Prime notation is used to denote only
Answer: C
Prime notation is convenient only for low orders:
\[ y', \quad y'', \quad y''' \]
Beyond the third derivative the primes become hard to count, and the numeral form \( y^{(4)} \) or the Leibniz form \( \tfrac{d^{4}y}{dx^{4}} \) is used instead.
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Ln(2) is
Answer: B
The notation \( \ln \) always means the logarithm to base \( e \), so \( \ln 2 \) is a natural logarithm - the power to which \( e \) must be raised to give 2:
\[ \ln 2 \;=\; \log_e 2 \;\approx\; 0.693 \]
The trap is the numeral 2 in the bracket, which suggests base 2. The 2 is the argument, not the base. A binary logarithm is written \( \log_2 x \) and takes 2 as its base; the two are different, and indeed \( \log_2 2 = 1 \) whereas \( \ln 2 \approx 0.693 \).
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Laplace transform of function f(t)=sin(wt) is
Answer: D
The standard transform pair is
\[ \mathcal{L}\{\sin \omega t\} = \frac{\omega}{s^{2}+\omega^{2}} \]
while its companion is
\[ \mathcal{L}\{\cos \omega t\} = \frac{s}{s^{2}+\omega^{2}} \]
The two differ only in the numerator: \( \omega \) for the sine, \( s \) for the cosine. That is the pair most often confused.
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Model in which every set of variable states is uniquely determined by parameters in model and by sets of previous states of these variables is termed as
Answer: A
A deterministic model is one in which the present state is fixed entirely by the parameters and the earlier states: run it twice from the same start and it gives the same answer twice.
A stochastic or probabilistic model includes randomness, so repeated runs can differ.
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If rate of flow of water of tank's inlet and outlet is constant so rate of change of volume w.r.t time i.e dV/dt will be
Answer: D
Volume changes at the difference between what flows in and what flows out.
\[ \frac{dV}{dt} = \text{rate in} - \text{rate out} \]
The two rates are equal and constant, so the difference is zero:
\[ \frac{dV}{dt} = 0 \]
The volume is therefore constant - the tank is in steady state, neither filling nor draining.
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Initial condition is also termed as
Answer: D
The value prescribed at the starting point of a problem is the initial condition, also called the seed value.
\[ y(t_{0}) = y_{0} \]
It is what picks a single solution out of the general family, each member of which differs only by the arbitrary constant.
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For a gas at low pressure p and constant temperature T has volume V equals to
Answer: A
Boyle's law states that at constant temperature the pressure and volume of a gas are inversely proportional:
\[ pV = \text{constant} \quad\Longrightarrow\quad V = \frac{\text{constant}}{p} \]
Halving the pressure therefore doubles the volume. The approximation holds best at low pressure, where the gas is close to ideal.
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Linear accelerators are used in physics to accelerate
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.
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Napier's logarithm is also called
Answer: A
Napier's logarithms are the natural logarithms, taken to base \( e \) and written \( \ln x \).
Common logarithms, to base 10, came slightly later and are named after Briggs. The mantissa is only the fractional part of a logarithm, and the antilogarithm is the inverse operation, so neither names a system of logarithms at all.
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Laplace transform of f(x)=tanx where tanx has limit of infinity is
Answer: D
A function has a Laplace transform only if the integral
\[ \int_{0}^{\infty} e^{-st}f(t)\,dt \]
converges. That requires \( f \) to be piecewise continuous and of exponential order.
\( \tan t \) has infinite discontinuities at \( t = \tfrac{\pi}{2}, \tfrac{3\pi}{2}, \ldots \), where it blows up faster than any exponential can damp. The integral diverges, so no Laplace transform exists.
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