Question

Consider \(\mathrm{Q}=\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}\) where \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) are real and distinct. Also, \(\mathrm{Q}=0\) has real roots Column 1 \(\quad\) Column II (a) If \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) are in \(\mathrm{AP}\), then the least value of \(\left|\frac{\mathrm{d}}{\mathrm{b}}\right|\) (where \(\mathrm{d}\) is the common (p) \(-\sec ^{2} \frac{\pi}{4}\) difference), is (b) If \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) are in \(\mathrm{GP}\), then \(\mathrm{b}\) cannot be equal to (q) \(\cos \frac{\pi}{3}\) (c) If \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) are in AP with common difference 'd', and if zero is a root of (r) \(\sin \frac{\pi}{3}\) \(\mathrm{Q}=0\), then \(\left|\frac{\mathrm{d}}{\mathrm{a}}\right|\) equals (d) If \(a, b, c\) are in \(A P\), and \(b, c\), a are in GP, then the common ratio \(r\) equals (s) \(\sin \frac{\pi}{2}\)

Short answer

In summary: a) The least value of |d/b| for the arithmetic progression is 2 (p). b) For geometric progression, b cannot be equal to q. c) For arithmetic progression with zero as a root, |d/a| is equal to 1/2 (r). d) When the coefficients are in both arithmetic and geometric progressions, the common ratio r is equal to 3 (s).

Step-by-step solution

Condition 1: Arithmetic Progression

If a, b, and c are in AP, then their common difference d is given by \(b-a=c-b\), or \(\dfrac{d}{b}=2\). To find the least value of \(\left|\dfrac{d}{b}\right|\), we can simply apply the fact that the minimum value of the absolute value is when the value is equal to zero. Thus, \(\left|\dfrac{d}{b}\right|=\sec^2(\dfrac{\pi}{4})=2\). Therefore, the answer for (a) is (p).

Condition 2: Geometric Progression

If a, b, and c are in GP, then their common ratio r is given by \(\dfrac{b}{a}=\dfrac{c}{b}\). To show that b cannot be equal to q, which is \(\cos(\dfrac{\pi}{3})=\dfrac{1}{2}\), we will rewrite the equation as \(b^2-ac=0\). Since the roots of the quadratic are real, the discriminant \(\Delta = b^2 - 4ac\) must be greater than or equal to 0. Substituting \(b=\dfrac{1}{2}\) and the given GP condition: $$ \left(\frac{1}{2}\right)^2 - ac = 0 \\ \frac{1}{4} = ac \\ c = \frac{1}{4a} $$ Now, substituting c back into the discriminant condition, we have: $$ \left(\frac{1}{2}\right)^2 - 4a\left(\frac{1}{4a}\right) \ge 0 \\ \frac{1}{4} - 1 \ge 0 $$ Since this inequality does not hold true, b cannot be equal to q. Therefore, the answer for (b) is (q).

Condition 3: Arithmetic Progression, Zero is a Root

If a, b, and c are in AP with common difference d and zero is a root of the quadratic equation Q=0, then we can write the progression as \(a, a+d, a+2d\). Replacing the coefficients in the equation, we now have \(a^2x^2+(a+d)x+(a+2d)=0\). Since zero is a root, we can let x=0, which results in: $$ a+2d = 0 \\ d = -\frac{a}{2} $$ Now, the question asks for the value of \(\left|\dfrac{d}{a}\right|\): $$ \left|\frac{-\frac{a}{2}}{a}\right| = \frac{1}{2} $$ Therefore, the answer for (c) is (r).

Condition 4: Arithmetic and Geometric Progression

If a, b, c are in AP and b, c, and a are in GP, then their common ratio r can be found by combining the two conditions. Writing down the equations for both progressions, we can see that \(b-a = c-b\) and \(\dfrac{c}{b} = \dfrac{a}{c}\). From the arithmetic progression condition, we find that \(c=2b-a\) and substitute this into the geometric progression condition: $$ \frac{2b-a}{b} = \frac{a}{2b-a} $$ Cross-multiplying and simplifying, we obtain: $$ a^2 = b^2 \\ a = \pm b $$ Since a ≠ b, we conclude that a=-b. Now, substituting this into the condition for the common ratio, we get: $$ r = \frac{2b-(-b)}{b} = 3 $$ So, the answer for (d) is (s).

Key concepts

Arithmetic Progression

The concept of an arithmetic progression (AP) is fundamental in algebra and forms the basis for various mathematical problems, including those in IIT-JEE examinations. An arithmetic progression is a sequence of numbers in which the difference of any two successive members is a constant, known as the common difference (d).

When dealing with AP in the context of quadratic equations, as in our original exercise, we translate the relationships between the terms into algebraic equations. For instance, if the terms a, b, c are in AP, then the common difference is expressed as b - a or c - b. To illustrate, let's say we have a quadratic equation with these terms as coefficients. Their relationship, based on the AP property, can provide insight into the characteristics of the equation's roots. For learners, understanding that the systematic structure of AP can decode complex relationships between terms is crucial to grasping more challenging algebra concepts.

This concept intersects with the exercise in Step 1, where it's inferred that the lowest value for |d/b| is when the common difference is zero, which cannot be the case for a true AP. Instead, using trigonometric identities and the characteristics of an AP helps us find the correct least value. Moreover, when zero is a root, using the AP property allows us to simplify the equation drastically, affirming the practicality and power of a strong grasp on arithmetic progressions.

Geometric Progression

In contrast to arithmetic progression, a geometric progression (GP) is defined by a sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (r). This common ratio is a critical aspect of GPs as it provides a multiplicative step between consecutive numbers in the sequence.

When the coefficients of a quadratic equation are in geometric progression, as mentioned in Step 2 of the original solution, it implies a relationship expressed by the formula r = b/a = c/b. Here, the value of b, or the common ratio, can be used to determine the nature of the next terms. It is particularly insightful when we are asked to prove whether a certain value for b is possible given the real roots of a quadratic equation. By applying the condition that the discriminant (Δ = b² - 4ac) must be non-negative for real roots to exist, we can investigate possible values of b.

The mastery of a GP is not just learning its definition, but being able to manipulate and apply it to determine possible outcomes and the nature of the roots of a quadratic equation. Solving problems related to GPs reinforces the algebraic intuition required for various quantitative fields.

Quadratic Equations Roots

The roots of a quadratic equation, which take the standard form ax² + bx + c = 0, are fundamental in solving many different types of math problems, including those in the IIT-JEE syllabus. The roots can give us pivotal information about the graph of a quadratic equation, such as where it intersects the x-axis. A crucial determinant for the nature of these roots is the discriminant, represented by Δ = b² - 4ac.

The discriminant can tell us whether the roots are real or complex. If Δ > 0, the roots are real and distinct; if Δ = 0, the roots are real and the same (also known as repeated or double roots); and if Δ < 0, the roots are complex conjugates. In our original exercise's Step 3, we used the fact that one root is zero, which can only happen when c, the constant term, is zero. This provides a direct connection to the coefficients being in arithmetic progression.

Understanding how the discriminant influences the roots helps students predict the behavior of quadratic equations without necessarily finding the precise roots. This fundamental knowledge of quadratic equation roots is indispensable not only for solving problems on paper but also for grasping more advanced mathematics concepts and their real-world applications.