JEE Main 2023 · 10 April · morning · Numerical answer · +4 / −0
The number of elements in the set { n ∈ Z:|n² - 10n + 19| < 6} is _________.
Answer and solution
Answer 6
Solution
Maths sir · 4 steps
Complete the square first: the expression is (n-5)² - 6, and the whole question becomes easy.
- Complete the square. n² - 10n + 19 = (n-5)² - 25 + 19 = (n-5)² - 6. Put t = (n-5)², which is a perfect square of an integer.
|t - 6| < 6 ⇔ -6 < t - 6 < 6 ⇔ 0 < t < 12
- Pick the squares. The perfect squares strictly between 0 and 12 are 1, 4 and 9. The value t = 0 is excluded because the inequality is strict, so n = 5 is out.
- Find n. n - 5 = ± 1, ± 2, ± 3, which gives n = 2, 3, 4, 6, 7, 8. As a check, n = 2 gives |4 - 20 + 19| = 3 < 6.
- So. The set has 6 elements.
The common slip: 7. Counting n = 5 as well gives 7. At n = 5 the expression equals -6, and |-6| = 6 is not less than 6.
The idea to keep. When a quadratic sits inside a modulus and n must be an integer, complete the square. Then you only need to count perfect squares in an interval, and you must watch whether its ends are strict.
Answer from the NTA final answer key. Solution by PrepWiser.