Molarity $$(\mathrm{M})$$ of an aqueous solution containing $$x \mathrm{~g}$$ of anhyd. $$\mathrm{CuSO}_4$$ in $$500 \mathrm{~mL}$$ solution at $$32^{\circ} \mathrm{C}$$ is $$2 \times 10^{-1} \mathrm{M}$$. Its molality will be _________ $$\times 10^{-3} \mathrm{~m}$$. (nearest integer). [Given density of the solution $$=1.25 \mathrm{~g} / \mathrm{mL}$$]

If the domain of the function $$f(x)=\sin ^{-1}\left(\frac{x-1}{2 x+3}\right)$$ is $$\mathbf{R}-(\alpha, \beta)$$, then $$12 \alpha \beta$$ is equal to :

Let three vectors ,$$\overrightarrow{\mathrm{a}}=\alpha \hat{i}+4 \hat{j}+2 \hat{k}, \overrightarrow{\mathrm{b}}=5 \hat{i}+3 \hat{j}+4 \hat{k}, \overrightarrow{\mathrm{c}}=x \hat{i}+y \hat{j}+z \hat{k}$$ form a triangle such that $$\vec{c}=\vec{a}-\vec{b}$$ and the area of the triangle is $$5 \sqrt{6}$$. If $$\alpha$$ is a positive real number, then $$|\vec{c}|^2$$ is equal to:

Let $$f(x)=a x^3+b x^2+c x+41$$ be such that $$f(1)=40, f^{\prime}(1)=2$$ and $$f^{\prime \prime}(1)=4$$. Then $$a^2+b^2+c^2$$ is equal to: