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IITM BS Week 7 Mathematics 1 Questions | Prasnya
Mathematics 1 > Week 7
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Difficulty:
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Year:
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23Q
01
Let
f
1
,
f
2
f_1, f_2
f
1
,
f
2
be real-valued functions defined on a domain
D
D
D
. Let
Z
1
Z_1
Z
1
and
Z
2
Z_2
Z
2
denote the set of zeroes of
f
1
f_1
f
1
and
f
2
f_2
f
2
, respectively, i.e., for
i
∈
{
1
,
2
}
i \in \{1, 2\}
i
∈
{
1
,
2
}
,
Z
i
=
{
x
∈
D
∣
f
i
(
x
)
=
0
}
Z_i = \{x \in D \mid f_i(x) = 0\}
Z
i
=
{
x
∈
D
∣
f
i
(
x
)
=
0
}
. Which of the following is a valid domain for the real-valued function
f
1
/
f
2
f_1 / f_2
f
1
/
f
2
?
Single correct
MEDIUM
3 marks
10 May 2026
02
Find
lim
n
→
∞
a
n
\lim_{n \to \infty} a_n
lim
n
→
∞
a
n
for the sequence
{
a
n
}
\{a_n\}
{
a
n
}
such that
a
n
=
3
n
+
2
−
7
n
3
3
n
+
6
n
3
a_n = \frac{3^{n+2} - 7n^3}{3^n + 6n^3}
a
n
=
3
n
+
6
n
3
3
n
+
2
−
7
n
3
.
Numerical
EASY
2 marks
10 May 2026
03
Find
lim
n
→
∞
a
n
\lim_{n \to \infty} a_n
lim
n
→
∞
a
n
for the sequence
{
a
n
}
\{a_n\}
{
a
n
}
such that
a
n
=
n
5
−
3
n
3
+
sin
(
n
)
2
n
5
+
ln
(
n
)
+
n
2
a_n = \frac{n^5 - 3n^3 + \sin(n)}{2n^5 + \ln(n) + n^2}
a
n
=
2
n
5
+
l
n
(
n
)
+
n
2
n
5
−
3
n
3
+
s
i
n
(
n
)
.
Numerical
EASY
2 marks
21 December 2025
04
If the function
f
(
x
)
=
{
A
x
−
B
,
if
x
≤
−
1
2
x
2
+
3
A
x
+
B
,
if
−
1
<
x
≤
1
4
,
if
x
>
1
f(x) = \begin{cases} Ax - B, & \text{if } x \le -1 \\ 2x^2 + 3Ax + B, & \text{if } -1 < x \le 1 \\ 4, & \text{if } x > 1 \end{cases}
f
(
x
)
=
⎩
⎨
⎧
A
x
−
B
,
2
x
2
+
3
A
x
+
B
,
4
,
if
x
≤
−
1
if
−
1
<
x
≤
1
if
x
>
1
is continuous for all
x
∈
R
x \in \mathbb{R}
x
∈
R
, then find the value of
A
+
B
A + B
A
+
B
.
Numerical
MEDIUM
4 marks
21 December 2025
05
Find
lim
n
→
∞
a
n
\lim_{n \to \infty} a_n
lim
n
→
∞
a
n
for the sequence
{
a
n
}
\{a_n\}
{
a
n
}
such that
a
n
=
15
n
3
+
2
n
2
−
1
n
3
+
3
n
a_n = \frac{15n^3 + 2n^2 - 1}{n^3 + 3n}
a
n
=
n
3
+
3
n
15
n
3
+
2
n
2
−
1
.
Comprehension
EASY
2 marks
13 April 2025
06
Find
lim
n
→
∞
a
n
\lim_{n \to \infty} a_n
lim
n
→
∞
a
n
for the sequence
{
a
n
}
\{a_n\}
{
a
n
}
such that
a
n
=
1
8
+
(
−
1
)
n
n
a_n = \frac{1}{8} + \frac{(-1)^n}{n}
a
n
=
8
1
+
n
(
−
1
)
n
. Enter your answer correctly to two decimal places.
Comprehension
EASY
2 marks
13 April 2025
07
f
f
f
is continuous at
x
=
0
x = 0
x
=
0
.
Comprehension
EASY
1 marks
13 April 2025
08
Consider the function
f
:
R
→
R
f : \mathbb{R} \to \mathbb{R}
f
:
R
→
R
defined by
f
(
x
)
=
{
x
2
−
∣
x
∣
,
if
x
<
0
x
2
−
[
x
]
,
if
x
≥
0
f(x) = \begin{cases} x^2 - |x|, & \text{if } x < 0 \\ x^2 - [x], & \text{if } x \ge 0 \end{cases}
f
(
x
)
=
{
x
2
−
∣
x
∣
,
x
2
−
[
x
]
,
if
x
<
0
if
x
≥
0
. Which of the following option(s) is(are) correct?
Multiple correct
MEDIUM
3 marks
22 December 2024
09
Find the limit of the sequence
{
a
n
}
\{a_n\}
{
a
n
}
where
a
n
=
11
n
3
+
2
n
2
−
1
n
3
+
3
n
a_n = \frac{11n^3 + 2n^2 - 1}{n^3 + 3n}
a
n
=
n
3
+
3
n
11
n
3
+
2
n
2
−
1
as
n
→
∞
n \to \infty
n
→
∞
.
Comprehension
EASY
2 marks
22 December 2024
10
Find the limit of the sequence
{
u
n
}
\{u_n\}
{
u
n
}
where
u
n
=
1
8
+
(
−
1
)
n
n
u_n = \frac{1}{8} + \frac{(-1)^n}{n}
u
n
=
8
1
+
n
(
−
1
)
n
as
n
→
∞
n \to \infty
n
→
∞
. Enter your answer correctly to two decimal places.
Comprehension
EASY
2 marks
22 December 2024
11
Consider the following function
f
(
x
)
f(x)
f
(
x
)
:
f
(
x
)
=
{
x
(
x
+
1
)
(
x
+
2
)
,
if
x
≥
1
1
x
−
5
,
if
x
<
1
f(x) = \begin{cases} \frac{x}{(x+1)(x+2)}, & \text{if } x \ge 1 \\ \frac{1}{x-5}, & \text{if } x < 1 \end{cases}
f
(
x
)
=
{
(
x
+
1
)
(
x
+
2
)
x
,
x
−
5
1
,
if
x
≥
1
if
x
<
1
Which of the following options is (are) correct?
Multiple correct
MEDIUM
3 marks
01 September 2024
12
Choose the correct option(s).
Single correct
EASY
3 marks
01 September 2024
13
Note: Enter your answer correctly to two decimal places. Find
lim
n
→
∞
a
n
\lim_{n \to \infty} a_n
lim
n
→
∞
a
n
for
{
a
n
}
\{a_n\}
{
a
n
}
such that
a
n
=
n
2
+
5
n
6
n
2
+
1
a_n = \frac{n^2 + 5n}{6n^2 + 1}
a
n
=
6
n
2
+
1
n
2
+
5
n
.
Comprehension
EASY
2 marks
28 April 2024
14
Note: Enter your answer correctly to two decimal places. Find
lim
n
→
∞
a
n
\lim_{n \to \infty} a_n
lim
n
→
∞
a
n
for
{
a
n
}
\{a_n\}
{
a
n
}
such that
a
n
=
1
10
+
(
−
1
)
n
n
3
a_n = \frac{1}{10} + \frac{(-1)^n}{n^3}
a
n
=
10
1
+
n
3
(
−
1
)
n
.
Comprehension
EASY
1 marks
28 April 2024
15
The limit of
f
(
x
)
f(x)
f
(
x
)
at
x
=
0
x = 0
x
=
0
does not exist.
Comprehension
MEDIUM
1 marks
28 April 2024
16
The limit of
f
(
x
)
f(x)
f
(
x
)
at
x
=
2
x = 2
x
=
2
exists and it is equal to
f
(
2
)
=
−
1
f(2) = -1
f
(
2
)
=
−
1
, i.e.,
f
(
x
)
f(x)
f
(
x
)
is continuous at
x
=
2
x = 2
x
=
2
.
Comprehension
MEDIUM
1 marks
28 April 2024
17
The limit of
f
(
x
)
f(x)
f
(
x
)
at
x
=
−
2
x = -2
x
=
−
2
exists and it is equal to
f
(
−
2
)
=
−
1
f(-2) = -1
f
(
−
2
)
=
−
1
, i.e.,
f
(
x
)
f(x)
f
(
x
)
is continuous at
x
=
−
2
x = -2
x
=
−
2
.
Comprehension
HARD
1 marks
28 April 2024
18
f
(
x
)
f(x)
f
(
x
)
is continuous on the entire real line.
Comprehension
EASY
1 marks
28 April 2024
19
Find the value of
lim
x
→
2
f
(
x
)
\lim_{x \to 2} f(x)
lim
x
→
2
f
(
x
)
.
Comprehension
MEDIUM
1 marks
24 December 2023
20
Is this statement True or False: Let
a
n
=
1
f
(
n
+
1
/
2
)
a_n = \frac{1}{f(n + 1/2)}
a
n
=
f
(
n
+
1/2
)
1
. Then
lim
n
→
∞
a
n
\lim_{n \to \infty} a_n
lim
n
→
∞
a
n
does not exist.
Comprehension
MEDIUM
2 marks
24 December 2023
21
Suppose the
n
n
n
-th term of a sequence
{
a
n
}
\{a_n\}
{
a
n
}
is
n
2
+
3
n
−
sin
(
n
)
2
−
n
2
\frac{n^2 + 3n - \sin(n)}{2 - n^2}
2
−
n
2
n
2
+
3
n
−
s
i
n
(
n
)
. Let
{
b
n
}
\{b_n\}
{
b
n
}
be another sequence defined as
b
n
=
a
n
2
+
2
a
n
−
5
b_n = a_n^2 + 2a_n - 5
b
n
=
a
n
2
+
2
a
n
−
5
. Find the limit of the sequence
{
a
n
b
n
}
\{a_n b_n\}
{
a
n
b
n
}
.
Numerical
MEDIUM
3 marks
11 December 2022
22
Choose the correct options from the following.
Multiple correct
MEDIUM
3 marks
07 August 2022
23
If the piecewise function f(x) = m*x for x < 3 f(x) = n for x = 3 f(x) = -2x + 9 for x > 3 is continuous, then the value of m + n is
Numerical
EASY
2 marks
07 August 2022
Showing 23 questions.
Complete question index for this section:
Question 1 (single):
Question 2 (integer):
Question 3 (integer):
Question 4 (integer):
Question 5 (comprehension):
Question 6 (comprehension):
Question 7 (comprehension):
Question 8 (multiple):
Question 9 (comprehension):
Question 10 (comprehension):
Question 11 (multiple):
Question 12 (single):
Question 13 (comprehension):
Question 14 (comprehension):
Question 15 (comprehension):
Question 16 (comprehension):
Question 17 (comprehension):
Question 18 (comprehension):
Question 19 (comprehension):
Question 20 (comprehension):
Question 21 (integer):
Question 22 (multiple):
Question 23 (integer):
