Determine which of the following binary operations are associative and which are commutative: '*' on N defined by a * b = 1 for all a, b .

Clearly, by defination a * b = 1 = b * a, a, b Also, (a * b) * c = (1 * c) = 1 and a * (b * c) = (a * 1) = 1 a, b, c Hence, N is both associative and commutative.

Determine which of the following binary operations are associativ…

Compute $\left[\begin{array}{ll} {a^{2}+b^{2}} & {b^{2}+c^{2}} \\ {a^{2}+c^{2}} & {a^{2}+b^{2}} \end{array}\right]+\left[\begin{array}{cc} {2 a b} & {2 b c} \\ {-2 a c} & {-2 a b} \end{array}\right]$

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Write the adjoint of the matrix $\text{A}=\begin{bmatrix} -3 & 4 \\ 7 & -2 \end{bmatrix}$.

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Differentiate the functions with respect to x.
$\sin(\text{ax + b})$

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Find the general solution of the differential equation $\frac{d y}{d x}+y=1~(y \neq 1)$

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Let * be a binary operation on N given by a * b = HCF (a, b), a, b ∈ N. Write the value of 22 * 4.

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A stone is dropped into a quiet lake and waves move in circles at the speed of 5 cm/s. At the instant when the radius of the circular wave is 8 cm, how fast is the enclosed area increasing?

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Find the derivative of the function given by f(x) = sin (x²).

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Find the value of $2\sec^{-1}2+\sin^{-1}\Big(\frac{1}{2}\Big)$

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A random variable has the following probability distribution:

X = X_i	1	2	3	4
P(X = X_i)	k	2k	3k	4k

Write the value of $\text{P}(\text{X}\geq3).$

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Verify that the function $y = \sqrt {{a^2} - {x^2}} , $ $x\in \left( { - a,a} \right)$ (explicit or implicit) is a solution of differential equation $x + y\frac{{dy}}{{dx}} = 0\left( {y \ne 0} \right)$

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