x 23.
Question 3
What is the remainder when 250 is divided by 7?
A. 1
B. 2
C. 3
🟢 D. 4
🔴 Explanation: By Fermat's Little Theorem, 26 ≡ 1 (mod 7). Since
50 = 6(8) + 2, 250 ≡ (26 )8 × 22 ≡ 18 × 4 ≡ 4 (mod 7).
Question 4
Find the multiplicative inverse of 15 modulo 26.
A. 3
B. 5
🟢 C. 7
D. 9
🔴 Explanation: We look for an integer x such that 15x ≡ 1 (mod 26).
Testing values shows 15 × 7 = 105, and 105 = 4(26) + 1, so 15(7) ≡
1 (mod 26).
Question 5
How many positive divisors does the number 720 have?
A. 20
B. 24
🟢 C. 30
D. 36
🔴 Explanation: The prime factorization of 720 is 24 × 32 × 51 . The
,number of divisors is found by adding 1 to each exponent and multiplying:
(4 + 1)(2 + 1)(1 + 1) = 5 × 3 × 2 = 30.
Question 6
What is the units digit of 32026 ?
A. 1
🟢 B. 9
C. 3
D. 7
🔴 Explanation: The powers of 3 follow a repeating units digit pattern: 3,
9, 7, 1 (period of 4). Since 2026 ÷ 4 = 506 with a remainder of 2, the
units digit corresponds to the second term in the cycle, which is 9.
Question 7
Which of the following integers satisfies the linear congruence 4x ≡ 6
(mod 10)?
A. 2
B. 3
🟢 C. 4
D. 5
🔴 Explanation: Testing x = 4 gives 4(4) = 16. Since 16 ≡ 6 (mod 10)
, x = 4 is a valid solution.
Question 8
What is the sum of all solutions to the equation ∣2x − 3∣ = 7?
A. 2
🟢 B. 3
C. 4
D. 7
🔴 Explanation: The equation splits into 2x − 3 = 7 (giving x = 5) and
2x − 3 = −7 (giving x = −2). The sum of the solutions is 5 + (−2) = 3.
Question 9
If f (x) = 3x2 − 5x + 2, what is the value of f (x + 1) − f (x)?
A. 3x − 2
B. 3x + 1
🟢 C. 6x − 2
D. 6x + 1
🔴 Explanation: f (x + 1) = 3(x + 1)2 − 5(x + 1) + 2 = 3x2 + 6x + 3 −
5x − 5 + 2 = 3x2 + x. Subtracting f (x): (3x2 + x) − (3x2 − 5x + 2) =
6x − 2.
, Question 10
x2 −4
What are all the real values of x for which x−2 = x + 2 is an identity?
A. All real numbers
B. x = 2
🟢 C. All real numbers except x =2
D. No real numbers
x2 −4
🔴 Explanation: The expression x−2 simplifies to
x + 2 for all x = 2
because division by zero is undefined at x = 2.
Question 11
What is the sum of the roots of the quadratic equation 3x2 − 12x + 7 = 0
?
A. −4
🟢 B. 4
7
C. − 3
D. 73
🔴 Explanation: By Vieta's formulas, the sum of the roots of a quadratic
equation ax2 + bx + c = 0 is given by −b/a. Here, −(−12)/3 = 4.
Question 12
If log2 (x) + log2 (x − 2) = 3, what is the value of x?
A. −2
B. 2
🟢 C. 4
D. 8
🔴 Explanation: Using logarithm properties, log2 (x(x − 2)) = 3, meaning
x(x − 2) = 23 = 8. Thus, x2 − 2x − 8 = 0, factoring to (x − 4)(x + 2) = 0
. Since domain requires x > 2, x = 4.
Question 13
What is the coefficient of x3 in the expansion of (2x − 3)5 ?
A. −720
B. −1080
🟢 C. 720
D. 1080
🔴 Explanation: Using the Binomial Theorem, the term is
(53)(2x)2 (−3)3 = 10 ⋅ 4x2 ⋅ (−27) = −1080x3 . Wait, let's recalculate:
(53) = 10. (2x)2 = 4x2 . (−3)3 = −27. 10 ⋅ 4 ⋅ (−27) = −1080. Let's
select option B.
CORE DOMAINS:
1. Number Theory
2. Algebra and Complex Numbers
3. Matrices and Vectors
4. Functions and Polynomials
5. Sequences, Series, and Binomial Theorem
6. Mathematical Induction and Proof Techniques
INTRODUCTION:
The California Subject Examinations for Teachers (CSET) Mathematics
Subtest 1 is designed to measure foundational mathematical knowledge
and pedagogical readiness for prospective secondary educators. This
assessment rigorously evaluates critical competencies including number
theory, algebra, matrices, vectors, complex numbers, and advanced proof
techniques. Featuring a comprehensive blend of multiple-choice and
scenario-based items, the exam tests both theoretical comprehension
and practical application. Candidates must demonstrate advanced critical
thinking, rigorous problem-solving skills, and the ability to make sound
pedagogical and mathematical decisions in real-world educational
contexts.
Question 1
What is the greatest common divisor of 1071 and 462 using the Euclidean
algorithm?
A. 21
🟢 B. 42
C. 63
D. 84
🔴 Explanation: Applying the Euclidean algorithm: 1071 = 2(462) + 147;
462 = 3(147) + 21; 147 = 7(21) + 0. The last non-zero remainder is 21.
Wait, let's recalculate: 2462 = 924. 1071 - 924 = 147. 3147 = 441. 462 -
441 = 21. 7*21 = 147. So the GCD is 21. Let's fix option A as the correct
answer.
Question 1
What is the greatest common divisor of 1071 and 462 using the Euclidean
algorithm?
🟢 A. 21
,B. 42
C. 63
D. 84
🔴 Explanation: Applying the Euclidean algorithm: 1071 = 2(462) + 147,
then 462 = 3(147) + 21, and 147 = 7(21) + 0. The last non-zero remainder
is 21.
Question 2
Which of the following numbers is prime?
A. 143
B. 221
🟢 C. 311
D. 391
🔴 Explanation: 311 cannot be divided evenly by prime numbers up to its
square root (approx 17.6), making it prime. 143 = 11 x 13, 221 = 13 x 17,
and 391 = 17 x 23.
Question 3
What is the remainder when 250 is divided by 7?
A. 1
B. 2
C. 3
🟢 D. 4
🔴 Explanation: By Fermat's Little Theorem, 26 ≡ 1 (mod 7). Since
50 = 6(8) + 2, 250 ≡ (26 )8 × 22 ≡ 18 × 4 ≡ 4 (mod 7).
Question 4
Find the multiplicative inverse of 15 modulo 26.
A. 3
B. 5
🟢 C. 7
D. 9
🔴 Explanation: We look for an integer x such that 15x ≡ 1 (mod 26).
Testing values shows 15 × 7 = 105, and 105 = 4(26) + 1, so 15(7) ≡
1 (mod 26).
Question 5
How many positive divisors does the number 720 have?
A. 20
B. 24
🟢 C. 30
D. 36
🔴 Explanation: The prime factorization of 720 is 24 × 32 × 51 . The
,number of divisors is found by adding 1 to each exponent and multiplying:
(4 + 1)(2 + 1)(1 + 1) = 5 × 3 × 2 = 30.
Question 6
What is the units digit of 32026 ?
A. 1
🟢 B. 9
C. 3
D. 7
🔴 Explanation: The powers of 3 follow a repeating units digit pattern: 3,
9, 7, 1 (period of 4). Since 2026 ÷ 4 = 506 with a remainder of 2, the
units digit corresponds to the second term in the cycle, which is 9.
Question 7
Which of the following integers satisfies the linear congruence 4x ≡ 6
(mod 10)?
A. 2
B. 3
🟢 C. 4
D. 5
🔴 Explanation: Testing x = 4 gives 4(4) = 16. Since 16 ≡ 6 (mod 10)
, x = 4 is a valid solution.
Question 8
What is the sum of all solutions to the equation ∣2x − 3∣ = 7?
A. 2
🟢 B. 3
C. 4
D. 7
🔴 Explanation: The equation splits into 2x − 3 = 7 (giving x = 5) and
2x − 3 = −7 (giving x = −2). The sum of the solutions is 5 + (−2) = 3.
Question 9
If f (x) = 3x2 − 5x + 2, what is the value of f (x + 1) − f (x)?
A. 3x − 2
B. 3x + 1
🟢 C. 6x − 2
D. 6x + 1
🔴 Explanation: f (x + 1) = 3(x + 1)2 − 5(x + 1) + 2 = 3x2 + 6x + 3 −
5x − 5 + 2 = 3x2 + x. Subtracting f (x): (3x2 + x) − (3x2 − 5x + 2) =
6x − 2.
, Question 10
x2 −4
What are all the real values of x for which x−2 = x + 2 is an identity?
A. All real numbers
B. x = 2
🟢 C. All real numbers except x =2
D. No real numbers
x2 −4
🔴 Explanation: The expression x−2 simplifies to
x + 2 for all x = 2
because division by zero is undefined at x = 2.
Question 11
What is the sum of the roots of the quadratic equation 3x2 − 12x + 7 = 0
?
A. −4
🟢 B. 4
7
C. − 3
D. 73
🔴 Explanation: By Vieta's formulas, the sum of the roots of a quadratic
equation ax2 + bx + c = 0 is given by −b/a. Here, −(−12)/3 = 4.
Question 12
If log2 (x) + log2 (x − 2) = 3, what is the value of x?
A. −2
B. 2
🟢 C. 4
D. 8
🔴 Explanation: Using logarithm properties, log2 (x(x − 2)) = 3, meaning
x(x − 2) = 23 = 8. Thus, x2 − 2x − 8 = 0, factoring to (x − 4)(x + 2) = 0
. Since domain requires x > 2, x = 4.
Question 13
What is the coefficient of x3 in the expansion of (2x − 3)5 ?
A. −720
B. −1080
🟢 C. 720
D. 1080
🔴 Explanation: Using the Binomial Theorem, the term is
(53)(2x)2 (−3)3 = 10 ⋅ 4x2 ⋅ (−27) = −1080x3 . Wait, let's recalculate:
(53) = 10. (2x)2 = 4x2 . (−3)3 = −27. 10 ⋅ 4 ⋅ (−27) = −1080. Let's
select option B.
