Bissoy.com
Doctors Medicines MCQs Q&A

Consider the following statements in respect of the quadratic equation 4(x - p)(x - q) - r2 = 0, where p, q and r are real numbers: 1. The roots are real 2. The roots are equal if p = q and r = 0 Which of the above statements is/are correct?

Consider the following statements in respect of the quadratic equation 4(x - p)(x - q) - r2 = 0, where p, q and r are real numbers: 1. The roots are real 2. The roots are equal if p = q and r = 0 Which of the above statements is/are correct? Correct Answer Both 1 and 2

Concept:

For any quadratic equation, ax2 + bx + c = 0. We have discriminant, D = b2 - 4ac then the given quadratic equation has:

  1. Distinct and real roots if D > 0.
  2. Real and repeated roots, if D = 0.
  3. Complex roots and conjugate of each other, D < 0.

Calculation:

Given: 4(x - p)(x - q) - r2 = 0

⇒ x2 - (p + q) x + pq - r2 /4 = 0

By comparing the quadratic equation x2 - (p + q) x + (pq - (r /2)2) = 0 with the standard quadratic equation ax2 + bx + c = 0. We get, a = 1, b = - (p + q) and c =  pq - r2 /4

⇒ D = b2 - 4ac = (p + q)2 - 4pq + r2 = (p - q)2 + r2 ≥ 0

Hence statement I is true

The given quadratic equation 4(x - p)(x - q) - r2 = 0 will have repeated roots if D = 0.

⇒ D = (p - q)2 + r2 = 0 if and only if p = q and r = 0.

Hence statement II is true.

Related Questions

Consider the statements given below: A: Sum of roots of a quadratic equation ax2 + bx + c = 0 is (c/a). B: If one root of a quadratic equation is (p + qi), then root must be (q + pi). C: Roots of the equation 3x2 + 4x = 5 are all real. Which of the statement(s) is/are TRUE?
Which of the following is true? A: If the sum of cube of three numbers is thrice the product of the three numbers, then the sum of three numbers will be non-zero. B: For quadratic equation ax2 + bx + c = 0, If b2 < 4ac, then the given equation has only one real root. C: In the equation, x2 – (a + b)x + ab = 0, the roots of this equation will be ‘a’, ‘b’.
A and B are positive roots of quadratic equation and (A + B)2 = 400 and (A – B)2 = 4. Find the quadratic equation whose roots are A and B.
If the roots of the quadratic equation x2 – x + 1 = 0 are α and β, then find the quadratic equation whose roots are α4 and β4.
The product of the roots of a quadratic equation is 20 and the sum of its roots is (-8). What is the quadratic equation?
If sum of the roots of a quadratic equation is ­-7 and product of the roots is 12. Find the quadratic equation.
In the following question, two statements are given each followed by two conclusions I and II. You have to consider the statements to be true even if theyseem to be at variance from commonly known facts. You have to decide which of the given conclusions, if any, follows from the given statements. Statements:
(I) Some polynomials are linear equations.
(II) Some linear equations are quadratic. Conclusion:
(I) Polynomials are quadratic.
(II) Linear equations are quadratic.
Given two quadratic equations, I. x2 - 31x + 240 = 0 Equation I has two roots x1, x2. (x1 > x2) II. 3y2 – 7y – 3 = 8y - 3y2 - 9 Equation II has two roots y1, y2. (y2 > y1) Find the equation whose roots are (– 2y1 + x1), - (y2 – x2).
Consider the following statements in respect of the roots of the equation x3 - 8 = 0: 1. The roots are non-collinear. 2. The roots lie on a circle of unit radius. Which of the above statements is/are correct?
Given below are two statements, followed by four conclusions. Assume the facts in the statements to be true and decide which of the conclusions logically follow(s) from the statements. Statements: All roots are fruits. All fruits are guns. Conclusions: I. All roots are guns,. II. All guns are roots. III. All fruits are roots. IV. Some guns are roots.