Which of the following statements are true?
A. The same Bernoulli's equation is applicable to all the points in the flow field if the flow is irrotational.
B. The value of "Constant in the Bernoulli's equation" is different for different streamlines if the flow is rotational.
C. When a nozzle is fitted at the end of a long pipeline, the discharge increases.
D. The velocity of flow at the nozzle end is more than that in the case of a pipe without a nozzle, the head in both cases being the same.
Choose the most appropriate answer from the options given below:
Step 1: Check each statement.
- (A) True: For irrotational flow, Bernoulli's equation applies universally in the flow field.
- (B) True: For rotational flow, the constant differs along different streamlines.
- (C) True: A nozzle reduces area, so velocity increases, leading to increased discharge.
- (D) True: With same head, nozzle increases velocity compared to a pipe without nozzle.
Step 2: Conclusion.
All four statements are correct $\Rightarrow$ Option (3).
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: