1. **Problem:** Determine the number of zeroes of a polynomial graphically. The zeroes of a polynomial are the values of $x$ for which the polynomial equals zero. Graphically, these correspond to the points where the graph intersects the $x$-axis. **Answer:** (b) intersects x-axis 2. **Problem:** Find two successive integral multiples of 5 whose product is 300. Let the two numbers be $5n$ and $5(n+1)$. Their product is: $$5n \times 5(n+1) = 25n(n+1) = 300$$ Simplify: $$n(n+1) = \frac{300}{25} = 12$$ Solve the quadratic: $$n^2 + n - 12 = 0$$ Factor: $$(n+4)(n-3) = 0$$ So, $n = 3$ or $n = -4$. Taking positive $n=3$, the numbers are: $$5 \times 3 = 15, \quad 5 \times 4 = 20$$ **Answer:** (d) 15, 20 3. **Problem:** Find the ratio $a:b$ for the parallelogram with vertices $(-3,-1)$, $(a,b)$, $(3,3)$, and $(4,3)$. In a parallelogram, the diagonals bisect each other. Midpoint of diagonal 1: $$M_1 = \left(\frac{-3 + 3}{2}, \frac{-1 + 3}{2}\right) = (0, 1)$$ Midpoint of diagonal 2: $$M_2 = \left(\frac{a + 4}{2}, \frac{b + 3}{2}\right)$$ Since midpoints are equal: $$\frac{a + 4}{2} = 0 \Rightarrow a + 4 = 0 \Rightarrow a = -4$$ $$\frac{b + 3}{2} = 1 \Rightarrow b + 3 = 2 \Rightarrow b = -1$$ Ratio: $$a : b = -4 : -1 = 4 : 1$$ **Answer:** (a) 4 : 1 4. **Problem:** A man on a ship 10 m above water observes the angle of elevation of the hill top as $45^\circ$ and angle of depression of the hill base as $30^\circ$. Find the distance to the hill and its height. Let the distance from the ship to the hill base be $d$. From angle of depression $30^\circ$: $$\tan 30^\circ = \frac{10}{d} \Rightarrow d = \frac{10}{\tan 30^\circ} = \frac{10}{\frac{1}{\sqrt{3}}} = 10\sqrt{3} \approx 17.32$$ From angle of elevation $45^\circ$: Let height of hill be $h$. $$\tan 45^\circ = \frac{h - 10}{d} = 1 \Rightarrow h - 10 = d \Rightarrow h = d + 10 = 17.32 + 10 = 27.32$$ **Answer:** (a) 17.32 m, 27.3 m 5. **Problem:** Find the area of the circle inscribed in a square of side 6 cm. The inscribed circle touches all sides, so its diameter equals the side of the square. Radius $r = \frac{6}{2} = 3$ cm. Area: $$\pi r^2 = \pi \times 3^2 = 9\pi$$ **Answer:** (c) 9\pi cm² 6. **Problem:** Mean of $x - 5y$, $x - 3y$, $x - y$, $x + y$, $x + 3y$, $x + 5y$ is 22. Find $x$. Sum: $$6x + (-5y - 3y - y + y + 3y + 5y) = 6x + 0 = 6x$$ Mean: $$\frac{6x}{6} = x = 22$$ **Answer:** (b) 22 7. **Problem:** Probability that a leap year has 53 Fridays or 53 Saturdays. A leap year has 366 days = 52 weeks + 2 days. The 2 extra days can be: - Sunday & Monday - Monday & Tuesday - Tuesday & Wednesday - Wednesday & Thursday - Thursday & Friday - Friday & Saturday - Saturday & Sunday For 53 Fridays, the extra days must include Friday. For 53 Saturdays, the extra days must include Saturday. Possible pairs with Friday or Saturday: - Thursday & Friday - Friday & Saturday - Saturday & Sunday Number of favorable pairs = 3 Total possible pairs = 7 Probability: $$\frac{3}{7}$$ **Answer:** (d) 3/7