1. Problem: Find the total number of students in the radiology class given the numbers who understand X-ray, USG, both, and neither. Formula: Use the principle of inclusion-exclusion for sets. Let $A$ be students understanding X-ray, $B$ be students understanding USG. Total students $= |A| + |B| - |A \cap B| +$ students who understand neither. Given: $|A|=25$, $|B|=18$, $|A \cap B|=10$, neither $=5$. Calculate: $$25 + 18 - 10 + 5 = 38$$ So, total students $= 38$. 2. Problem: Find the roots of the quadratic equation $x^2 + 7x + 10 = 0$. Formula: Quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a=1$, $b=7$, $c=10$. Calculate discriminant: $$\Delta = 7^2 - 4 \times 1 \times 10 = 49 - 40 = 9$$ Roots: $$x = \frac{-7 \pm \sqrt{9}}{2} = \frac{-7 \pm 3}{2}$$ So, $$x_1 = \frac{-7 + 3}{2} = \frac{-4}{2} = -2$$ $$x_2 = \frac{-7 - 3}{2} = \frac{-10}{2} = -5$$ 3. Problem: Given roots $x_1$ and $x_2$ of $x^2 + 6x + 5 = 0$, find $x_1^2 + x_2^2$. Formula: Use $x_1 + x_2 = -\frac{b}{a}$ and $x_1 x_2 = \frac{c}{a}$. Here, $a=1$, $b=6$, $c=5$. Calculate: $$x_1 + x_2 = -6$$ $$x_1 x_2 = 5$$ Recall: $$x_1^2 + x_2^2 = (x_1 + x_2)^2 - 2x_1 x_2$$ Calculate: $$(-6)^2 - 2 \times 5 = 36 - 10 = 26$$ 4. Problem: Find the derivative $f'(x)$ of $f(x) = 4x^4 + 3x^3 + 6x + 7$. Formula: Power rule $\frac{d}{dx} x^n = n x^{n-1}$. Calculate: $$f'(x) = 4 \times 4x^{3} + 3 \times 3x^{2} + 6 \times 1 + 0 = 16x^3 + 9x^2 + 6$$ 5. Problem: Find the total surface area of a cylindrical infectious waste bin with diameter 10 cm and height 25 cm. Formula: Surface area $= 2\pi r^2 + 2\pi r h$ where $r$ is radius, $h$ is height. Given diameter $= 10$ cm, so radius $r = 5$ cm, height $h = 25$ cm, and $\pi = 3.14$. Calculate: $$2 \times 3.14 \times 5^2 + 2 \times 3.14 \times 5 \times 25 = 2 \times 3.14 \times 25 + 2 \times 3.14 \times 125 = 157 + 785 = 942$$ So, total surface area $= 942$ cm$^2$. Final answers: 1. Total students = 38 2. Roots: $x_1 = -2$, $x_2 = -5$ 3. $x_1^2 + x_2^2 = 26$ 4. $f'(x) = 16x^3 + 9x^2 + 6$ 5. Surface area = 942 cm$^2$