Basics & LogarithmsmediumPYQ · JEE Main · 10 Apr 2023 · Shift 1 (Morning)Free

Basics & Logarithms: Number Elements Set 10n (JEE Main 2023)

JEE Maths question with a full step-by-step solution.

Question
The number of elements in the set {n∈Z:∣n2−10n+19∣<6}\{n\in\mathbb{Z}:|n^{2}-10n+19|<6\} is
Solution
Answer: 6
Step 1: The inequality ∣n2−10n+19∣<6|n^{2}-10n+19|<6 is equivalent to:
−6<n2−10n+19<6-6<n^{2}-10n+19<6
 Step 2: Right inequality: n2−10n+19<6⇒n2−10n+13<0n^{2}-10n+19<6\Rightarrow n^{2}-10n+13<0. Roots: n=10±100−522=10±482=5±23n=\dfrac{10\pm\sqrt{100-52}}{2}=\dfrac{10\pm\sqrt{48}}{2}=5\pm 2\sqrt{3}. So 5−23<n<5+235-2\sqrt{3}<n<5+2\sqrt{3}, i.e., approximately 1.54<n<8.461.54<n<8.46. Step 3: Left inequality: n2−10n+19>−6⇒n2−10n+25>0⇒(n−5)2>0n^{2}-10n+19>-6\Rightarrow n^{2}-10n+25>0\Rightarrow(n-5)^{2}>0. This holds for all n≠5n\neq 5. Step 4: Combine: n∈(5−23,5+23)n\in(5-2\sqrt{3},5+2\sqrt{3}) and n≠5n\neq 5. Integer values in this interval: n=2,3,4,5,6,7,8n=2,3,4,5,6,7,8. Excluding n=5n=5: n∈{2,3,4,6,7,8}n\in\{2,3,4,6,7,8\}. Step 5: Count: 66 elements. Answer: 66
Still stuck on this question?Ask your doubt on WhatsApp
Similar questions
Basics & Logarithms · easy
If x=∏n=12026nx = \prod_{n=1}^{2026} n, then the value of the expression 11log⁡2x+1log⁡3x+⋯+1log⁡2026x\frac{1}{\dfrac{1}{\log_2 x} + \dfrac{1}{\log_3 x} + \cdots + \dfrac{1}{\log_{2026} x}} is
Basics & Logarithms · easy
Number of integral values of xx satisfying the inequality (34)6x+10−x2<2764\left(\dfrac{3}{4}\right)^{6x+10-x^2} < \dfrac{27}{64} is:
Basics & Logarithms · easy
Number of natural numbers for which the number log⁡4−x(x2−14x+45)\log_{4-x}(x^2-14x+45) is defined is:
Basics & Logarithms · easy
The product of all solutions of the equation e5(log⁡ex)2+3=x8e^{5(\log_{e}x)^{2}+3}=x^{8}, x>0x>0, is

Solve more, learn faster

Sign up free to solve more JEE Maths questions and explore doMath — timed drills, mastery sprints, bookmarks, and chapter-wise progress tracking.

Sign up freeExplore doMath