Q1. Use dimensional analysis to determine the dimensions of resistance (R) given the formula for power dissipated by a resistor: .

Background

Topic: Dimensional Analysis & Algebraic Manipulation

This question tests your ability to use dimensional analysis to derive the physical dimensions of resistance based on the formula relating power, voltage, and resistance.

Key Terms and Formulas:

• = Power (dimensions: )

• = Voltage (dimensions: )

• = Resistance (unknown dimensions)

• Dimensional analysis: Equate the dimensions on both sides of the equation to solve for .

Step-by-Step Guidance

1. Write the formula for power: .

2. Express the dimensions of each variable: , .

3. Calculate the dimensions of : .

4. Set up the dimensional equation: .

5. Rearrange to solve for : .

Try solving on your own before revealing the answer!

Q2. Use dimensional analysis to show that the proposed formula for the period of vibration of a guitar string is incorrect: .

Background

Topic: Dimensional Analysis

This question tests your ability to check the dimensional consistency of a formula involving mass, length, tension, and period.

Key Terms and Formulas:

• = Mass ($M$)

• = Length ($L$)

• = Tension ()

• = Period ()

• Dimensional analysis: Compare the dimensions on both sides of the equation.

Step-by-Step Guidance

1. Write the formula: (ignore as it is dimensionless).

2. Express the dimensions: , .

3. Set up the dimensional equation: .

4. Simplify the dimensions: .

5. Compare with the expected dimension for period (): Does match $T$?

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Q3. Calculate the sum of 15 voltage samples (3, 6, 9, ..., 45) using the arithmetic progression formula.

Background

Topic: Arithmetic Progression

This question tests your ability to use the formula for the sum of an arithmetic sequence.

Key Terms and Formulas:

• First term (): 3

• Common difference (): 3

• Number of terms (): 15

• Sum formula:

Step-by-Step Guidance

1. Identify the sequence: .

2. Determine , , .

3. Write the sum formula: .

4. Substitute the values: .

5. Calculate and .

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Q4. Find the 11th term in the geometric progression: 512, 1024, 2048, 4096, ...

Background

Topic: Geometric Progression

This question tests your ability to use the formula for the nth term of a geometric sequence.

Key Terms and Formulas:

• First term (): 512

• Common ratio (): 2

• nth term formula:

Step-by-Step Guidance

1. Identify the sequence: .

2. Determine , , .

3. Write the nth term formula: .

4. Substitute the values: .

5. Calculate the exponent: .

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Q5. Given , with V after 4 seconds, V, MΩ, find the approximate value for capacitance .

Background

Topic: Exponential Functions & Algebraic Manipulation

This question tests your ability to rearrange and solve an exponential equation for capacitance in an RC circuit.

Key Terms and Formulas:

• = Voltage across capacitor

• = Supply voltage

• = Resistance

• = Capacitance

• Formula:

Step-by-Step Guidance

1. Write the formula: .

2. Substitute known values: .

3. Divide both sides by 21: .

4. Rearrange to isolate the exponential term: .

5. Take the natural logarithm of both sides to solve for .

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Q6. Make time () the subject of the formula and determine the first point in time when V, given MHz.

Background

Topic: Trigonometric Functions & Algebraic Manipulation

This question tests your ability to rearrange a sine function and solve for time, given a specific voltage and frequency.

Key Terms and Formulas:

• = Instantaneous signal voltage

• = Frequency

• Formula:

• Inverse sine function () to solve for .

Step-by-Step Guidance

1. Write the formula: .

2. Divide both sides by 8: .

3. Take the inverse sine: .

4. Add to both sides: .

5. Solve for : .

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Q7. For the curve , calculate: (i) when ; (ii) when .

Background

Topic: Hyperbolic Functions & Algebraic Manipulation

This question tests your ability to evaluate hyperbolic functions and solve for variables in engineering contexts.

Key Terms and Formulas:

• = Hyperbolic cosine function

• Formula:

• Inverse hyperbolic cosine () to solve for .

Step-by-Step Guidance

1. For (i): Substitute into the formula: .

2. Calculate .

3. Evaluate using the definition: .

4. Multiply by 80 to find .

5. For (ii): Set and solve for using the inverse hyperbolic cosine.

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Q8. Use dimensional analysis to find a formula for the period () of a pendulum involving mass (), length (), and gravity ().

Background

Topic: Dimensional Analysis & Algebraic Manipulation

This question tests your ability to derive a formula for the period of a pendulum using dimensional analysis.

Key Terms and Formulas:

• = Mass ()

• = Length ()

• = Gravity ()

• = Period ()

Step-by-Step Guidance

1. Assume a general formula: .

2. Write the dimensions: , , , .

3. Set up the dimensional equation: .

4. Expand: .

5. Equate powers of , , and to solve for , , .

Try solving on your own before revealing the answer!

Q9. Convert the voltage into a complex number.

Background

Topic: Complex Numbers & Polar to Rectangular Conversion

This question tests your ability to convert a phasor (polar form) into its rectangular (complex) form.

Key Terms and Formulas:

• Polar form:

• Rectangular form:

• = Imaginary unit

Step-by-Step Guidance

1. Write the conversion formula: .

2. Calculate and .

3. Multiply: and .

4. Write the result in the form .

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Q10. Find the inverse of the matrix .

Background

Topic: Matrix Algebra

This question tests your ability to find the inverse of a 2x2 matrix.

Key Terms and Formulas:

• Matrix

• Inverse formula:

Step-by-Step Guidance

1. Identify , , , .

2. Calculate the determinant: .

3. Write the adjugate matrix: .

4. Divide each element by the determinant to get the inverse.

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Q11. Calculate the mean, mode, median, and standard deviation for the transmit power samples: 22.3, 24.1, 22.8, 22.3, 24.9, 22.3, 21.5, 23.9, 22.3, 21.5.

Background

Topic: Statistics – Measures of Central Tendency and Dispersion

This question tests your ability to calculate mean, mode, median, and standard deviation for a data set.

Key Terms and Formulas:

• Mean:

• Mode: Most frequent value

• Median: Middle value when sorted

• Standard deviation:

Step-by-Step Guidance

1. List the data: 22.3, 24.1, 22.8, 22.3, 24.9, 22.3, 21.5, 23.9, 22.3, 21.5.

2. Calculate the mean: Add all values and divide by 10.

3. Find the mode: Identify the value that appears most frequently.

4. Sort the data and find the median: Middle value or average of two middle values.

5. Calculate the standard deviation using the formula.

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Q12. Determine the probability that (i) two of seven bolts exceed the tolerable diameter, (ii) more than two exceed the tolerable diameter, given 96% are within tolerance.

Background

Topic: Binomial Probability

This question tests your ability to apply the binomial probability formula to calculate probabilities for discrete events.

Key Terms and Formulas:

• Probability of exceeding tolerance (): 0.04

• Probability of within tolerance (): 0.96

• Binomial formula:

Step-by-Step Guidance

1. For (i): Use , , , .

2. Calculate .

3. For (ii): Calculate .

4. Set up the calculations for each probability.

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Q13. For 500 capacitors with mean 200μF and standard deviation 12μF, normally distributed, determine how many are likely to have values between 185μF and 215μF.

Background

Topic: Normal Distribution & Z-Score

This question tests your ability to use z-scores and the normal distribution to estimate the number of items within a range.

Key Terms and Formulas:

• Mean (): 200μF

• Standard deviation (): 12μF

• Z-score:

• Use z-table to find probabilities.

Step-by-Step Guidance

1. Calculate z-scores for 185μF and 215μF: , .

2. Find the probability for each z-score using the z-table.

3. Calculate the total probability between the two z-scores.

4. Multiply the probability by 500 to estimate the number of capacitors.

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Q14. Interpret the results of a fuel additive test using hypothesis testing and graphical representation.

Background

Topic: Hypothesis Testing & Statistical Analysis

This question tests your ability to perform hypothesis testing and communicate results graphically.

Key Terms and Formulas:

• Null hypothesis (): No effect

• Alternative hypothesis (): Effect present

• Z-score:

• Compare z-score to critical value (e.g., 1.96 for 5% significance)

Step-by-Step Guidance

1. State the hypotheses: : mean mpg = 48; : mean mpg ≠ 51.

2. Calculate the sample standard deviation: .

3. Calculate the z-score: .

4. Compare the z-score to the critical value (1.96).

5. Interpret whether to reject or accept the null hypothesis.

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