Here is a breakdown of the mathematical problems from your uploaded image:

1. Q.9 (a): Fit a second-degree parabola to the given data points.

   • Given data:
     • $x = 0, 1, 2, 3, 4$
     • $y = 1, 5, 10, 22, 38$
   • The goal is to find a parabolic equation $y = ax^2 + bx + c$ that best fits this data using the method of least squares.

2. Q.9 (b): Using the method of least squares, fit a curve of the form $y = a + bx^2$.

   • Given data:
     • $x = -1, 0, 1, 2$
     • $y = 2, 5, 3, 8$
   • The goal is to determine values of $a$ and $b$ that fit this model using the least squares method.

3. Q.10 (a): Fit a curve of the form $y = ab^x$ to the data provided.

   • Given data:
     • $x = 2, 3, 4, 5$
     • $y = 144, 172.8, 207.4, 248.8, 298.5$
   • For this exponential model, logarithmic transformation can be applied to linearize the relationship, then apply least squares to determine $a$ and $b$.

4. Q.10 (b): Fit a straight line using the method of least squares to the provided data points.

   • Given data:
     • $x = 1, 2, 3, 4, 5, 6, 7, 8, 9$
     • $y = 9, 8, 10, 12, 11, 13, 14, 16, 15$
   • The goal is to fit a linear model $y = mx + c$ using least squares.

5. Q.8 (b) (at the top): Find the percentage of students who scored more than 60 marks given a normal distribution with a mean of 60 and a standard deviation of 5.

Would you like step-by-step solutions for any specific part of this question set?

Here are five questions for deeper exploration:

1. How does the least squares method work for parabolic fits as compared to linear fits?
2. What transformations are typically used to fit non-linear curves like $y = ab^x$ to data?
3. Can the least squares method be applied directly to exponential functions without transformation?
4. What are the assumptions of normal distribution when calculating probabilities for data scores?
5. How do we calculate and interpret residuals in the least squares method?

Tip: In least squares fitting, the goal is to minimize the sum of squared residuals, which represents the discrepancy between the observed data and the fitted model.