Which of the following statements are correct?
A. Malleability is the ability of a material to absorb strain energy till the elastic limit.
B. Toughness is the ability of a material to absorb energy till the rupture.
C. Resilience is the area under the load deformation curve within the elastic limit.
D. Stress-strain diagram of highly brittle material has no plastic zone.
Choose the most appropriate answer from the options given below:
Step 1: Evaluate each statement.
- (A) Incorrect: Malleability is the ability to withstand plastic deformation under compression (hammering into sheets), not limited to elastic energy.
- (B) Correct: Toughness is the capacity to absorb energy up to fracture (entire stress-strain area).
- (C) Correct: Resilience is the elastic energy stored, i.e., the area under the stress-strain curve up to the elastic limit.
- (D) Correct: Brittle materials fracture almost immediately after the elastic limit, showing negligible/no plastic zone.
Step 2: Conclusion.
Correct set of statements = (B, C, 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:
A steel wire of $20$ mm diameter is bent into a circular shape of $10$ m radius. If modulus of elasticity of wire is $2\times10^{5}\ \text{N/mm}^2$, then the maximum bending stress induced in wire is:
The degree of static indeterminacy of the beam (as shown below) for general case of loading is: