If $a + b + c = 1$ and $a < b < c , a , b , c \in R$ and $a ^ { \mathbf { 2 } } , 2 b ^ { \mathbf { 2 } } , c ^ { \mathbf { 2 } }$ are in G.P. and $a , b , c$ are in A.P. then find the value of $9 \left( a ^ { 2 } + b ^ { 2 } + c ^ { 2 } \right) =$ ?
Let $\alpha , \beta$ be the roots of quadratic equation $12 \mathrm { x } ^ { 2 } - 20 \mathrm { x } + 3 \lambda = 0$, $\lambda \in \mathbf { z }$. If $1 / 2 \leq | \beta - \alpha | \leq 3 / 2$ then the sum of all possible valued of $\lambda$ is $\_\_\_\_$ -
Let $\mathrm { f } ( \mathrm { x } ) = \min \left\{ \sqrt { 2 } \mathrm { x } , \mathrm { x } ^ { 2 } \right\}$ and $\mathrm { g } ( \mathrm { x } ) = | x | \left[ x ^ { 2 } \right ]$ If $x \in ( - 2,2 )$ then sum of all values of $f ( x )$ at those $x$ values where $g ( x )$ is non-differentiable ([.] denotes GIF). (A) $2 - \sqrt { 3 }$ (B) $1 / - \sqrt { 3 }$ (C) [answer] (D) $2 - \sqrt { 2 }$
$x - n y + z = 6$ $\mathbf { x } - ( \mathbf { n } - \mathbf { 2 } ) \mathbf { y } + ( \mathbf { n } + \mathbf { 1 } ) \mathbf { z } = \mathbf { 8 }$ $( \mathrm { n } - 1 ) \mathrm { y } + \mathrm { z } = 1$ Let $\mathbf { n } \boldsymbol { = }$ number on the dies when rolled randomly then $\mathbf { P }$ (that system equation has unique solution) $= \left( \frac { \mathrm { k } } { 6 } \right)$ then sum of value of k and all possible value of n is (A) 22 (B) 24 (C) 20 (D) 21