(A) rewrites as $\exp[\log\theta+(\theta-1)\log(1-x)]$ on a $\theta$ -free support $(0,1)$ : a valid one-parameter exponential family with natural statistic $T(x)=\log(1-x)$ .

(B), the Bernoulli law, is the textbook exponential-family example: $\theta^x(1-\theta)^{1-x}=(1-\theta)\exp\left[x\log\frac{\theta} {1-\theta}\right]$ .

(D): expand $-\dfrac{(x-\theta)^2}{2\theta}=-\dfrac{x^2}{2\theta}+x -\dfrac\theta2$ ; the coefficient of $x$ is the constant $1$ , not $\theta$ -dependent, so it simply folds into $h(x)=e^x$ , leaving a genuine one-parameter family $f(x,\theta)=h(x)\exp[\eta(\theta)T(x) -A(\theta)]$ with $\eta(\theta)=-1/(2\theta)$ , $T(x)=x^2$ .

(C) is the classic counterexample: here both the mean and the variance are tied to the same single parameter $\theta$ , and expanding $-\dfrac{(x-\theta)^2}{2\theta^2}=-\dfrac{x^2}{2\theta^2} +\dfrac x\theta-\dfrac12$ produces two $\theta$ -dependent coefficients (on $x^2$ and on $x$ ) driven by one scalar parameter. The natural parameters $(\eta_1,\eta_2)=(-1/2\theta^2, 1/\theta)$ trace out a one-dimensional curve inside the two-dimensional natural parameter space rather than filling it, which is exactly the definition of a curved exponential family — not a (full/regular) exponential family.