1. Problem: An object thrown vertically upward has height given by $$h = -16t^2 + 96t$$. Find when the height is 144 feet. 2. Set equation: $$144 = -16t^2 + 96t$$. 3. Rearrange to standard quadratic form: $$-16t^2 + 96t - 144 = 0$$ Dividing all terms by -16: $$t^2 - 6t + 9 = 0$$ 4. Factor quadratic: $$t^2 - 6t + 9 = (t - 3)^2 = 0$$ 5. Solve for $$t$$: $$t - 3 = 0 \Rightarrow t = 3$$ seconds. Answer: (a) 3 sec --- 1. Problem: Total cost $$C = 0.5x^2 - 16x + 466$$, find $$x$$ if $$C=338$$. 2. Set equation: $$338 = 0.5x^2 - 16x + 466$$ 3. Rearranged: $$0.5x^2 - 16x + 466 - 338 = 0$$ $$0.5x^2 - 16x + 128 = 0$$ Multiply both sides by 2: $$x^2 - 32x + 256 = 0$$ 4. Factor: $$x^2 - 32x + 256 = (x - 16)^2 = 0$$ 5. Solve for $$x$$: $$x - 16 = 0 \Rightarrow x = 16$$ items. Answer: (b) 16 items --- 1. Problem: Invest 8000 in two enterprises paying 5.5% and 5%. Total income 425. Find amount at 5%. 2. Let $$x$$ be amount at 5%, then $$8000 - x$$ at 5.5%. 3. Equation: $$0.05x + 0.055(8000 - x) = 425$$ 4. Simplify: $$0.05x + 440 - 0.055x = 425$$ $$-0.005x + 440 = 425$$ 5. Solve: $$-0.005x = -15 \Rightarrow x = \frac{-15}{-0.005} = 3000$$ Answer: (b) 3000 --- 1. Problem: 20,000 invested at 6% and 7%. Total interest at 6.75%. Find amount at 7%. 2. Let $$x$$ be amount at 7%, so $$20,000 - x$$ at 6%. 3. Interest equation: $$0.06(20,000 - x) + 0.07x = 0.0675 \times 20,000$$ 4. Simplify: $$1200 - 0.06x + 0.07x = 1350$$ $$1200 + 0.01x = 1350$$ 5. Solve: $$0.01x = 150 \Rightarrow x = 15,000$$ Answer: (c) 15,000 --- 1. Problem: Company runs 3 lines with outputs $$x,y,z$$. Conditions: $$x + y + z = 45$$ Twice first equals sum of others: $$2x = y + z$$ Output second is 4 more than third: $$y = z + 4$$ 2. Write equations: $$\begin{cases} x + y + z = 45 \\ 2x - y - z = 0 \\ y - z = 4 \end{cases}$$ Answer: (d) --- 1. Problem: Solve $$x^2 + ax = bx + ab$$ for $$x$$. 2. Rearrange: $$x^2 + ax - bx - ab = 0$$ $$x^2 + (a - b)x - ab = 0$$ 3. Factor: $$ (x - b)(x + a) = 0$$ 4. Solutions: $$x = b \text{ or } x = -a$$ Answer: (c) b or -a --- 1. Problem: Blend 3 types coffee: Rs 65, 70, 75 per gram. Total 100g worth Rs 71/g. Uses equal amounts of 70 and 75. Find amount of 65 g coffee. 2. Let $$x$$ be grams of 65 coffee, and $$y$$ grams each of 70 and 75 coffees. 3. Equation for weight: $$x + 2y = 100$$ 4. Value equation: $$65x + 70y + 75y = 71 \times 100 = 7100$$ $$65x + 145y = 7100$$ 5. From weight equation: $$x = 100 - 2y$$ 6. Substitute in value equation: $$65(100 - 2y) + 145y = 7100$$ $$6500 - 130y + 145y = 7100$$ $$15y = 600 \Rightarrow y = 40$$ 7. Calculate $$x$$: $$x = 100 - 2 \times 40 = 20$$ grams. Answer: (b) 20 gram --- 1. Problem: Two investments with same % return. (3/10) of total + 600 invested in one venture. Return from this venture: 384. Total return: 1120. Find total amount invested, $$T$$. 2. Return rate: $$\text{rate} = \frac{384}{(3/10)T + 600} = \frac{1120}{T}$$ 3. Equate rates: $$\frac{384}{(3/10)T + 600} = \frac{1120}{T}$$ 4. Cross multiply: $$384 T = 1120 ((3/10)T + 600)$$ 5. Simplify: $$384T = 1120 \times \frac{3T}{10} + 1120 \times 600$$ $$384T = 336T + 672000$$ 6. Bring terms together: $$384T - 336T = 672000$$ $$48T = 672000$$ 7. Solve for $$T$$: $$T = \frac{672000}{48} = 14000$$ Answer: Total invested amount is 14000.