1
JEE Advanced 2017 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-2
f : R $$\to$$ R is a differentiable function such that f'(x) > 2f(x) for all x$$\in$$R, and f(0) = 1 then
A
f(x) > e2x in (0, $$\infty$$)
B
f'(x) < e2x in (0, $$\infty$$)
C
f(x) is increasing in (0, $$\infty$$)
D
f(x) is decreasing in (0, $$\infty$$)
2
JEE Advanced 2017 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-2
If $$f(x) = \left| {\matrix{ {\cos 2x} & {\cos 2x} & {\sin 2x} \cr { - \cos x} & {\cos x} & { - \sin x} \cr {\sin x} & {\sin x} & {\cos x} \cr } } \right|$$,

then
A
f(x) attains its minimum at x = 0
B
f(x) attains its maximum at x = 0
C
f'(x) = 0 at more than three points in ($$-$$$$\pi$$, $$\pi$$)
D
f'(x) = 0 at exactly three points in ($$-$$$$\pi$$, $$\pi$$)
3
JEE Advanced 2016 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-2
Let f: R $$\to \left( {0,\infty } \right)$$ and g : R $$\to$$ R be twice differentiable functions such that f'' and g'' are continuous functions on R. Suppose f'$$(2)$$ $$=$$ g$$(2)=0$$, f''$$(2)$$$$\ne 0$$ and g'$$(2)$$ $$\ne 0$$. If
$$\mathop {\lim }\limits_{x \to 2} {{f\left( x \right)g\left( x \right)} \over {f'\left( x \right)g'\left( x \right)}} = 1,$$ then
A
$$f$$ has a local minimum at $$x=2$$
B
$$f$$ has a local maximum at $$x=2$$
C
$$f''(2)>f(2)$$
D
$$f(x)-f''(x)=0$$ for at least one $$x \in R$$
4
JEE Advanced 2015 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-1
Let $$f, g :$$ $$\left[ { - 1,2} \right] \to R$$ be continuous functions which are twice differentiable on the interval $$(-1, 2)$$. Let the values of f and g at the points $$-1, 0$$ and $$2$$ be as given in the following table:
X = -1 X = 0 X = 2
f(x) 3 6 0
g(x) 0 1 -1

In each of the intervals $$(-1, 0)$$ and $$(0, 2)$$ the function $$(f-3g)''$$ never vanishes. Then the correct statement(s) is (are)

A
$$f'\left( x \right) - 3g'\left( x \right) = 0$$ has exactly three solutions in $$\left( { - 1,0} \right) \cup \left( {0,2} \right)$$
B
$$f'\left( x \right) - 3g'\left( x \right) = 0$$ has exactly one solution in $$(-1, 0)$$
C
$$f'\left( x \right) - 3g'\left( x \right) = 0$$ has exactly one solution in $$(0, 2)$$
D
$$f'\left( x \right) - 3g'\left( x \right) = 0$$ has exactly two solutions in $$(-1, 0)$$ and exactly two solutions in $$(0, 2)$$
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