The number of distinct real values of $$\lambda$$, for which the vectors $$ - {\lambda ^2}\widehat i + \widehat j + \widehat k,\widehat i - {\lambda ^2}\widehat j + \widehat k$$ and $$\widehat i + \widehat j - {\lambda ^2}\widehat k$$ are coplanar, is :
Let the vector $$\overrightarrow {PQ} ,\overrightarrow {QR} ,\overrightarrow {RS} ,\overrightarrow {ST} ,\overrightarrow {TU} $$ and $$\overrightarrow {UP} $$, represent the sides of a regular hexagon.
Statement 1 : $$\overrightarrow {PQ} \times \left( {\overrightarrow {RS} + \overrightarrow {ST} } \right) \ne \overrightarrow 0 $$
Statement 2 : $$\overrightarrow {PQ} \times \overrightarrow {RS} = \overrightarrow 0 $$ and $$\overrightarrow {PQ} \times \overrightarrow {ST} \ne \overrightarrow 0 $$
| (i) | Two rays in the first quadrant $x+y=|a|$ and $a x-y=1$ Intersects each other in the interval $a \in\left(a_0, \infty\right)$, the value of $a_0$ is | (A) | 2 |
|---|---|---|---|
| (ii) | Point $(\alpha, \beta, \gamma)$ lies on the plane $x+y+z=2$. Let $\vec{a}=\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}, \hat{k} \times(\hat{k} \times \vec{a})=0$, then $\gamma=$ |
(B) | 4/3 |
| (iii) | $$ \left|\int_0^1\left(1-y^2\right) d y\right|+\left|\int_1^0\left(y^2-1\right) d y\right| $$ |
(C) | $$ \left|\int_0^1 \sqrt{1-x} d x\right|+\left|\int_1^0 \sqrt{1+x} d x\right| $$ |
| (iv) | If $\sin A \sin B \sin C+\cos A \cos B=1$, then the value of $\sin C=$ | (D) | 1 |
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