If a = i -2 j +3 k , and b =2 i +3 j -5 k , then find a b . verify that a and a b are perpendicular to each other.

Given: a = i -2 j +3 k b =2 i +3 j -5 k a b = i & j & k 1&-2&3 2&3&-5 = i +11 j +7 k Now, a . ( a b )=1-22+21 =0 Thus, a is perpendicular to a b .

If a = i -2 j +3 k , and b =2 i +3 j -5 k , then find a b . verif…

If $\text{A}=\begin{bmatrix}\cos\alpha&\sin\alpha\\-\sin\alpha&\cos\alpha\end{bmatrix},$ and $\text{A}^{-1}=\text{A}',$ find value of $\alpha.$

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What inference can you draw if $\vec{\text{a}}\times\vec{\text{b}}=\vec{0}$ and $\vec{\text{a}}.\vec{\text{b}}=0.$

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Find the distance of the point $(2, 3, -5)$ from the plane $x + 2y - 2z - 9 = 0$.

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Let * be a binary operation on Z defined by a * b = a + b - 4 for all a, b ∈ Z.
Show that '*' is both commutative and associative.

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Differentiate the following functions with respect to x:
$3^{\text{x}\log\text{x}}$

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$\int\sin^3(2\text{x}+1)\text{dx}$

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Evaluate the following:
$\int\limits^1_0\frac{\text{x}}{\sqrt{1+\text{x}^2}}\text{dx}$

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Find fog and gof if:$f(x) = e^x, g(x) = log_ex$

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Solve the following differential equation
$\frac{\text{dy}}{\text{dx}}=\text{x}^5+\text{x}^2-\frac{2}{\text{x}},\text{x}\ne0$

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Evaluate the following intregals:
$\int\frac{1}{\cos2\text{x}+3\sin^2\text{x}}\ \text{dx}$

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