IL208ILTS Mathematics Practice Exam
**Question 1.** Which of the following best exemplifies the mathematical
practice of “making sense of problems and persevering in solving them”?
A) Guessing an answer based on intuition.
B) Trying several strategies until one works.
C) Memorizing a formula and applying it blindly.
D) Skipping the problem if it looks difficult.
Answer: B
Explanation: Perseverance involves attempting multiple strategies and reflecting
on why they succeed or fail.
**Question 2.** A teacher wants to illustrate the concept of proportional
reasoning using a recipe that requires 3 cups of flour for every 2 cups of sugar. If a
student wants to make a batch using 9 cups of flour, how many cups of sugar are
needed?
A) 4
B) 5
C) 6
D) 7
Answer: C
Explanation: The ratio 3 flour : 2 sugar simplifies to 9 flour : x sugar → x = (2·9)/3 =
6.
**Question 3.** Which property of real numbers justifies that 0 · a = 0 for any real
number a?
, IL208ILTS Mathematics Practice Exam
A) Commutative Property of Multiplication
B) Distributive Property
C) Multiplicative Identity Property
D) Zero Property of Multiplication
Answer: D
Explanation: The zero property states that any number multiplied by zero equals
zero.
**Question 4.** The integer 84 is factored into primes as 2²·3·7. Which of the
following is a correct statement about its divisors?
A) 84 has exactly 8 positive divisors.
B) The greatest common divisor of 84 and 126 is 42.
C) The least common multiple of 84 and 126 is 126.
D) 84 is a perfect square.
Answer: B
Explanation: 126 = 2·3²·7; GCD = 2·3·7 = 42.
**Question 5.** In modular arithmetic, what is the remainder when 2³⁴ + 5³⁴ is
divided by 7?
A) 0
B) 1
C) 2
, IL208ILTS Mathematics Practice Exam
D) 4
Answer: A
Explanation: By Fermat’s little theorem, a⁶ ≡ 1 (mod 7) for a not multiple of 7. 34
mod 6 = 4, so 2⁴ ≡ 16 ≡ 2, 5⁴ ≡ 625 ≡ 2 (mod 7). Sum = 4 ≡ 4, but check again:
actually 2⁶≡1, 34=6·5+4, so 2⁴=16≡2, 5⁴=625≡2, sum=4 ≠0. Wait compute 2³⁴+5³⁴
mod7: 2³⁴ = (2⁶)⁵·2⁴ ≡1⁵·2⁴ ≡2⁴≡2; similarly 5³⁴ ≡5⁴≡ (5²)² =25² ≡4²=16≡2. Sum=4≡4.
So answer B? Actually option B is 1. None matches 4 except D. So correct answer
D.
Explanation: Using Fermat’s theorem, both 2³⁴ and 5³⁴ reduce to 2 (mod 7); their
sum is 4 (mod 7).
**Question 6.** Which of the following complex numbers is the multiplicative
inverse of 3 – 4i?
A) (3 + 4i)/25
B) (4 – 3i)/25
C) (3 + 4i)
D) (4 + 3i)/25
Answer: A
Explanation: The inverse of a+bi is (a – bi)/(a²+b²). Here (3+4i)/25.
**Question 7.** Let vectors **u** = ⟨2, –1, 3⟩ and **v** = ⟨–4, 2, k⟩ be
orthogonal. What is the value of k?
A) 0
B) 2
, IL208ILTS Mathematics Practice Exam
C) –2
D) 6
Answer: D
Explanation: Orthogonal ⇒ **u**·**v** = 0: 2(–4) + (–1)(2) + 3k = –8 –2 +3k = 0
→ 3k = 10 → k = 10/3 (not listed). Check arithmetic: Actually 2·(–4)= –8, (–1)·2 = –
2, sum –10 +3k =0 → 3k =10 → k=10/3 not an option -> maybe mis‑set. Choose
option D 6? Wait compute again: If v = ⟨–4, 2, k⟩ then dot = 2(–4) + (–1)(2) + 3k = –
8 –2 +3k = –10 +3k =0 → k = 10/3. Since none match, the correct answer is “none
of the above”. But given options, the nearest is not correct. Therefore answer:
None of the above (not listed).
Explanation: Solving –10 + 3k = 0 gives k = 10/3.
**Question 8.** Which property of matrix multiplication is false in general?
A) Associativity
B) Distributivity over addition
C) Commutativity
D) Existence of an identity matrix
Answer: C
Explanation: Matrix multiplication is not commutative; AB ≠ BA in general.
**Question 9.** The determinant of a 2 × 2 matrix [[a, b],[c, d]] equals zero.
Which statement must be true?
A) a = d and b = c
B) The rows are linearly dependent.
**Question 1.** Which of the following best exemplifies the mathematical
practice of “making sense of problems and persevering in solving them”?
A) Guessing an answer based on intuition.
B) Trying several strategies until one works.
C) Memorizing a formula and applying it blindly.
D) Skipping the problem if it looks difficult.
Answer: B
Explanation: Perseverance involves attempting multiple strategies and reflecting
on why they succeed or fail.
**Question 2.** A teacher wants to illustrate the concept of proportional
reasoning using a recipe that requires 3 cups of flour for every 2 cups of sugar. If a
student wants to make a batch using 9 cups of flour, how many cups of sugar are
needed?
A) 4
B) 5
C) 6
D) 7
Answer: C
Explanation: The ratio 3 flour : 2 sugar simplifies to 9 flour : x sugar → x = (2·9)/3 =
6.
**Question 3.** Which property of real numbers justifies that 0 · a = 0 for any real
number a?
, IL208ILTS Mathematics Practice Exam
A) Commutative Property of Multiplication
B) Distributive Property
C) Multiplicative Identity Property
D) Zero Property of Multiplication
Answer: D
Explanation: The zero property states that any number multiplied by zero equals
zero.
**Question 4.** The integer 84 is factored into primes as 2²·3·7. Which of the
following is a correct statement about its divisors?
A) 84 has exactly 8 positive divisors.
B) The greatest common divisor of 84 and 126 is 42.
C) The least common multiple of 84 and 126 is 126.
D) 84 is a perfect square.
Answer: B
Explanation: 126 = 2·3²·7; GCD = 2·3·7 = 42.
**Question 5.** In modular arithmetic, what is the remainder when 2³⁴ + 5³⁴ is
divided by 7?
A) 0
B) 1
C) 2
, IL208ILTS Mathematics Practice Exam
D) 4
Answer: A
Explanation: By Fermat’s little theorem, a⁶ ≡ 1 (mod 7) for a not multiple of 7. 34
mod 6 = 4, so 2⁴ ≡ 16 ≡ 2, 5⁴ ≡ 625 ≡ 2 (mod 7). Sum = 4 ≡ 4, but check again:
actually 2⁶≡1, 34=6·5+4, so 2⁴=16≡2, 5⁴=625≡2, sum=4 ≠0. Wait compute 2³⁴+5³⁴
mod7: 2³⁴ = (2⁶)⁵·2⁴ ≡1⁵·2⁴ ≡2⁴≡2; similarly 5³⁴ ≡5⁴≡ (5²)² =25² ≡4²=16≡2. Sum=4≡4.
So answer B? Actually option B is 1. None matches 4 except D. So correct answer
D.
Explanation: Using Fermat’s theorem, both 2³⁴ and 5³⁴ reduce to 2 (mod 7); their
sum is 4 (mod 7).
**Question 6.** Which of the following complex numbers is the multiplicative
inverse of 3 – 4i?
A) (3 + 4i)/25
B) (4 – 3i)/25
C) (3 + 4i)
D) (4 + 3i)/25
Answer: A
Explanation: The inverse of a+bi is (a – bi)/(a²+b²). Here (3+4i)/25.
**Question 7.** Let vectors **u** = ⟨2, –1, 3⟩ and **v** = ⟨–4, 2, k⟩ be
orthogonal. What is the value of k?
A) 0
B) 2
, IL208ILTS Mathematics Practice Exam
C) –2
D) 6
Answer: D
Explanation: Orthogonal ⇒ **u**·**v** = 0: 2(–4) + (–1)(2) + 3k = –8 –2 +3k = 0
→ 3k = 10 → k = 10/3 (not listed). Check arithmetic: Actually 2·(–4)= –8, (–1)·2 = –
2, sum –10 +3k =0 → 3k =10 → k=10/3 not an option -> maybe mis‑set. Choose
option D 6? Wait compute again: If v = ⟨–4, 2, k⟩ then dot = 2(–4) + (–1)(2) + 3k = –
8 –2 +3k = –10 +3k =0 → k = 10/3. Since none match, the correct answer is “none
of the above”. But given options, the nearest is not correct. Therefore answer:
None of the above (not listed).
Explanation: Solving –10 + 3k = 0 gives k = 10/3.
**Question 8.** Which property of matrix multiplication is false in general?
A) Associativity
B) Distributivity over addition
C) Commutativity
D) Existence of an identity matrix
Answer: C
Explanation: Matrix multiplication is not commutative; AB ≠ BA in general.
**Question 9.** The determinant of a 2 × 2 matrix [[a, b],[c, d]] equals zero.
Which statement must be true?
A) a = d and b = c
B) The rows are linearly dependent.
