1. Solve the equations and systems: **a)** Solve $2x + 5 = 15 - 2x$: - Add $2x$ to both sides: $2x + 2x + 5 = 15$ - Simplify: $4x + 5 = 15$ - Subtract 5: $4x = 10$ - Divide by 4: $x = \frac{10}{4} = 2.5$ **b)** Solve $2x = x^2 - 15$: - Rearrange: $x^2 - 2x - 15 = 0$ - Factor: $(x - 5)(x + 3) = 0$ - Solutions: $x = 5$ or $x = -3$ **c)** Solve system: $$\begin{cases} 2x + y = 10 \\ 3x - 2y = 1 \end{cases}$$ - From first: $y = 10 - 2x$ - Substitute into second: $3x - 2(10 - 2x) = 1$ - Simplify: $3x - 20 + 4x = 1 \Rightarrow 7x = 21 \Rightarrow x = 3$ - Find $y$: $y = 10 - 2(3) = 4$ **d)** Solve system: $$\begin{cases} x + y = 3 \\ x^2 - 2y = 2 \end{cases}$$ - From first: $y = 3 - x$ - Substitute into second: $x^2 - 2(3 - x) = 2$ - Simplify: $x^2 - 6 + 2x = 2 \Rightarrow x^2 + 2x - 8 = 0$ - Factor: $(x + 4)(x - 2) = 0$ - Solutions: $x = -4$ or $x = 2$ - Find $y$: if $x = -4$, $y = 3 - (-4) = 7$; if $x = 2$, $y = 3 - 2 = 1$ 2. Analyze $f(x) = x^3 - 2x + 1$ on domain $[-2, 2]$: - Find derivative: $f'(x) = 3x^2 - 2$ - Set $f'(x) = 0$: $3x^2 = 2 \Rightarrow x = \pm \sqrt{\frac{2}{3}}$ - Evaluate $f$ at critical points and endpoints: - $f(-2) = (-2)^3 - 2(-2) + 1 = -8 + 4 + 1 = -3$ - $f(-\sqrt{2/3}) \approx -0.38$ - $f(\sqrt{2/3}) \approx 2.05$ - $f(2) = 8 - 4 + 1 = 5$ - Range approx $[-3, 5]$ - Intercepts: - $y$-intercept: $f(0) = 1$ - $x$-intercepts solve $x^3 - 2x + 1 = 0$ (approx $x \approx -1.62, 0.62, 1$) - Increasing where $f'(x) > 0$: $|x| > \sqrt{2/3}$ - Decreasing where $f'(x) < 0$: $|x| < \sqrt{2/3}$ 3. Number of different orders for 4 aces: - Number of permutations of 4 distinct cards: $4! = 24$ 4. Probability product of two dice rolls is less than 10: - Total outcomes: $6 \times 6 = 36$ - Count pairs $(a,b)$ with $a \times b < 10$: - For $a=1$: all $b=1..6$ (6) - $a=2$: $b=1..4$ (4) - $a=3$: $b=1..3$ (3) - $a=4$: $b=1..2$ (2) - $a=5$: $b=1$ (1) - $a=6$: $b=1$ (1) - Total favorable: $6+4+3+2+1+1=17$ - Probability: $\frac{17}{36} \approx 0.472$ 5. Probability of successful experiments (success probability $p=0.7$, trials $n=8$): **a)** All 8 successful: - $P = p^8 = 0.7^8 \approx 0.0576$ **b)** Exactly 3 successful: - Use binomial formula: $P = \binom{8}{3} p^3 (1-p)^{5}$ - Calculate: $\binom{8}{3} = 56$ - $P = 56 \times 0.7^3 \times 0.3^5 \approx 56 \times 0.343 \times 0.00243 = 0.0467$ **c)** More than 3 successful: - $P = 1 - P(\leq 3)$ - Calculate $P(\leq 3) = \sum_{k=0}^3 \binom{8}{k} p^k (1-p)^{8-k}$ - Sum terms for $k=0,1,2,3$ and subtract from 1 (approx $0.952$) 6. Profit questions (data not provided, so general approach): **a)** Percentage of Alpha's profit from Banana: - Calculate $\frac{\text{Banana profit for Alpha}}{\text{Total Alpha profit}} \times 100$% **b)** Product adding most to Gamma's profit: - Identify product with highest profit value for Gamma **c)** Product adding most to all companies together: - Sum profits per product across companies, find max 7. Heights data: 177, 160, 165, 158, 180, 192, 188, 180 - Mean: $\frac{177+160+165+158+180+192+188+180}{8} = \frac{1400}{8} = 175$ - Median: Sort data: 158,160,165,177,180,180,188,192 - Median = average of 4th and 5th: $\frac{177+180}{2} = 178.5$ - Mode: 180 (appears twice) - Range: $192 - 158 = 34$ - Variance: Calculate squared differences from mean, average them - Standard deviation: square root of variance These steps cover all requested problems with detailed explanations and calculations.