SATHEE: Theory of Equations 1 Question 10

Theory of Equations 1 Question 10

11. The number of all possible positive integral values of $α$ for which the roots of the quadratic equation, $6 x^{2} - 11 x + α = 0$ are rational numbers is

(2019 Main, 9 Jan II)

(a) 5

(b) 2

(d) 3

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Answer:

Correct Answer: 11. (d)

Solution:

For the roots of quadratic equation $a x^{2} + b x + c = 0$ to be rational $D = (b^{2} - 4 a c)$ should be perfect square.

In the equation $6 x^{2} - 11 x + α = 0$

$a = 6, b = - 11 and c = α$

$∴$ For roots to be rational

$D = (- 11)^{2} - 4 (6) (α)$ should be a perfect square.

$\Rightarrow D (α) = 121 - 24 α$ should be a perfect square Now,

$D (1) = 121 - 24 = 97$ is not a perfect square.

$D (2) = 121 - 24 \times 2 = 73$ is not a perfect square.

$D (3) = 121 - 24 \times 3 = 49$ is a perfect square.

$D (4) = 121 - 24 \times 4 = 25$ is a perfect square.

$D (5) = 121 - 24 \times 5 = 1$ is a perfect square.

and for $α \geq 6, D (α) < 0$ , hence imaginary roots.

$∴$ For 3 values of $α (α = 3, 4, 5)$ , the roots are rational.

Theory of Equations 1 Question 10

11. The number of all possible positive integral values of $α$ for which the roots of the quadratic equation, $6 x^{2} - 11 x + α = 0$ are rational numbers is

Answer:

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Theory of Equations 1 Question 10

11. The number of all possible positive integral values of α for which the roots of the quadratic equation, 6x2−11x+α=0 are rational numbers is

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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11. The number of all possible positive integral values of $α$ for which the roots of the quadratic equation, $6 x^{2} - 11 x + α = 0$ are rational numbers is