# GATE CE 2003

 Question 1
Given Matrix $[A]=\begin{bmatrix} 4 & 2 & 1 &3 \\ 6&3 &4 & 7\\ 2 &1 & 0& 1 \end{bmatrix}$, the rank of the matrix is
 A 4 B 3 C 2 D 1
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
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}
 Question 2
A box contains 10 screws, 3 of which are defective. Two screws are drawn at random with replacement. The probability that none of the two screws is defective will be
 A 100% B 50% C 49% D None of these
Engineering Mathematics   Probability and Statistics
Question 2 Explanation:
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}
 Question 3
If P, Q and R are three points having coordinates (3,-2,-1), (1,3,4), (2,1,-2) in XYZ space, then the distance from point P to plane OQR (O being the origin of the coordinate system) is given by
 A 3 B 5 C 7 D 9
Engineering Mathematics   Calculus
Question 3 Explanation:
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,
$\left | \frac{2\times 3-2\times \left ( -2 \right )+\left ( -1 \right )}{\sqrt{2^{2}+\left ( -2 \right )^{2}+1^{2}}} \right |=3$
 Question 4
A bar of varying square cross-section is loaded symmetrically as shown in the figure. Loads shown are placed on one of the axes of symmetry of cross-section. Ignoring self weight, the maximum tensile stress in $N/mm^{2}$ anywhere is
 A 16 B 20 C 25 D 30
Solid Mechanics   Properties of Metals, Stress and Strain
Question 4 Explanation:
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 $25 \mathrm{N} / \mathrm{mm}^{2}.$
 Question 5
Muller Breslau's principle in structural analysis is used for
 A drawing influence line diagram for any force function B writing virtual work equation C super position of load effects D none of these
Structural Analysis   Influence Line Diagram and Rolling Loads
Question 5 Explanation:
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."
 Question 6
The effective length of a column in a reinforced concrete building frame, as per IS:456-2000, is independent of the
 A frame type i.e. braced (no sway) or un-braced (with sway) B span of the beam C height of the column D loads acting on the frame
RCC Structures   Footing, Columns, Beams and Slabs
Question 6 Explanation:
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).
 Question 7
A curved member with a straight vertical leg is carrying a vertical load at Z , as shown in the figure. The stress resultants in the XY segment are
 A bending moment, shear force and axial force B bending moment and axial force only C bending moment and shear force only D axial force only
Solid Mechanics   Shear Force and Bending Moment
Question 7 Explanation:
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.
 Question 8
The working stress method of design specifies the value of modular ratio, m=280/( 3$\sigma _{cbc}$), where $\sigma _{cbc}$ is the allowable stress in bending compression in concrete. To what extent does the above value of 'm' make any allowance for the creep of concrete ?
 A No compensation B Full compensation C Partial compensation D The two are unrelated
RCC Structures   Working Stress and Limit State Method
 Question 9
In the design of lacing system for a built-up steel column, the maximum allowable slenderness ratio of a lacing bar is
 A 120 B 145 C 180 D 250
Design of Steel Structures   Compression Member
 Question 10
Which of the following elements of a pitched roof industrial steel building primarily resists lateral load parallel to the ridge ?
 A bracings B purlins C truss D columns
Design of Steel Structures   Plate Griders and Industrial Roofs
There are 10 questions to complete.