GATE CE 2014 SET-2

Question 1
A fair (unbiased) coin was tossed four times in succession and resulted in the following outcomes: (i) Head, (ii) Head, (iii) Head, (iv) Head. The probability of obtaining a "Tail" when the coin is tossed again is
A
0
B
\frac{1}{2}
C
\frac{4}{5}
D
\frac{1}{5}
Engineering Mathematics   Probability and Statistics
Question 2
The determinant of matrix \begin{bmatrix} 0 &1 &2 &3 \\ 1 &0 &3 &0 \\ 2 &3 &0 &1 \\ 3 &0 &1 &2 \end{bmatrix} is__________.
A
88
B
44
C
66
D
22
Engineering Mathematics   Linear Algebra
Question 2 Explanation: 
\begin{aligned} \Delta &=\begin{vmatrix} 0 & 1 & 2 & 3\\ 1 & 0 & 3 & 0\\ 2 & 3 & 0 & 1\\ 3 & 0 & 1 & 2 \end{vmatrix} \\ R_{4} & \rightarrow R_{4}-R_{2}-R_{3} \\ \Delta &=\begin{vmatrix} 0 & 1 & 2 & 3\\ 1 & 0 & 3 & 0\\ 2 & 3 & 0 & 1\\ 0 & -3 & -2 &1 \end{vmatrix} \\ R_{4} & \rightarrow R_{4}+3R_{1} \\ \Delta &=\begin{vmatrix} 0 & 1 & 2 & 3\\ 1 & 0 & 3 & 0\\ 2 & 3 & 0 & 1\\ 0 & 0 & 4 & 10 \end{vmatrix} \\ R_{3} & \rightarrow R_{3}-3R_{1} \\ \Delta &=\begin{vmatrix} 0 & 1 & 2 & 3\\ 1 & 0 & 3 & 0\\ 2 & 0 & -6 & -8\\ 0 & 0 & 4 & 10 \end{vmatrix} \end{aligned}
interchanging column 1 and column 2 and taking transpose,
\begin{aligned} \Delta &=-\begin{vmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 2 & 0\\ 2 & 3 & -6 & 4\\ 3 & 0 & -8 & 10 \end{vmatrix} \\ &=-1\times \begin{vmatrix} 1 & 2 & 0\\ 3 & -6 & 4\\ 0 & -8 & 10 \end{vmatrix} \\ &=1\times \left \{ 1\left ( -60+32 \right )+2\left ( 0-30 \right ) \right \} \\ &=-\left ( -28-60 \right )=88\end{aligned}
Question 3
z=\frac{2-3i}{-5+i} can be expressed as
A
-0.5-0.5i
B
-0.5+0.5i
C
0.5-0.5i
D
0.5+0.5i
Engineering Mathematics   Calculus
Question 3 Explanation: 
\begin{aligned} \frac{\left ( 2-3i \right )}{\left ( -5+i \right )}&=\frac{\left ( 2-3i \right )}{\left ( -5+i \right )}\times \frac{\left ( -5-i \right )}{\left ( -5-i \right )} \\ &=\frac{-10-2i+15i-3}{25+1} \\ &=\frac{-13+13i}{26} \\ &=-0.5+0.5i \end{aligned}
Question 4
The integrating factor for the differential equation \frac{\mathrm{d} P}{\mathrm{d} t}+k_{2}P=k_{1}L_{o}e^{-k_{1}t} is
A
e^{-k_{1}t}
B
e^{-k_{2}t}
C
e^{k_{1}t}
D
e^{k_{2}t}
Engineering Mathematics   Calculus
Question 5
If {x} is a continuous, real valued random variable defined over the interval (-\infty ,+\infty ) and its occurrence is defined by the density function given as: f(x)=\frac{1}{\sqrt{2\pi*b}}e^{\frac{-1}{2}(\frac{x-a}{b})^{2}} where 'a' and 'b' are the statistical attributes of the random variable {x}. The value of the integral \int_{-\infty }^{a}\frac{1}{\sqrt{2\pi*b}}e^{\frac{-1}{2}(\frac{x-a}{b})^{2}} dx is:
A
1
B
0.5
C
\pi
D
\frac{\pi}{2}
Engineering Mathematics   Probability and Statistics
Question 6
Group I contains representative stress-strain curves as shown in the figure, while Group II given the list of materials. Match the stress-strain curves with the corresponding materials.

A
P-1; Q-3; R-2
B
P-2; Q-3; R-1
C
P-3; Q-1; R-2
D
P-3; Q-2; R-1
RCC Structures   Working Stress and Limit State Method
Question 7
The first moment of area about the axis of bending for a beam cross-section is
A
moment of inertia
B
section modulus
C
shape factor
D
polar moment of inertia
RCC Structures   Working Stress and Limit State Method
Question 8
Polar moment of inertia (I_{p}), in cm^{4}, of a rectangular section having width, b=2 and depth, d=6 cm is _____
A
22
B
44
C
20
D
40
Engineering Mechanics   
Question 8 Explanation: 
Polar moment of inertia,
\begin{aligned} I_p&= I_x+I_y\\ &=\frac{bd^3}{12}+\frac{db^3}{12}\\ &=\frac{bd}{12}(b^2+d^2) \\ &= \frac{2 \times 6}{12}(2^2+6^2)\\ &=40\; cm^4 \end{aligned}
Question 9
The target means strength f_{cm} for concrete mix design obtained from the characteristic strength f_{ck} and standard deviation \sigma, as defined in IS:456-2000, is
A
f_{ck}+1.35\sigma
B
f_{ck}+1.45\sigma
C
f_{ck}+1.55\sigma
D
f_{ck}+1.65\sigma
RCC Structures   Working Stress and Limit State Method
Question 10
The flexural tensile strength of M25 grade of concrete, in N/mm^{2}, as per IS:456-2000 is
A
0.5
B
5
C
3.5
D
2.5
RCC Structures   Working Stress and Limit State Method
Question 10 Explanation: 
\text { Flexural strength }=0.7 \sqrt{f_{c k}}=3.5 \mathrm{N} / \mathrm{mm}^{2}
There are 10 questions to complete.

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