Problem 76 - A Complete Resource Book in Mathematics for JEE Main

Question

The sum of two positive quantities is equal to \(2 n\). The probability that their product is not less than \(\frac{3}{4}\) times their greatest product is (A) \(\frac{3}{4}\) (B) \(\frac{1}{2}\)(C) \(\frac{1}{4}\) (D) none of these

Step-by-Step Solution

Answer

The probability that their product is at least \(\frac{3}{4}\) times the maximum product is \(\frac{1}{2}\).

Step 1: Define the Problem

Consider two positive quantities, \(x\) and \(2n-x\) whose sum is equal to \(2n\). We need to find the probability that their product is at least \(\frac{3}{4}\) times their maximum possible product.

Step 2: Find the Expression for the Product

The product of the two quantities \(x\) and \(2n-x\) is given by:\[p = x(2n - x) = 2nx - x^2.\]

Step 3: Determine the Maximum Product

The expression for the product \(p = 2nx - x^2\) is a quadratic function. The maximum occurs at the vertex since it opens downwards (negative coefficient of \(x^2\)). The vertex form occurs at \(x = \frac{2n}{2} = n\). Substitute \(x = n\) into the product expression:\[ p_{\text{max}} = n(2n - n) = n^2.\]

Step 4: Constraint on the Product

We want the product to be at least \(\frac{3}{4}\) times the maximum product. Therefore, we set the inequality:\[ p = x(2n - x) \geq \frac{3}{4}n^2. \]

Step 5: Solve the Inequality

Substitute the expression for \(p\):\[ 2nx - x^2 \geq \frac{3}{4}n^2. \]Rearrange the inequality:\[ x^2 - 2nx + \frac{3}{4}n^2 \leq 0. \]

Step 6: Find the Roots of the Quadratic

Solve the inequality \(x^2 - 2nx + \frac{3}{4}n^2 \leq 0\) by finding its roots using the quadratic formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \]where \(a = 1\), \(b = -2n\), and \(c = \frac{3}{4}n^2\).Substitute into the formula:\[ x = \frac{2n \pm \sqrt{(2n)^2 - 4 \cdot 1 \cdot \frac{3}{4}n^2}}{2}. \]\[ x = \frac{2n \pm \sqrt{4n^2 - 3n^2}}{2}. \]\[ x = \frac{2n \pm \sqrt{n^2}}{2}. \]\[ x = \frac{2n \pm n}{2}. \]The roots are \(x = \frac{3n}{2}\) and \(x = \frac{n}{2}\).

Step 7: Analyze the Feasible Range

The product will satisfy the inequality when \(\frac{n}{2} \leq x \leq \frac{3n}{2}\). This is because \(x(2n - x)\) gives a maximum in this range.

Step 8: Calculate the Probability

The total interval for \(x\) is from \(0\) to \(2n\) giving a length of \(2n\). The interval that satisfies the inequality is \(\frac{3n}{2} - \frac{n}{2} = n\). Therefore, the probability is:\[ \frac{n}{2n} = \frac{1}{2}. \]

Key Concepts

Inequalities

Inequalities are expressions where one side is not necessarily equal to the other. Instead, one side is either greater than, less than, or equal to the other side. In mathematics, particularly when working with probabilities or algebraic expressions, we often encounter inequalities.
When solving inequalities, we aim to find all possible values that make the inequality true. For example, if we have an inequality such as \(x^2 - 2nx + \frac{3}{4}n^2 \leq 0\), we need to find the range of values for \(x\) that satisfy this condition.
Here are some steps to solve quadratic inequalities:

• Rearrange the inequality into the standard quadratic form \(ax^2 + bx + c \leq 0\).
• Identify the coefficients \(a\), \(b\), and \(c\).
• Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the roots.
• Determine the intervals where the inequality holds true by testing values or drawing a number line.

Inequalities are pivotal in determining ranges over which certain conditions, like maximum or minimum values of functions, are maintained.

Quadratic Functions

Quadratic functions are polynomial functions of degree 2 and have the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). These functions graph as parabolas on a coordinate plane.
Quadratic functions can model various real-world situations and are especially significant when discussing motions, like projectile paths or maximizing area problems.
Key features of quadratic functions include:

• The vertex, which is the highest or lowest point on the parabola, depending on if it opens upward (\(a > 0\)) or downward (\(a < 0\)).
• The axis of symmetry, a vertical line that passes through the vertex, dividing the parabola into two symmetric halves.
• The roots or x-intercepts are points where the parabola crosses the x-axis. These are found using the quadratic formula or by factoring the quadratic equation if possible.

In the context of the problem, the maximum product of two numbers summing to a constant can be determined by identifying the vertex of this quadratic function. This vertex provides an optimal solution for maximizing or minimizing a real-world quantity.

Vertex of Parabola

The vertex of a parabola is a crucial point that identifies the parabola's peak or trough. In a quadratic function \(ax^2 + bx + c\), the vertex can be a maximum or minimum point, depending on whether the parabola opens upwards (\(a > 0\)) or downwards (\(a < 0\)).
To find the vertex of a quadratic function, use the formula for the x-coordinate: \(x = -\frac{b}{2a}\). Once you have the x-coordinate, substitute it back into the original quadratic equation to find the corresponding y-coordinate.
In the exercise, we are interested in the product of two quantities, which is modeled by a quadratic function. The vertex of the quadratic \(x(2n-x)\) occurs at \(x = n\). At this point, the product is maximized. This is crucial when determining probabilities, as we relate constraints, like being \(\frac{3}{4}\) of the greatest product (which is found at the vertex).
The vertex not only aids in maximizing or minimizing values but also provides insight into the behavior of the parabola in broader problem-solving situations. By understanding this concept, we can better analyze and predict outcomes in mathematical modeling.