# GATE ME 2005

 Question 1
Strokes theorem connects
 A a line integral and a surface integral B a surface integral and a volume integral C a line integral and a volume integral D gradient of a function and its surface integral
Engineering Mathematics   Calculus
Question 1 Explanation:
A line integral and a surface integral is related by stroke's theorem
 Question 2
A lot has 10% defective items. Ten items are chosen randomly from this lot. The probability that exactly 2 of the chosen items are defective is
 A 0.0036 B 0.1937 C 0.2234 D 0.3874
Engineering Mathematics   Probability and Statistics
Question 2 Explanation:
Probability of defective item
$P=0.1$
Probability of non-defective item
$Q=1-p=1-0.1=0.9$
Probability that exactly 2 of the chosen items are defective
$=^{10} C_{2}(P)^{2}(Q)^{8}$
$=^{10} C_{2}(0.1)^{2}(0.9)^{8}=0.1937$
 Question 3
$\int_{-a}^{a}(\sin ^{6} x + \sin ^{7} x)dx$ is equal to
 A $2\int_{0}^{a}(\sin ^{6} x)dx$ B $2\int_{0}^{a}(\sin ^{7} x)dx$ C $2\int_{0}^{a}(\sin ^{6} x + \sin ^{7} x)dx$ D zero
Engineering Mathematics   Calculus
Question 3 Explanation:
$I=\int_{-a}^{a}\left(\sin ^{6} x+\sin ^{7} x \right) dx$
$= 2 \int_{0}^{a} \sin ^{6} x d x+0$
( because $\int_{0}^{a} \sin ^{7}x\cdot d x=0$)
 Question 4
A is a 3 x 4 real matrix and Ax=b is an inconsistent system of equations. The highest possible rank of A is
 A 1 B 2 C 3 D 4
Engineering Mathematics   Linear Algebra
Question 4 Explanation:
$C =[A: B]_{3 \times 5}$
$\therefore \rho\left[C_{3 \times 5}\right] \leq \min \{3,5\}$
$\because$ The system is inconsistent
$\rho(A) \lt \rho(C)$
$\therefore \rho(A)\lt 3$
Hence maximum possible rank of
$A=2$
 Question 5
Changing the order of the integration in the double integral $I=\int_{0}^{8}\int_{\frac{x}{4}}^{2}f(x,y)dydx$ leads to $I=\int_{r}^{s}\int_{p}^{q}f(x,y)dxdy$. What is q?
 A 4y B $16y^{2}$ C x D 8
Engineering Mathematics   Calculus
Question 5 Explanation:
$\text { When } \quad I=\int_{0}^{8} \int_{x / 4}^{2} f(x \cdot y) dydx$  $I=\int_{0}^{2} \int_{0}^{4 y} f(x)dydx$
 Question 6
The time variation of the position of a particle in rectilinear motion is given by $x-2t^{3}+t^{2}+2t$. If v is the velocity and a the acceleration of the particle in consistent units, the motion started with
 A v=0, a=0 B v=0, a=2 C v=2 a=0 D v=2 a=2
Engineering Mechanics   Translation and Rotation
Question 6 Explanation:
\begin{aligned} \text{Given }\quad x&=2 t^{3}+t^{2}+2 t \\ Velocity: \quad V&=\frac{d x}{d t}=6 t^{2}+2 t+2 \\ \text{At }\quad t&=0 \quad (\text{start of motion}) \\ V &=0+0+2 \\ V &=2 \\ \text { Acceleration: } a&=\frac{d V}{d t}=12 t+2 \\ \text{At }\quad t&=0 \\ a&=0+2 \\ \therefore a&=2 \\ \end{aligned}
 Question 7
A simple pendulum of length 5m, with a bob of mass 1 kg, is in simple harmonic motion. As it passes through its mean position, the bob has a speed of 5 m/s. the net force on the bob at the mean position is
 A zero B 2.5 N C 5 N D 25 N
Engineering Mechanics   Translation and Rotation
Question 7 Explanation:
Mean position $\rightarrow$ It is the position where acceleration is zero and hence force is zero
 Question 8
A uniform, slender cylindrical rod is made of a homogeneous and isotropic material. The rod rests on a frictionless surface. The rod is heated uniformly. If the radial and longitudinal thermal stresses are represented by $\sigma _{r}$ and $\sigma _{z}$, respectively, then
 A $\sigma _{r}=0,\sigma _{z}=0$ B $\sigma _{r}\neq 0,\sigma _{z}=0$ C $\sigma _{r}= 0,\sigma _{z}\neq 0$ D $\sigma _{r}\neq 0,\sigma _{z}\neq 0$
Strength of Materials   Strain Energy and Thermal Stresses
Question 8 Explanation:
If a body is allowed to expand or contract freely
with rise or fall in temperature then no stress are
induced in body. i.e.
$\sigma_{r}=0 \text { and } \sigma_{z}=0$
 Question 9
Two identical cantilever beams are supported as shown, with their free ends in contact through a rigid roller. After the load P is applied, the free ends will have A equal deflections but not equal slopes B equal slopes but not equal deflections C equal slopes as well as equal deflections D neither equal slopes nor equal deflections
Strength of Materials   Bending of Beams
Question 9 Explanation:
From the figure, we can say that load P applies a force on upper cantilever and the reaction force also applied on upper cantilever by the rigid roller. Due to this, deflections are occur in both the cantilever, which are equal in amount. But because of different forces applied by the P and rigid roller, the slopes are unequal.
 Question 10
The number of degrees of freedom of a planar linkage with 8 links and 9 simple revolute joints is
 A 1 B 2 C 3 D 4
Theory of Machine   Planar Mechanisms
Question 10 Explanation:
$J=9$
$h=0$
$\therefore$ Number of degree of freedom,
\begin{aligned} n &=3(1-1)-2 J-h \\ &=3(8-1)-2 \times 9-0\\ \therefore \quad &=3 \end{aligned} 