If $\to a = x^i + (x - 1)^j +^k$ and $\to b = (x + 1)^i +^j + a^k$ always make an acute angle with each other for every value of $x \in R$ , then

a \in (- \infty, 2)

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a \in (2, \infty)

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a \in (- \infty, 1)

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a \in (1, \infty)

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Solution

The correct option is D $a \in (2, \infty)$
Lets take
$→a.→b={x^i+(x−1)^j+^k}.{(x+1)^i+^ak}$

$\Rightarrow \to a . \to b = x (x + 1) + x - 1 + a$
$= x^{2} + 2 x + a - 1$

We must have $\to a . \to b > 0 \forall x \in R$
$x^{2} + 2 x + a - 1 > 0$ $\forall$ $x \in R$
$\Rightarrow 4 - 4 (a - 1) < 0$
$\Rightarrow a > 2$

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Similar questions

If $\to a = x^i + (x - 1)^j +^k$ and $\to b = (x + 1)^i +^j + a^k$ always make an acute angle with each other for every value of $x ϵ R$ , then

\to a = x^i + (x - 1)^j +^k

\to b = (x + 1)^i +^j + a^k

\forall x \in R

A

2^i - m^j + 3 m^k

(1 + m)^i - 2 m^j +^k

m

\notin [- 2, - \frac{1}{2}] .

R :

(a, b)

a . b > 0

\to p = (a + 1)^i + a^j + a^k, \to q = a^i + (a + 1)^j + a^k

\to r = a^i + a^j + (a + 1)^k, (a \in R)

3 (\to p . \to q)^{2} - λ | \to r \times \to q |^{2} = 0,

λ

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