Problem 100 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(p\) and \(q\) are chosen randomly from the set \((1,2,3,4,\), \(5,6,7,8,9,10\) ) with replacement then the probability that the roots of the equation \(x^{2}+p x+q=0\) are real, is (A) \(0.62\) (B) \(0.32\) (C) \(0.44\) (D) none of these

Step-by-Step Solution

Answer

The answer is (D) none of these.

Step 1: Understand the Condition for Real Roots

For a quadratic equation \(ax^2 + bx + c = 0\) to have real roots, the discriminant \(b^2 - 4ac\) must be non-negative. In this scenario, the equation is \(x^2 + px + q = 0\), hence the discriminant is \(p^2 - 4q\). We require \(p^2 - 4q \geq 0\).

Step 2: Analyze Possible Values for p and q

Both \(p\) and \(q\) are chosen from the set \(\{1,2,3,4,5,6,7,8,9,10\}\). Thus, there are \(10\) possible values for both \(p\) and \(q\), implying there are \(10 \times 10 = 100\) possible \((p, q)\) pairs.

Step 3: Calculate the Valid (p, q) Pairs

For each value of \(p\), calculate the maximum integer value of \(q\) such that \(p^2 \geq 4q\). Count all valid \((p, q)\) combinations:- \(p = 1\): 1 pair (\(q = 1\))- \(p = 2\): 1 pair (\(q = 1\))- \(p = 3\): 2 pairs (\(q = 1, 2\))- \(p = 4\): 4 pairs (\(q = 1, 2, 3, 4\))- \(p = 5\): 6 pairs (\(q = 1, 2, 3, 4, 5, 6\))- \(p = 6\): 9 pairs (\(q = 1, 2, 3, 4, 5, 6, 7, 8, 9\))- \(p = 7\): 10 pairs (\(q = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\))- \(p = 8\): 10 pairs (\(q = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\))- \(p = 9\): 10 pairs (\(q = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\))- \(p = 10\): 10 pairs (\(q = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\))

Step 4: Calculate Probability of Real Roots

Counting all valid pairs from step 3 gives us \(1 + 1 + 2 + 4 + 6 + 9 + 10 + 10 + 10 + 10 = 63\) pairs.The probability is the ratio of valid pairs to the total pairs: \(\frac{63}{100} = 0.63\).

Step 5: Choose the Correct Answer

The calculated probability of \(0.63\) does not match any of the given options. Thus, the answer is (D) none of these.

Key Concepts

Real Roots of Quadratic Equation

In mathematics, a quadratic equation is one of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents an unknown variable. Real roots mean that the solutions for \( x \) are real numbers. For the roots to be real, the equation must not involve any imaginary numbers, which ensures that the solutions are tangible and can be plotted on a regular number line. While solving quadratic equations with real roots, these solutions can either be two distinct real numbers or a repeated real number (in which case the roots are said to be a 'double root'). Quadratic equations are key to various applications across science and engineering due to their ability to model a variety of physical phenomena. When you encounter roots in quadratic equations, always try to first see if they are real by evaluating the discriminant condition. This simple step can save a lot of unnecessary algebraic computation if the roots turn out to be imaginary.

Discriminant Condition

The discriminant is a crucial entity when dealing with quadratic equations. For any quadratic equation \( ax^2 + bx + c = 0 \), the formula for the discriminant is \( b^2 - 4ac \). This single expression tells us about the nature of roots without having to solve the equation fully.

• If the discriminant is positive \((b^2 - 4ac > 0)\), the quadratic equation will have two distinct real roots.
• If the discriminant is zero \((b^2 - 4ac = 0)\), the quadratic equation will have exactly one real root, known as a double root or repeated root.
• If the discriminant is negative \((b^2 - 4ac < 0)\), the quadratic equation will have no real roots, but rather two complex conjugate roots.

In this exercise, for equation \( x^2 + px + q = 0 \), the discriminant is \( p^2 - 4q \). We want \( p^2 - 4q \geq 0 \) to ensure real roots. This is why understanding the discriminant condition is crucial. Knowing this allows you to determine the feasibility of real roots just by reviewing the problem's conditions.

Random Selection

Random selection is a process whereby elements are chosen from a set without any specific pattern, ensuring that each element has an equal probability of being selected. In the context of probability, this is fundamental as it ensures fairness and unpredictability in experiments.In the original exercise, both \( p \) and \( q \) are chosen randomly from the set \( \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \) with replacement. "With replacement" means that after choosing a number, it is "replaced" back into the original set, so it can be chosen again. This keeps the total number of possible outcomes consistent for each draw.Random selection influences our understanding of probability, as the selections hence define the set of possible outcomes and subsequent calculation of likelihoods. A good rule of thumb for problems involving random selection is to first determine total possible outcomes and then ascertain the number of desired outcomes to calculate probability effectively. For this solution, the total number of \((p, q)\) combinations is \(10 \times 10 = 100\).