IIT Jee maths

1. Graph, Function and Limit
1 Single correct answer type
1. The range of sin−1
√
x2 + x + 1 is
(a)(0, π/2] (b)(0, π/3] (c)[π/3, π/2] (d)[π/6, π/3].
2. The domain of
log2(x + 3)
x2 + 3x + 2
is
(a)R−{−1, −2} (b)R−(−2, ∞)∞ (c)R−{−1, −2, −3} (d)(−3, ∞)−{−1, −2}.
3. If the graph of
ax
− 1
xn(ax + 1)
is symmetric about y-axis then n equals
(a)2 (b)2/3 (c)1/4 (d) − 1/3.
4. The equation ||x − 2| + a| = 4 have four distinct real roots for x then a belongs to
interval
(a)(−∞, −4) (b)(−∞, 0] (c)[4, ∞) (d)NOT.
5. If f is a periodic function, g is polynomial, f(g(x)) is periodic, g(2) = 3, g(4) = 7
then g(6) is
(a)13 (b)15 (c)11 (d)NOT.
6. If
f(x) =
(
x2
sin πx
2
, |x| < 1
x|x|, |x| ≥ 1
(a) even function (b) odd function (c) periodic function (d) NOT
7. A function f(x) satisfies the functional equation x2
f(x) + f(1 − x) = 2x − x4
for all
real x then f(x) is
(a)x2
(b)1 − x2
(c)1 + x2
(d)x2
+ x + 1.
8. The period of function [6x + 7] + cosπx − 6x, where [.] is greatest integer function.
(a)3 (b)2π (c)2 (d)NOT.
9. If f(x) is a real valued function defined as f(x) = ln(1 − sin x), then the graph of
f(x) is symmetrical about
(a)line x = π (b)y − axis (c)line x = π (d)origin.

2. 10. The number of roots of the equation x sin x = 1 in x ∈ [−2π, 0) ∪ (0, 2π)
(a)2 (b)3 (c)4 (d)0.
11. If f(x) = sin x + cos x, g(x) = x2
−, then g(f(x)) is invertible in the domain
(a)[0, π/2] (b)[−π/4, π/4] (c)[−π/2, π/2] (d)[0, π].
12. The number of solutions of [x]2
= x + 2{x}, where [.] is greatest integer function
and {.} is fractional part function
(a)2 (b)4 (c)6 (d)NOT.
13. The domain of the function f(x) = sin−1 8(3)x−2
1 − 32(x−1)
(a)[2, ∞] (b)[−4, 0] (c)(−4, 0) (d)NOT.
14. Let f(
2x − 3
x − 2
) = 5x − 2, then f−1
(13) is
(a)2 (b)3 (c)4 (d)NOT.
15. The value of
lim
x→a
√
a2 − x2 cot
π
2
r
a − x
a + x
.
is
(a)2a/π (b) − 2a/π (c)4a/π (d) − 4a/π.
16. If
f(x) =
(
x + 1, x > 0
2 − x, x ≤ 0
g(x) =







x + 3, x < 1
x2
− 2x − 2, 1 ≤ x < 2
x − 5, x ≥ 2
then
lim
x→0
g(f(x)).
is

3. (a)2 (b)1 (c) − 3 (d)DoesNotExist.
17. The value of
lim
x→0
{tan(π/4 + x)1/x
}).
is
(a)e2
(b)1 (c)e3
(d)e.
18. If
lim
x→∞
(
x2
+ x + 1
x + 1
− ax − b) = 4.
then
(a)a = 1, b = 4 (b)a = 1, b = −4 (c)a = 2, b = −3 (d)a = 2, b = 3.
19.
lim
x→0
sin(π cos2
x)
x2
.
is equal to
(a) − π (b)π (c)π/2 (d)1.
20. The integral value of n for which
lim
x→0
cos2
x − cos x − ex
cos x + ex
− x3
/2
xn
.
is a finite number
(a)2 (b)3 (c)4 (d)1.
2 More than one correct answer type
21. Let f(x) = sgn(cot−1
x) + tan(π
2
[x]), where [.] is greatest integer function(GIF).
Then which of the following is true for f(x)
(a) many-one but not even function
(b) periodic function
(c) bounded function
(d) Graph remains above x- axis

4. 22. f : R → [−1, ∞) and f(x) = ln([| sin 2x| + | cos 2x|]))(where [.] is GIF then
(a) f(x) has range integers(Z)
(b) f(x) is periodic function with fundamental period π/4
(c) f(x) is invertible in [0, π/4]
(d) f(x) is into function
23. Which of the function given by following functional equation have the graph sym-
metrical about origin
(a) f(x) + f(y) = f( x+y
1−xy
)
(b) f(x) + f(y) = f(x
p
1 − y2 + y
√
1 − x2)
(c) f(x + y) = f(x) + f(y)
(d) f(x)f(y) = f(x) + f(y)
24. f(x) = cos[π2
]x + cos[−π2
]x, where [.] is GIF then
(a) f(π/2) = −1
(b) f(π) = 1
(c) f(−π) = 0
(d) f(π/4) = 1
25. Which of the following function is periodic. (Here [.] is GIF)
(a) (−1)[2x/π]
,
(b) x − [x + 3] + tan(πx/2)
(c) esinx
(d) eπ2
26. Consider the function satisfies 2f(sinx) + f(cosx) = x, then
(a) domain of f(x) is R,
(b) domain of f(x) is [-1, 1],
(c) range of of f(x) is −π/3, π/3],
(d) range of f(x) is R,
27.
lim
n→∞
1
1 + n sin2
nx
.
is equal to
(a) − 1 (b)0 (c)1 (d)∞.
28. If
L = lim
x→0
a −
√
a2 − x2 − x2
/4
x4
.
. If L is finite then

5. (a)a = 2 (b)a = 1 (c)L = 1/64 (d)L = 1/32.
29. If
L = lim
x→0
f(x)
x2
= 2.
Here [.] is GIF then
(a) lim
x→0
[f(x)] = 0
(b) lim
x→0
[f(x)] = 1
(c) lim
x→0
[
f(x)
x
] does not exist
(d) lim
x→0
[
f(x)
x
] exist
30. If
L = lim
n→∞
x
x2n + 1
.
then
(a) f(1+
) + f(1−
) = 0
(b) f(1+
) + f(1−
) + f(1) = 3/2
(c) f(−1+
) + f(−1−
) = −1
(d) f(1+
) + f(−1−
) = 0