# GATE CE 2002

 Question 1
Eigen values of the following matrix are:
$\begin{bmatrix} -1 &4 \\ 4&-1 \end{bmatrix}$
 A 3 and -5 B -3 and 5 C -3 and -5 D 3 and 5
Engineering Mathematics   Linear Algebra
 Question 2
The value of the following definite integral is:
$\int_{\frac{-\pi }{2}}^{\frac{\pi }{2}}\frac{sin2x}{1+cosx}dx$
 A -2 ln2 B 2 C 0 D $(ln 2)^{2}$
Engineering Mathematics   Calculus
Question 2 Explanation:
Since the givem function is odd function therefore in the given limits the integral will be zero.

 Question 3
The following function has a local minima at which value of $x$?
$f(x)=x\sqrt{5-x^{2}}$
 A $-\frac{\sqrt{5}}{2}$ B $\sqrt{5}$ C $\sqrt{\frac{5}{2}}$ D $-\sqrt{\frac{5}{2}}$
Engineering Mathematics   Calculus
Question 3 Explanation:
\begin{aligned} f\left ( x \right )&=x\sqrt{5-x^{2}} \\ {f}'\left ( x \right )&=\sqrt{5-x^{2}}+\frac{x\times -2x}{2\sqrt{5-x^{2}}} \\ 5-x^{2}&=x^{2} \\ 2x^{2}&=5 \\ x&=\pm \sqrt{\frac{5}{2}}\end{aligned}
 Question 4 I. Member AB Carries shear force and bending moment.
II. Member BC carries axial load and shear force.

Which of the following is true?
 A Statement I is true but II is false B Statement I is false but II is true C Both statements I and II are true D Both statements I and II are false
Solid Mechanics   Shear Force and Bending Moment
Question 4 Explanation: Member AB carries shear force and bending moment both. Hence statement-I is true Member BC carries axial load and bending moment only. Hence statement-II is false.
 Question 5