A ball is chosen randomly from Box-I ; call this ball $b$. If $b$ is red then a ball is chosen randomly from Box-$II$, if $b$ is blue then a ball is chosen randomly from Box-$III$, and if $b$ is green then a ball is chosen randomly from Box-$IV$. The conditional probability of the event 'one of the chosen balls is white' given that the event 'at least one of the chosen balls is green' has happened, is equal to

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Choose the correct answer from the given four options.Which of the following is the general solution of $\frac{\text{d}^2\text{y}}{\text{d}\text{x}^2}-2\frac{\text{d}\text{y}}{\text{d}\text{x}}+\text{y}=0?$

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Area of the region bounded by rays |x| + y = 1 and X - axis is ___________.

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If ${\Delta _1} = \left| {\begin{array}{*{20}{c}}
{{b^5}{c^6}\left( {{c^3} - {b^3}} \right)}&{{a^4}{c^6}\left( {{a^3} - {c^3}} \right)}&{{a^4}{b^5}\left( {{b^3} - {a^3}} \right)} \\
{{b^2}{c^3}\left( {{b^6} - {c^6}} \right)}&{a{c^3}\left( {{c^6} - {a^6}} \right)}&{a{b^2}\left( {{a^6} - {b^6}} \right)} \\
{{b^2}{c^3}\left( {{c^3} - {b^3}} \right)}&{a{c^3}\left( {{a^3} - {c^3}} \right)}&{a{b^2}\left( {{b^3} - {a^3}} \right)}
\end{array}} \right|$ and ${\Delta _2} = \left| {\begin{array}{*{20}{c}}
a&{{b^2}}&{{c^3}} \\
{{a^4}}&{{b^5}}&{{c^6}} \\
{{a^7}}&{{b^8}}&{{c^9}}
\end{array}} \right|$ then ${\Delta _1}{\Delta _2}$ is equal to

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If $A = \left[ {\begin{array}{*{20}{c}}
1&1\\
1&1
\end{array}} \right]$ and $\det ({A^n} - I) = 1 - {\lambda ^n}\,,\,n \in N$ then $\lambda $ is

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If $\int \frac{d x}{\left(x^{2}+x+1\right)^{2}}=a \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)+b\left(\frac{2 x+1}{x^{2}+x+1}\right)+C$ $x>0$ where $C$ is the constant of integration, then the value of $9(\sqrt{3} \mathrm{a}+\mathrm{b})$ is equal to ... .

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Choose the correct answer from the given four options : Area of the region in the first quadrant enclosed by the $x-$ axis, the line $y = x $ and the circle $x^2 + y^2 = 32$ is: