Questions -

Q1. Verify numerically the following relation.

153 = 1³ + 5³ + 3³

1634 = 1⁴ + 6⁴ + 3⁴ + 4⁴

548834 = 5⁶ + 4⁶ + 8⁶ + 8⁶ + 3⁶ + 4⁶

Q2. Demonstrate numerically the validity of the following expressions.

cot(π/5) = 1/5 √(25+10 √(5))

sin(π/15) = ¼ √(7- √5- √(30-6 √5))

π = 16 tan^-1 1/5 - 4 tan^-1 1/239

Hint: use the Matlab intrinsic functions sin, sqrt, cot and atan; and the built-in Matlab constant pi.

Q3. Show numerically the validity of the following expression.

Sin(π/17) = (√2/8)√(γ2- √2(α+γ))

where

γ² = 17 - √(17)

α = √(34+√2(√17-1) γ - 8√2√(√17+17)+6 √17)

Hint: Use the Matlab intrinsic functions sin and sqrt and the built-in Matlab constant pi.

Q4. The acceleration of a reciprocating engine piston is given by

a = rω²[cosθ + (n²cos2θ+sin⁴θ/(n²-sin²θ)^3/2)]

where n = l/r , r is the crank arm length, l is the length of the connecting rod, ω is the angular velocity of the crank arm, and θ is the rotation angle of the crank arm. Write a Matlab code that takes n and θ as inputs and display the value of the ratio a/(rω²) on the screen. Test your code for n = 3 and θ = pi/7.

Q5. The second order differential equation:

d²y/dt² + 2 dy/dt + y = x(t)

could describe the behavior of an electrical system, where x(t) is the input voltage and y(t) is the output volatage and dy/dt is the current. The complex ratio

y(ω)/x(ω) = 1/(-ω²+2ωj+1)

is called the frequency response of the system because it describes the relationship between input and output for sinusoidal excitation at a frequency of ω and where j is √-1. Write a Matlab script program that reads in a value of ω and evaluates the frequency response for this value of ω together with its polar form (magnitude and phase).

Hint: Use the built-in Matlab functions real (extract the real part of a complex quantity), imag (extract the imaginary part of a complex quantity), abs (returns the magnitude of a complex quantity) and atan2 (tan^-1). Also, you may need to use the built-in Matlab constant i or j, where i = j = √-1.