Basics & LogarithmseasyPYQ · JEE Main · 30 Jan 2024 · Shift 1 (Morning)Free

Basics & Logarithms: Number Real Solutions Equation (JEE Main 2024)

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

Question
The number of real solutions of the equation x(x2+3∣x∣+5∣x−1∣+6∣x−2∣)=0x\left(x^{2}+3|x|+5|x-1|+6|x-2|\right)=0 is
Solution
Answer: 1
Step 1: A product is zero when at least one factor is zero. So either x=0x=0 or:
x2+3∣x∣+5∣x−1∣+6∣x−2∣=0x^{2}+3|x|+5|x-1|+6|x-2|=0
 Step 2: Examine the second factor. All terms are non-negative: x2≥0x^{2}\ge 0, 3∣x∣≥03|x|\ge 0, 5∣x−1∣≥05|x-1|\ge 0, 6∣x−2∣≥06|x-2|\ge 0. Step 3: For the sum to be zero, each term must be zero simultaneously: x2=0⇒x=0x^{2}=0\Rightarrow x=0, but then 5∣x−1∣=5≠05|x-1|=5\neq 0. So the second factor cannot be zero anywhere. Step 4: Therefore, the only solution is x=0x=0.
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