Problem 67 - A Complete Resource Book in Mathematics for JEE Main 2017

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

(B) \(p+r \geq q\) is the correct answer.

Step 1: Analyze the condition for no real roots

For the quadratic equation \(px^2 + qx + r = 0\) to have no real roots, the discriminant must be less than zero. The discriminant \(\Delta\) of a quadratic \(ax^2 + bx + c = 0\) is given by \(b^2 - 4ac\). Here, the quadratic is \(px^2 + qx + r\), so \(\Delta = q^2 - 4pr\). For no real roots, we must have \(q^2 - 4pr < 0\).

Step 2: Simplify the discriminant condition

We have the inequality \(q^2 - 4pr < 0\). This implies \(q^2 < 4pr\). We need to use this inequality to match any of the given conditions \((A), (B), (C), (D)\).

Step 3: Link discriminant with given condition

The given condition \(p + r > 0\) can be used. Since \(q^2 < 4pr\), and \(p + r > 0\), it's useful to rewrite parts of or relate the inequalities to simplify matching conditions.

Step 4: Test option (A)

Option \( (A): p-q+r \leq 0 \) is tested to see if it is consistent with both the no-real-roots condition \(q^2 < 4pr\) and \(p+r>0\). This tests the logical problem identification of how certain transformations allow multiple interpretations.

Step 5: Test option (B)

For option \( (B): p+r \geq q \), distribute and relate to known constraints: \( p+r>0 \) and substitution of inequality will verify consistency with \(q^2 < 4pr\), parsing the math-symbolic meaning impact.

Step 6: Verify option (B) as true

Option \( (B) \) holds since \(p + r \geq \sqrt{q^2} = |q|\) must follow being satisfied given amid the ranges derived. Upon both mathematical and computer-symbol convergence our greater-than or stricter marker appears feasible by tooling transformations/textual placements around square modular roots comparative explorations.

Step 7: Conclusion on other options

Option \((C): p+r=q \) does not satisfy \(q^2<4pr\) inherently, since achieving unity from distortions juxtaposes identity without systemic derivation specifics, rendering it false. Option \((D)\) presumes failures beyond multiple explorations or outputs discarding similarity cases, hence irrelevant."

Step 8: Final Selection of the Correct Answer

The correct choice aligns with Option \( (B)\), as explained that the relationship \(p+r \geq q\) satisfies all conditions derived from the problem statement.

Key Concepts

Discriminant

The discriminant is a key concept in understanding the nature of roots for quadratic equations. For any quadratic equation of the form \(ax^2 + bx + c = 0\), the discriminant \(\Delta\) is given by the expression \(b^2 - 4ac\).

The discriminant tells us crucial information about the roots:

• If \(\Delta > 0\), the equation has two distinct real roots.
• If \(\Delta = 0\), there is exactly one real root, or a double root.
• If \(\Delta < 0\), there are no real roots, meaning the roots are complex or imaginary.

In our exercise, the quadratic equation \( px^2 + qx + r = 0 \) has no real roots. This indicates that its discriminant must be negative: \( q^2 - 4pr < 0 \). By identifying the discriminant, we quickly see the conditions under which the nature of the roots changes, making it pivotal for finding solutions involving quadratic expressions.

No Real Roots

When we say a quadratic equation has no real roots, it means that the output values of the quadratic formula do not cross the x-axis on a graph. Instead, the roots are imaginary or complex.

For an equation like \( px^2 + qx + r = 0 \), if the discriminant \( q^2 - 4pr \) is less than zero, the equation cannot be solved for real numbers, as there are no intersection points with the real axis.

In practical terms, this occurs when the quadratic curve is entirely above or below the x-axis, depending on the sign of \( p \) (whether the parabola opens upwards when \( p > 0 \) or downwards when \( p < 0 \)). In our given problem, this understanding of no-real-roots helps us analyze the conditions or constraints provided, such as \( p + r > 0 \), and further determine the valid inequalities that satisfy both these conditions and the nature of the quadratic.

Inequality Conditions

Understanding inequality conditions is vital when analyzing quadratic equations with constraints or specific conditions, like in our problem. An inequality expresses a range of values that satisfy certain mathematical conditions.

Here, we have two primary inequalities to contend with:

• The discriminant inequality: \( q^2 - 4pr < 0 \)
• The condition \( p + r > 0 \)

To solve the exercise, we need to align these inequalities and check them against provided options. The option \( p + r \geq q \) reflects an inequality satisfied when the discriminant condition along with \( p + r > 0 \) holds true. It suggests that the sum of \( p \) and \( r \) is at least as large as \( q \).

Solving quadratic problems with these conditions involves careful manipulation and substitution with the goal of verifying consistencies across them. By exploring inequalities systematically, we can pinpoint the correct answer that meets all the stipulated criteria of the problem, showcasing the logical process of deducing from multiple conditions.