Which of the following parameters are required for the design of a transition curve for a highway system?
(A) Rate of change of grade
(B) Rate of change of radial acceleration
(C) Rate of change of super elevation
(D) Rate of change of curvature
Choose the most appropriate answer from the options given below:
Step 1: Transition curve design criteria.
Transition curves are introduced in highways to provide gradual change in curvature and superelevation. The key parameters are:
- Rate of change of radial acceleration.
- Rate of change of superelevation.
- Rate of change of curvature.
Step 2: Eliminate unnecessary parameter.
Rate of change of grade (longitudinal slope) is not a governing factor in transition curve design.
Step 3: Conclusion.
Hence, the required parameters are (B), (C), and (D). Therefore, the correct answer is (D).
The solution(s) of the ordinary differential equation $y'' + y = 0$, is:
(A) $\cos x$
(B) $\sin x$
(C) $1 + \cos x$
(D) $1 + \sin x$
Choose the most appropriate answer from the options given below:
For the matrix, $A = \begin{bmatrix} -4 & 0 \\ -1.6 & 4 \end{bmatrix}$, the eigenvalues ($\lambda$) and eigenvectors ($X$) respectively are:
The value of $\iint_S \vec{F} \cdot \vec{N} \, ds$ where $\vec{F} = 2x^2y \hat{i} - y^2 \hat{j} + 4xz^2 \hat{k}$ and $S$ is the closed surface of the region in the first octant bounded by the cylinder $y^2 + z^2 = 9$ and the planes $x = 0, x = 2, y = 0, z = 0$, is:
The value of the integral $\displaystyle \oint_C \frac{z^3 - 6}{2z - i} \, dz$, where $C: |z| \leq 1$, is:
Match LIST-I with LIST-II (adopting standard notations):\[\begin{array}{|c|c|} \hline \textbf{LIST-I (Parameter)} & \textbf{LIST-II (Formula)} \\ \hline \\ \text{A. Cubic parabola equation} & \text{IV. $\dfrac{X^3}{6RL}$} \\ \\ \hline \\ \text{B. Shift in transition curve} & \text{II. $\dfrac{L^2}{24R}$} \\ \\ \hline \\ \text{C. Length of valley curve} & \text{III. $\dfrac{N S^2}{(1.50 + 0.035S)}$} \\ \\ \hline \\ \text{D. Length of summit curve} & \text{I. $\dfrac{N S^2}{4.4}$} \\ \\ \hline \end{array}\] Choose the most appropriate match from the options given below: