# 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:
$\text { When } \quad I=\int_{0}^{8} \int_{x / 4}^{2} f(x \cdot y) dydx$

$I=\int_{0}^{2} \int_{0}^{4 y} f(x)dydx$

There are 5 questions to complete.