JEE MainMathematics

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+4 marks−1 if incorrectSingle correctPrevious-year question

Finding Unknown Parameters in a Polynomial and Computing Divisor Sums

Q: Why do we not need to find the value of lambda?

Because lambda cancels out in differences of the form f(a) - f(b) = 2(a^n - b^n).

Let f(x) = 2x^n + λ, λ ∈ ℝ, n ∈ ℕ, and f(4) = 133, f(5) = 255. Then the sum of all the positive integer divisors of (f(3) - f(2)) is

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A60B59C61D58

Step-by-step solutionView

Correct answer

The value of $f (3) - f (2)$ is $38$ , and the sum of all its positive divisors is $60$ .

Option analysis

Why each option works or fails

A · 60

The student solved for $n = 3$ , found $f (3) - f (2) = 2 (27 - 8) = 38$ , and correctly summed its divisors $1 + 2 + 19 + 38 = 60$ . Correctly determined $n = 3$ , evaluated the difference, and summed all positive divisors.

B · 59

The student computed the sum of proper divisors (excluding $1$ or forgetting to include $1$ ) instead of all positive divisors. Remember that the set of positive integer divisors of $N$ includes $1$ and $N$ itself. For $38$ , the divisors are $1, 2, 19, 38$ , which sum to $60$ , not $59$ .

C · 61

The student made an arithmetic error when adding the divisors, adding $1$ extra or misidentifying a divisor. Check the divisor sum carefully: $σ (38) = (1 + 2) (1 + 19) = 3 \times 20 = 60$ .

D · 58

The student made an arithmetic slip in evaluating $f (3) - f (2)$ or when summing the divisors. Ensure $f (3) - f (2) = 2 (3^{3} - 2^{3}) = 2 (27 - 8) = 38$ , whose prime factorization is $2 \times 19$ .

Reviewed route

Solution

StepWorking

01Given

$f (x) = 2 x^{n} + λ$ where $λ \in R, n \in N$ , $f (4) = 133$ , and $f (5) = 255$ .

02Goal

Find the sum of all positive integer divisors of $f (3) - f (2)$ .

03Approach

Subtract $f (4)$ from $f (5)$ to eliminate $λ$ and solve for $n \in N$ . Then calculate $f (3) - f (2) = 2 (3^{n} - 2^{n})$ and compute the divisor sum formula $σ_{1} (N)$ .

04Execute

Subtracting the equations gives $f (5) - f (4) = 2 (5^{n} - 4^{n}) = 255 - 133 = 122$ , hence $5^{n} - 4^{n} = 61$ . For $n = 3$ , $5^{3} - 4^{3} = 125 - 64 = 61$ , so $n = 3$ .

05Execute

Evaluate $f (3) - f (2) = 2 (3^{3} - 2^{3}) = 2 (27 - 8) = 2 \times 19 = 38$ .

06Execute

The prime factorization of 38 is $2^{1} \times 19^{1}$ . The sum of divisors is $σ (38) = (1 + 2) (1 + 19) = 3 \times 20 = 60$ .

✓Verify

Divisors of 38 are 1, 2, 19, 38. Their direct sum is $1 + 2 + 19 + 38 = 60$ , matching the result.

Hints that build this answer step by step

What equation results from subtracting $f (4)$ from $f (5)$ to eliminate $λ$ ?
$2 (5^{n} - 4^{n}) = 122$
From $5^{n} - 4^{n} = 61$ with $n \in N$ , what is the value of $n$ ?
$n = 3$
What is the value of $f (3) - f (2)$ ?
$38$
What is the sum of all positive integer divisors of $38$ ?
$60$

✓ Source and academic review

Question type: Single correct
Exam relevance: JEE Main · Mathematics
Academic status: Answer verified against the official NTA key
Source: Previous-year question
Editorial review: 9 September 2026

Quick checks

Students also ask

Why do we not need to find the value of lambda?

Because lambda cancels out in differences of the form $f (a) - f (b) = 2 (a^{n} - b^{n})$ .

Answer

The value of $f (3) - f (2)$ is $38$ , and the sum of all its positive divisors is $60$ .

Why each option works or fails

A · correct: 60 - The student solved for $n = 3$ , found $f (3) - f (2) = 2 (27 - 8) = 38$ , and correctly summed its divisors $1 + 2 + 19 + 38 = 60$ . Correctly determined $n = 3$ , evaluated the difference, and summed all positive divisors.
B: 59 - The student computed the sum of proper divisors (excluding $1$ or forgetting to include $1$ ) instead of all positive divisors. Remember that the set of positive integer divisors of $N$ includes $1$ and $N$ itself. For $38$ , the divisors are $1, 2, 19, 38$ , which sum to $60$ , not $59$ .
C: 61 - The student made an arithmetic error when adding the divisors, adding $1$ extra or misidentifying a divisor. Check the divisor sum carefully: $σ (38) = (1 + 2) (1 + 19) = 3 \times 20 = 60$ .
D: 58 - The student made an arithmetic slip in evaluating $f (3) - f (2)$ or when summing the divisors. Ensure $f (3) - f (2) = 2 (3^{3} - 2^{3}) = 2 (27 - 8) = 38$ , whose prime factorization is $2 \times 19$ .

Step-by-step solution

Given: $f (x) = 2 x^{n} + λ$ where $λ \in R, n \in N$ , $f (4) = 133$ , and $f (5) = 255$ .
Goal: Find the sum of all positive integer divisors of $f (3) - f (2)$ .
Approach: Subtract $f (4)$ from $f (5)$ to eliminate $λ$ and solve for $n \in N$ . Then calculate $f (3) - f (2) = 2 (3^{n} - 2^{n})$ and compute the divisor sum formula $σ_{1} (N)$ .
Execute: Subtracting the equations gives $f (5) - f (4) = 2 (5^{n} - 4^{n}) = 255 - 133 = 122$ , hence $5^{n} - 4^{n} = 61$ . For $n = 3$ , $5^{3} - 4^{3} = 125 - 64 = 61$ , so $n = 3$ .
Execute: Evaluate $f (3) - f (2) = 2 (3^{3} - 2^{3}) = 2 (27 - 8) = 2 \times 19 = 38$ .
Execute: The prime factorization of 38 is $2^{1} \times 19^{1}$ . The sum of divisors is $σ (38) = (1 + 2) (1 + 19) = 3 \times 20 = 60$ .
Verify: Divisors of 38 are 1, 2, 19, 38. Their direct sum is $1 + 2 + 19 + 38 = 60$ , matching the result.

Finding Unknown Parameters in a Polynomial and Computing Divisor Sums

What feels right?

The value of $f (3) - f (2)$ is $38$ , and the sum of all its positive divisors is $60$ .

Why each option works or fails

Solution

01Given

02Goal

03Approach

04Execute

05Execute

06Execute

✓Verify

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