MCQ
If $A =$ $\left[ {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right]$ (where $bc \ne 0$) satisfies the equations $x^2 + k = 0$, then
- A$a + d = 0$
- B$k = -|A|$
- C$k = |A|$
- ✓both $(A)$ and $(C)$
As $A$ satisfies, $x^2 + k = 0, A^2 + kI = O$
==>$\left[ {\begin{array}{*{20}{c}}{{a^2} + bc + k}&{(a + d)b}\\{(a + d)c}&{bc + {d^2} + k}\end{array}} \right]$
==>$a^2 + bc + k = 0 = bc + d^2 + k = 0$ and $(a + d)b = (a + d) c = 0$
As $bc \ne 0, b \ne 0, c \ne 0$ ==> $a + d = 0$ ==> $a = -d$
Also, $k = -(a^2 + bc)$ $= -(d^2 + bc)$ $= - ( (-ad) + bc ) = |A|$
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$f_1(x)=\int_0^x \prod_{j=1}^{21}( t - j )^{ j } dt , x >0$
and
$f_2(x)=98(x-1)^{50}-600(x-1)^{39}+2450, x>0,$
where, for any positive integer $n$ and real numbers $a _1, a _2, \ldots, a _{ n }, \prod_{i=1}^{ n } a _i$ denotes the product of $a _1, a _2, \ldots, a _{ n }$. Let $m _i$ and $n _i$, respectively, denote the number of points of local minima and the number of points of local maxima of function $f_i, i=1,2$, in the interval $(0, \infty)$
($2$) The value of $2 m_1+3 n_1+m_1 n_1$ is. . . . . .
($2$) The value of $6 m _2+4 n _2+8 m _2 n _2$ is. . . . . .
Give the answer or quetion ($1$) and ($2$)