Question

Given \(1<\mathrm{a}, \mathrm{c}<5\) and \(2 \mathrm{~b}=\mathrm{a}+\mathrm{c}\) Column I \(\quad\) Column II Quadratic Equation Nature of the roots (a) \((x-a)(x-c)=0\) (p) Equal roots (b) \(x[x+(a+c)]+b^{2}=0\) (q) Sum of the roots equals \(2 \mathrm{~b}\) (c) \((\mathrm{x}-\mathrm{a})(\mathrm{x}-\mathrm{c})+(2 \mathrm{~b}+1) \mathrm{x}=0\) (r) Roots lie in \([1,5]\) (d) \(2 x^{2}-(a+c) x+5(a+c)=0\) (s) Complex roots

Short answer

Based on the step-by-step solution, analyze the quadratic equations in column I and match them with the options in column II. a) has roots lying in the interval [1, 5], so it matches with r) b) has a sum of roots equal to 2b, so it matches with q) c) has roots that are neither equal nor equal to 2b, so it matches with r) d) has complex roots, so it matches with s) The matching pairs are as follows: a) - r) b) - q) c) - r) d) - s)

Step-by-step solution

Given information and understanding quadratic equations

We are given that \(1< a,c<5\), and \(2b=a+c\). A quadratic equation is represented as \(ax^2+bx+c=0\) or in other forms with the same meaning, having the roots \(x_1\) and \(x_2\). The nature of the roots depends on the discriminant of the equation, denoted by D, where \(D=b^2-4ac\). It is determinant of roots nature:• If \(D> 0\), the roots are real and distinct.• If \(D=0\), the roots are real and equal.• If \(D< 0\), the roots are complex. Now, let's analyze each equation.

Analyzing equation (a)

Equation \((x-a)(x-c)=0\) has roots \(x_1=a\) and \(x_2=c\). Since both roots are real, we can deduce that its corresponding item in column II is option (r), roots lie in \([1, 5]\).

Analyzing equation (b)

Equation \(x[x+(a+c)]+b^{2}=0\) can be rewritten as \(x^2+ (a+c)x + b^2 =0\). We are given that \(2b=a+c\), hence the sum of roots \(x_1 + x_2 = -(a+c) = -2b\). Therefore, the corresponding item in column II is option (q).

Analyzing equation (c)

Equation \((x-a)(x-c)+(2b+1)x=0\) can be represented as \(x^2 - (a+c)x + ac +(2b +1)x = 0\). Now, simplifying, we get \(x^2+(2b+1-a-c)x+ac=0\). The sum of roots \(x_1 + x_2 = -(2b+1-a-c) = a+c-2b-1\) which is not \(2b\) and not equal as well. Hence, the corresponding item in column II is option (r).

Analyzing equation (d)

Equation \(2x^2 -(a+c)x+5(a+c)=0\). To check for the nature of roots, let's calculate D. \(D=(-(a+c))^2-4(2)(5(a+c)) = (a+c)^2 - 40(a+c)\) Now, observe that we have the expression given in terms of \(a+c\), but we know that \(2b=a+c\). So, let's convert D into a function of b. \(D=(2b)^2-40(2b) = 4b^2 -80b = 4b(b-20)\) Now since \(b\) can be only in the range (1, 5), \(20-b\) will always be a positive number. So \(D<0\). Hence, the corresponding item in column II is option (s). The matching pairs are: a) - r) b) - q) c) - r) d) - s)

Key concepts

Nature of Roots

The nature of the roots of a quadratic equation refers to whether the roots are real and distinct, real and equal, or complex. A root of a quadratic equation is a solution to the equation, typically represented as \(x_1\) and \(x_2\) when the quadratic is in the form \(ax^2+bx+c=0\).

Understanding the nature of roots helps us in figuring out not just the number, but also the interrelationship between the roots, as well as how they are situated on the number line. An equation like \((x-a)(x-c)=0\) leads us directly to the roots \(x_1=a\) and \(x_2=c\), both of which are real numbers and within the range of 1 to 5 in the given problem. As they stem from the factors of the quadratic, they indicate where the graph of the quadratic equation crosses the x-axis. In essence, the nature of the roots is a foundational concept for both understanding the behavior of quadratic functions and solving quantitative problems across various fields of study.

Discriminant Analysis

Discriminant analysis in the context of quadratic equations involves evaluating the discriminant, represented by \(D\), of the quadratic equation \(ax^2+bx+c=0\). The discriminant is given by the formula \(D=b^2-4ac\) and serves as a crucial tool in determining the nature of the roots.

The significance of the discriminant lies in the information it provides:

• If \(D>0\), the quadratic equation has two distinct real roots.
• If \(D=0\), there is exactly one real root, known as a repeated or double root.
• If \(D<0\), the equation has two complex roots, which are complex conjugates of each other.

For the equation \(2x^2 -(a+c)x+5(a+c)=0\), the discriminant analysis reveals that \(D<0\) indicating complex roots. This analysis is fundamental, not only for this specific problem but for comprehending the behavior of any quadratic equation.

Root Intervals

Root intervals refer to the range within which the roots of a quadratic equation lie. In many cases, especially in standardized tests like the IIT-JEE, finding the interval of possible values for the roots can be as important as finding the roots themselves.

When given conditions as in the problem above, where \(1< a,c <5\), we can directly infer that the roots of an equation like \((x-a)(x-c)=0\) must fall within the interval \([1, 5]\). Understanding root intervals assists students in visualizing the number line and the segments where the roots reside. It also aids in graphing the quadratic function and predicting the behavior of the function within specific intervals. For example, in competitive exams, knowing that roots lie in \([1,5]\) can save a lot of time as opposed to calculating the exact roots. Therefore, grasping the concept of root intervals is instrumental for solving and graphing quadratic equations efficiently.