If a function \( f \) satisfies \( f(m + n) = f(m) + f(n) \) for all \( m, n \in \mathbb{N} \) and \( f(1) = 1 \), then the largest natural number \( \lambda \) such that \[ \sum_{k=1}^{2022} f(\lambda + k) \leq (2022)^2 \] is equal to __________.
Correct Answer: 1010
Solution and Explanation
Step 1: Analyze the functional equation The functional equation \( f(m+n) = f(m) + f(n) \) suggests that \( f(x) \) is linear. Assume:
\[ f(x) = kx. \]
Substitute \( f(1) = 1 \):
\[ f(1) = k \cdot 1 \implies k = 1. \]
Thus:
\[ f(x) = x. \]
Step 2: Expand the summation We are given:
\[ \sum_{k=1}^{2022} f(\lambda + k) \leq (2022)^2. \]
Substitute \( f(x) = x \):
\[ \sum_{k=1}^{2022} (\lambda + k) \leq (2022)^2. \]
Split the summation:
\[ \sum_{k=1}^{2022} (\lambda + k) = \sum_{k=1}^{2022} \lambda + \sum_{k=1}^{2022} k. \]
Step 2.1: Simplify each term
- \[ \sum_{k=1}^{2022} \lambda = 2022 \cdot \lambda. \]
- \[ \sum_{k=1}^{2022} k = \frac{2022 \cdot 2023}{2} \quad \text{(sum of the first 2022 natural numbers)}. \]
Thus:
\[ \sum_{k=1}^{2022} (\lambda + k) = 2022 \cdot \lambda + \frac{2022 \cdot 2023}{2}. \]
Step 3: Solve the inequality Substitute into the inequality:
\[ 2022\lambda + \frac{2022 \cdot 2023}{2} \leq (2022)^2. \]
Simplify:
\[ 2022\lambda \leq (2022)^2 - \frac{2022 \cdot 2023}{2}. \]
Factor 2022 out:
\[ 2022\lambda \leq 2022 \left( 2022 - \frac{2023}{2} \right). \]
Simplify further:
\[ \lambda \leq 2022 - \frac{2023}{2}. \]
Calculate:
\[ \lambda \leq 2022 - 1011.5 = 1010.5. \]
Step 4: Largest natural number Since \( \lambda \) must be a natural number:
\[ \lambda = 1010. \]
Final Answer:- \( 1010. \)
Top Questions on Functions
- The sum of all local minimum values of the function \( f(x) \) as defined below is:
\[ f(x) = \left\{ \begin{array}{ll} 1 - 2x & \text{if } x < -1 \\ \frac{1}{3}(7 + 2|x|) & \text{if } -1 \leq x \leq 2 \\ \frac{11}{18} (x-4)(x-5) & \text{if } x > 2 \end{array} \right. \]
- In an arithmetic progression, if \( S_{40} = 1030 \) and \( S_{12} = 57 \), then \( S_{30} - S_{10} \) is equal to:
- The area of the region enclosed by the curves \( y = e^x \), \( y = |e^x - 1| \), and the y-axis is:
- The least value of \( n \) for which the number of integral terms in the Binomial expansion of \( \left( \sqrt{7} + \sqrt{11} \right)^n \) is 183, is:
- The integral \[ 80 \int_0^{\frac{\pi}{4}} \frac{(\sin \theta + \cos \theta)}{(9 + 16 \sin 2\theta)} \, d\theta \] is equal to:
Questions Asked in JEE Main exam
- Suppose that the number of terms in an A.P. is \( 2k, k \in \mathbb{N} \). If the sum of all odd terms of the A.P. is 40, the sum of all even terms is 55, and the last term of the A.P. exceeds the first term by 27, then \( k \) is equal to:
- JEE Main - 2025
- Arithmetic Progression
- Two iron solid discs of negligible thickness have radii \( R_1 \) and \( R_2 \) and moment of inertia \( I_1 \) and \( I_2 \), respectively. For \( R_2 = 2R_1 \), the ratio of \( I_1 \) and \( I_2 \) would be \( \frac{1}{x} \), where \( x \) is:
- JEE Main - 2025
- System of Particles & Rotational Motion
- Given below are two statements:
Statement (I): Corrosion is an electrochemical phenomenon in which pure metal acts as an anode and impure metal as a cathode.
Statement (II): The rate of corrosion is more in alkaline medium than in acidic medium.
In the light of the above statements, choose the correct answer from the options given below:- JEE Main - 2025
- Electrochemistry
- The number of solutions of the equation \[ \left( \frac{9}{x} - \frac{9}{\sqrt{x}} + 2 \right) \left( \frac{2}{x} - \frac{7}{\sqrt{x}} + 3 \right) = 0 \] is:
- JEE Main - 2025
- permutations and combinations
- Let \( (2, 3) \) be the largest open interval in which the function \( f(x) = 2 \log_e (x - 2) - x^2 + ax + 1 \) is strictly increasing, and \( (b, c) \) be the largest open interval, in which the function \( g(x) = (x - 1)^3 (x + 2 - a)^2 \) is strictly decreasing. Then \( 100(a + b - c) \) is equal to:
- JEE Main - 2025
- Quadratic Equations