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Week 8 Mathematics 1 Questions | Prasnya
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Type:
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33Q
01
f
(
x
)
=
a
sin
∣
x
∣
+
b
e
∣
x
∣
f(x)=a\sin|x|+be^{|x|}
f
(
x
)
=
a
sin
∣
x
∣
+
b
e
∣
x
∣
is differentiable at
x
=
0
x=0
x
=
0
, if and only if
Single correct
MEDIUM
4 marks
03 August 2025
02
If
y
=
m
x
+
c
y=mx+c
y
=
m
x
+
c
is the equation of the tangent of
f
f
f
at
x
=
3
x=3
x
=
3
, choose the correct options.
Comprehension
MEDIUM
3 marks
03 August 2025
03
Find the number of points where
∣
f
(
x
)
∣
|f(x)|
∣
f
(
x
)
∣
is not differentiable.
Comprehension
MEDIUM
3 marks
03 August 2025
04
f
(
x
)
=
a
sin
∣
x
∣
+
b
e
∣
x
∣
f(x)=a\sin|x|+be^{|x|}
f
(
x
)
=
a
sin
∣
x
∣
+
b
e
∣
x
∣
is differentiable at
x
=
0
x=0
x
=
0
, if and only if
Single correct
MEDIUM
4 marks
16 March 2025
05
Consider the function
f
:
R
→
R
f:\mathbb{R}\to\mathbb{R}
f
:
R
→
R
defined by
f
(
x
)
=
{
x
2
−
∣
x
∣
,
x
<
0
,
x
2
+
∣
x
∣
,
x
≥
0.
f(x)=\begin{cases}x^2-|x|, & x<0,\\ x^2+|x|, & x\ge0.\end{cases}
f
(
x
)
=
{
x
2
−
∣
x
∣
,
x
2
+
∣
x
∣
,
x
<
0
,
x
≥
0.
Which of the following option(s) is(are) correct?
Multiple correct
MEDIUM
4 marks
16 March 2025
06
Let
f
f
f
be a differentiable function at
x
=
2
x=2
x
=
2
. The tangent line to the curve represented by the function
f
f
f
at the point
(
2
,
6
)
(2,6)
(
2
,
6
)
passes through the point
(
6
,
−
18
)
(6,-18)
(
6
,
−
18
)
. What will be the value of
f
′
(
2
)
f'(2)
f
′
(
2
)
?
Numerical
EASY
3 marks
16 March 2025
07
If
f
(
x
)
=
g
(
x
2
+
7
x
)
×
h
(
x
3
+
2
x
)
f(x)=g(x^2+7x)\times h(x^3+2x)
f
(
x
)
=
g
(
x
2
+
7
x
)
×
h
(
x
3
+
2
x
)
,
g
′
(
0
)
=
g
(
0
)
≠
0
g'(0)=g(0)\ne0
g
′
(
0
)
=
g
(
0
)
=
0
, and
h
′
(
0
)
=
h
(
0
)
≠
0
h'(0)=h(0)\ne0
h
′
(
0
)
=
h
(
0
)
=
0
, then find the value of
f
′
(
0
)
f
(
0
)
\frac{f'(0)}{f(0)}
f
(
0
)
f
′
(
0
)
.
Numerical
MEDIUM
4 marks
16 March 2025
08
Consider the following function:
f
(
x
)
=
{
sin
x
x
,
x
≠
0
,
1
,
x
=
0.
f(x)=\begin{cases}\frac{\sin x}{x}, & x\ne0,\\ 1, & x=0.\end{cases}
f
(
x
)
=
{
x
s
i
n
x
,
1
,
x
=
0
,
x
=
0.
Which of the following option(s) is/are true about
f
(
x
)
f(x)
f
(
x
)
?
Multiple correct
MEDIUM
4 marks
01 December 2024
09
Let
f
f
f
be a polynomial of degree
5
5
5
, given by
f
(
x
)
=
a
5
x
5
+
a
4
x
4
+
a
3
x
3
+
a
2
x
2
+
a
1
x
+
a
0
f(x)=a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0
f
(
x
)
=
a
5
x
5
+
a
4
x
4
+
a
3
x
3
+
a
2
x
2
+
a
1
x
+
a
0
. Let
f
′
(
b
)
f'(b)
f
′
(
b
)
denote the derivative of
f
f
f
at
x
=
b
x=b
x
=
b
. Choose the set of correct options.
Multiple correct
MEDIUM
4 marks
01 December 2024
10
Let
f
f
f
be a differentiable function such that
f
′
(
9
)
=
4
f'(9)=4
f
′
(
9
)
=
4
and
f
(
9
)
=
−
14
f(9)=-14
f
(
9
)
=
−
14
. If
y
=
a
x
+
b
y=ax+b
y
=
a
x
+
b
denotes the tangent of the function
f
f
f
at
x
=
9
x=9
x
=
9
, then find the value of
a
−
b
a-b
a
−
b
.
Numerical
EASY
4 marks
01 December 2024
11
Consider the graphs given below. Choose the set of correct options.
Multiple correct
MEDIUM
3 marks
04 August 2024
12
If
f
(
x
)
=
9
−
x
2
f(x)=\sqrt{9-x^2}
f
(
x
)
=
9
−
x
2
, then find out the value of
5
lim
x
→
2
f
(
x
)
−
f
(
2
)
x
−
2
\sqrt{5}\,\lim_{x\to2}\frac{f(x)-f(2)}{x-2}
5
lim
x
→
2
x
−
2
f
(
x
)
−
f
(
2
)
.
Numerical
MEDIUM
3 marks
04 August 2024
13
Is the statement True or False: Left-hand derivative (LHD) and right-hand derivative (RHD) of the function
f
(
x
)
f(x)
f
(
x
)
exist and are equal to each other at
x
=
−
1
x=-1
x
=
−
1
.
Comprehension
MEDIUM
2 marks
04 August 2024
14
Is the statement True or False: The derivative of the function
f
f
f
at
x
=
0
x=0
x
=
0
is
1
1
1
.
Comprehension
MEDIUM
2 marks
04 August 2024
15
Calculate the limit of the following function at
x
=
1
x=1
x
=
1
:
f
(
x
)
=
x
4
−
3
x
3
+
2
x
4
−
5
x
3
+
3
x
2
+
1
f(x)=\frac{x^4-3x^3+2}{x^4-5x^3+3x^2+1}
f
(
x
)
=
x
4
−
5
x
3
+
3
x
2
+
1
x
4
−
3
x
3
+
2
, if
x
≠
1
x\ne1
x
=
1
.
Comprehension
MEDIUM
2 marks
04 August 2024
16
Choose the set of correct options.
Multiple correct
EASY
4 marks
24 March 2024
17
If
f
(
x
)
=
9
−
x
2
f(x)=\sqrt{9-x^2}
f
(
x
)
=
9
−
x
2
, then find the value of
8
8
×
lim
x
→
1
f
(
x
)
−
f
(
1
)
x
−
1
8\sqrt{8}\times \lim_{x\to1}\frac{f(x)-f(1)}{x-1}
8
8
×
lim
x
→
1
x
−
1
f
(
x
)
−
f
(
1
)
.
Numerical
MEDIUM
3 marks
24 March 2024
18
Let
f
f
f
be a differentiable function such that
f
′
(
4
)
=
1
f'(4)=1
f
′
(
4
)
=
1
and
f
(
4
)
=
−
3
f(4)=-3
f
(
4
)
=
−
3
. If
y
=
m
x
+
b
y=mx+b
y
=
m
x
+
b
denotes the tangent of the function
f
f
f
at
x
=
4
x=4
x
=
4
, then find the value of
b
b
b
.
Numerical
EASY
4 marks
24 March 2024
19
Find the following limit. Let
f
(
2
)
=
10
f(2)=10
f
(
2
)
=
10
and
f
′
(
2
)
=
4
f'(2)=4
f
′
(
2
)
=
4
. Calculate
lim
x
→
2
x
f
(
2
)
−
2
f
(
x
)
x
−
2
\lim_{x\to2}\frac{x f(2)-2f(x)}{x-2}
lim
x
→
2
x
−
2
x
f
(
2
)
−
2
f
(
x
)
.
Comprehension
MEDIUM
3 marks
03 December 2023
20
Find the following limit. Calculate
lim
x
→
9
2
(
f
(
x
)
−
3
)
x
−
3
\lim_{x\to9}\frac{2(\sqrt{f(x)}-3)}{\sqrt{x}-3}
lim
x
→
9
x
−
3
2
(
f
(
x
)
−
3
)
, given that
f
(
9
)
=
9
f(9)=9
f
(
9
)
=
9
and
f
′
(
9
)
=
4
f'(9)=4
f
′
(
9
)
=
4
.
Comprehension
MEDIUM
2 marks
03 December 2023
21
Choose the set of INCORRECT options.
Single correct
EASY
4 marks
03 December 2023
22
Consider the graphs given below. Choose the set of correct options.
Multiple correct
MEDIUM
3 marks
03 December 2023
23
Let
f
f
f
be a differentiable function such that
f
(
4
)
=
6
f(4)=6
f
(
4
)
=
6
and
f
′
(
4
)
=
−
2
f'(4)=-2
f
′
(
4
)
=
−
2
. What is the approximated value of
f
(
4.2
)
f(4.2)
f
(
4.2
)
using the linear approximation of
f
f
f
at
x
=
4
x=4
x
=
4
?
Single correct
EASY
3 marks
03 December 2023
24
Which of the following options is/are true?
Comprehension
MEDIUM
5 marks
06 August 2023
25
Which of the following options is/are true?
Comprehension
MEDIUM
3 marks
06 August 2023
26
Find the value of
f
′
(
0
)
g
(
0
)
+
lim
x
→
0
f
(
x
)
+
lim
x
→
0
g
(
x
)
+
lim
x
→
0
h
(
x
)
\frac{f'(0)}{g(0)}+\lim_{x\to0}f(x)+\lim_{x\to0}g(x)+\lim_{x\to0}h(x)
g
(
0
)
f
′
(
0
)
+
lim
x
→
0
f
(
x
)
+
lim
x
→
0
g
(
x
)
+
lim
x
→
0
h
(
x
)
.
Comprehension
MEDIUM
2 marks
06 August 2023
27
Which of the following options is/are true for the function
f
(
x
)
f(x)
f
(
x
)
at
x
=
1
x=1
x
=
1
?
Comprehension
MEDIUM
4 marks
06 August 2023
28
Which of the following options is/are true?
Comprehension
MEDIUM
5 marks
02 April 2023
29
Find the value of
f
′
(
1
)
f'(1)
f
′
(
1
)
.
Comprehension
EASY
2 marks
02 April 2023
30
If
f
′
(
3
)
=
1
f'(3)=1
f
′
(
3
)
=
1
, then which of the following is the linear approximation
L
f
(
x
)
L_f(x)
L
f
(
x
)
of the function at
x
=
3
x=3
x
=
3
?
Comprehension
MEDIUM
2 marks
02 April 2023
31
Find the number of points where the function
f
(
x
)
f(x)
f
(
x
)
is not differentiable.
Comprehension
MEDIUM
2 marks
02 April 2023
32
If
f
(
x
)
=
g
(
x
2
+
5
x
)
×
h
(
x
3
+
x
)
f(x)=g(x^2+5x)\times h(x^3+x)
f
(
x
)
=
g
(
x
2
+
5
x
)
×
h
(
x
3
+
x
)
,
g
′
(
0
)
=
g
(
0
)
≠
0
g'(0)=g(0)\ne0
g
′
(
0
)
=
g
(
0
)
=
0
, and
h
′
(
0
)
=
h
(
0
)
≠
0
h'(0)=h(0)\ne0
h
′
(
0
)
=
h
(
0
)
=
0
, then find the value of
f
′
(
0
)
f
(
0
)
\frac{f'(0)}{f(0)}
f
(
0
)
f
′
(
0
)
.
Numerical
MEDIUM
5 marks
20 November 2022
33
Let
f
f
f
be a differentiable function at
x
=
2
x=2
x
=
2
. The tangent line to the curve represented by the function
f
f
f
at the point
(
2
,
6
)
(2,6)
(
2
,
6
)
passes through the point
(
4
,
−
14
)
(4,-14)
(
4
,
−
14
)
. Find the value of
f
′
(
2
)
f'(2)
f
′
(
2
)
.
Numerical
EASY
3 marks
20 November 2022
Showing 33 questions.
Mathematics 1 > Week 8
All PYQs
Topic-Wise PYQs
Start Weekly Test
Mock tests
Week mock
Topic mock
Subject mock
Type:
All
Difficulty:
All
Year:
All
33Q
01
f
(
x
)
=
a
sin
∣
x
∣
+
b
e
∣
x
∣
f(x)=a\sin|x|+be^{|x|}
f
(
x
)
=
a
sin
∣
x
∣
+
b
e
∣
x
∣
is differentiable at
x
=
0
x=0
x
=
0
, if and only if
Single correct
MEDIUM
4 marks
03 August 2025
02
If
y
=
m
x
+
c
y=mx+c
y
=
m
x
+
c
is the equation of the tangent of
f
f
f
at
x
=
3
x=3
x
=
3
, choose the correct options.
Comprehension
MEDIUM
3 marks
03 August 2025
03
Find the number of points where
∣
f
(
x
)
∣
|f(x)|
∣
f
(
x
)
∣
is not differentiable.
Comprehension
MEDIUM
3 marks
03 August 2025
04
f
(
x
)
=
a
sin
∣
x
∣
+
b
e
∣
x
∣
f(x)=a\sin|x|+be^{|x|}
f
(
x
)
=
a
sin
∣
x
∣
+
b
e
∣
x
∣
is differentiable at
x
=
0
x=0
x
=
0
, if and only if
Single correct
MEDIUM
4 marks
16 March 2025
05
Consider the function
f
:
R
→
R
f:\mathbb{R}\to\mathbb{R}
f
:
R
→
R
defined by
f
(
x
)
=
{
x
2
−
∣
x
∣
,
x
<
0
,
x
2
+
∣
x
∣
,
x
≥
0.
f(x)=\begin{cases}x^2-|x|, & x<0,\\ x^2+|x|, & x\ge0.\end{cases}
f
(
x
)
=
{
x
2
−
∣
x
∣
,
x
2
+
∣
x
∣
,
x
<
0
,
x
≥
0.
Which of the following option(s) is(are) correct?
Multiple correct
MEDIUM
4 marks
16 March 2025
06
Let
f
f
f
be a differentiable function at
x
=
2
x=2
x
=
2
. The tangent line to the curve represented by the function
f
f
f
at the point
(
2
,
6
)
(2,6)
(
2
,
6
)
passes through the point
(
6
,
−
18
)
(6,-18)
(
6
,
−
18
)
. What will be the value of
f
′
(
2
)
f'(2)
f
′
(
2
)
?
Numerical
EASY
3 marks
16 March 2025
07
If
f
(
x
)
=
g
(
x
2
+
7
x
)
×
h
(
x
3
+
2
x
)
f(x)=g(x^2+7x)\times h(x^3+2x)
f
(
x
)
=
g
(
x
2
+
7
x
)
×
h
(
x
3
+
2
x
)
,
g
′
(
0
)
=
g
(
0
)
≠
0
g'(0)=g(0)\ne0
g
′
(
0
)
=
g
(
0
)
=
0
, and
h
′
(
0
)
=
h
(
0
)
≠
0
h'(0)=h(0)\ne0
h
′
(
0
)
=
h
(
0
)
=
0
, then find the value of
f
′
(
0
)
f
(
0
)
\frac{f'(0)}{f(0)}
f
(
0
)
f
′
(
0
)
.
Numerical
MEDIUM
4 marks
16 March 2025
08
Consider the following function:
f
(
x
)
=
{
sin
x
x
,
x
≠
0
,
1
,
x
=
0.
f(x)=\begin{cases}\frac{\sin x}{x}, & x\ne0,\\ 1, & x=0.\end{cases}
f
(
x
)
=
{
x
s
i
n
x
,
1
,
x
=
0
,
x
=
0.
Which of the following option(s) is/are true about
f
(
x
)
f(x)
f
(
x
)
?
Multiple correct
MEDIUM
4 marks
01 December 2024
09
Let
f
f
f
be a polynomial of degree
5
5
5
, given by
f
(
x
)
=
a
5
x
5
+
a
4
x
4
+
a
3
x
3
+
a
2
x
2
+
a
1
x
+
a
0
f(x)=a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0
f
(
x
)
=
a
5
x
5
+
a
4
x
4
+
a
3
x
3
+
a
2
x
2
+
a
1
x
+
a
0
. Let
f
′
(
b
)
f'(b)
f
′
(
b
)
denote the derivative of
f
f
f
at
x
=
b
x=b
x
=
b
. Choose the set of correct options.
Multiple correct
MEDIUM
4 marks
01 December 2024
10
Let
f
f
f
be a differentiable function such that
f
′
(
9
)
=
4
f'(9)=4
f
′
(
9
)
=
4
and
f
(
9
)
=
−
14
f(9)=-14
f
(
9
)
=
−
14
. If
y
=
a
x
+
b
y=ax+b
y
=
a
x
+
b
denotes the tangent of the function
f
f
f
at
x
=
9
x=9
x
=
9
, then find the value of
a
−
b
a-b
a
−
b
.
Numerical
EASY
4 marks
01 December 2024
11
Consider the graphs given below. Choose the set of correct options.
Multiple correct
MEDIUM
3 marks
04 August 2024
12
If
f
(
x
)
=
9
−
x
2
f(x)=\sqrt{9-x^2}
f
(
x
)
=
9
−
x
2
, then find out the value of
5
lim
x
→
2
f
(
x
)
−
f
(
2
)
x
−
2
\sqrt{5}\,\lim_{x\to2}\frac{f(x)-f(2)}{x-2}
5
lim
x
→
2
x
−
2
f
(
x
)
−
f
(
2
)
.
Numerical
MEDIUM
3 marks
04 August 2024
13
Is the statement True or False: Left-hand derivative (LHD) and right-hand derivative (RHD) of the function
f
(
x
)
f(x)
f
(
x
)
exist and are equal to each other at
x
=
−
1
x=-1
x
=
−
1
.
Comprehension
MEDIUM
2 marks
04 August 2024
14
Is the statement True or False: The derivative of the function
f
f
f
at
x
=
0
x=0
x
=
0
is
1
1
1
.
Comprehension
MEDIUM
2 marks
04 August 2024
15
Calculate the limit of the following function at
x
=
1
x=1
x
=
1
:
f
(
x
)
=
x
4
−
3
x
3
+
2
x
4
−
5
x
3
+
3
x
2
+
1
f(x)=\frac{x^4-3x^3+2}{x^4-5x^3+3x^2+1}
f
(
x
)
=
x
4
−
5
x
3
+
3
x
2
+
1
x
4
−
3
x
3
+
2
, if
x
≠
1
x\ne1
x
=
1
.
Comprehension
MEDIUM
2 marks
04 August 2024
16
Choose the set of correct options.
Multiple correct
EASY
4 marks
24 March 2024
17
If
f
(
x
)
=
9
−
x
2
f(x)=\sqrt{9-x^2}
f
(
x
)
=
9
−
x
2
, then find the value of
8
8
×
lim
x
→
1
f
(
x
)
−
f
(
1
)
x
−
1
8\sqrt{8}\times \lim_{x\to1}\frac{f(x)-f(1)}{x-1}
8
8
×
lim
x
→
1
x
−
1
f
(
x
)
−
f
(
1
)
.
Numerical
MEDIUM
3 marks
24 March 2024
18
Let
f
f
f
be a differentiable function such that
f
′
(
4
)
=
1
f'(4)=1
f
′
(
4
)
=
1
and
f
(
4
)
=
−
3
f(4)=-3
f
(
4
)
=
−
3
. If
y
=
m
x
+
b
y=mx+b
y
=
m
x
+
b
denotes the tangent of the function
f
f
f
at
x
=
4
x=4
x
=
4
, then find the value of
b
b
b
.
Numerical
EASY
4 marks
24 March 2024
19
Find the following limit. Let
f
(
2
)
=
10
f(2)=10
f
(
2
)
=
10
and
f
′
(
2
)
=
4
f'(2)=4
f
′
(
2
)
=
4
. Calculate
lim
x
→
2
x
f
(
2
)
−
2
f
(
x
)
x
−
2
\lim_{x\to2}\frac{x f(2)-2f(x)}{x-2}
lim
x
→
2
x
−
2
x
f
(
2
)
−
2
f
(
x
)
.
Comprehension
MEDIUM
3 marks
03 December 2023
20
Find the following limit. Calculate
lim
x
→
9
2
(
f
(
x
)
−
3
)
x
−
3
\lim_{x\to9}\frac{2(\sqrt{f(x)}-3)}{\sqrt{x}-3}
lim
x
→
9
x
−
3
2
(
f
(
x
)
−
3
)
, given that
f
(
9
)
=
9
f(9)=9
f
(
9
)
=
9
and
f
′
(
9
)
=
4
f'(9)=4
f
′
(
9
)
=
4
.
Comprehension
MEDIUM
2 marks
03 December 2023
21
Choose the set of INCORRECT options.
Single correct
EASY
4 marks
03 December 2023
22
Consider the graphs given below. Choose the set of correct options.
Multiple correct
MEDIUM
3 marks
03 December 2023
23
Let
f
f
f
be a differentiable function such that
f
(
4
)
=
6
f(4)=6
f
(
4
)
=
6
and
f
′
(
4
)
=
−
2
f'(4)=-2
f
′
(
4
)
=
−
2
. What is the approximated value of
f
(
4.2
)
f(4.2)
f
(
4.2
)
using the linear approximation of
f
f
f
at
x
=
4
x=4
x
=
4
?
Single correct
EASY
3 marks
03 December 2023
24
Which of the following options is/are true?
Comprehension
MEDIUM
5 marks
06 August 2023
25
Which of the following options is/are true?
Comprehension
MEDIUM
3 marks
06 August 2023
26
Find the value of
f
′
(
0
)
g
(
0
)
+
lim
x
→
0
f
(
x
)
+
lim
x
→
0
g
(
x
)
+
lim
x
→
0
h
(
x
)
\frac{f'(0)}{g(0)}+\lim_{x\to0}f(x)+\lim_{x\to0}g(x)+\lim_{x\to0}h(x)
g
(
0
)
f
′
(
0
)
+
lim
x
→
0
f
(
x
)
+
lim
x
→
0
g
(
x
)
+
lim
x
→
0
h
(
x
)
.
Comprehension
MEDIUM
2 marks
06 August 2023
27
Which of the following options is/are true for the function
f
(
x
)
f(x)
f
(
x
)
at
x
=
1
x=1
x
=
1
?
Comprehension
MEDIUM
4 marks
06 August 2023
28
Which of the following options is/are true?
Comprehension
MEDIUM
5 marks
02 April 2023
29
Find the value of
f
′
(
1
)
f'(1)
f
′
(
1
)
.
Comprehension
EASY
2 marks
02 April 2023
30
If
f
′
(
3
)
=
1
f'(3)=1
f
′
(
3
)
=
1
, then which of the following is the linear approximation
L
f
(
x
)
L_f(x)
L
f
(
x
)
of the function at
x
=
3
x=3
x
=
3
?
Comprehension
MEDIUM
2 marks
02 April 2023
31
Find the number of points where the function
f
(
x
)
f(x)
f
(
x
)
is not differentiable.
Comprehension
MEDIUM
2 marks
02 April 2023
32
If
f
(
x
)
=
g
(
x
2
+
5
x
)
×
h
(
x
3
+
x
)
f(x)=g(x^2+5x)\times h(x^3+x)
f
(
x
)
=
g
(
x
2
+
5
x
)
×
h
(
x
3
+
x
)
,
g
′
(
0
)
=
g
(
0
)
≠
0
g'(0)=g(0)\ne0
g
′
(
0
)
=
g
(
0
)
=
0
, and
h
′
(
0
)
=
h
(
0
)
≠
0
h'(0)=h(0)\ne0
h
′
(
0
)
=
h
(
0
)
=
0
, then find the value of
f
′
(
0
)
f
(
0
)
\frac{f'(0)}{f(0)}
f
(
0
)
f
′
(
0
)
.
Numerical
MEDIUM
5 marks
20 November 2022
33
Let
f
f
f
be a differentiable function at
x
=
2
x=2
x
=
2
. The tangent line to the curve represented by the function
f
f
f
at the point
(
2
,
6
)
(2,6)
(
2
,
6
)
passes through the point
(
4
,
−
14
)
(4,-14)
(
4
,
−
14
)
. Find the value of
f
′
(
2
)
f'(2)
f
′
(
2
)
.
Numerical
EASY
3 marks
20 November 2022
Showing 33 questions.
