Evaluate the following: [2 i j k ]+ [ i k j ]+ [ k j 2 i ]

If $f : R → R$ is defined by $f(x) = x^2$, write $f^{-1}(25)$.

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Write the value of the determinant $\begin{vmatrix}2&3&4\\2\text{x}&3\text{x}&4\text{x}\\5&6&8\end{vmatrix}$

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Show that points:

A (a, b + c), B (b, c + a), C (c, a + b) are collinear.

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For the following differntial equations verify that the accompanying function is a solution:

Differential equation	Function
$\text{x}\frac{\text{dy}}{\text{dx}}=\text{y}$	$\text{y}=\text{ax}$

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If $|\vec{\text{a}}|=10,\big|\vec{\text{b}}\big|=2$ and $\big|\vec{\text{a}}\times\vec{\text{b}}\big|=16,$ find $\vec{\text{a}}.\vec{\text{b}}.$

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Differentiate the following w.r.t.x: $\frac{\text{e}^\text{x}}{\sin\text{x}}$

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If $\vec{\text{a}}.\vec{\text{a}}=0$ and $\vec{\text{a}}.\vec{\text{b}}=0,$ what can you conclude about the vector $\vec{\text{b}}$?

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If the area of a triangle is 18 sq units and vertices are $(x, 7) ;(2,2)$ and $(10,8)$, then find the value of $x$.

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Show that $\text{f}(\text{x})=\text{x}+\cos\text{x}-\text{a}$ is an increasing function on R for all values of a.

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Integrate the rational function $\frac{1}{\left(e^{x}-1\right)}$ [Hint: $e^x = t$]

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Answer