1. The RMS (Root Mean Square) value of a function $f(x)$ over the interval $a \leq x \leq b$ is defined as the square root of the average of the square of the function over that interval. Mathematically, it is given by: $$\text{RMS} = \sqrt{\frac{1}{b - a} \int_a^b [f(x)]^2 \, dx}$$ So, the correct option is (B). 2. If $F[f(x)] = F(s)$ is the Fourier transform of $f(x)$, then the Fourier transform of $x f(x)$ is given by: $$F[x f(x)] = i \frac{d}{ds} F(s)$$ Hence, the correct option is (C). 3. The constant term $a_0$ of the Fourier series for the function $f(x) = k$ on $0 \leq x \leq 2\pi$ is the average value of the function over one period: $$a_0 = \frac{1}{2\pi} \int_0^{2\pi} k \, dx = \frac{1}{2\pi} (k \times 2\pi) = k$$ So, the correct option is (A). 4. The steady state temperature distribution $u(x)$ in a rod of length $l$ with ends kept at temperatures 30 and 40 is a linear function satisfying boundary conditions: $$u(0) = 30, \quad u(l) = 40$$ The linear function is: $$u(x) = \frac{40 - 30}{l} x + 30 = \frac{10}{l} x + 30$$ So, the correct option is (A). 5. The Fourier cosine transform of $f(ax)$ is related to the transform of $f(x)$ by: $$F_c[f(ax)] = \frac{1}{a} F_c\left(\frac{s}{a}\right)$$ So, the correct option is (D). 6. The Z-transform of $a^n f(n)$ is: $$Z[a^n f(n)] = F\left(\frac{z}{a}\right)$$ So, the correct option is (C). 7. The PDE $u_{xx} - 5 u_{yy} = 0$ has discriminant $D = B^2 - 4AC$ with $A=1$, $B=0$, $C=-5$: $$D = 0^2 - 4(1)(-5) = 20 > 0$$ So, it is hyperbolic. Correct option is (C). 8. The one-dimensional heat equation in steady state is: $$u_{xx} = 0$$ So, correct option is (D). 9. Eliminating $a$ and $b$ from $z = (x^2 + a)(y^2 + b)$ leads to: $$4xyz = pq$$ So, correct option is (C). 10. The complete integral of $p = q$ is: $$z = a x + b y + c$$ So, correct option is (C). 11. The solution of $(D^2 - D D') z = 0$ is: $$z = f_1(y) + x f_2(y)$$ So, correct option is (A). 12. The particular integral of $(D^2 - 2 D D' - D'^2) z = \sin(x + y)$ is: $$\frac{1}{2} \sin(x + y)$$ So, correct option is (C). 13. The Z-transform of $\cos\left(\frac{n \pi}{2}\right)$ is: $$\frac{z}{z^2 + 1}$$ So, correct option is (A). 14. The Fourier cosine transform of $e^{-4x}$ is: $$\sqrt{\frac{2}{\pi}} \frac{4}{s^2 + 16}$$ So, correct option is (C). 15. The Z-transform of $f(n+2)$ is: $$z^2 F(z) - z f(0) - f(1)$$ So, correct option is (A). 16. For half range cosine series of $f(x) = \cos x$ in $(0, \pi)$, the value of $a_0$ is: $$1$$ So, correct option is (A). 17. If $x = a$ is an interior point of discontinuity in $(a,b)$, the sum of the Fourier series at $x=a$ is: $$\frac{f(a^+) + f(a^-)}{2}$$ So, correct option is (B). 18. If $F[f(x)] = f(s)$, then $f(x)$ is called a self-reciprocal function. So, correct option is (A). 19. The Z-transform of $e^{-5n}$ is: $$\frac{z}{z - e^{-5}}$$ So, correct option is (B). 20. The boundary conditions of a one-dimensional wave equation for a string of length $l$ are: $$y(0,t) = 0, \quad y(l,t) = 0$$ So, correct option is (C).