Let a-x-x2 -2k- b-3-k- c-3x-11-x-9-3k- be three vectors such that the angle between a- and b- is acute and between c- and a- is obtuse- then x lies between

Since, angle between a⃗ and b⃗ is acute. ∴ 3x+x^2+2 gt;0 ie, x ∈ ( ∞,1)∪ (2, ∞) Since, the angle between a⃗ and c⃗ is obtuse. ∴ 3x^2+11x +x^3 9x^2 6 lt;0 ...

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Byju's Answer

Standard XI

Mathematics

Applications of Dot Product

Let a⃗=xî+x2 ...

Question

Let $\to a = x^i + x^{2}^j + 2^k, \to b = - 3^i +^j +^k, \to c = (3 x + 11)^i + (x - 9)^j - 3^k$ be three vectors such that the angle between $\to a$ and $\to b$ is acute and between $\to c$ and $\to a$ is obtuse, then x lies between

(- \infty, 1) \cup (2, 3)

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(- \infty, 0)

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(4, 5)

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None of these

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Solution

The correct option is A $(- \infty, 1) \cup (2, 3)$

Since, angle between $\to a$ and $\to b$ is acute.
$∴ - 3 x + x^{2} + 2 > 0$ ie, $x \in (- \infty, 1) \cup (2, \infty)$
Since, the angle between $\to a$ and $\to c$ is obtuse.
$∴ 3 x^{2} + 11 x + x^{3} - 9 x^{2} - 6 < 0$
ie, $x^{3} - 6 x^{2} + 11 x - 6 < 0$
ie, $(x - 1) (x - 2) (X - 3) < 0$
$∴ x \in (- \infty, 1) \cup (2, 3)$

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Similar questions

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Let →a=x^i+x2^j+2^k,→b=−3^i+^j+^k,→c=(3x+11)^i+(x−9)^j−3^k be three vectors such that the angle between →a and →b is acute and between →c and →a is obtuse, then x lies between

The correct option is A (−∞,1)∪(2,3) Since, angle between →a and →b is acute. ∴−3x+x2+2>0 ie, x∈(−∞,1)∪(2,∞) Since, the angle between →a and →c is obtuse. ∴3x2+11x+x3−9x2−6<0 ie, x3−6x2+11x−6<0 ie, (x−1)(x−2)(X−3)<0 ∴x∈(−∞,1)∪(2,3)

Let $\to a = x^i + x^{2}^j + 2^k, \to b = - 3^i +^j +^k, \to c = (3 x + 11)^i + (x - 9)^j - 3^k$ be three vectors such that the angle between $\to a$ and $\to b$ is acute and between $\to c$ and $\to a$ is obtuse, then x lies between