# 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

There are 5 questions to complete.