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.

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