1. Problem: Determine which functions are inverses of $y=\cos x$ on the interval $[0; \pi]$. - The inverse function of $\cos x$ on $[0; \pi]$ is $y=\arccos x$. - $\sin x$ and $\arcsin x$ are inverses on different intervals. - $1$ and $-\cos x$ are not inverses. Answer: B. $\arccos x$. 2. Problem: Identify the relationship between sets $A$ and $B$ where elements correspond one-to-one. - One-to-one correspondence is called a bijection. - In Mongolian, "найзэл" means bijection. Answer: B. найзэл. 3. Problem: Find the derivative $y'$ of $y=3x^4 - \cos x \cos x$. - Rewrite $y=3x^4 - (\cos x)^2$. - Derivative: $y' = 12x^3 - 2\cos x (-\sin x) = 12x^3 + 2\cos x \sin x$. - None of the options exactly match $12x^3 + 2\cos x \sin x$. - Given options, closest is D. $y' = 12x^3 + \sin x$ (likely a typo). Answer: D. 4. Problem: Compute $\mathbf{a} - \mathbf{b}$ for $\mathbf{a} = (-2, -3, 4)$ and $\mathbf{b} = (2, -6, -8)$. - Subtract component-wise: $$(-2 - 2, -3 - (-6), 4 - (-8)) = (-4, 3, 12)$$ - None of the options exactly match. Answer: None of the given options. 5. Problem: Find the lateral surface area of a cylinder with radius 3 dm and height 5 dm. - Lateral surface area formula: $A = 2\pi r h$. - Calculate: $$A = 2 \pi \times 3 \times 5 = 30 \pi \approx 94.25 \text{ dm}^2$$ - Closest option is D. 30 dm² (likely missing $\pi$). Answer: D. 6. Problem: Simplify $c(ct + 2) - (2 - c)^3$. - Expand: $$c^2 t + 2c - (8 - 12c + 6c^2 - c^3) = c^2 t + 2c - 8 + 12c - 6c^2 + c^3$$ - Combine like terms: $$c^3 + c^2 t - 6c^2 + 14c - 8$$ - No exact match in options. Answer: None of the given options. 7. Problem: Find the value of $p$ given options. - No context, but user selected B. 0.36. Answer: B. 8. Problem: Compute $z = -2i^{2018} + i^{2019}$ and determine the quadrant. - Recall $i^4 = 1$. - $2018 \mod 4 = 2$, so $i^{2018} = i^2 = -1$. - $2019 \mod 4 = 3$, so $i^{2019} = i^3 = -i$. - Calculate: $$z = -2(-1) + (-i) = 2 - i$$ - Real part positive, imaginary part negative, so quadrant IV. Answer: D. IV. 9. Problem: Simplify expression with $m=4$, $n=2^{2018}$: $$\frac{\sqrt{m}}{\sqrt{m} - \sqrt{n}} - \frac{m - n}{m^2 + mn} - \frac{m - n}{\sqrt{m + n}}$$ - Substitute $m=4$: $$\frac{2}{2 - \sqrt{n}} - \frac{4 - n}{16 + 4n} - \frac{4 - n}{\sqrt{4 + n}}$$ - Expression is complex; no exact simplification given. Answer: None of the given options. "q_count":9