The set of values of - for which vectors a-x-x-1-k- and b-x-1-k- make an acute angle for - x- is

Given : nbsp;a=xî+(x 1)ĵ+k̂ and b =(x+1)î+ĵ+αk̂ ⇒a·b=x^2+2x+α 1 For acute angle : a·b gt;0 ,∀ x∈ℝ ⇒ x^2+2x+α 1 gt;0, ∀ x∈ℝ As cofficient of x^2 gt;0 : ...

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Byju's Answer

Standard XII

Mathematics

Test for Coplanarity

The set of va...

Question

The set of value(s) of $α \in R$ for which vectors $\to a = x^i + (x - 1)^j +^k$ and $\to b = (x + 1)^i +^j + α^k$ make an acute angle for $\forall x \in R,$ is

α \in [0, \infty)

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α \in (- \infty, 0)

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α \in (- 2, 2)

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α \in (2, \infty)

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Solution

The correct option is D $α \in (2, \infty)$
Given : $\to a = x^i + (x - 1)^j +^k$ and $\to b = (x + 1)^i +^j + α^k$
$\Rightarrow \to a \cdot \to b = x^{2} + 2 x + α - 1$
For acute angle : $\to a \cdot \to b > 0, \forall x \in R$
$\Rightarrow x^{2} + 2 x + α - 1 > 0, \forall x \in R$
As cofficient of $x^{2} > 0 : D < 0$
$\Rightarrow 4 - 4 (α - 1) < 0 \Rightarrow α > 2$

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The set of value(s) of α∈R for which vectors →a=x^i+(x−1)^j+^k and →b=(x+1)^i+^j+α^k make an acute angle for ∀x∈R, is

The correct option is D α∈(2,∞)Given : →a=x^i+(x−1)^j+^k and →b=(x+1)^i+^j+α^k ⇒→a⋅→b=x2+2x+α−1 For acute angle : →a⋅→b>0,∀x∈R ⇒x2+2x+α−1>0,∀x∈R As cofficient of x2>0:D<0 ⇒4−4(α−1)<0⇒α>2

The set of value(s) of $α \in R$ for which vectors $\to a = x^i + (x - 1)^j +^k$ and $\to b = (x + 1)^i +^j + α^k$ make an acute angle for $\forall x \in R,$ is