GATE CE 2003

Consider first 3\times 3 minors, since maximum possible rank is 3
\begin{aligned} \begin{vmatrix} 4 & 2 &1 \\ 6 & 3 & 4\\ 2 & 1 & 0 \end{vmatrix}&=0 \\ \begin{vmatrix} 2 & 1 &3 \\ 3 & 4& 7\\ 1 & 0& 1 \end{vmatrix}&=0 \\ \begin{vmatrix} 4 & 1 & 3\\ 6 & 4 & 7\\ 2 & 0 & 1 \end{vmatrix}&=0\\ \begin{vmatrix} 4 & 2 &3 \\ 6 & 3 & 7\\ 2 & 1 & 1 \end{vmatrix}&=0 \end{aligned}
Since all 3\times 3 minors are zero, now try 2\times 2 minors.
\begin{aligned} \begin{vmatrix} 4 &2 \\ 6 & 3 \end{vmatrix}&=0 \\ \begin{vmatrix} 2 &1 \\ 3 & 4 \end{vmatrix}&=8-3=5\neq 0 \\ \therefore\;\; \text{rank }&=2 \end{aligned}

This problem is to be solved by hypergeometric distribution, since population is finite.



\therefore p(none of 2 screws is defective)
\begin{aligned} &=\frac{3C_{0}\times 7C_{2}}{10C_{2}}\times 100\% \\ &=\frac{7}{15}\times 100\%=46.6\simeq 47\ \% \end{aligned}

Given, \begin{aligned} P\left ( 3,-2,-1 \right ) &\\ Q\left ( 1,3,4 \right ) &\\ R\left ( 2,1,-2 \right ) &\\ O\left ( 0,0,0 \right )&\\ \text{Equation of plane OQR is,}&\\ \begin{vmatrix} x-x_{1} & y-y_{1} & z-z_{1}\\ x_{2}-x_{1} & y_{2}-y_{1} & z_{2}-z_{1}\\ x_{3}-x_{1} & y_{3}-y_{1} & z_{3}-z_{1} \end{vmatrix}&=0 \\ \begin{vmatrix} x-0 & y-0 & z-0\\ 1 & 3 & 4\\ 2 & 1 & -2 \end{vmatrix}&=0\\ \text{i.e., }2x-2y+z&=0 \end{aligned}
Now \perp distance of \left ( x_{1},\: y_{1},\: z_{1} \right ) from ax + by + cz + d=0 is given by,
\left | \frac{ax_{1}+by_{1}+cz_{1}+d}{\sqrt{a^{2}+b^{2}+c^{2}}} \right |
Therefore, \perp distance of (3, -2, 1) from plane 2x-2y+z=0 is given by,
\left | \frac{2\times 3-2\times \left ( -2 \right )+\left ( -1 \right )}{\sqrt{2^{2}+\left ( -2 \right )^{2}+1^{2}}} \right |=3

The stress in lower bar
=\frac{50 \times 1000}{50 \times 50}=20 \mathrm{N} / \mathrm{mm}^{2}
The stress in upper bar
=\frac{250 \times 1000}{100 \times 100}=25 \mathrm{N} / \mathrm{mm}^{2}
Thus the maximum tensile stress any where in the bar is 25 \mathrm{N} / \mathrm{mm}^{2}.

Muller Breslau's principle is a principle on the basis of which it is possible to draw influence lines for various quantities pertaining to a structure. It states that:
"The ordinates of the influence lines for any stress element (such as axial stress, moment or reaction) of any structure are proportional to those of deflection curve, which is obtained by removing the restraint corresponding to that element from the structure and introducing in its place a deformation into the primary structure which remains."

Effective length of a column depends upon:
(i) Flexural stiffness ( El/L ) of beams joining at a point
(ii) Flexural stiffness ( EI/L ) of columns joining at a point.
(iii) Type of frame (sway or non-sway).

There is no eccentricity between the XY segment and the load. So, it is subjected to axial force only. But the curved YZ segment is subjected to axial force, shear force and bending moment.