Let coefficient of x2 and x3 in the expansion of (3+a x)9 are equal. If a lies between the roots of the equation 49 x2 +7 p x-9=0, then the range of p is (-∞,-λ). Find the value of λ.

Question

Let coefficient of x2 and x3 in the expansion of (3+a x)9 are equal. If a lies between the roots of the equation 49 x2 +7 p x-9=0, then the range of p is (-∞,-λ). Find the value of λ. (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Coefficient of x2= coefficient of x3 9 C2 37 a2= 9 C3 36 a3 ∴ a=(9/7) Now, f(a) < 0 ⇒ 81+9 p-9 < 0 9 p < -72 ⇒ p < -8

Q. Let coefficient of $x^{2}$ and $x^{3}$ in the expansion of $(3 + a x)^{9}$ are equal. If a lies between the roots of the equation $49 x^{2}$ $+ 7 p x - 9 = 0$ , then the range of $p$ is $(- \infty, - λ)$ . Find the value of $λ$ .

Coefficient of $x^{2} =$ coefficient of $x^{3}$
$^{9} C_{2} 3^{7} a^{2} =^{9} C_{3} 3_{6} a^{3}$
$∴ a = \frac{9}{7}$
Now, $f (a) < 0$
$\Rightarrow 81 + 9 p - 9 < 0$
$9 p < - 72 \Rightarrow p < - 8$

Q. Let coefficient of x2 and x3 in the expansion of (3+ax)9 are equal. If a lies between the roots of the equation 49x2 +7px−9=0, then the range of p is (−∞,−λ). Find the value of λ.

Coefficient of x2= coefficient of x3 9C2​37a2=9C3​36​a3 ∴a=79​ Now, f(a)<0 ⇒81+9p−9<0 9p<−72⇒p<−8

Q. Let coefficient of $x^{2}$ and $x^{3}$ in the expansion of $(3 + a x)^{9}$ are equal. If a lies between the roots of the equation $49 x^{2}$ $+ 7 p x - 9 = 0$ , then the range of $p$ is $(- \infty, - λ)$ . Find the value of $λ$ .

Coefficient of $x^{2} =$ coefficient of $x^{3}$
$^{9} C_{2} 3^{7} a^{2} =^{9} C_{3} 3_{6} a^{3}$
$∴ a = \frac{9}{7}$
Now, $f (a) < 0$
$\Rightarrow 81 + 9 p - 9 < 0$
$9 p < - 72 \Rightarrow p < - 8$