1. The document contains 30 multiple choice questions about limits, functions, and graphs. Questions cover topics like domains, ranges, limits, periodicity, symmetry, and functional equations.
2. Several questions involve greatest integer functions, fractional parts of numbers, trigonometric functions like sine, cosine, and tangent, and inverse trigonometric functions.
3. Correct answers are requested for questions involving evaluating limits, determining properties of functions, finding function values, and identifying function domains and ranges.
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Iit jee question_paper
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