1
GATE ECE 2017 Set 1
+1
-0.3
Consider the following statements about the linear dependence of the real valued functions $${y_1} = 1,\,\,{y_2} = x$$ and $${y_3} = {x^2}$$. Over the field of real numbers.

$${\rm I}.\,\,\,\,\,$$ $${y_1},{y_2}$$ and $${y_3}$$ are linearly independent on $$- 1 \le x \le 0$$
$${\rm II}.\,\,\,\,\,$$ $${y_1},{y_2}$$ and $${y_3}$$ are linearly dependent on $$0 \le x \le 1$$
$${\rm III}.\,\,\,\,\,$$ $${y_1},{y_2}$$ and $${y_3}$$ are linearly independent on $$0 \le x \le 1$$
$${\rm IV}.\,\,\,\,\,$$ $${y_1},{y_2}$$ and $${y_3}$$ are linearly dependent on $$- 1 \le x \le 0$$

Which one among the following is correct?

A
Both $${\rm I}$$ and $${\rm I}$$$${\rm I}$$ are true
B
Both $${\rm I}$$ and $${\rm I}$$$${\rm I}$$$${\rm I}$$ are true
C
Both $${\rm I}$$$${\rm I}$$ and $${\rm IV}$$ are true
D
Both $${\rm I}$$$${\rm I}$$$${\rm I}$$ and $${\rm IV}$$ are true
2
GATE ECE 2015 Set 2
+1
-0.3
The general solution of the differential equation $$\,\,{{dy} \over {dx}} = {{1 + \cos 2y} \over {1 - \cos 2x}}\,\,$$ is
A
$$\,\,\tan \,y - \cot \,x = C\,\,$$
B
$$\tan \,x - \cot \,y = C\,$$
C
$$\,\,\tan \,y + \cot \,x = C\,\,$$
D
$$\tan \,x + \cot \,y = C\,$$
3
GATE ECE 2015 Set 2
+1
-0.3
Consider the differential equation $$\,\,{{dx} \over {dt}} = 10 - 0.2\,x$$ with initial condition $$x(0)=1.$$ The response $$x(t)$$ for $$t > 0$$ is
A
$$2 - {e^{ - 0.2t}}$$
B
$$2 - {e^{ 0.2t}}$$
C
$$50 - 49\,{e^{ - 0.2t}}$$
D
$$50 - 49\,{e^{ 0.2t}}$$
4
GATE ECE 2014 Set 4
+1
-0.3
If $$a$$ and $$b$$ are constants, the most general solution of the differential equation $$\,{{{d^2}x} \over {d{t^2}}} + 2{{dx} \over {dt}} + x = 0$$ is
A
$$a{e^{ - t}}$$
B
$$a{e^{ - t}} + bt{e^{ - t}}$$
C
$$a{e^t} + bt{e^{ - t}}$$
D
$$a{e^{ - 2t}}$$
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