Question

Using properties of determinants, prove that : $\begin{vmatrix}x²+1 & xy & xz \\ xy & y²+1 & yz \\ xz & yz & z²+1\end{vmatrix}$

Assume that each born child is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls ? Given that : (i) the youngest is a girl. (ii) atleast one is a girl.

Solve the differential equation given as : $\left(\frac{e^{-2\sqrt{x}}}{\sqrt{x}}-\frac{y}{\sqrt{x}}\right)\frac{d x}{d y}\;=1,x\neq0$

If A = $\left(\begin{array}{l l l }1 & -2 & 3 \\ 0 & -1 & 4 \\ -2 & 2 & 1\end{array}\right)$, find $\left(A^{\prime}\right)^{-1}.$

Show that a function f : R → R given by f(x) =ax + b,a, b ∈ R, a ≠ 0 is a bijective.

If the function f : R → R is given by f(x) = $\frac{x+3}{3}$ and g : R → R is given by g(x) = 2x – 3, then find (i) fog (ii) gof Is f⁻¹ = g?

A and B throw a pair of dice alternately. A wins the game if he gets a total of 9 and B wins if he gets a total of 7. If A starts the game, find the probability of winning the game by B.