# GATE CE 2007

 Question 1
The minimum and the maximum eigen values of the matrix $\begin{bmatrix} 1 & 1&3 \\ 1& 5 & 1\\ 3&1 & 1 \end{bmatrix}$ are -2 and 6, respectively. What is the other eigen value ?
 A 5 B 3 C 1 D -1
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
\begin{aligned} \sum \lambda _{i}&= \text{ Trace(A)}\\ \lambda _{1}+\lambda _{2}+\lambda _{3}&=1+5+1=7\\ \text{Now } \lambda _{1}&=-2,\; \lambda _{2}=6 \\ \therefore \;\; -2+6+\lambda _{3}&=7 \\ \lambda _{3}&=3\end{aligned}
 Question 2
The degree of the differential equation $\frac{d^{2}x}{dt^{2}}+2x^{3}=0$ is
 A 0 B 1 C 2 D 3
Engineering Mathematics   Ordinary Differential Equation
Question 2 Explanation:
Degree of a differential equation is the power of its highest order derivative after the differential equation is made free of radicals and fractions if any, in derivative power.
Hence, here the degree is 1, which is power of $\frac{d^{2}x}{dt^{2}}$

 Question 3
The solution for the differential equation $\frac{dy}{dx}=x^{2}y$ with the condition that y = 1 at x = 0 is
 A $y=e^{\frac{1}{2x}}$ B $ln(y)=\frac{x^{3}}{3}+4$ C $ln(y)=\frac{x^{2}}{2}$ D $y=e^{\frac{x^{3}}{3}}$
Engineering Mathematics   Ordinary Differential Equation
Question 3 Explanation:
$\frac{dy}{dx}=x^{2}y$
This is variable separable from,
\begin{aligned} \frac{dy}{y}&=x^{2}\, dx \\ \int \frac{dy}{y}&=\int x^{2}dx \\ \Rightarrow \;\; \log_{e}y&=\frac{x^{3}}{3}+C_{1} \\ \Rightarrow \;\; y&=^{\frac{x^{3}}{3}+C_{1}} \\ &=e^{C_{1}}\times e^{\frac{x^{3}}{3}} \\ y&=C\times e^{\frac{x^{3}}{3}} \\ \text{Now at }& x=0, \;y=1 \\ 1&=C\times e^\frac{0}{3} \\ \Rightarrow \;\; C&=1 \\ \therefore \;\; y&=e^{\frac{x^{3}}{3}} \end{aligned}
 Question 4
An axially loaded bar is subjected to a normal stress of 173 MPa. The stress in the bar is
 A 75MPa B 86.5MPa C 100MPa D 122.3MPa
Solid Mechanics   Principal Stress and Principal Strain
Question 4 Explanation:
\begin{aligned} \text { Shear stress }&=\frac{\sigma_{1}-\sigma_{2}}{2} \\ \therefore \quad&=\frac{173-0}{2}=86.5 \mathrm{MPa} \end{aligned}
 Question 5
A steel column, pinned at both end, has a buckling load of 200 kN. If the column is restrained against lateral movement at its mid-height, it buckling load will be
 A 200kN B 283kN C 400kN D 800kN
Solid Mechanics   Theory of Columns and Shear Centre
Question 5 Explanation:
When both ends are hinged, the buckling load is given by,
$P_{c r}=\frac{\pi^{2} E I}{L^{2}} \Rightarrow 200=\frac{\pi^{2} E l}{L^{2}}$
When the lateral movement at the mid-height is not available, then buckling load
\begin{aligned} &=\frac{\pi^{2} E \mid}{\left(L_{1}\right)^{2}} \quad \text { where } L_{1}=\frac{L}{2} \\ &=\frac{4 \pi^{2} E \mid}{L^{2}}=4 \times 200=800 \mathrm{kN} \end{aligned}

There are 5 questions to complete.