1. **Stating the problem:** We are given a frequency table of marks obtained by students and asked to represent this data on a histogram. 2. **Understanding a histogram:** A histogram is a graphical representation of data where the x-axis represents class intervals (marks ranges) and the y-axis represents frequencies (number of students). 3. **Data from the table:** - Marks intervals: $8 \leq X < 9$, $9 \leq X < 11$, $11 \leq X < 13$, $13 \leq X < 16$, $16 \leq X < 20$, $20 \leq X < 21$ - Frequencies: 2, 6, 8, 3, 2, 1 respectively 4. **Plotting the histogram:** - On the x-axis, mark the intervals as given. - On the y-axis, mark the number of students. - For each interval, draw a bar whose height corresponds to the frequency. 5. **Important notes:** - The width of each bar corresponds to the class interval width. - Bars are adjacent with no gaps since data is continuous. --- **Next problem:** 17. (a) Calculate the value of $n$ given the sum of interior angles of two regular polygons with sides $n-1$ and $n$ are in ratio 2:3. (b) Calculate the interior angle of each polygon. --- **Step 1: Formula for sum of interior angles of a polygon** $$\text{Sum of interior angles} = (s - 2) \times 180^\circ$$ where $s$ is the number of sides. **Step 2: Express sums for polygons with sides $n-1$ and $n$** $$S_1 = (n-1 - 2) \times 180 = (n-3) \times 180$$ $$S_2 = (n - 2) \times 180$$ **Step 3: Given ratio** $$\frac{S_1}{S_2} = \frac{2}{3}$$ Substitute: $$\frac{(n-3) \times 180}{(n-2) \times 180} = \frac{2}{3}$$ Simplify: $$\frac{n-3}{n-2} = \frac{2}{3}$$ **Step 4: Solve for $n$** Cross multiply: $$3(n-3) = 2(n-2)$$ $$3n - 9 = 2n - 4$$ $$3n - 2n = -4 + 9$$ $$n = 5$$ **Step 5: Calculate interior angles of each polygon** - Polygon with $n-1 = 4$ sides (square): $$\text{Interior angle} = \frac{(4-2) \times 180}{4} = \frac{2 \times 180}{4} = 90^\circ$$ - Polygon with $n = 5$ sides (pentagon): $$\text{Interior angle} = \frac{(5-2) \times 180}{5} = \frac{3 \times 180}{5} = 108^\circ$$ --- **Final answers:** - (a) $n = 5$ - (b) Interior angles are $90^\circ$ for the polygon with $n-1=4$ sides and $108^\circ$ for the polygon with $n=5$ sides.