# GATE EC 2014 SET-3

 Question 1
The maximum value of the function $f(x)=ln(1+x)-x(\; where \; x>-1)$ occurs at x = ____.
 A 0 B 1 C 0.5 D -1
Engineering Mathematics   Calculus
Question 1 Explanation:
\begin{aligned} f^{\prime}(x) &=\frac{1}{1+x}-1=0 \\ \frac{1-1-x}{1+x} &=0 \\ \frac{x}{1+x} &=0 \\ x &=0\\ f^{\prime \prime}(x)&=\frac{-1}{(1+x)^{2}} \\ f^{\prime}(0)&=-1 \lt 0 \end{aligned}
f(x) have maximum value at x=0
\begin{aligned} f(0)&=\ln (1+0)-0=0 \\ t_{\max }&=0 \end{aligned}
 Question 2
Which ONE of the following is a linear non-homogeneous differential equation, where x and y are the independent and dependent variables respectively ?
 A $\frac{dy}{dx}+xy=e^{-x}$ B $\frac{dy}{dx}+xy=0$ C $\frac{dy}{dx}+xy=e^{-y}$ D $\frac{dy}{dx}+e^{-y}=0$
Engineering Mathematics   Differential Equations
Question 2 Explanation:
General form of linear differential equation
$\frac{d y}{d x}+p y=\theta$ when P and $\theta$ can be function of x
Only option (A) is in this form.

 Question 3
Match the application to appropriate numerical method. A P1-M3, P2-M2, P3-M4, P4-M1 B P1-M3, P2-M1, P3-M4, P4-M2 C P1-M4, P2-M1, P3-M3, P4-M2 D P1-M2, P2-M1, P3-M3, P4-M4
Engineering Mathematics   Numerical Methods
 Question 4
An unbiased coin is tossed an infinite number of times. The probability that the fourth head appears at the tenth toss is
 A 0.067 B 0.073 C 0.082 D 0.091
Engineering Mathematics   Probability and Statistics
Question 4 Explanation:
It means 3-head appears in $1^{\text {st }}$ 9 trials.
Probability of getting exactly 3 head in 1^{\text {st }} 9 trials
\begin{aligned} &=\text{ Coefficient } p^{3}\text{ in }(4+p) 9 \\ \text{When, }&[\overline{p+q=1}] \\ &={ }^{9} C_{3} q^{6} p^{3}\\ \text{when, }\quad \rho&= \text{ probability of occure of head}\\ q &=\text { probability of occure of tail } \\ &={ }^{9} C_{3} \times\left(\frac{1}{2}\right)^{6}\left(\frac{1}{2}\right)^{3} \\ &={ }^{9} C_{3} \times\left(\frac{1}{2}\right)^{9} \end{aligned}
and in $10^{\text {th }}$ trial head must appears.
So required probability
\begin{aligned} &={ }^{9} C_{3}\left(\frac{1}{2}\right)^{9} \times \frac{1}{2} \\ &=\frac{9 \times 8 \times 7}{3 !} \times\left(\frac{1}{2}\right)^{10}=\frac{84}{1024}=0.082 \end{aligned}
 Question 5
If z = xy ln(xy), then
 A $x\frac{\partial z}{\partial x}+y\frac{\partial z}{\partial y}=0$ B $y\frac{\partial z}{\partial x}=x\frac{\partial z}{\partial y}$ C $x\frac{\partial z}{\partial x}=y\frac{\partial z}{\partial y}$ D $y\frac{\partial z}{\partial x}+x\frac{\partial z}{\partial y}=0$
Engineering Mathematics   Calculus
Question 5 Explanation:
\begin{aligned} \frac{\partial z}{\partial x}&=yln(x y)+\frac{x y}{x y} y\\ \frac{\partial z}{\partial x}&=y[\ln (x y)+1] &\ldots(i)\\ \frac{\partial z}{\partial y}&=x \ln (x y)+\frac{x y}{x y} \times x\\ \frac{\partial z}{\partial y}&=x[\ln (x y)+1]\\ \text{Here}\quad x\frac{\partial z}{\partial x}&=y \frac{\partial z}{\partial y} \end{aligned}

There are 5 questions to complete.