Let $$\vec{a}=3 \hat{i}+\hat{j}-\hat{k}$$ and $$\vec{c}=2 \hat{i}-3 \hat{j}+3 \hat{k}$$. If $$\vec{b}$$ is a vector such that $$\vec{a}=\vec{b} \times \vec{c}$$ and $$|\vec{b}|^{2}=50$$, then $$|72-| \vec{b}+\left.\vec{c}\right|^{2} \mid$$ is equal to __________.
Let $$\alpha$$ be the constant term in the binomial expansion of $$\left(\sqrt{x}-\frac{6}{x^{\frac{3}{2}}}\right)^{n}, n \leq 15$$. If the sum of the coefficients of the remaining terms in the expansion is 649 and the coefficient of $$x^{-n}$$ is $$\lambda \alpha$$, then $$\lambda$$ is equal to _____________.
Let the image of the point $$\left(\frac{5}{3}, \frac{5}{3}, \frac{8}{3}\right)$$ in the plane $$x-2 y+z-2=0$$ be P. If the distance of the point $$Q(6,-2, \alpha), \alpha > 0$$, from $$\mathrm{P}$$ is 13 , then $$\alpha$$ is equal to ___________.
Match List - I with List - II
List - I (Layer of atmosphere) |
List - II (Approximate height over earth's surface) |
||
---|---|---|---|
(A) | $$\mathrm{F_1}$$ - Layer | (I) | 10 km |
(B) | $$\mathrm{D}$$ - Layer | (II) | 170 - 190 km |
(C) | Troposphere | (III) | 100 km |
(D) | $$\mathrm{E}$$ - Layer | (IV) | 65 - 75 km |
Choose the correct answer from the options given below: