Let S be the set of all possible integral values of λ in the interval (-3,7) for which the roots of the quadratic equation λ x2+13 x+7=0 are all rational numbers. Then the sum of the elements in S is

Question

Let S be the set of all possible integral values of λ in the interval (-3,7) for which the roots of the quadratic equation λ x2+13 x+7=0 are all rational numbers. Then the sum of the elements in S is (A) 4 (B) 2 (C) 3 (D) 1

Tardigrade Admin · Accepted Answer

We have λ x2+13 x+7=0 ∴ D=(13)2-4(λ)(7)=169-28 λ For rational roots, D= perfect square So, λ ∈(-3,7) has values -2,0,6 so that D become perfect square. ∴ Required sum =-2+0+6=4

Q. Let $S$ be the set of all possible integral values of $λ$ in the interval $(- 3, 7)$ for which the roots of the quadratic equation $λ x^{2} + 13 x + 7 = 0$ are all rational numbers. Then the sum of the elements in $S$ is

4

2

3

1

We have
$λ x^{2} + 13 x + 7 = 0$
$∴ D = (13)^{2} - 4 (λ) (7) = 169 - 28 λ$
For rational roots, $D =$ perfect square
So, $λ \in (- 3, 7)$ has values $- 2, 0, 6$
so that $D$ become perfect square.
$∴$ Required sum $= - 2 + 0 + 6 = 4$

Q. Let S be the set of all possible integral values of λ in the interval (−3,7) for which the roots of the quadratic equation λx2+13x+7=0 are all rational numbers. Then the sum of the elements in S is

4

2

3

1

We have λx2+13x+7=0 ∴D=(13)2−4(λ)(7)=169−28λ For rational roots, D= perfect square So, λ∈(−3,7) has values −2,0,6 so that D become perfect square. ∴ Required sum =−2+0+6=4

Q. Let $S$ be the set of all possible integral values of $λ$ in the interval $(- 3, 7)$ for which the roots of the quadratic equation $λ x^{2} + 13 x + 7 = 0$ are all rational numbers. Then the sum of the elements in $S$ is

We have
$λ x^{2} + 13 x + 7 = 0$
$∴ D = (13)^{2} - 4 (λ) (7) = 169 - 28 λ$
For rational roots, $D =$ perfect square
So, $λ \in (- 3, 7)$ has values $- 2, 0, 6$
so that $D$ become perfect square.
$∴$ Required sum $= - 2 + 0 + 6 = 4$