1
GATE EE 2017 Set 1
+2
-0.6
A function $$f(x)$$ is defined as
$$f\left( x \right) = \left\{ {\matrix{ {{e^x},x < 1} \cr {\ln x + a{x^2} + bx,x \ge 1} \cr } \,\,,\,\,} \right.$$ where $$x \in R.$$

Which one of the following statements is TRUE?

A
$$f(x)$$ is NOT differentiable at $$x=1$$ for any values of $$a$$ and $$b.$$
B
$$f(x)$$ is differentiable at $$x=1$$ for the unique values of $$a$$ and $$b.$$
C
$$f(x)$$ is differentiable at $$x=1$$ for all values of $$a$$ and $$b$$ such that $$a+b=c.$$
D
$$f(x)$$ is differentiable at $$x=1$$ for all values of $$a$$ and $$b.$$
2
GATE EE 2017 Set 2
+2
-0.6
Let $$g\left( x \right) = \left\{ {\matrix{ { - x} & {x \le 1} \cr {x + 1} & {x \ge 1} \cr } } \right.$$ and
$$f\left( x \right) = \left\{ {\matrix{ {1 - x,} & {x \le 0} \cr {{x^{2,}}} & {x > 0} \cr } } \right..$$
Consider the composition of $$f$$ and $$g,$$ i.e., $$\left( {f \circ g} \right)\left( x \right) = f\left( {g\left( x \right)} \right).$$ The number of discontinuities in $$\left( {f \circ g} \right)\left( x \right)$$ present in the interval $$\left( { - \infty ,0} \right)$$ is
A
$$0$$
B
$$1$$
C
$$2$$
D
$$4$$
3
GATE EE 2016 Set 2
+2
-0.6
The value of the integral $$\,\,2\int_{ - \infty }^\infty {\left( {{{\sin \,2\pi t} \over {\pi t}}} \right)} dt\,\,$$ is equal to
A
$$0$$
B
$$0.5$$
C
$$1$$
D
$$2$$
4
GATE EE 2016 Set 1
Numerical
+2
-0
Let $$\,\,S = \sum\limits_{n = 0}^\infty {n{\alpha ^n}} \,\,$$ where $$\,\,\left| \alpha \right| < 1.\,\,$$ The value of $$\alpha$$ in the range $$\,\,0 < \alpha < 1,\,\,$$ such that $$\,\,S = 2\alpha \,\,$$ is ___________.