Problem 114 - A Complete Resource Book in Mathematics for JEE Main

Question

Each coefficient of the equation \(a x^{2}+b x+c=0\) is determined by throwing an ordinary die. The probability that the equation has non-real complex roots is (A) \(\frac{173}{216}\) (B) \(\frac{43}{216}\) (C) \(\frac{54}{216}\) (D) none of these

Step-by-Step Solution

Answer

The probability that the equation has non-real complex roots is \(\frac{43}{216}\). Option (B).

Step 1: Determine Conditions for Non-real Roots

For a quadratic equation \( ax^2 + bx + c = 0 \) to have complex (non-real) roots, the discriminant \( b^2 - 4ac \) must be less than zero.

Step 2: Assign Possible Values for Coefficients

Each coefficient \( a \), \( b \), and \( c \) can take any integer value from 1 to 6, since they are determined by rolling a die.

Step 3: Calculate Total Possible Outcomes

The total number of outcomes for choosing \( a \), \( b \), and \( c \) is \( 6 \times 6 \times 6 = 216 \) because each coefficient can be any of the 6 possible values.

Step 4: Determine Outcomes for Complex Roots

Calculate the number of outcomes where \( b^2 < 4ac \). Compute \( b^2 \) for each \( b \) from 1 to 6, then check for which values of \( a \) and \( c \), the condition \( b^2 < 4ac \) holds true. Count the total number of cases satisfying this condition.

Step 5: Evaluate Probability

The probability is the ratio of the number of outcomes where \( b^2 < 4ac \) to the total possible outcomes. You compute the specific number of cases (43) where \( b^2 < 4ac \), giving the probability as \( \frac{43}{216} \).

Step 6: Compare with Options

Compare the computed probability \( \frac{43}{216} \) with the given choices. The correct choice matches option (B).

Key Concepts

Quadratic Equation Coefficients

In a quadratic equation of the form \( ax^2 + bx + c = 0 \), the coefficients \( a \), \( b \), and \( c \) play crucial roles in defining the equation's graph and its roots. In this exercise, each of these coefficients is determined by rolling a die. This means that:

• Each of \( a \), \( b \), and \( c \) can take on integer values from 1 to 6.
• The possible combinations for these coefficients amount to throwing three dice, giving us \( 6 \times 6 \times 6 = 216 \) possible outcomes.

The values of \( a \), \( b \), and \( c \) determine the nature of the roots of the quadratic equation, whether they are real or complex. This is directly influenced by these coefficients, highlighting their importance as they can entirely dictate the behavior of the quadratic graph.

Discriminant for Complex Roots

One key feature of quadratic equations is the discriminant, which is used to determine the nature of the roots. The discriminant \( \Delta \) is given by the formula \( b^2 - 4ac \). For a quadratic equation to have complex roots, the discriminant must be less than zero. This indicates the absence of real roots and a pair of complex conjugates instead.

• If \( b^2 - 4ac < 0 \), the equation has non-real complex roots.
• The discriminant is highly sensitive to the values of the coefficients \( a \), \( b \), and \( c \).
• We evaluate \( b^2 \) for each value of \( b \) and determine when \( 4ac \) exceeds this square, thus verifying the condition \( b^2 < 4ac \).

Understanding this discriminant's role is essential in investigating equations within mathematics, especially in determining root behavior. It offers a clear criterion by which we can assess whether the roots of an equation are real or complex.

Probability Calculation in Mathematics

Probability is a critical component that helps us understand the likelihood of certain outcomes. When dealing with our quadratic equation exercise, we aim to find the probability of getting complex roots when \( a \), \( b \), and \( c \) are based on the roll of dice.

• The total possible outcomes are determined by the dice rolls: \( 6 \times 6 \times 6 = 216 \).
• The number of favorable outcomes, or the combinations where \( b^2 < 4ac \), is found through systematic calculation and counting, which results in 43 such scenarios.
• The probability of complex roots occurring is then calculated by dividing these favorable cases by the total possible outcomes, or \( \frac{43}{216} \).

Probability provides a quantifiable means to predict the behavior of quadratic equations under random conditions. By analyzing the outcomes, students can gain insights into both theoretical and practical applications of probability in mathematics.