GATE ME 2005

A line integral and a surface integral is related by stroke's theorem

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

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)

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

\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

\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}

Mean position \rightarrow It is the position where acceleration is zero and hence force is zero

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

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.

No. of links: 1=8
No. of revolute joints,
J=9
No. of higher pair,
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}