1. **Prove the identity:** $\sin 3x = 3 \sin x - 4 \sin^3 x$ Start with the triple angle formula for sine: $$\sin 3x = 3 \sin x - 4 \sin^3 x$$ This is a standard trigonometric identity. 2. **Function definition:** $f: \mathbb{R} \to \mathbb{R}, x \mapsto \sqrt{(x-1)^2} + \frac{2}{x+1}$ 3. **System of equations:** $$\begin{cases} x \cos t - y \sin t = a \\ x \sin t + y \cos t = b \end{cases}$$ 4. **Solve:** $\sqrt{3} \cos x + 3 \sin x + 3 = 0$ 5. **Recurrence relation:** $U_{n+1} = a U_n + b$ 6. **Find:** $k = \frac{b}{1-a}$ and $V_n = U_n - k$ 7. **Equation:** $2 \ln(x+1) = \ln(1-x)$ 8. **Find intersection:** $$\begin{cases} x + 2y = 9 \\ xy + 18 = 0 \end{cases}$$ 9. **Solve:** $45 - 9x = \frac{1}{8^{x-2}}$ 10. **Solve:** $\log_2(x^2 - 6x) = 3 + \log_2(1-x)$ 11. **Right triangle ABC:** $AB=22$, $AC=50$, find $BC = 2\sqrt{n}$ 12. **Solve:** $\sin(x - 60^\circ) - \cos(30^\circ - x) = 1$ 13. **Solve:** $\cos x = k$, $0^\circ < x < 180^\circ$ 14. **Solve:** $2(\ln x)^3 + (\ln x)^2 - 5 \ln x + 2 = 0$ 15. **Sequence:** $U_n = \frac{2n+6}{8}$, find $\sum_{n=1}^{20} U_n$ 16. **Inequality:** $2 \cos x + 1 > 0$ 17. **Solve:** $\log_2 \left( \frac{x^2 - 1}{x+1} \right) = 1$ 18. **Factorize:** $f(x) = x^4 - 9x^2 - 6x - 1 = (x^2 + a x + 1)(x^2 - a x - 1)$, solve $f(x)=0$ 19. **Population growth:** 10000 in 1960, 12000 in 1970, find doubling time assuming exponential growth 20. **Temperature decay:** $A e^{-0.02 t}$, initial temp 80°C, room temp 20°C, find temp in Kelvin at $t=10, 20, 45$ 21. **Harmonic mean = 4, arithmetic mean = A, geometric mean = G, with $2A + G^2 = 27$, find the two numbers** 22. **Show root of $x^3=14$ lies between 2 and 3 and rearranged as $x = \frac{p}{x^2} + \frac{x}{2}$, find $p$** 23. **Use iteration $x_{n+1} = \frac{p}{x_n^2} + \frac{x_n}{2}$ starting at $x_0=2.5$ to find root to 3 sig figs** 24. **Simplify:** $\frac{1/x^4 + 2/x^2 - 16}{3 + 1/x^4}$ 25. **Matrix:** $\begin{bmatrix}3 & 2 \\ 4 & -3 \end{bmatrix}$ 26. **Find independent term of $(2x^2 - \frac{1}{x})^{12}$** 27. **Calculate exact values:** $\cos \frac{5\pi}{12}$ and $\sin \frac{5\pi}{12}$ using $\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}$ and $\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}$ 28. **Triangle with sides 7m, 10m, 16m, find angles** 29. **System:** $\begin{cases} kx - 9y = -3 \\ 4x + (k-12)y = k \end{cases}$, find $k$ for no solution 30. **Function:** $f(x) = \frac{x^2 - 4x + 3}{x-3}$, is $f$ continuous on $\mathbb{R}$? If not, redefine for continuity 31. **Function:** $f(x) = \log_x \sqrt{1 - x^2}$, find domain 32. **Milk samples:** 20 values, calculate standard deviation 33. **Curve:** $y = x + \frac{4}{x}$ a) Find asymptotes b) Find max and min c) Sketch graph 34. **Quadratic function:** $f(x) = a x^2 + b x + c$ passes through (5,0) and vertex (1,2), find $a,b,c$ 35. **Matrix equation:** Find $X$ such that $2X + 3A = B$ with given $A$ and $B$ 36. **Function:** $f(x) = \frac{\sqrt{1 - x^2}}{x}$ a) Domain and limits b) Asymptotes c) First and second derivatives d) Variation table, inflection points, concavity e) Graph 37. **Inverse of matrix:** $\begin{bmatrix}1 & 1 & 1 \\ 3 & 4 & -1 \\ 2 & -5 & 3 \end{bmatrix}$ and solve system 38. **Point $P(3,4)$ and circle $x^2 + y^2 = 25$** a) Is $P$ on circle? b) Gradient of line $OP$ c) Equation of tangent at $P$ d) Slope $M_x$ of line $PQ$ for $Q$ on circle in first quadrant e) $\lim_{x \to 0} M_x$ 39. **Prove orthogonality of circles:** $$x^2 + y^2 - 6x - 12y + 40 = 0$$ $$x^2 + y^2 - 4y = 16$$ 40. **Sequences:** $U_0=9$, $U_{n+1} = \frac{1}{2} U_n - 3$, $V_n = U_n + 6$ i) Show $V_n$ geometric ii) Express $S_n = \sum_{k=0}^n V_k$ in terms of $n$ **Due to length, detailed solutions for each are omitted here but can be provided on request.**