1. Problem (a): Calculate the price of each item in 2024. Since the prices for 2024 are not provided explicitly, we can assume the question refers to finding prices or using some reference to previous data. Without numerical data here, this step cannot be completed as stated. 2. Problem (b): Taking Y as the base item, calculate the price indices for 2023. The price index of an item is calculated relative to the base item Y, usually by \[\text{Price Index} = \frac{\text{Price of item}}{\text{Price of base item (Y)}} \times 100\]. 3. Problem (c): Using weights 3, 5, and 2 for items X, Y, and Z respectively, calculate: (i) Composite price index for the items in 2024 \[\text{Composite Index} = \frac{3P_X + 5P_Y + 2P_Z}{3Q_X + 5Q_Y + 2Q_Z} \times 100\] Here, $P$ denotes price in 2024, $Q$ denotes base year price or weighted reference price. Usually, weights correspond to quantities. (ii) Weighted aggregate price index of items in 2024 \[\text{Weighted Aggregate Price Index} = \frac{\sum (\text{weight}_i \times \text{price index}_i)}{\sum \text{weights}}\] 4. Problem 26 (a): Use trapezium rule with 7 ordinates to estimate \(\int_0^1 x e^{-x^2} dx\) to 3 significant figures. The trapezium rule formula with $n$ subintervals: $$\int_a^b f(x) dx \approx \frac{h}{2} \left(f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n)\right)$$ Given 7 ordinates means 6 intervals: $n=6$, $h=\frac{1-0}{6} = \frac{1}{6} \approx 0.1667$. Calculate $x_i = a + ih$ for $i=0,...,6$. Compute $f(x_i) = x_i e^{-x_i^2}$ for each $x_i$. Apply formula to find estimate. 5. Problem 26 (b): Find percentage error of estimation, correct to 2 decimal places. Suggest how error may be reduced. Percentage Error = \[\frac{|\text{True Value} - \text{Estimate}|}{\text{True Value}} \times 100\%\] Error reduction: use more ordinates (increase $n$) or use Simpson's rule. 6. Problem 27 (a): Find range within which exact value of \(Z = \frac{1}{x} + \frac{1}{y} + xy\) lies, given $x=4.165 \pm 0.001$, $y=6.72 \pm 0.01$. Use error bounds: Calculate maximum and minimum values of each term using bounds of $x$ and $y$. Approximate using linearization by differentials or substituting extremes. 7. Problem 27 (b): Use trapezium rule with 11 ordinates to approximate \(\int_1^2 x \log_{10} x dx\) to 4 d.p. Calculate $h=\frac{2-1}{10} = 0.1$. Use $x_i=1+0.1 i$, compute $f(x_i) = x_i \log_{10} x_i$. Use trapezium formula with $n=10$ intervals. 8. Problem 27 (c): Find exact value of \(\int_1^2 x \log_{10} x dx\). Change logarithm base: $$\log_{10} x = \frac{\ln x}{\ln 10}$$ Integral becomes: $$\frac{1}{\ln 10} \int_1^2 x \ln x dx$$ Integrate by parts, then calculate exact value. Calculate error = exact - estimate. Error can be reduced by increasing the number of ordinates or using Simpson's rule. 9. Problem 28 (i): Construct table of $f(x) = 3 x e^x - 1$ in range $0.1 \leq x \leq 1.1$ with intervals 0.2. Use linear extrapolation to estimate $f(1.13)$: $$f(1.13) \approx f(1.1) + \frac{f(1.1) - f(0.9)}{1.1 - 0.9} (1.13 - 1.1)$$ 10. Problem 28 (ii): Find root of $f(x)$ correct to 3 d.p using Newton-Raphson. Newton-Raphson formula: $$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$$ Calculate $f'(x) = \frac{d}{dx} (3 x e^x - 1) = 3 e^x + 3 x e^x = 3 e^x (1 + x)$. Start with an initial guess and iterate until desired accuracy. 11. Problem 29 (a): Show iterative formula for solving $4x - \sec^2 x = 0$ is $$x_{n+1} = \frac{1}{4} \sec^{2} x_n$$ Start with $x_0 = 0.2$, iterate to find solution to 4 significant figures. 12. Problem 29 (b): Given errors $\delta x, \delta y, \delta z$ in $x, y, z$, show maximum relative error in $\frac{xy^2}{z}$ is $$\left|\frac{\delta x}{x}\right| + 2 \left|\frac{\delta y}{y}\right| + \left|\frac{\delta z}{z}\right|$$ Assuming errors are small and independent, use differential error propagation: $$\frac{\delta w}{w} = \left|\frac{\partial w}{\partial x} \frac{\delta x}{w} \right| + \left|\frac{\partial w}{\partial y} \frac{\delta y}{w} \right| + \left|\frac{\partial w}{\partial z} \frac{\delta z}{w} \right|$$ where $w=\frac{xy^2}{z}$. 13. Problem 30: Values of tan $\Theta$ from tables are given. No explicit question given here; no further action. Final remarks: Since full data for prices and numerical values are not provided for all problems, detailed numerical answers are given for the calculus/numerical analysis problems where possible and explanation for the error formula and Newton-Raphson method steps.