91Ó°ÊÓ

If \(\mathbf{a} \cdot \mathbf{b}\) and \(\mathbf{c}\) are any three non-zero vectors, then the component of \(a \times(b \times c)\) perpendicular to \(b\) is (a) \(a \times(b \times c)+\frac{(a \times b) \cdot(c \times a)}{|b|^{2}}\) (b) \(a \times(b \times c)+\frac{(a \times c) \cdot(a \times b)}{|b|^{2}}\) (c) \(a \times(b \times c)+\frac{(b \times c) \cdot(b \times a)}{|b|^{2}} b\) (d) \(a \times(b \times c)+\frac{(a \times b) \cdot(b \times c)}{|b|^{2}} b\)

The shortest distance between a diagonal of a unit cube and a diagonal of a face skew to it is (a) \(\frac{1}{2}\) (b) \(\frac{1}{\sqrt{2}}\) (c) \(\frac{1}{\sqrt{3}}\) (d) \(\frac{1}{\sqrt{6}}\)

If the angle between the vectors \(\mathbf{a}=\hat{\mathbf{i}}+(\cos x) \hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(\mathbf{b}=\left(\sin ^{2} x-\sin x\right) \hat{\mathbf{i}}-(\cos x) \hat{j}+(3-4 \sin x) \hat{k}\) is obtuse and \(x \in\left(0, \frac{\pi}{2}\right)\), then the exhaustive set of values of \(^{\prime} x^{\prime}\) is equal to (a) \(x \in\left(0, \frac{\pi}{6}\right)\) (b) \(x \in\left(\frac{\pi}{6}, \frac{\pi}{2}\right)\) (c) \(x \in\left(\frac{\pi}{6}, \frac{\pi}{3}\right)\) (d) \(x \in\left(\frac{\pi}{3}, \frac{\pi}{2}\right)\)

If vectors a and \(\mathbf{b}\) are non-collinear, than \(\frac{\mathrm{a}}{|\mathrm{a}|}+\frac{\mathrm{b}}{|\mathbf{b}|}\) is (a) a unit vector (b) in the plane of \(\mathbf{a}\) and \(\mathbf{b}\) (c) equally inclined to \(\mathbf{a}\) and \(\mathbf{b}\) (d) perpendicular to \(\mathbf{a} \times \mathbf{b}\)

If \(M\) and \(N\) are the mid-point of the diagonals \(A C\) and \(B D\), respectively of a quadrilateral \(A B C D\), then \(A B+A D+C B+C D=k M N\), where \(k=\ldots \ldots .\)

If \(\alpha\) and \(\beta\) are two perpendicular unit vectors such that \(\mathbf{x}=\hat{\boldsymbol{\beta}}-(\alpha \times \mathbf{x})\), then the value of \(4|\mathbf{x}|^{2}\) is.

Two points \(P\) and \(Q\) are given in the rectangular cartesian coordinates in the curve \(y=2^{x+2}\), such that OP. \(\hat{\mathbf{i}}=-1\) and \(\mathbf{O Q} \cdot \hat{\mathbf{i}}=2\), where \(\hat{\mathbf{i}}\) is a unit vector along the \(X\)-axis. Find the magnitude of \(\mathrm{OQ}-4 \mathrm{OP}\).

If \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) are non-coplanar vectors and \(\lambda\) is a real number, then \(\left[\lambda(a+b) \quad \lambda^{2} b \quad \lambda c\right]=\left[\begin{array}{ll}a b+c b] \text { for } \\ \text { [AIEEE 2005] }\end{array}\right.\) (a) exactly two values of \(\lambda\) (b) exactly three values of \(\lambda\) (c) no value of \(\lambda\) (d) exactly one value of \(\lambda\)

Let \(\mathbf{u}=\hat{\mathbf{i}}+\hat{\mathbf{j}}, \mathbf{v}=\hat{\mathbf{i}}-\hat{\mathbf{j}}\) and \(\mathbf{w}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}\). If \(\mathbf{n}\) is a unit vector such that \(\mathbf{u} \cdot \mathbf{n}=0\) and \(\mathbf{v} \cdot \mathbf{n}=0\), then \(|\mathbf{w} \cdot \mathbf{n}|\) is equal to [AIEEE 2003] (a) 0 (b) 1 (c) 2 (d) 3