Step 1: Evaluate the numerator. $$ 8-2+12 \times 9-4 \times 6 $$ Perform multiplications first: $$ 8-2+108-24 $$ Perform additions and subtractions from left to right: $$ 6+108-24 $$ $$ 114-24 $$ $$ 90 $$ The numerator is $90$. Step 2: Evaluate the denominator. $$ 56 \div 7(2) $$ Perform division first: $$ 8(2) $$ Perform multiplication: $$ 16 $$ The denominator is $16$. Step 3: Divide the numerator by the denominator. $$ \frac{90}{16} $$ Simplify the fraction by dividing both numerator and denominator by their greatest common divisor, which is 2: $$ \frac{90 \div 2}{16 \div 2} = \frac{45}{8} $$ The fraction can also be expressed as a mixed number: $$ 5\frac{5}{8} $$ The question asks to evaluate, so the simplified fraction is acceptable. The final answer is $\boxed{\frac{45}{8}}$. Step 1: Find the greatest common divisor (GCD) of the dimensions of the land. The dimensions are $840$ m and $396$ m. The side length of the largest possible square plot will be the GCD of these two numbers. Using prime factorization: $$ 840 = 2^3 \times 3 \times 5 \times 7 $$ $$ 396 = 2^2 \times 3^2 \times 11 $$ The GCD is the product of the lowest powers of common prime factors: $$ \text{GCD}(840, 396) = 2^2 \times 3 = 4 \times 3 = 12 $$ So, the side length of each square plot is $12$ m. Step 2: Calculate the number of plots along each dimension. Number of plots along the $840$ m side: $$ \frac{840 \text{ m}}{12 \text{ m/plot}} = 70 \text{ plots} $$ Number of plots along the $396$ m side: $$ \frac{396 \text{ m}}{12 \text{ m/plot}} = 33 \text{ plots} $$ Step 3: Calculate the total number of plots. Total number of plots = (plots along length) $\times$ (plots along width) $$ 70 \times 33 = 2310 $$ The final answer is $\boxed{2310}$. Step 1: Identify the relevant formula for the circumradius. The distance OS is the radius ($R$) of the circumcircle. According to the sine rule, for any triangle with sides $a, b, c$ and opposite angles $A, B, C$, the following relationship holds: $$ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R $$ In triangle RSQ, we are given side $QR = 12$ cm and its opposite angle $\angle RSQ = 50^\circ$. Step 2: Substitute the given values into the sine rule to find $R$. $$ \frac{QR}{\sin(\angle RSQ)} = 2R $$ $$ \frac{12 \text{ cm}}{\sin(50^\circ)} = 2R $$ Solve for $R$: $$ R = \frac{12 \text{ cm}}{2 \sin(50^\circ)} $$ $$ R = \frac{6 \text{ cm}}{\sin(50^\circ)} $$ Step 3: Calculate the value of $R$ and round to 2 decimal places. Using a calculator for $\sin(50^\circ)$: $$ \sin(50^\circ) \approx 0.76604444 $$ Now, calculate $R$: $$ R = \frac{6}{0.76604444} \approx 7.832409 \text{ cm} $$ Rounding to 2 decimal places: $$ R \approx 7.83 \text{ cm} $$ The distance OS is $R$. The final answer is $\boxed{7.83 \text{ cm}}$.