1. The problem involves multiple expressions and questions related to algebra, calculus, and geometry. 2. For the ellipse equation $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$, this represents an ellipse where $a$ and $b$ are the semi-major and semi-minor axes respectively. 3. The second derivative expression $$\frac{d^2y}{dx^2} + \frac{dy}{dx}$$ is a differential calculus expression involving derivatives of $y$ with respect to $x$. 4. The derivative of $$\sin(\ln x)$$ is found using the chain rule: $$\frac{d}{dx} \sin(\ln x) = \cos(\ln x) \cdot \frac{d}{dx}(\ln x) = \cos(\ln x) \cdot \frac{1}{x} = \frac{\cos(\ln x)}{x}$$ 5. The hyperbolic functions expressions: $$\frac{e^x - e^{-x}}{2} = \sinh x$$ $$\frac{e^x + e^{-x}}{2} = \cosh x$$ $$\frac{e^x e^{-x}}{e^x + e^{-x}} = \frac{1}{e^x + e^{-x}}$$ 6. Vector $\mathbf{v} = -\mathbf{i} + \mathbf{j}$ has magnitude: $$|\mathbf{v}| = \sqrt{(-1)^2 + 1^2} = \sqrt{1 + 1} = \sqrt{2}$$ 7. Length of latus rectum of parabola $$y^2 = 4ax$$ is known to be $$4a$$. 8. The general second degree equation $$ax^2 + 2hxy + by^2 = 0$$ represents a pair of straight lines. 9. Integral $$\int \cot \alpha \, d\gamma$$ is an indefinite integral with respect to $\gamma$ treating $\alpha$ as constant, so the integral is: $$\int \cot \alpha \, d\gamma = \cot \alpha \cdot \gamma + C$$ 10. Function value at $$f(\pi/2)$$ depends on the function definition, which is not provided. 11. Limit: $$\lim_{x \to a} \frac{x^3 - a^3}{x - a}$$ Using factorization: $$x^3 - a^3 = (x - a)(x^2 + ax + a^2)$$ So limit becomes: $$\lim_{x \to a} (x^2 + ax + a^2) = a^2 + a \cdot a + a^2 = 3a^2$$ 12. Square root expression $$\sqrt{x}$$ is the principal square root of $x$. 13. Points $$-\frac{1}{\sqrt{a}}, -2\sqrt{a}, 2\sqrt{a}$$ are given, possibly roots or critical points. 14. Integral: $$\int \frac{\ln x}{x} \, dx$$ Use substitution $t = \ln x$, then $dt = \frac{1}{x} dx$, so integral becomes: $$\int t \, dt = \frac{t^2}{2} + C = \frac{(\ln x)^2}{2} + C$$ 15. Expression $$\frac{x}{a} + \frac{y}{b}$$ is a linear form. 16. Slope-intercept form of a line is: $$y = mx + c$$ where $m$ is slope and $c$ is y-intercept. 17. Function: $$f(x) = \frac{2 + 3x}{2x} = \frac{2}{2x} + \frac{3x}{2x} = \frac{1}{x} + \frac{3}{2}$$ 18. Expression $$\frac{1}{6} [u v w]$$ likely refers to volume of a parallelepiped formed by vectors $u,v,w$. 19. Point $(2,1)$ is a solution to an inequality (not specified). 20. Derivative: $$\frac{d}{dx} \left[ \frac{1}{g(x)} \right] = -\frac{g'(x)}{(g(x))^2}$$ 21. Distance of point $(\cos 3x, \sin 3x)$ from origin is: $$\sqrt{(\cos 3x)^2 + (\sin 3x)^2} = \sqrt{1} = 1$$ Answer: (D) 1 --- Final answers: - Derivative of $\sin(\ln x)$ is $$\frac{\cos(\ln x)}{x}$$ - Length of latus rectum of parabola $y^2 = 4ax$ is $$4a$$ - Limit $$\lim_{x \to a} \frac{x^3 - a^3}{x - a} = 3a^2$$ - Integral $$\int \frac{\ln x}{x} dx = \frac{(\ln x)^2}{2} + C$$ - Distance of point $(\cos 3x, \sin 3x)$ from origin is 1