Let f(x)=(√sgn (α x2 + α x + 1)/cot- 1 (x2 - α )) . If f(x) is continuous for all x∈ R , then number of integers in the range of α is

Question

Let f(x)=(√sgn (α x2 + α x + 1)/cot- 1 (x2 - α )) . If f(x) is continuous for all x∈ R , then number of integers in the range of α is (A) 0 (B) 4 (C) 5 (D) 6

Tardigrade Admin · Accepted Answer

∵ f(x) is continuous for all x∈ R ∴ α x2+α x+1>0∀ x∈ R (i) If α =0 then it is true. (ii) If α ≠ 0 then D < 0 α 2-4α < 0⇒ 0 < α < 4 ∴ Integral values of α =0,1,2,3 ∴ Number of integral values of α =4

Q. Let $f (x) = \frac{s g n ( α x ^{2} + αx + 1 )}{co t ^{- 1} ( x ^{2} - α )}$ . If $f (x)$ is continuous for all $x \in R$ , then number of integers in the range of $α$ is

$0$

$4$

$5$

$6$

$∵$ $f (x)$ is continuous for all $x \in R$
$∴$ $α x^{2} + αx + 1 > 0\forall x \in R$
(i) If $α = 0$ then it is true.
(ii) If $α \neq = 0$ then $D < 0$
$α^{2} - 4 α < 0 \Rightarrow 0 < α < 4$
$∴$ Integral values of $α = 0, 1, 2, 3$
$∴$ Number of integral values of $α = 4$

Q. Let f(x)=cot−1(x2−α)sgn(αx2+αx+1)​​ . If f(x) is continuous for all x∈R , then number of integers in the range of α is

0

4

5

6

∵ f(x) is continuous for all x∈R ∴ αx2+αx+1>0∀x∈R (i) If α=0 then it is true. (ii) If α=0 then D<0 α2−4α<0⇒0<α<4 ∴ Integral values of α=0,1,2,3 ∴ Number of integral values of α=4

Q. Let $f (x) = \frac{s g n ( α x ^{2} + αx + 1 )}{co t ^{- 1} ( x ^{2} - α )}$ . If $f (x)$ is continuous for all $x \in R$ , then number of integers in the range of $α$ is

$0$

$4$

$5$

$6$

$∵$ $f (x)$ is continuous for all $x \in R$
$∴$ $α x^{2} + αx + 1 > 0\forall x \in R$
(i) If $α = 0$ then it is true.
(ii) If $α \neq = 0$ then $D < 0$
$α^{2} - 4 α < 0 \Rightarrow 0 < α < 4$
$∴$ Integral values of $α = 0, 1, 2, 3$
$∴$ Number of integral values of $α = 4$