JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

Let $f(x) = 2x^n + \lambda, \lambda \in \mathbb{R}, n \in \mathbb{N}$ , and $f(4) = 133, f(5) = 255$ . Then the sum of all the positive integer divisors of $(f(3) - f(2))$ is

Choose the correct answer:

Explanation

Solving

1. Finding $n$

Using the given values:

f(5) = 2(5^n) + \lambda = 255

f(4) = 2(4^n) + \lambda = 133

Subtracting the two equations to eliminate $\lambda$ :

(2 \cdot 5^n + \lambda) - (2 \cdot 4^n + \lambda) = 255 - 133

2(5^n - 4^n) = 122

5^n - 4^n = 61

By inspection/trial:

If $n=1$ : $5 - 4 = 1 \neq 61$
If $n=2$ : $25 - 16 = 9 \neq 61$
If $n=3$ : $125 - 64 = 61$

So, $n = 3$ .

2. Calculating $(f(3) - f(2))$

The expression for $(f(3) - f(2))$ is:

(2 \cdot 3^n + \lambda) - (2 \cdot 2^n + \lambda)

2(3^n - 2^n)

Substitute $n = 3$ :

2(3^3 - 2^3)

2(27 - 8)

2(19) = 38

3. Finding the sum of divisors of $38$

The prime factorization of $38$ is:

38 = 2^1 \times 19^1

The divisors of $38$ are: $1, 2, 19, 38$ .

Sum of divisors:

1 + 2 + 19 + 38 = 60

Correct Option: (1)

Choose the correct answer:

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