Rankbit System
JEE Physics: Waves (+15.5%) | Electrostatics: Concentric Shells (-29.7%) | Modern Physics: Photoelectric Clones (+34.2%) | Mathematics: Definite Integrals (+18.1%) | Chemistry: Coordination Splitting (-11.4%) | JEE Physics: Waves (+15.5%) | Electrostatics: Concentric Shells (-29.7%) | Modern Physics: Photoelectric Clones (+34.2%) | Mathematics: Definite Integrals (+18.1%) | Chemistry: Coordination Splitting (-11.4%)

Let [x] denote the greatest integer function, and let m and n respectively be the numbers of the points, where the function f(x)=[x]+|x-2|, -2

Solution & Explanation

### Related Formula The greatest integer function [x] is discontinuous at all integer points. The absolute value function |x-x_0| is continuous everywhere but non-differentiable at its corner tip x = x_0. ### Core Logic Break down the function f(x) = [x] + |x-2| in the open domain (-2, 3) across sub-intervals between integers [cite: 3278, 3951]: f(x) = begincases -2 - (x-2) = -x & -2 < x < -1 \\ -1 - (x-2) = -x+1 & -1 le x < 0 \\ 0 - (x-2) = -x+2 & 0 le x < 1 \\ 1 - (x-2) = -x+3 & 1 le x < 2 \\ 2 + (x-2) = x & 2 le x < 3 endcases ### Step 1: Count Discontinuity Points (m) Evaluate the limits at internal integers \-1, 0, 1, 2\: - At x = -1: textLHL = 1, textRHL = 2 Rightarrow Discontinuous. - At x = 0: textLHL = 1, textRHL = 2 Rightarrow Discontinuous. - At x = 1: textLHL = 1, textRHL = 2 Rightarrow Discontinuous. - At x = 2: textLHL = 1, textRHL = 2 Rightarrow Discontinuous. Thus, f(x) is discontinuous at exactly 4 integer locations , meaning m = 4. ### Step 2: Count Non-Differentiability Points (n) Since discontinuity automatically implies non-differentiability, the points \-1, 0, 1, 2\ are non-differentiable. Let\'s check if there are other sharp corners. The modulus part |x-2| turns sharp at x=2, which is already covered in our discontinuity list. Hence, there are no additional non-differentiable points. Thus, n = 4. ### Step 3: Total Evaluation Calculate the \sum requested : m + n = 4 + 4 = 8 ### Pattern Recognition For expressions containing [x], the discontinuity at integers usually drives the overall non-differentiability tally, making any coincidental sharp points from continuous elements redundant if they happen at the exact same integers. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Mathematics: Limits, Continuity and Differentiability

Reference Study Guides

Limits Continuity And Differentiability

More Limits, Continuity and Differentiability Previous-Year Questions — Page 5

Q65 2025 Limits of Special Series
The value of lim_nto inftyleft(sum_K = 1^nfrack^3 + 6k^2 + 11k + 5(k + 3)!right) is:
  • A. \frac{4}{3}
  • B. 2
  • C. \frac{7}{3}
  • D. \frac{5}{3}

Solution

### Related Formula sum_k=1^infty left( frac1k! - frac1(k+3)! right) implies textTelescoping Series simplification ### Core Logic Rewrite the numerator polynomial to establish factor terms matching the factorial expansion base (k+3): k^3 + 6k^2 + 11k + 5 = (k^3 + 6k^2 + 11k + 6) - 1 = (k+1)(k+2)(k+3) - 1 ### Step 1: Simplify General Term T_k = frac(k+1)(k+2)(k+3)(k+3)! - frac1(k+3)! T_k = frac1k! - frac1(k+3)! This creates a clean telescoping layout format structure. ### Step 2: Sum the Series Writing out expanded \partial sums up to infinity: S = left( frac11! + frac12! + frac13! + frac14! + dots right) - left( frac14! + frac15! + frac16! + dots right) All higher terms cancel out systematically, leaving exactly the leading remaining fragments: S = frac11! + frac12! + frac13! = 1 + frac12 + frac16 = frac106 = frac53 ### Pattern Recognition Whenever factorials dominate fraction denominators, manipulate structural terms to align components via Telescoping sums (V_n - V_n-k). ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Mathematics: Limits, Continuity and Differentiability Class 11 Mathematics: Sequences and Series

More Limits, Continuity and Differentiability Questions — jee_main_2025_24_jan_evening

Practice all Limits, Continuity and Differentiability previous-year questions →

YOUR FIRST PREP STEP STARTS HERE

We Map Every Repeating Question in Competitive Exams.

Say goodbye to generic mock test fatigue. RankBit uses smart analysis to group past exam questions into their foundational Repeating Question Types. Find chapter weightage, track repeating questions, and score higher with targeted practice.

Select Your Target Exam

Choose an exam track below to find formulas per chapter and patterns.

Syncing Exam Intelligence

Mapping formulas and patterns across all tracks…

PATH A — FULL LENGTH PRACTICE

Full Mock Test Hub

Simulate real NTA exam conditions with fully tracked mocks. Time yourself against past papers.

Under Development
PATH B — TARGETED PRACTICE

Topic-wise Practice Hub

Practice past-year questions one chapter at a time. Pick an exam → subject → chapter and get every PYQ for that topic — pulled together from all past papers — with the chapter's key formulas alongside.

Loading Questions... Browse Topics
Latest from the Blog
View all →

Loading articles...