Logarithms Questions and Answers

Answer: B

The general solution of a linear homogeneous equation is called the complementary function.

Every such equation has the trivial solution \( y = 0 \), which is why one solution is always "automatic"; the complementary function collects all of them.

For an inhomogeneous equation the full solution is the complementary function plus a particular integral.

Answer: C

Read the two features separately.

The parenthesis at \( a \) excludes the endpoint (\( x > a \)), so the interval is open. It runs to \( +\infty \) with no upper endpoint, so it is unbounded.

Hence open and unbounded.

Answer: D

A linear accelerator drives particles forward with oscillating electric fields, and an electric field can only exert force on something carrying charge.

\[ F = qE \]

So only charged particles - electrons, protons, ions - can be accelerated. A neutral particle has \( q = 0 \) and feels no force at all.

Answer: B

Expressing a physical situation in variables, functions and equations produces a mathematical model - the bridge between the engineering problem and the mathematics used to solve it.

Solving the model then gives an answer that has to be interpreted back in the original setting, which is the last step of the modelling cycle.

Answer: C

The standard transform pair is

\[ \mathcal{L}\{\cos \omega t\} = \frac{s}{s^{2}+\omega^{2}} \]

while its companion is

\[ \mathcal{L}\{\sin \omega t\} = \frac{\omega}{s^{2}+\omega^{2}} \]

The denominators are identical; only the numerator distinguishes them - \( s \) for cosine, \( \omega \) for sine.

Answer: A

The Laplace transform is defined by an integral:

\[ F(s) = \int_{0}^{\infty} e^{-st}f(t)\,dt \]

The function of \( t \) is multiplied by \( e^{-st} \) and integrated over all positive time, producing a new function of \( s \). Its power comes from turning differentiation in \( t \) into multiplication in \( s \).

Answer: A

The graph of a function \( y = h(x) \) that satisfies a differential equation is its solution curve.

The equation prescribes the slope at every point, and a solution curve is one whose tangent matches that prescription everywhere. It is also called an integral curve.

Answer: A

A deductive model is built downward from an accepted theory: the governing equations are derived from established principles rather than fitted to data.

An inductive model works the other way, generalising from observations, and a floating model rests on neither.

Answer: B

The Laplace method runs in three steps:

1. Transform the ODE into an algebraic subsidiary equation in \( s \)
2. Solve that equation for \( Y(s) \) by algebra
3. Take the inverse transform to recover \( y(t) \)

The initial conditions enter at step 1, which is why no arbitrary constants remain at the end.

Answer: A

A homogeneous system \( AX = 0 \) always has the trivial solution \( X = 0 \). It has only that solution when \( A \) is invertible, which happens exactly when

\[ \det A \neq 0 \]

If the determinant were zero, the matrix would be singular and infinitely many non-trivial solutions would exist.