If the minimum value of the expression E(x)=4 operatornamecosec2 x+ sin 2 x lies between the roots of the quadratic equation x2+2 k x+k2-9=0, then least integral value of k is

Question

If the minimum value of the expression E(x)=4 operatornamecosec2 x+ sin 2 x lies between the roots of the quadratic equation x2+2 k x+k2-9=0, then least integral value of k is (A) -8 (B) -7 (C) -9 (D) -10

Tardigrade Admin · Accepted Answer

E(x)= operatornamecosec2 x+ sin 2 x-2+3 operatornamecosec2 x+2 =( operatornamecosec x- sin x)2+3 operatornamecosec2 x+2 ⇒ E min =5 f(x)=x2+2 k x+k2-9 f(5)< 0 ⇒ k2+10 k+16< 0 k ∈(-8,-2)

Q. If the minimum value of the expression $E (x) = 4 cosec^{2} x + sin^{2} x$ lies between the roots of the quadratic equation $x^{2} + 2 k x + k^{2} - 9 = 0$ , then least integral value of $k$ is

-8

-7

-9

-10

$E (x) = cosec^{2} x + sin^{2} x - 2 + 3 cosec^{2} x + 2$
$= (cosec x - sin x)^{2} + 3 cosec^{2} x + 2 \Rightarrow E_{m i n} = 5$
$f (x) = x^{2} + 2 k x + k^{2} - 9$
$f (5) < 0 \Rightarrow k^{2} + 10 k + 16 < 0$
$k \in (- 8, - 2)$

Q. If the minimum value of the expression E(x)=4cosec2x+sin2x lies between the roots of the quadratic equation x2+2kx+k2−9=0, then least integral value of k is

-8

-7

-9

-10

E(x)=cosec2x+sin2x−2+3cosec2x+2 =(cosecx−sinx)2+3cosec2x+2⇒Emin​=5 f(x)=x2+2kx+k2−9 f(5)<0⇒k2+10k+16<0 k∈(−8,−2)

Q. If the minimum value of the expression $E (x) = 4 cosec^{2} x + sin^{2} x$ lies between the roots of the quadratic equation $x^{2} + 2 k x + k^{2} - 9 = 0$ , then least integral value of $k$ is

$E (x) = cosec^{2} x + sin^{2} x - 2 + 3 cosec^{2} x + 2$
$= (cosec x - sin x)^{2} + 3 cosec^{2} x + 2 \Rightarrow E_{m i n} = 5$
$f (x) = x^{2} + 2 k x + k^{2} - 9$
$f (5) < 0 \Rightarrow k^{2} + 10 k + 16 < 0$
$k \in (- 8, - 2)$