1. **Problem 38:** Find which equation is equivalent to $y - 34 = x(x - 12)$. Start by expanding the right side: $$ y - 34 = x^2 - 12x $$ Add 34 to both sides: $$ y = x^2 - 12x + 34 $$ Now check each option: (a) $y = (x - 17)(x + 2) = x^2 + 2x - 17x - 34 = x^2 - 15x - 34$ (not equal) (b) $y = (x - 6)^2 + 2 = (x^2 - 12x + 36) + 2 = x^2 -12x + 38$ (not equal) (c) $y = (x - 6)^2 - 2 = (x^2 - 12x + 36) - 2 = x^2 - 12x + 34$ (matches) (d) $y = (x - 17)(x - 2) = x^2 - 2x - 17x + 34 = x^2 - 19x + 34$ (not equal) Answer: (c) 2. **Problem 39:** Identify which pair of equations could not solve $$\begin{cases} 4x + 2y = 22 \\ -2x + 2y = -8 \end{cases}$$ Check if each pair is equivalent or can be transformed into the original. (a) Changes second equation sign: $2x - 2y=8$ is not equivalent to $-2x + 2y = -8$ (b) Second equation multiplied by 2: $-4x + 4y = -16$ from $-2x + 2y = -8$ (valid) (c) Both equations multiplied by 3: $12x + 6y = 66$ and $6x - 6y = 24$ (valid) (d) First doubled: $8x + 4y = 44$ and double second with sign flipped: $-8x +8y = -16$ vs given $-8x + 8y = -8$ (not matching) Answer: (a) and (d) pairs do not preserve equivalency, but question asks which could not be used. (a) is incorrect because second equation is reversed sign and value. So answer: (a) 3. **Problem 40:** Translate 'sixty more than 9 times a number is 375' This means $9h + 60 = 375$ Answer: (a) 4. **Problem 41:** Solve $\frac{3}{5}(x + 2) = x - 4$ Multiply both sides by 5: $$3(x + 2) = 5x - 20$$ Expand: $$3x + 6 = 5x - 20$$ Bring terms together: $$6 + 20 = 5x - 3x$$ $$26 = 2x$$ $$x = 13$$ Answer: (b) 5. **Problem 42:** Solve $x^2 - 6x = 0$ Factor: $$x(x - 6) = 0$$ Solutions: $$x = 0$$ or $$x = 6$$ Answer: (c) 6. **Problem 43:** Three brothers ages are consecutive even integers. Let youngest be $x$. Oldest is $x + 4$ (since even consecutives, step 2). Condition: $3x = x + 4 + 48$ Solve: $$3x = x + 52$$ $$2x = 52$$ $$x = 26$$ Answer: (d) 7. **Problem 44:** Two numbers sum to 47 and difference is 15. Let larger = $x$, smaller = $y$ $$x + y = 47$$ $$x - y = 15$$ Add equations: $$2x = 62 ightarrow x = 31$$ Answer: (c) 8. **Problem 45:** Expression undefined if denominator zero: $$2n - 1 = 0 ightarrow n = \frac{1}{2}$$ Answer: (d) 9. **Problem 46:** Define $C$ = Chartered class students. Professional = $C + 60$ Graduate = $2C - 50$ Fundamental = $3C$ Total: $$C + (C + 60) + (2C - 50) + 3C = 1424$$ Sum: $$C + C + 60 + 2C - 50 + 3C = 1424$$ $$7C + 10 = 1424$$ $$7C = 1414$$ $$C = 202$$ Answer: (b) 10. **Problem 47:** Solve $f(x) = h(x)$: $$\frac{1}{2}x + 3 = |x|$$ Consider cases: Case 1: $x \geq 0$ $$\frac{1}{2}x + 3 = x$$ $$3 = x - \frac{1}{2}x = \frac{1}{2}x$$ $$x = 6$$ (not in options) Case 2: $x < 0$ $$\frac{1}{2}x + 3 = -x$$ $$\frac{1}{2}x + x = -3$$ $$\frac{3}{2}x = -3$$ $$x = -2$$ (in options) Answer: (a) Final answers: 38: (c) 39: (a) 40: (a) 41: (b) 42: (c) 43: (d) 44: (c) 45: (d) 46: (b) 47: (a)