[10×2] A binary operation defined on is given by . Find the identity element. Without expanding at…

A binary operation $*$ defined on $\mathbb{Q} - \{1\}$ is given by $a * b = a + b - ab$. Find the identity element.

Without expanding at any stage, find the value of the determinant: $\Delta = \begin{vmatrix} 2 & x & y+z \\ 2 & y & z+x \\ 2 & z & x+y \end{vmatrix}$

Solve: $\sin^{-1}(\cos(\sin^{-1} x)) = \frac{\pi}{3}$

Find the value of $k$ if $M = \begin{pmatrix} 1 & 2 \\ 2 & 3 \end{pmatrix}$ and $M^2 - kM - I_2 = 0$.

Evaluate: $\int_{0}^{\frac{\pi}{2}} \frac{\sin^{\frac{3}{2}} x}{\sin^{\frac{3}{2}} x + \cos^{\frac{3}{2}} x} \, dx$

Find $\frac{dy}{dx}$, if $x = at^2$ and $y = 2at$.

Find the differential equation of the family of curves $y = Ae^x + Be^{-x}$, where $A$ and $B$ are arbitrary constants.

Find the intervals in which the function $f(x)$ is strictly increasing where, $f(x) = 10 - 6x - 2x^2$.

A family has two children. What is the probability that both children are boys, given that at least one of them is a boy?

Given that the events $A$ and $B$ are such that $P(A) = \frac{1}{2}$, $P(A \cup B) = \frac{3}{5}$ and $P(B) = k$. Find $k$ if: (a) $A$ and $B$ are mutually exclusive. (b) $A$ and $B$ are independent.