Problem 67 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(p x^{2}+q x+r=0\) has no real roots and \(p, q, r\) are real such that \(p+r>0\), then (A) \(p-q+r \leq 0\) (B) \(p+r \geq q\) (C) \(p+r=q\) (D) None of these

Step-by-Step Solution

Answer

The correct option is (B).

Step 1: Understand Discriminant of a Quadratic Equation

A quadratic equation of the form \( ax^2 + bx + c = 0 \) has no real roots if its discriminant \( b^2 - 4ac \) is less than zero.

Step 2: Apply Discriminant Condition

For the given quadratic equation \( p x^2 + q x + r = 0 \), the discriminant is \( q^2 - 4pr \). Since the equation has no real roots, we have \( q^2 - 4pr < 0 \).

Step 3: Investigate Given Condition

We are given that \( p + r > 0 \). This condition will be used in conjunction with other options to find a relation among \( p, q, \) and \( r \).

Step 4: Analyze Each Option

- For option (A), if \( p - q + r \leq 0 \), this implies \( p + r \leq q \). However, this does not necessarily lead to \( q^2 < 4pr \) because the opposite is considered.- For option (B), if \( p + r \geq q \), it implies \( p + r > q \) when combined with \( q^2 - 4pr < 0 \), making sense since it could lead to \( q^2 < (p+r)^2 \).- For option (C), if \( p + r = q \), \( q^2 - 4pr < 0 \) doesn't always hold.- Option (D) would imply none of these conditions satisfy all given conditions.

Step 5: Conclude the Correct Option

By deducing from the inequalities, option (B): \( p+r \geq q \) leads naturally to fulfilling the conditions set by having no real roots and \( p + r > 0 \). Therefore, the correct answer is option (B).

Key Concepts

Discriminant of Quadratic Equation

The discriminant is a critical part of every quadratic equation, helping us determine the nature of its roots. Given a quadratic equation in the standard form \( ax^2 + bx + c = 0 \), the discriminant is expressed as \( b^2 - 4ac \). This part of the quadratic formula helps to indicate the type of roots the equation has.

• If the discriminant \( b^2 - 4ac > 0 \), the equation has two distinct real roots.
• If \( b^2 - 4ac = 0 \), it has exactly one real root or a repeated root.
• If \( b^2 - 4ac < 0 \), there are no real roots; instead, the roots are complex conjugates.

In the exercise, we know there are no real roots, which is why the condition \( q^2 - 4pr < 0 \) holds. Understanding the discriminant's role allows us to explore root characteristics and solve quadratic equations effectively.

Inequalities with Quadratics

Quadratic inequalities involve expressions where a quadratic function is set in relation to another value, like solving for \( ax^2 + bx + c > 0 \). These inequalities require understanding how quadratics behave over intervals.
For our exercise, given \( q^2 - 4pr < 0 \), we know it aligns with quadratics having complex roots. This inequality tells us about the relationship and sums of \( p \), \( q \), and \( r \) in our quadratic. It means not only are there no real roots, but graphically, the parabola doesn't cross the x-axis at all. Combined with \( p + r > 0 \), these inequalities suggest viable solutions must meet simultaneous conditions.

• Checking each proposed condition in the exercise, we found that \( p + r \geq q \) fits well with no real roots, as it supports the condition \( q^2 < 4pr \).

Recognizing these inequalities allows deeper insights into the behavior of quadratic expressions.

Real and Complex Roots

The nature of the roots of a quadratic equation is crucial to understanding its solutions and graph. Real roots are where the quadratic can intersect the x-axis, while complex roots indicate no intersection.
For quadratic equations such as \( p x^2 + q x + r = 0 \), if \( q^2 - 4pr < 0 \), the roots are complex. Let's break down complex roots:

• Complex numbers are in the form \( a + bi \), where \( i \) is the imaginary unit satisfying \( i^2 = -1 \).
• Complex roots always occur in conjugate pairs, such as \( a + bi \) and \( a - bi \), ensuring the roots are symmetric about the real axis.

In the exercise context, having \( q^2 - 4pr < 0 \) and understanding complex roots mean we anticipate no real solution intersections on the graph. The exercise uses this understanding to validate which condition is true, knowing any viable condition must respect these non-real solutions.