1 TIFR 2020 Mathematics The Eleventh Ray
1
1. Consider the sequences {an}∞ ∞ n n n
n=1 and {bn}n=1 de ned by an = (2 + 3 ) and
n
bn = n
. What is the limit of {bn}∞
n=1?
∑i=1 a1
i
(a) 2 (b) 3 (c) 5 (d) The limit does not exist.
Hint:
1
lim (2n + 3n) n = L
n→∞
log(2n + 3n)
⟹ lim = log(L)
n→∞ n
3
log(1 + ( 2 )n)
⟹ lim log(3) + = log(L)
n→∞ n
3 n
Since, log(1 + ( ) ) is bounded.
2
⟹L =3
Use Stolz theorem to get the limit of {bn}∞
n=1.
————————————————————————————————————————
2. Consider the set of continuous function f : [0,1] → ℝ that satisfy:
1
∫
f (x)(1 − f (x))d x = . Then the cordiality of this set is :
4
(a) 0 (b) 1 (c) 2 (d) more than 2
Hint:
1
1
∫0
f (x)(1 − f (x)) d x =
4
1
1
∫0
⟹ f (x)(1 − f (x)) − dx = 0
4
1
∫0
⟹ (2 f (x) − 1)2 d x = 0
1
⟹ f (x) =
2
————————————————————————————————————————
(x)
1
{
x 2 sin if x ≠ 0
3. Let f : ℝ → ℝ be de ned as: f (x) =
0 if x = 0
Which of the following statements is correct?
fi fi
, 2 TIFR 2020 Mathematics The Eleventh Ray
(a) f is a surjective function (b) f is bounded (c) The derivative of f exists and is
continuous on ℝ (d) {x ∈ ℝ | f (x) = 0} is a nite set.
Hint:
(x)
1
| x 2 sin | ≤ |x|
(h)
1
As lim h sin = 1 ≠ 0, we conclude derivate doest exist at 0.
h→0
1
Clearly, ∈ {x ∈ ℝ | f (x) = 0}
nπ
————————————————————————————————————————
4. Let {an}∞
n=1 be a strictly increasing bounded sequence of real numbers such that
lim an = A. Let f : [ai, A] → ℝ be a continuous function such that for each positive
n→∞
integer i, f /[ai, ai+1] : [ai, ai+1] → ℝ is either strictly increasing or strictly decreasing.
Consider the set B = {M ∈ ℝ | there exists in nitely many x ∈ [a1, A] such that f (x) = M .
Then the cardinality of B is :
(a) Necessarily 0. (b) at most 1. (c) possibly greater that 1, but nite. (d) possibly in nite.
Hint:
Observe, f assumes every value utmost once at each [ai, ai+1]. If there exist xi such that
ai ≤ xi ≤ ai+1and f (xi ) = M ∈ B for in nitely many i′s then by squeeze theorem
lim xn = A. Then f (xi ) = M = f (A).
n→∞
————————————————————————————————————————
5. Let f : ℝ → ℝ be a function that satis es: | f (x) − f (y) | ≤ | x − y | | sin(x − y) | , for all
x, y ∈ ℝ. Which of the following statement is correct?
(a) f is continuous but need not be uniformly continuous. (b) f is uniformly continuous
but not necessarily differentiable (c) f is differentiable. But its derivative may not be
continuous. (d) f is constant.
Hint:
| f (x) − f (y) | ≤ | x − y | | sin(x − y) | , can also be as: | f (x) − f (y) | ≤ ( | x − y | )2,
which inequality can be used to prove uniform continuity. Inequality
| f (x) − f (y) |
≤ | x − y | proves derivative exist everywhere and is a zero.
|x − y|
————————————————————————————————————————
6. Let C = {f : ℝ → ℝ | f is differentiable and lim (2f (x) + f′(x)) = 0}. Which of the
x→∞
following statements is correct?
(a) For each L with 0 ≠ L < ∞, there exists f ∈ C such that lim f (x) = L.
x→∞
 fi 
fifi fi fi fi
1
1. Consider the sequences {an}∞ ∞ n n n
n=1 and {bn}n=1 de ned by an = (2 + 3 ) and
n
bn = n
. What is the limit of {bn}∞
n=1?
∑i=1 a1
i
(a) 2 (b) 3 (c) 5 (d) The limit does not exist.
Hint:
1
lim (2n + 3n) n = L
n→∞
log(2n + 3n)
⟹ lim = log(L)
n→∞ n
3
log(1 + ( 2 )n)
⟹ lim log(3) + = log(L)
n→∞ n
3 n
Since, log(1 + ( ) ) is bounded.
2
⟹L =3
Use Stolz theorem to get the limit of {bn}∞
n=1.
————————————————————————————————————————
2. Consider the set of continuous function f : [0,1] → ℝ that satisfy:
1
∫
f (x)(1 − f (x))d x = . Then the cordiality of this set is :
4
(a) 0 (b) 1 (c) 2 (d) more than 2
Hint:
1
1
∫0
f (x)(1 − f (x)) d x =
4
1
1
∫0
⟹ f (x)(1 − f (x)) − dx = 0
4
1
∫0
⟹ (2 f (x) − 1)2 d x = 0
1
⟹ f (x) =
2
————————————————————————————————————————
(x)
1
{
x 2 sin if x ≠ 0
3. Let f : ℝ → ℝ be de ned as: f (x) =
0 if x = 0
Which of the following statements is correct?
fi fi
, 2 TIFR 2020 Mathematics The Eleventh Ray
(a) f is a surjective function (b) f is bounded (c) The derivative of f exists and is
continuous on ℝ (d) {x ∈ ℝ | f (x) = 0} is a nite set.
Hint:
(x)
1
| x 2 sin | ≤ |x|
(h)
1
As lim h sin = 1 ≠ 0, we conclude derivate doest exist at 0.
h→0
1
Clearly, ∈ {x ∈ ℝ | f (x) = 0}
nπ
————————————————————————————————————————
4. Let {an}∞
n=1 be a strictly increasing bounded sequence of real numbers such that
lim an = A. Let f : [ai, A] → ℝ be a continuous function such that for each positive
n→∞
integer i, f /[ai, ai+1] : [ai, ai+1] → ℝ is either strictly increasing or strictly decreasing.
Consider the set B = {M ∈ ℝ | there exists in nitely many x ∈ [a1, A] such that f (x) = M .
Then the cardinality of B is :
(a) Necessarily 0. (b) at most 1. (c) possibly greater that 1, but nite. (d) possibly in nite.
Hint:
Observe, f assumes every value utmost once at each [ai, ai+1]. If there exist xi such that
ai ≤ xi ≤ ai+1and f (xi ) = M ∈ B for in nitely many i′s then by squeeze theorem
lim xn = A. Then f (xi ) = M = f (A).
n→∞
————————————————————————————————————————
5. Let f : ℝ → ℝ be a function that satis es: | f (x) − f (y) | ≤ | x − y | | sin(x − y) | , for all
x, y ∈ ℝ. Which of the following statement is correct?
(a) f is continuous but need not be uniformly continuous. (b) f is uniformly continuous
but not necessarily differentiable (c) f is differentiable. But its derivative may not be
continuous. (d) f is constant.
Hint:
| f (x) − f (y) | ≤ | x − y | | sin(x − y) | , can also be as: | f (x) − f (y) | ≤ ( | x − y | )2,
which inequality can be used to prove uniform continuity. Inequality
| f (x) − f (y) |
≤ | x − y | proves derivative exist everywhere and is a zero.
|x − y|
————————————————————————————————————————
6. Let C = {f : ℝ → ℝ | f is differentiable and lim (2f (x) + f′(x)) = 0}. Which of the
x→∞
following statements is correct?
(a) For each L with 0 ≠ L < ∞, there exists f ∈ C such that lim f (x) = L.
x→∞
 fi 
fifi fi fi fi
