Using cofactors of elements of third column, evaluate =

=a_ 13 A_ 13 +a_ 23 A_ 23 +a_ 33 A_ 33 =y z(z-y)+z x(x-z)+x y(y-x) =y z^2-y^2 z+z x^2-z^2 x+x y^2-x^2 y =z x^2-x^2 y+x y^2-z^2 x+y z^2-y^2 z =x^2(z-y)+x (y^2-z^2 )+y z(z-y) =(z-y) [x^2-x(z+y)+y z ] =(z-y) [x^2-x z-x y+y z ] =(z-y)[x(x-y)-z(x-y)] =(z-y)[(x-y)(x-z)] =(z-y)(x-y)(x-z)

Find the interval for which $f(x)=x^4-2 x^2$ is increasing or decreasing.

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If $\text{x}\begin{bmatrix}2\\3 \end{bmatrix}+\text{y}\begin{bmatrix}-1\\1 \end{bmatrix}=\begin{bmatrix}10\\5 \end{bmatrix},$ find the value of x.

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If in a binomial distribution n = 4 and P(X = 0) $=\frac{16}{81},$ find q.

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Evaluate the following integrals:
$\int^\limits4_{-4}|\text{x}+2|\text{dx}$

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The cartesian equation of a line AB are $\frac{2\text{x}-1}{\sqrt{3}}=\frac{\text{y}+2}{2}=\frac{\text{z}-3}{3}.$ Find the direction cosines of a line parallel to AB.

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Find $\mathrm{f}^{-1}$ if it exists: $\mathrm{f}: \mathrm{A} \rightarrow \mathrm{B}$, where, $\mathrm{A}=\{1,3,5,7,9\} ; B=\{0,1,9,25,49,81\}$ and $f(x)=x^2$.

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Evaluate the following integrals:$\int\text{e}^{\text{}x}\Big(\frac{1+\sin\text{x}}{1+\cos\text{x}}\Big)\text{dx}$

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Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem.
$f(x) = 2x^2 - 3x + 1 on [1, 3]$

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For each of the differential equations in find a particular solution satisfying the given condition:
$\text{cos}\bigg(\frac{\text{dy}}{\text{dx}}\bigg) = \text{a}(\text{a} \in \text{R}); \ \text{y}=1\text{when x = }0$

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Verify A (adj. A) = (adj. A) A= |A|I:
$\left[ {\begin{array}{*{20}{c}} 2&3 \\ { - 4}&{ - 6} \end{array}} \right]$

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