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JEE MATHEMATICS MCQ: ADDITION OF VECTORS

1 Easy
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Analyzing +4 | -1
If a, b and c are any three vectors forming a linearly independent system, then θ∈R,\theta \in R,
[acos⁡θ+bsin⁡θ+cos⁡2θ,   acos⁡(2π3+θ)+bsin⁡(2π3+θ)+c cos⁡2(2π3+θ)[a\cos \theta + b\sin \theta + \cos 2\theta ,\,\,\,a\cos \left( {\frac{{2\pi }}{3} + \theta } \right) + b\sin \left( {\frac{{2\pi }}{3} + \theta } \right) + c\,\cos 2\left( {\frac{{2\pi }}{3} + \theta } \right) acos⁡(θ−2π3)+bsin⁡(θ−2π3)+c cos⁡2(θ−2π3)a\cos \left( {\theta - \frac{{2\pi }}{3}} \right) + b\sin \left( {\theta - \frac{{2\pi }}{3}} \right) + c\,\cos 2\left( {\theta - \frac{{2\pi }}{3}} \right)
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Correct option: D
None of these
Option D (None of these) is correct because the determinant simplifies to 0, making the scalar triple product zero.

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Let a=i^−j^,b=j^−k^\mathbf{a}=\hat{\mathbf{i}}-\hat{\mathbf{j}}, \mathbf{b}=\hat{\mathbf{j}}-\hat{\mathbf{k}} and c=k^−i^\mathbf{c}=\hat{\mathbf{k}}-\hat{\mathbf{i}}. If d\mathbf{d} is aa unit vector such that a⋅d=0=(bc\mathbf{a} \cdot \mathbf{d}=0=(b \mathbf{c} ) , then d\mathbf{d} is equal to
 A vector a⃗{\mathbf{\vec a}} has components 2  p2\;p and 1 with respect to a rectangular cartesian system. This system is rotated through a certain angle about the origin in the counter clockwise sense, if this respect to new system a⃗{\mathbf{\vec a}} has components p+1p + 1 and 1, then If a⃗,b⃗\vec{a}, \vec{b} and c⃗\vec{c} are three mutually perpendicular unit vetors and d⃗\vec{d} is a unit vector which makes equal angles with a→,b→\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}} ānd c→\overrightarrow{\mathrm{c}} then the value of ∣a→+b→+c→+d→∣2|\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{d}}|^2 The non-zero vectors are a⃗,b⃗\vec{a}, \vec{b} and c⃗\vec{c} are related by a⃗=8b⃗\vec{a}=8 \vec{b} and c⃗=−7b⃗\vec{c}=-7 \vec{b}. Then the angle between a⃗\vec{a} and c⃗\vec{c} is If the vectors 2i+j−k,−i+2j+λk2 \mathbf{i}+\mathbf{j}-\mathbf{k},-\mathbf{i}+2 \mathbf{j}+\lambda \mathbf{k} and −5i+2j−k-5 \mathbf{i}+2 \mathbf{j}-\mathbf{k} ar coplanar, then the value of λ\lambda is equal The medium ADAD of the triangle ABCABC is bisected at E,  BEE,\;BE meets ACAC in FF, then AF:AC=AF:AC =
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