Problem 8 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

A number \(x\) is chosen at random from the set \(\\{1,2,3,4, \ldots .\), 100\\}. Define the event: \(\mathrm{A}=\) the chosen number \(\mathrm{x}\) satisfies \(\frac{(x-10)(x-50)}{(x-30)} \geq 0\) Then \(\mathrm{P}(\mathrm{A})\) is: \(\quad[\) Online April 12, 2014] (a) \(0.71\) (b) \(0.70\) (c) \(0.51\) (d) \(0.20\)

Step-by-Step Solution

Answer

The probability \( P(A) \) is 0.80, which matches none of the given choices.

Step 1: Analyze the Inequality

Given the inequality \( \frac{(x-10)(x-50)}{(x-30)} \geq 0 \), we need to find the values of \(x\) that satisfy it. Consider the zeros of the numerator and the point where the denominator changes sign: \(x=10\), \(x=50\), and the vertical asymptote at \(x=30\). These critical points divide the number line into intervals: \((1,10)\), \((10,30)\), \((30,50)\), \((50,100)\).

Step 2: Test Each Interval

Test the inequality in each interval:- For \(x\in (1,10)\), choose \(x=5\): \(\frac{(-5)(-45)}{-25} > 0\). Inequality holds.- For \(x\in (10,30)\), choose \(x=20\): \(\frac{(10)(-30)}{-10} > 0\). Inequality holds.- For \(x\in (30,50)\), choose \(x=40\): \(\frac{(30)(-10)}{10} < 0\). Inequality does not hold.- For \(x\in (50,100)\), choose \(x=60\): \(\frac{(50)(10)}{30} > 0\). Inequality holds.

Step 3: Include Critical Points

Consider the critical points \(x=10\), \(x=30\), and \(x=50\):- \(x=10\): Inequality is \(0/(-20) = 0 \); holds.- \(x=30\): Denominator is zero; inequality undefined.- \(x=50\): Inequality is \(0/20 = 0 \); holds.

Step 4: List Satisfying Values

The intervals and points satisfying \( \frac{(x-10)(x-50)}{(x-30)} \geq 0 \) are \([1,30) \,\cup \, [50,100]\). Excluding 30.

Step 5: Calculate the Probability

Count the satisfying values:- Values in \([1,30)\) are \(1\) to \(29\), totaling 29 values.- Values in \([50,100]\) are \(50\) to \(100\), totaling 51 values.- Total favorable outcomes = \(29 + 51 = 80\).The sample space is \(100\), so the probability is \( \frac{80}{100} = 0.80\).

Key Concepts

Inequalities

Inequalities are statements that compare two values or expressions using symbols such as \( > \), \( < \), \( \geq \), and \( \leq \). In this exercise, we are looking at an inequality involving a fraction: \( \frac{(x-10)(x-50)}{(x-30)} \geq 0 \). This means we need the product of the expressions \((x-10)\), \((x-50)\), and \( \frac{1}{x-30} \) to be non-negative.

• The numerator \((x-10)(x-50)\) will be positive or zero when \(x\) is exactly 10 or 50, and positive in between or beyond these points if the same factor combination provides a positive product.
• The expression \(\frac{1}{x-30}\) creates a vertical asymptote at \(x=30\), where the entire fraction is undefined.

Understanding these points helps determine where the inequality holds true on a number line. We have to analyze sections of the expression to decide if their signs would keep the full fraction non-negative.

Number Line Test

The number line test is a visual tool that helps us analyze inequalities, especially those involving fractions and complex algebraic expressions. We plot critical points on the number line to divide it into several intervals to test which sections meet the inequality's conditions.
With the given inequality \( \frac{(x-10)(x-50)}{(x-30)} \geq 0 \), we identify critical points at \(x=10\), \(x=30\), and \(x=50\). These points divide the number line into four sections:

• \((1, 10)\)
• \((10, 30)\)
• \((30, 50)\)
• \((50, 100)\)

To use the number line method, choose a random test point in each interval, substitute it into the inequality, and check whether the inequality is true or false in that interval. This gives a clear snapshot of where the inequality holds.

Critical Points

Critical points are values of \(x\) where the inequality might change its sign. They can be found by setting the numerator and denominator equal to zero. For our inequality \( \frac{(x-10)(x-50)}{(x-30)} \geq 0 \):

• Set \(x-10 = 0\), so \(x = 10\). This is where the numerator is zero.
• Set \(x-50 = 0\), so \(x = 50\). Again, the numerator is zero here.
• Set \(x-30 = 0\), so \(x = 30\). This makes the denominator zero, creating a vertical asymptote, and thus the expression is undefined here.

Critical points are key because they potentially alter the sign of the inequality, helping to mark off intervals where the entire expression might be positive or negative. We include the critical points in our number line test, except where the expression becomes undefined.

Intervals

Intervals in a mathematical sense are ranges between two numbers that can be either inclusive or exclusive of those numbers. After identifying critical points for the inequality \( \frac{(x-10)(x-50)}{(x-30)} \geq 0 \), the real number line gets split into intervals that need to be tested.
For this exercise, the intervals are \((1,10)\), \((10,30)\), \((30,50)\), and \((50,100)\). Each of these intervals represents a segment of values for \(x\) where we test our inequality:

• In \((1,10)\), the inequality holds.
• In \((10,30)\), the inequality holds.
• In \((30,50)\), the inequality does not hold.
• In \((50,100)\), the inequality holds.

Adding the exceptions at points \(x = 10\), \(x = 50\) (where the inequality is zero, so it holds), and avoiding \(x = 30\) (undefined), ultimately tells us which values of \(x\) are part of event \(A\). Thus, intervals are essential to precisely determine the sections where these complex inequalities apply.