Page 28 - ISC-12

Multiple Correct Choices Type Jai Baba Ki5
 
n n
n
n (x + n) x + . . . x +
 2 n 
19. Let f(x) = lim  , for allx > 0. Then

n→∞  n 2 n 2 
2
2
2
2
n!(x + n ) x + . . . x +
4 n 2

1 1 2
(A) f ≥ f(1). (B) f ≤ f .
2 3 3
0
0
f (3) f (2)
0
(C) f (2) ≤ 0. (D) ≥ .
f(3) f(2)
[JEE 2016]
3
3
20. Let a, b ∈ R and f : R → R be defined by f(x) = a cos (|x − x|) + b |x| sin (|x + x|). Then f is
(A) differentiable at x = 0 if a = 0 and b = 1.
(B) differentiable at x = 1 if a = 1 and b = 0.
(C) NOT differentiable at x = 0 if a = 1 and b = 0.
(D) NOT differentiable at x = 1 if a = 1 and b = 1.

[JEE 2016]

00
00
21. Let f : R → (0, ∞) and g : R → R be twice differentiable functions such that f and g are
0
0
00
continuous functions on R. Suppose f (2) = g(2) = 0, f (2) 6= 0 and g (2) 6= 0.
f(x)g(x)
If lim = 1, then
0
0
x→2 f (x)g (x)
(A) f has a local minimum at x = 2.
(B) f has a local maximum at x = 2.
00
(C) f (2) > f(2).
00
(D) f(x) − f (x) = 0 for at least one x ∈ R.
[JEE 2016]

1 1
22. Let f : − , 2 → R and g : − , 2 → R be functions defined by
2 2
2
f(x) = bx − 3c and g(x) = |x| f(x) + |4x − 7| f(x),
where byc denotes the greatest integer less than or equal to y for y ∈ R. Then

1
(A) f is discontinuous exactly at three points in − , 2 .
2

1
(B) f is discontinuous exactly at four points in − , 2 .
2

1
(C) g is NOT differentiable exactly at four points in − , 2 .
2

1
(D) g is NOT differentiable exactly at five points in − , 2 .
2
[JEE 2016]