### QUESTION 1A
The following data was obtained in an experiment to verify some certain relationship by the IJMB students in table 1 according to the empirical formula:
**p = ne^λx** ............(1)
where *n* and *λ* are constants to be determined.
| x | 0.21 | 0.46 | 0.61 | 0.8 | 0.99 | 1.25 | 1.3 | 1.6 | 1.81 | 1.9 |
|---|------|------|------|-----|------|------|-----|-----|------|-----|
| P | 1.61 | 1.75 | 1.73 | 1.96 | 2.08 | 2.27 | 2.26 | 2.51 | 2.90 | 3.00 |
**(i)** Transform equation (1) into a suitable straight line graph to determine the constants *n* and *λ*.
**(ii)** Prepare a composite table to plot a suitable graph to determine the constants *n* and *λ*.
**(iii)** Use your graph to determine the values of *n* and *λ*.
**(iv)** Use your graph to estimate the value of P when x = 0.92.
**(v)** Substitute your values of *n* and *λ* obtained from your graph in (iii) above into equation to determine the value of p when x = 0.92.
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**1b.** Given that z = √(r − y³). Find the percentage error in z when r = 2.01 ± 0.071 and y = 0.77 ± 0.0031.
### QUESTION 2A
*(Refers to Figure 1 — a plane mirror optics diagram showing object positions and eye position)*
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## solutions
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### 1A(i) — Linearising the equation
Starting with:
> **p = ne^λx**
Take natural log of both sides:
> **ln p = ln n + λx**
This is in the form **Y = C + mX**, where:
- Y = ln p
- X = x
- Gradient m = **λ**
- Y-intercept C = **ln n**, so **n = e^C**
---
### 1A(ii) — Composite Table
Compute ln P for each value:
| x | P | ln P |
|------|------|-------|
| 0.21 | 1.61 | 0.476 |
| 0.46 | 1.75 | 0.559 |
| 0.61 | 1.73 | 0.548 |
| 0.80 | 1.96 | 0.673 |
| 0.99 | 2.08 | 0.732 |
| 1.25 | 2.27 | 0.820 |
| 1.30 | 2.26 | 0.815 |
| 1.60 | 2.51 | 0.920 |
| 1.81 | 2.90 | 1.065 |
| 1.90 | 3.00 | 1.099 |
Plot **ln P** (y-axis) against **x** (x-axis). The graph should be a straight line.
---
### 1A(iii) — Finding n and λ from the graph
Using two well-separated points on the best-fit line, e.g.:
- Point 1: (0.21, 0.476)
- Point 2: (1.90, 1.099)
**Gradient = λ:**
$$\lambda = \frac{1.099 - 0.476}{1.90 - 0.21} = \frac{0.623}{1.69} \approx \boxed{0.369 \text{ per unit x}}$$
**Y-intercept (ln n):**
Using Y = mX + C:
> 0.476 = 0.369(0.21) + C
> C = 0.476 − 0.077 = **0.399**
$$n = e^{0.399} \approx \boxed{1.49}$$
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### 1A(iv) — Estimate P when x = 0.92 (from graph)
Read off the graph at x = 0.92:
> ln P ≈ 0.399 + 0.369(0.92) ≈ 0.399 + 0.340 = **0.739**
$$P = e^{0.739} \approx \boxed{2.09}$$
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### 1A(v) — Substitute into equation
$$p = ne^{\lambda x} = 1.49 \times e^{0.369 \times 0.92}$$
$$= 1.49 \times e^{0.340} = 1.49 \times 1.404 \approx \boxed{2.09}$$
This confirms the graphical estimate.
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### 1B — Percentage Error in z
Given: **z = √(r − y³)**, r = 2.01 ± 0.071, y = 0.77 ± 0.0031
**Step 1: Find z**
$$y^3 = 0.77^3 = 0.4565$$
$$r - y^3 = 2.01 - 0.4565 = 1.5535$$
$$z = \sqrt{1.5535} \approx 1.2464$$
**Step 2: Find the error propagation**
Let u = r − y³, so z = u^(1/2)
$$\delta u = \delta r + 3y^2 \cdot \delta y$$
$$= 0.071 + 3(0.77)^2(0.0031)$$
$$= 0.071 + 3(0.5929)(0.0031)$$
$$= 0.071 + 0.00551 = 0.07651$$
Then for z = √u:
$$\frac{\delta z}{z} = \frac{1}{2} \cdot \frac{\delta u}{u} = \frac{1}{2} \times \frac{0.07651}{1.5535}$$
$$= \frac{1}{2} \times 0.04924 = 0.02462$$
**Percentage error in z:**
$$= 0.02462 \times 100 \approx \boxed{2.46\%}$$
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### Question 2A
The figure shows a **plane mirror** optics setup. Without the full sub-questions visible, the standard approach for such diagrams involves:
- Locating the image in the plane mirror (same distance behind mirror as object is in front)
- Drawing rays from object points P₃ and P₄ to the eye
- Verifying the law of reflection
If you can share the specific sub-questions for Q2A, I'll answer them in full detail.
## 2025 IJMB Physics IIIA Continued
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### QUESTION 2A (continued)
Use the diagram in figure 1 as a guide to carry out the following instructions.
**(i)** ABCD the outline of the rectangular and transparent block on the ray-trace sheet, such that AB is a plane mirror mounted. NN' is the normal FE the incident ray at an angle of incidence i = 20°.
**(ii)** Fix two optical pins P1 and P2 into line FE and place the block on the sheet of drawing paper. Then observe the images of P1 and P2 through side CD of the block so that the images of P1 and P2 appear to be in a straight line with the others. The positions of P3 and P4 are marked as shown.
**(iii)** Continue the line FE so that it crosses CD and extends as far as side AB. Draw a line joining the positions of P4 and P3. Continue the line so that it crosses CD and extends as far as side AB. Label the point G where this line crosses the line from P1 and P2.
**(iv)** Measure the angle θ between the lines meeting at G, repeats the procedure using an angle of incidence i = 25, 30, 35 and 40 respectively. In each case, records the value of θ.
**(v)** Prepare a suitable composite table. Plot a graph of **i against ½θ**. Find the slope S of the graph and evaluate **k = ½ − S**. State the necessary precaution taken to ensure accurate results. *(Attach your traces to your answer booklet)*
## QUESTION 3A
Connect up the circuit as shown in figure 3.1, where R is the Rheostat (R_H) and E = 12V.
Set R_H to the minimum and record the corresponding ammeter and voltmeter readings respectively.
By gently adjusting R_H, record 6 different values corresponding to V and A readings.
Respectively, in each case evaluate **I⁻¹ and V⁻¹**.
Plot a graph of **I⁻¹ against V⁻¹**. Determine the slope S of the graph.
If I and V are related according to the equation **I⁻¹ = (r+2)V⁻¹**, use your slope to find the value of r.
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## solutions
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### QUESTION 2A — Plane Mirror / Refraction Experiment
#### (i)–(iii): Procedure (Ray Tracing)
These are practical steps performed on a ray-trace sheet. The key physics:
- The incident ray FE hits the mirror surface AB at normal NN'
- The refracted/reflected ray exits and is traced through pins P3, P4
- Point G is where the emergent ray meets the extended incident ray
#### (iv): Recording θ at each angle of incidence
A sample results table would look like:
| i (°) | θ (°) | ½θ (°) |
|--------|--------|---------|
| 20 | θ₁ | ½θ₁ |
| 25 | θ₂ | ½θ₂ |
| 30 | θ₃ | ½θ₃ |
| 35 | θ₄ | ½θ₄ |
| 40 | θ₅ | ½θ₅ |
#### (v): Graph of i against ½θ
**Theory:** For a glass block with refractive index k, it can be shown that:
$$i = k \cdot \frac{\theta}{2} + C$$
So plotting **i** (y-axis) vs **½θ** (x-axis) gives a straight line with:
- **Slope S = k** (the refractive index)
- Then **k = ½ − S** gives the deviation from ideal
**Precautions:**
1. Ensure pins P1, P2, P3, P4 are vertical and sharp for accurate sighting
2. Use a sharp pencil for marking pin positions
3. Ensure the block outline ABCD is drawn accurately without shifting the block
4. View pins from a sufficient distance to minimize parallax error
5. Ensure the ray-trace sheet is fixed firmly and does not shift during the experiment
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### QUESTION 3A — Internal Resistance Experiment
#### Setting Up
The circuit has:
- EMF, **E = 12 V**
- Internal resistance **r** (unknown)
- External series resistance of **2 Ω** (from equation)
- Rheostat R_H as variable resistor
- Ammeter (A) and Voltmeter (V)
#### Sample Data Table
| V (volts) | I (amps) | I⁻¹ (A⁻¹) | V⁻¹ (V⁻¹) |
|-----------|----------|------------|------------|
| 10.0 | 0.91 | 1.099 | 0.100 |
| 9.0 | 0.82 | 1.220 | 0.111 |
| 8.0 | 0.73 | 1.370 | 0.125 |
| 7.0 | 0.64 | 1.563 | 0.143 |
| 6.0 | 0.55 | 1.818 | 0.167 |
| 5.0 | 0.45 | 2.222 | 0.200 |
#### Graph of I⁻¹ against V⁻¹
Plot **I⁻¹** (y-axis) vs **V⁻¹** (x-axis) → straight line through origin area.
#### Finding r from the Slope
Given the relationship:
$$I^{-1} = (r + 2)V^{-1}$$
Comparing with **Y = mX**:
$$\text{Slope } S = (r + 2)$$
Therefore:
$$\boxed{r = S - 2}$$
**Example:** If the slope S = 2.8 from the graph:
$$r = 2.8 - 2 = 0.8 \ \Omega$$
#### Key Theory Explanation
From Kirchhoff's voltage law:
$$E = I(R_{ext} + r + 2)$$
$$V = IR_{ext} \Rightarrow I = \frac{V}{R_{ext}}$$
Rearranging gives the linear form I⁻¹ = (r+2)V⁻¹, confirming the slope directly yields **(r + 2)**, so subtracting 2 isolates the internal resistance **r**.
#### Precautions for 3A:
1. Start with R_H at maximum to protect the ammeter
2. Take readings quickly to avoid heating effects changing resistance
3. Ensure all connections are tight to minimize contact resistance
4. Zero the ammeter and voltmeter before starting