Week 7 Mathematics 1 Questions

Q1

The value of

\displaystyle \lim_{x\to 0}\frac{(1+x)^{1/2}-(1-x)^{1/2}}{x}

is:

Single correctAUGUST 03, 2025

AUGUST 03, 2025

Q2

Let Figure M2W2P1 represent the graph of a function

f

. The solid points denote the value of the function at the points, and the values denoted by the hollow points are not taken by the function. Choose the set of correct options.

Multiple correctAUGUST 03, 2025

AUGUST 03, 2025

Q3

Consider the functions

f(x)=\begin{cases}x-6,&x<6\\x-7,&x\ge 6\end{cases}

and

g(x)=\begin{cases}(x-6)^2,&x<6.5\\(x-7)^2,&x\ge 6.5\end{cases}

. Which of the following statements is/are correct?

Multiple correctAUGUST 03, 2025

AUGUST 03, 2025

Q4

Find the value of

\displaystyle \lim_{x\to\infty}\frac{(10x-1)(x+3)(x+2)(x+1)x}{(5x-1)(x+9)(x+4)(x+8)x}

.

Integer answerAUGUST 03, 2025

AUGUST 03, 2025

Q5

Let

\{a_n\}

be a sequence defined as

a_n=\frac{27n^4+25n^3+2n^5+n}{4n^2+2n+10n^3+5n^5+n}

. Consider a sequence

\{b_n\}

defined as

b_n=2^{5a_n}

. Find the limit of

\{b_n\}

.

Integer answerAUGUST 03, 2025

AUGUST 03, 2025

Q6

The value of

\displaystyle \lim_{x\to0}\frac{(1+x)^{1/2}-(1-x)^{1/2}}{x}

is:

Single correctMARCH 16, 2025

MARCH 16, 2025

Q7

\{a_n\}

is such that

a_n=\frac{3^{n+2}-7n}{3^n+6n}

. Find

\lim_{n\to+\infty}a_n

.

ComprehensionMARCH 16, 2025

MARCH 16, 2025

Q8

\{a_n\}

is such that

a_n=\frac{2n+5(-1)^n}{4n-3(-1)^n}

. Find

\lim_{n\to+\infty}a_n

.

ComprehensionMARCH 16, 2025

MARCH 16, 2025

Q9

If the function

f(x)=\begin{cases}Ax-B, & x\le-1,\\ 2x^2+3Ax+B, & -1<x\le1,\\ 4, & x>1\end{cases}

is continuous for all

x\in\mathbb{R}

, then find the value of

6(A+B)

.

Integer answerDECEMBER 01, 2024

DECEMBER 01, 2024

Q10

\{a_n\}

is such that

a_n=\frac{n^3-3n^2+\sin n}{2n^3+\ln n+n^2}

. Find

\lim_{n\to\infty}a_n

.

ComprehensionDECEMBER 01, 2024

DECEMBER 01, 2024

Q11

\{a_n\}

is such that

a_n=\frac{e^{3n}+n^4}{e^{5n}+n^5}

. Find

\lim_{n\to\infty}a_n

.

ComprehensionDECEMBER 01, 2024

DECEMBER 01, 2024

Q12

Is the statement True or False: The left-hand limit (LHL) and right-hand limit (RHL) of the given function

f(x)

exist at

x=0

and are equal to each other.

ComprehensionAUGUST 04, 2024

AUGUST 04, 2024

Q13

Is the statement True or False: The function

f(x)

is continuous at

x=0

.

ComprehensionAUGUST 04, 2024

AUGUST 04, 2024

Q14

Find the total number of points in

[-3,3]

at which

f(x)

is not continuous.

ComprehensionAUGUST 04, 2024

AUGUST 04, 2024

Q15

Is the statement True or False: The function

|x+1|f(x)

is continuous at

x=-1

.

Single correctAUGUST 04, 2024

AUGUST 04, 2024

Q16

Calculate the limit of the following function at

x=0

:

f(x)=x^2-2x+4

if

x\ge0

, and

f(x)=e^{x^2}+3

if

x<0

.

ComprehensionAUGUST 04, 2024

AUGUST 04, 2024

Q17

Find the limit of the sequence

\{a_n\}

such that

a_n=\frac{6+6\cdot2^2+6\cdot3^2+\cdots+6\cdot n^2}{\sqrt{4n^6+5}}

.

ComprehensionAUGUST 04, 2024

AUGUST 04, 2024

Q18

Find the limit of the sequence

\{a_n\}

such that

a_n=\frac{100n^2-11}{100n^3+7}

.

ComprehensionAUGUST 04, 2024

AUGUST 04, 2024

Q19

Consider the function

f(x)=\frac{2x^2}{|x|}

. Then

\lim_{x\to0} f(x)

is:

Integer answerMARCH 24, 2024

MARCH 24, 2024

Q20

Consider the function

f(x)=\begin{cases}\frac{3x}{(x+2)^2}, & x\le -1,\\ 2x-3, & -1<x\le 1,\\ -\frac{8}{x+1}, & x>1.\end{cases}

Find the total number of points in

(-2,2]

at which

f(x)

is not continuous.

Integer answerMARCH 24, 2024

MARCH 24, 2024

Q21

\{a_n\}

is such that

a_n=\frac{100n^2-11}{100n^3+7}

. Find

\lim_{n\to\infty} a_n

.

ComprehensionMARCH 24, 2024

MARCH 24, 2024

Q22

Evaluate the following limit:

\lim_{x\to2}\frac{x^6-24x-16}{x^3+2x-12}

.

ComprehensionMARCH 24, 2024

MARCH 24, 2024

Q23

Choose the correct option for

f(x)=\frac{1}{x-1}

.

Single correctDECEMBER 03, 2023

DECEMBER 03, 2023

Q24

Find the limit of the sequence

\{a_n\}

such that

a_n=\frac{n^3+2n^2-1}{n^3+3n+1}

.

ComprehensionDECEMBER 03, 2023

DECEMBER 03, 2023

Q25

Find the limit of the sequence

\{a_n\}

such that

a_n=\frac{n^3+2^n}{3^n+7^n}

.

ComprehensionDECEMBER 03, 2023

DECEMBER 03, 2023

Q26

Given a function

f(x)=\begin{cases}\frac{|x|}{x}, & x\ne0,\\ 1, & x=0.\end{cases}

Which of the following options is/are true?

Multiple correctDECEMBER 03, 2023

DECEMBER 03, 2023

Q27

Consider a function

f

defined as

f(x)=\begin{cases}3mx+n, & x<1,\\ 11, & x=1,\\ 5mx-2n, & x>1.\end{cases}

If

f

is continuous at

x=1

, then find the value of

m+n

.

Integer answerDECEMBER 03, 2023

DECEMBER 03, 2023

Q28

Which of the following options is/are true?

Multiple correctAUGUST 06, 2023

AUGUST 06, 2023

Q29

What is the limit of the sequence

\{a_n\}

?

ComprehensionAUGUST 06, 2023

AUGUST 06, 2023

Q30

What is the limit of the sequence

\{b_n\}

?

ComprehensionAUGUST 06, 2023

AUGUST 06, 2023

Q31

Find the value of

b

.

ComprehensionAUGUST 06, 2023

AUGUST 06, 2023

Q32

Define a function

f(x)=\begin{cases}\frac{x-3}{|x-3|}, & x\ne3,\\ 1, & x=3.\end{cases}

Which of the following options is/are true?

Multiple correctAUGUST 06, 2023

AUGUST 06, 2023

Q33

Find the limit of the sequence

\{a_n\}

such that

a_n=\frac{2022+8\cdot2023^n}{2021+4\cdot2023^n}

.

ComprehensionAPRIL 02, 2023

APRIL 02, 2023

Q34

Find the limit of the sequence

\{a_n\}

such that

a_n=\frac{8n^2+10n}{2n^2+6n-7}

.

ComprehensionAPRIL 02, 2023

APRIL 02, 2023

Q35

Find the limit

\lim_{x\to0}\frac{\sqrt{4+8x}-2}{x}

.

ComprehensionAPRIL 02, 2023

APRIL 02, 2023

Q36

Find the limit

\lim_{x\to0^+}\frac{\sin^2 x}{\sqrt{2x}}

.

ComprehensionAPRIL 02, 2023

APRIL 02, 2023

Q37

Find the number of points where the function

f(x)

is not continuous.

ComprehensionAPRIL 02, 2023

APRIL 02, 2023

Q38

Find the left limit

\lim_{x\to4^-}f(x)

.

ComprehensionAPRIL 02, 2023

APRIL 02, 2023

Q39

Find the right limit

\lim_{x\to4^+}10f(x)

.

ComprehensionAPRIL 02, 2023

APRIL 02, 2023

Q40

Consider a function

f

defined as

f(x)=\begin{cases}\frac{1}{e^{1/x}+1},&x\ne0\\0,&x=0.\end{cases}

Which of the following option(s) is/are true?

Multiple correctNOVEMBER 20, 2022

NOVEMBER 20, 2022

Q41

Consider a sequence

(1,2,3,4,5,6,\ldots)

, that is,

a_n=n

for all

n\in\mathbb{N}

. Which of the following sequences are subsequences of

\{a_n\}

?

Multiple correctNOVEMBER 20, 2022

NOVEMBER 20, 2022

Q42

Consider a function

f(x)=\begin{cases}mx^2-n,&x<1\\2,&x=1\\x+n,&x>1.\end{cases}

If

f

is continuous at

x=1

, then find the value of

m+n

.

Integer answerNOVEMBER 20, 2022

NOVEMBER 20, 2022

Q43

Evaluate

\lim_{x\to0}\frac{\sin^3 x}{\sin(x^3)}

.

ComprehensionNOVEMBER 20, 2022

NOVEMBER 20, 2022

Q44

Evaluate

\lim_{x\to0}\frac{x+\tan x}{\sin x}

.

ComprehensionNOVEMBER 20, 2022

NOVEMBER 20, 2022

Q45

Find the limit of the sequence

\{a_n\}

such that

a_n=\frac{4n-1}{3n^2+9}

.

ComprehensionNOVEMBER 20, 2022

NOVEMBER 20, 2022

Q46

Find the limit of the sequence

\{a_n\}

such that

a_n=\frac{10n+1+\sin n}{n}

if

n

is odd, and

a_n=\frac{10n-1+\cos n}{n}

if

n

is even.

ComprehensionNOVEMBER 20, 2022

NOVEMBER 20, 2022

All questions