CUET UG Applied Mathematics Booster Test 2 - Advanced Arithmetic Functions -

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CUET UG Applied Mathematics Booster Test 2 - Advanced Arithmetic Functions

⏱️ 30 Minutes

❓ 20 Questions

📌 Answers are locked once submitted — results and explanations appear at the end.

CUET UG Applied Mathematics Booster Test 2 - Advanced Arithmetic Functions

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QUESTION 1 OF 20

How does the concept of gcd (greatest common divisor) apply directly to calculating φ(n)?

Answer Review

1 How does the concept of gcd (greatest common divisor) apply directly to calculating φ(n)?

A It counts numbers where gcd(n, x) > 1.

B It sums all x where gcd(n, x) = 1.

C It counts all numbers x ≤ n where gcd(x, n) = 1.

D It divides n by its greatest common divisor.

💡 Short Explanation

�� Euler's totient uses coprimality �� gcd(x,n)=1 defines coprime numbers �� φ(n) counts such integers up to n

✅ Correct Answer Explanation

Euler's totient function is defined as: This means φ(n) counts all positive integers less than or equal to n that are coprime to n. For example: because only 1 and 5 satisfy: Hence Option C correctly describes the role of gcd.

❓ Why Other Options Are Incorrect

�� Option A → φ(n) counts coprime numbers, not numbers with gcd greater than 1.
�� Option B → φ(n) counts numbers; it does not sum them.
�� Option D → No division by gcd occurs in the definition.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Match the definition of coprimality with Euler's totient function.

Final Logic: Totient counts integers satisfying gcd(x,n)=1.

🧠 Memory Trick

"Totient = total coprimes."

📚 Class XI – NCERT Mathematics – Chapter: Relations and Functions – Topic: Euler's Totient Function

2 Match the prime numbers to their correct sum of divisors sigma(p) and number of divisors tau(p).

List 1	List 2
1. p = 2	a. tau = 2, sigma = 30
2. p = 5	b. tau = 2, sigma = 18
3. p = 17	c. tau = 2, sigma = 3
4. p = 29	d. tau = 2, sigma = 6

A 1-c, 2-d, 3-b, 4-a

B 1-a, 2-b, 3-c, 4-d

C 1-d, 2-c, 3-a, 4-b

D 1-c, 2-a, 3-d, 4-b

💡 Short Explanation

�� Every prime number has exactly two divisors. �� For any prime p, tau(p) = 2. �� For any prime p, sigma(p) = p + 1.

✅ Correct Answer Explanation

For every prime number, tau(p) = 2 sigma(p) = p + 1 Now evaluate each prime. • p = 2 sigma(2) = 1 + 2 = 3 Therefore, 1 → c. • p = 5 sigma(5) = 1 + 5 = 6 Therefore, 2 → d. • p = 17 sigma(17) = 1 + 17 = 18 Therefore, 3 → b. • p = 29 sigma(29) = 1 + 29 = 30 Therefore, 4 → a. Hence, the correct matching is: 1 → c 2 → d 3 → b 4 → a Therefore, Option A is correct.

❓ Why Other Options Are Incorrect

�� Option B → Incorrect because all four prime numbers are matched with incorrect divisor sums.
�� Option C → Incorrect because the divisor sums for 2, 5, 17 and 29 are interchanged.
�� Option D → Incorrect because sigma(5) = 6, not 30.

🎯 Strategy

Used: Substitution

Application:

Apply the formulas tau(p) = 2 and sigma(p) = p + 1 to each prime number and match the results.

Final Logic:

Using sigma(p) = p + 1 gives the sequence 1-c, 2-d, 3-b, 4-a, which matches Option A.

🧠 Memory Trick

Prime ⇒ tau = 2, sigma = p + 1.

📚 Class XI – Mathematics – Chapter: Number Theory – Topic: Arithmetic Functions of Prime Numbers

3 Which of the following evaluations hold true regarding the properties of simple arithmetic functions applied to prime numbers?
1. phi(5) = 4
2. tau(17) = 2
3. sigma(2) = 3
4. mu(7) = 0

A 1, 2

B 1, 2, 3

C 2, 3, 4

D 1, 3, 4

💡 Short Explanation

�� For a prime p, phi(p) = p - 1. �� Every prime has exactly two divisors. �� The Möbius function of a prime is -1, not 0.

✅ Correct Answer Explanation

Evaluate each statement. • Statement 1: phi(5) = 5 - 1 = 4. Hence, Statement 1 is correct. • Statement 2: The divisors of 17 are 1 and 17. Therefore, tau(17) = 2. Hence, Statement 2 is correct. • Statement 3: sigma(2) = 1 + 2 = 3. Hence, Statement 3 is correct. • Statement 4: Since 7 is a prime number, mu(7) = -1. Therefore, Statement 4 is incorrect. Thus, Statements 1, 2 and 3 are correct. Hence, Option B is correct.

❓ Why Other Options Are Incorrect

�� Option A → Incorrect because it omits Statement 3, which is also correct.
�� Option C → Incorrect because it includes Statement 4, which is false.
�� Option D → Incorrect because Statement 4 is false since mu(7) = -1.

🎯 Strategy

Used: Option Grouping

Application:

Evaluate each arithmetic function separately using the standard formulas for prime numbers.

Final Logic:

Only Statements 1, 2 and 3 are true, giving Option B.

🧠 Memory Trick

Prime ⇒ phi = p - 1, tau = 2, sigma = p + 1, mu = -1.

📚 Class XI – Mathematics – Chapter: Number Theory – Topic: Arithmetic Functions of Prime Numbers

4 Consider the multiplicative functions and properties. Identify the incorrect statement.

A phi(6) = phi(2) × phi(3)

B 12 = 2^2 × 3, so phi(12) = 12 × (1 - 1/2) × (1 - 1/3)

C phi(12) = 4

D phi(xy) = phi(x) × phi(y) is valid for all positive integers x and y regardless of their common divisors.

💡 Short Explanation

�� Euler's totient function is multiplicative only for coprime integers. �� The condition gcd(x, y) = 1 is essential. �� Without coprimality, the formula does not hold.

✅ Correct Answer Explanation

Euler's Totient Function satisfies phi(xy) = phi(x) × phi(y) only when gcd(x, y) = 1. Evaluate each option. • Option A: phi(2) = 1 phi(3) = 2 Therefore, phi(6) = 1 × 2 = 2. This is correct because 2 and 3 are coprime. • Option B: 12 = 2^2 × 3 Using Euler's product formula, phi(12) = 12 × (1 - 1/2) × (1 - 1/3) = 12 × 1/2 × 2/3 = 4. Hence, Option B is correct. • Option C: phi(12) = 4. This is the correct numerical value. • Option D: This statement is incorrect because Euler's multiplicative property is valid only when x and y are coprime. Therefore, Option D is the incorrect statement.

❓ Why Other Options Are Incorrect

�� Option A → Correct because 2 and 3 are coprime.
�� Option B → Correct application of Euler's product formula.
�� Option C → Correct numerical evaluation of phi(12).

🎯 Strategy

Used: Elimination

Application:

Check whether the required coprime condition is satisfied before applying Euler's multiplicative property.

Final Logic:

Euler's totient function is multiplicative only for coprime integers, making Option D incorrect.

🧠 Memory Trick

Multiplicative ⇒ Coprime compulsory.

📚 Class XI – Mathematics – Chapter: Number Theory – Topic: Euler's Totient Function and Multiplicative Functions

5 If n = 12 is decomposed into a mixture of its prime factors as 2^2 × 3, what is the evaluated value of φ(12) using the product formula?

A 6

B 4

C 2

D 8

💡 Short Explanation

�� Use Euler's product formula �� Distinct primes are 2 and 3 �� Final totient value is 4

✅ Correct Answer Explanation

Prime factorization: Apply Euler's formula: Simplifying: Hence Option B is correct.

❓ Why Other Options Are Incorrect

�� Option A → Incorrect multiplication.
�� Option C → Too small for φ(12).
�� Option D → Counts too many integers.

🎯 Strategy

Used: Substitution

Application: Substitute distinct prime factors into Euler's formula.

Final Logic: Product formula gives 4.

🧠 Memory Trick

"12 → halve then ×2/3."

📚 Class XI – NCERT Mathematics – Chapter: Relations and Functions – Topic: Euler Product Formula

6 For evaluating φ(12) using the formula n(1 - 1/x)(1 - 1/y), what constraint values do x and y represent?

A The composite factors of n

B The distinct prime factors of n (e.g., 2 and 3)

C The powers of the prime factors

D The total number of divisors

💡 Short Explanation

�� Euler's formula uses distinct primes �� Repeated powers are ignored separately �� For 12, primes are 2 and 3

✅ Correct Answer Explanation

Euler's totient product formula is: Here p represents distinct prime factors of n. For 12: The distinct primes are 2 and 3. Hence x and y represent distinct prime factors.

❓ Why Other Options Are Incorrect

�� Option A → Composite factors are not used in Euler's formula.
�� Option C → Powers do not appear directly in the product factors.
�� Option D → Total divisors are unrelated here.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Identify the meaning of variables in Euler's formula.

Final Logic: Product terms always use distinct primes.

🧠 Memory Trick

"Totient product uses unique primes."

📚 Class XI – NCERT Mathematics – Chapter: Relations and Functions – Topic: Totient Product Formula

7 The number of divisor function τ(n) equates the sum of evenly divided instances. If n = 12, what is τ(12)?

A 4

B 5

C 6

D 12

💡 Short Explanation

�� τ(n) counts divisors �� Divisors of 12 are six in number �� Count, not sum, is required

✅ Correct Answer Explanation

The positive divisors of 12 are: Counting them: Hence Option C is correct.

❓ Why Other Options Are Incorrect

�� Option A → Misses several divisors.
�� Option B → Incorrect divisor count.
�� Option D → τ(n) is not equal to n itself.

🎯 Strategy

Used: Substitution

Application: Explicitly list all divisors.

Final Logic: There are exactly six positive divisors.

🧠 Memory Trick

"12 has 6 divisors."

📚 Class XI – NCERT Mathematics – Chapter: Relations and Functions – Topic: Divisor Counting Function

8 Moving along the sequence of divisors for 12 {1, 2, 3, 4, 6, 12}, what does τ(12) correctly represent?

A The average of the divisors

B The sum of the divisors

C The count of these 6 divisors

D The prime factors only

💡 Short Explanation

�� τ(n) measures count only �� Divisor values are not added �� Six divisors exist for 12

✅ Correct Answer Explanation

The divisor function τ(n) counts the number of positive divisors. For 12: There are six divisors, so: Hence Option C is correct.

❓ Why Other Options Are Incorrect

�� Option A → τ(n) does not calculate averages.
�� Option B → Sum belongs to σ(n).
�� Option D → Prime factors alone are insufficient.

🎯 Strategy

Used: Option Grouping

Application: Distinguish divisor count from divisor sum.

Final Logic: τ(n) always counts divisors.

🧠 Memory Trick

"τ → tally divisors."

📚 Class XI – NCERT Mathematics – Chapter: Relations and Functions – Topic: Divisor Counting Function

9 When calculating σ(12), the probability of missing a divisor affects the sum. What is the correct calculated sum of positive divisors for 12?

A 16

B 28

C 12

D 24

💡 Short Explanation

�� σ(n) adds all divisors �� Divisors of 12 must all be included �� Total sum equals 28

✅ Correct Answer Explanation

The positive divisors of 12 are: Therefore: Hence Option B is correct.

❓ Why Other Options Are Incorrect

�� Option A → Misses several divisors.
�� Option C → Uses only n itself.
�� Option D → Incorrect total addition.

🎯 Strategy

Used: Substitution

Application: Add all divisors directly.

Final Logic: Divisor sum equals 28.

🧠 Memory Trick

"σ means sum divisors."

📚 Class XI – NCERT Mathematics – Chapter: Relations and Functions – Topic: Divisor Sum Function

10 If evaluating the sum vector σ(n) for n = 3, which components strictly constitute the result?

A 1 and 2

B 1 and 3

C 3 only

D 1, 2, and 3

💡 Short Explanation

�� σ(n) uses positive divisors �� Divisors of 3 are 1 and 3 �� 2 is not a divisor of 3

✅ Correct Answer Explanation

The positive divisors of 3 are: Thus σ(3) includes only these components: Hence Option B is correct.

❓ Why Other Options Are Incorrect

�� Option A → 2 is not a divisor of 3.
�� Option C → Omits divisor 1.
�� Option D → 2 again is incorrectly included.

🎯 Strategy

Used: Elimination

Application: Remove options containing non-divisors.

Final Logic: Only 1 and 3 divide 3 exactly.

🧠 Memory Trick

"Prime divisors: 1 and itself."

📚 Class XI – NCERT Mathematics – Chapter: Relations and Functions – Topic: Divisor Sum Function

11 If the domain area is restricted to prime numbers, how is τ(p) defined for any prime p?

A τ(p) = p

B τ(p) = p - 1

C τ(p) = 2

D τ(p) = 0

💡 Short Explanation

�� Prime numbers have exactly two divisors �� Divisors are 1 and p itself �� τ(p) counts total divisors

✅ Correct Answer Explanation

For any prime number p, the only positive divisors are: Hence the divisor counting function becomes: This is a standard property of prime numbers in number theory. Option C is therefore correct.

❓ Why Other Options Are Incorrect

�� Option A → τ(p) counts divisors, not the value of p itself.
�� Option B → p−1 represents φ(p), not τ(p).
�� Option D → A prime number always has divisors, so τ(p) cannot be zero.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Use the defining property of prime numbers having exactly two divisors.

Final Logic: Since primes have only 1 and p as divisors, τ(p)=2.

🧠 Memory Trick

"Prime → two divisors only."

📚 Class XI – NCERT Mathematics – Chapter: Relations and Functions – Topic: Divisor Function for Prime Numbers

12 Evaluate the integral sum of divisors σ(p) if the prime number p = 29.

A 29

B 30

C 28

D 2

💡 Short Explanation

�� σ(p)=1+p for primes �� Divisors of 29 are 1 and 29 �� Sum becomes 30

✅ Correct Answer Explanation

For a prime number p: Since: the divisors are 1 and 29. Therefore: Hence Option B is correct.

❓ Why Other Options Are Incorrect

�� Option A → Includes only the prime itself.
�� Option C → Incorrect subtraction.
�� Option D → Counts divisors instead of summing them.

🎯 Strategy

Used: Substitution

Application: Substitute p=29 into σ(p)=1+p.

Final Logic: 1+29 gives 30.

🧠 Memory Trick

"Sigma of prime = prime + 1."

📚 Class XI – NCERT Mathematics – Chapter: Relations and Functions – Topic: Divisor Sum Function

13 Evaluate the Mobius function μ(12) based on the repeated prime factor rule.

A 1

B -1

C 0

D 2

💡 Short Explanation

�� Möbius function checks repeated primes �� 12 contains 2² �� Repeated prime factor gives zero

✅ Correct Answer Explanation

Prime factorization of 12: The Möbius function rule states: • μ(n)=0 if n contains any repeated prime factor. Since 12 contains: it is not square-free. Therefore: Hence Option C is correct.

❓ Why Other Options Are Incorrect

�� Option A → μ(n)=1 only for even count of distinct primes without repetition.
�� Option B → μ(n)=−1 applies for odd count of distinct primes without repetition.
�� Option D → Möbius values are only −1, 0, or 1.

🎯 Strategy

Used: Elimination

Application: Check whether repeated prime factors exist.

Final Logic: Presence of 2² forces μ(12)=0.

🧠 Memory Trick

"Repeated prime → Möbius zero."

📚 Class XI – NCERT Mathematics – Chapter: Relations and Functions – Topic: Möbius Function

14 Why does μ(12) = 0?

A Because 12 is an even number.

B Because 12 = 2^2 × 3, which contains a repeated prime factor (2^2).

C Because 12 is a product of 3 distinct primes.

D Because 12 is greater than 10.

💡 Short Explanation

�� Möbius function depends on prime repetition �� 12 contains repeated factor 2² �� Repetition makes μ(n)=0

✅ Correct Answer Explanation

The Möbius function satisfies: • μ(n)=0 whenever n has a repeated prime factor. Since: the factor: shows repetition. Thus: Hence Option B is correct.

❓ Why Other Options Are Incorrect

�� Option A → Being even does not guarantee μ(n)=0.
�� Option C → 12 is not a product of distinct primes only.
�� Option D → Magnitude of number is irrelevant.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Match Möbius definition with repeated prime factors.

Final Logic: Repeated prime square directly gives μ(n)=0.

🧠 Memory Trick

"Square factor kills Möbius."

📚 Class XI – NCERT Mathematics – Chapter: Relations and Functions – Topic: Möbius Function Rules

15 Calculate the value of μ(70) given that 70 = 2 × 5 × 7.

A 1

B 0

C -1

D 3

💡 Short Explanation

�� 70 has distinct prime factors �� Number of primes is 3 �� μ(n)=(-1)^k

✅ Correct Answer Explanation

Prime factorization: There are 3 distinct prime factors and no repetition. Möbius function rule: where k is the number of distinct primes. Thus: Hence Option C is correct.

❓ Why Other Options Are Incorrect

�� Option A → Would occur for even number of distinct primes.
�� Option B → No repeated prime factor exists.
�� Option D → Möbius values cannot equal 3.

🎯 Strategy

Used: Substitution

Application: Count distinct prime factors and apply μ(n)=(-1)^k.

Final Logic: Odd number of distinct primes gives −1.

🧠 Memory Trick

"Odd distinct primes → −1."

📚 Class XI – NCERT Mathematics – Chapter: Relations and Functions – Topic: Möbius Function Evaluation

16 Determine the sum of arithmetic functions at n = 1: φ(1) + τ(1) + σ(1) + μ(1).

A 0

B 2

C 3

D 4

💡 Short Explanation

�� φ(1)=1 �� τ(1)=1 and σ(1)=1 �� μ(1)=1

✅ Correct Answer Explanation

Using standard arithmetic function values: Therefore: Hence Option D is correct.

❓ Why Other Options Are Incorrect

�� Option A → Ignores standard values at n=1.
�� Option B → Adds only two functions correctly.
�� Option C → Misses one function value.

🎯 Strategy

Used: Substitution

Application: Substitute standard arithmetic function values for n=1.

Final Logic: Total sum equals 4.

🧠 Memory Trick

"At n=1, all four equal 1."

📚 Class XI – NCERT Mathematics – Chapter: Relations and Functions – Topic: Basic Arithmetic Function Values

17 The function log(xy) = log x + log y (where x and y are coprime) is a classic example of which type of Simple Arithmetic Function?

A Multiplicative function

B Additive function

C Equivalence function

D Totient function

💡 Short Explanation

�� Additive functions convert products into sums �� Logarithm satisfies additive relation �� Multiplicative functions preserve products

✅ Correct Answer Explanation

An additive arithmetic function satisfies: The logarithm property is: Hence it behaves as an additive function. Therefore Option B is correct.

❓ Why Other Options Are Incorrect

�� Option A → Multiplicative functions satisfy f(xy)=f(x)f(y).
�� Option C → Equivalence functions are unrelated here.
�� Option D → Totient function is a specific arithmetic function, not logarithmic.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Compare the given functional identity with definitions.

Final Logic: Product transforming into sum implies additivity.

🧠 Memory Trick

"Log turns multiply into add."

📚 Class XI – NCERT Mathematics – Chapter: Relations and Functions – Topic: Additive Arithmetic Functions

18 Given that φ is a multiplicative function, calculate φ(35) using its prime factors.

A 35

B 12

C 24

D 6

💡 Short Explanation

�� 35=5×7 �� Use Euler product formula �� Final value becomes 24

✅ Correct Answer Explanation

Prime factorization: Apply Euler's formula: Simplifying: Hence Option C is correct.

❓ Why Other Options Are Incorrect

�� Option A → φ(n) is always less than n for n>1.
�� Option B → Incorrect multiplication result.
�� Option D → Too small for φ(35).

🎯 Strategy

Used: Substitution

Application: Substitute prime factors into Euler's product formula.

Final Logic: Computation gives φ(35)=24.

🧠 Memory Trick

"35 → ×4/5 ×6/7."

📚 Class XI – NCERT Mathematics – Chapter: Relations and Functions – Topic: Euler Totient Formula

📄 Passage

A Simple Arithmetic function is a function whose domain is positive integers and the range is real or complex numbers. These include the Euler's totient function φ(n), the Number of divisors function τ(n), and the Divisor sum function σ(n). Interestingly, these functions often interrelate when evaluated under specific conditions. For instance, if p is an odd prime number and n is defined such that n = 2p, a remarkable mathematical identity emerges linking the value of n with its corresponding arithmetic function values: n + φ(n) + τ(n) = σ(n).

Question:
Given n = 2p, where p is an odd prime number less than 14, evaluate the terms for p = 3. What is n?

A 3

B 5

C 6

D 9

💡 Short Explanation

�� n=2p is given �� Substitute p=3 �� Result becomes 6

✅ Correct Answer Explanation

The relation is: Substitute: Thus: Hence Option C is correct.

❓ Why Other Options Are Incorrect

�� Option A → Uses only p.
�� Option B → Incorrect arithmetic.
�� Option D → Squares p incorrectly

🎯 Strategy

Used: Substitution

Application: Directly replace p with 3.

Final Logic: 2×3=6.

🧠 Memory Trick

"Double the prime."

📚 Class XI – NCERT Mathematics – Chapter: Relations and Functions – Topic: Arithmetic Function Relations

📄 Passage

Question:
For n = 6 (where p = 3), verify the relation n + φ(n) + τ(n) = σ(n). What is the value of both the LHS and RHS?

A 10

B 12

C 14

D 16

💡 Short Explanation

�� n=6 �� φ(6)=2, τ(6)=4 �� σ(6)=12

✅ Correct Answer Explanation

For: We calculate: since coprimes are 1 and 5. Also: because divisors are 1,2,3,6. Now: For divisor sum: Thus both LHS and RHS equal 12. Hence Option B is correct.

❓ Why Other Options Are Incorrect

�� Option A → Uses incomplete divisor information.
�� Option C → Incorrect totient or divisor count.
�� Option D → Adds extra values incorrectly.

🎯 Strategy

Used: Substitution

Application: Evaluate each arithmetic function separately.

Final Logic: Both sides simplify to 12.

🧠 Memory Trick

"6 + 2 + 4 = 12."

📚 Class XI – NCERT Mathematics – Chapter: Relations and Functions – Topic: Relations Between Arithmetic Functions

CUET UG Applied Mathematics Booster Test 2 - Advanced Arithmetic Functions

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