JEE Exam > JEE Notes > Mathematics (Maths) Class 12 > Sample Solution Paper 2 - Math, Class 12
Sample Solution Paper 2 - Math, Class 12 | Mathematics (Maths) Class 12 - JEE PDF Download
| Download, print and study this document offline |
Please wait while the PDF view is loading
Page 1
CBSE XII | Mathematics
Sample Paper – 2 Solution
CBSE Board
Class XII Mathematics
Sample Paper – 2 Solution
SECTION – A
1. A = {1, 2, 3, 4, 5}
Now 1 * 2 = L.C.M. (1, 2) = 2
1 * 3 = 3, 1 * 4 = 4, 1 * 5 = 5
2 * 3 = 6 ? A
So * is not a binary operation on A
2. A’ =
23
12
???
??
??
A =
21
32
???
??
??
B =
10
12
???
??
??
? ?
41
A 2B
56
???
??
??
??
? ?
45
A 2B '
16
???
??
??
??
OR
Given that
? ??
??
??
??
??
??
0 a 3
A 2 0 1
b 1 0
is skew symmetric matrix.
? ? ?
? ? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ? ??
? ? ? ?
? ? ? ? ?
? ? ? ?
? ? ?
? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ?
?
? ? ?
T
T
AA
0 a 3 0 2 b
2 0 1 a 0 1
b 1 0 3 1 0
0 2 b 0 a 3
a 0 1 2 0 1
3 1 0 b 1 0
By equality of Matrices,
a 2 and b 3
Page 2
CBSE XII | Mathematics
Sample Paper – 2 Solution
CBSE Board
Class XII Mathematics
Sample Paper – 2 Solution
SECTION – A
1. A = {1, 2, 3, 4, 5}
Now 1 * 2 = L.C.M. (1, 2) = 2
1 * 3 = 3, 1 * 4 = 4, 1 * 5 = 5
2 * 3 = 6 ? A
So * is not a binary operation on A
2. A’ =
23
12
???
??
??
A =
21
32
???
??
??
B =
10
12
???
??
??
? ?
41
A 2B
56
???
??
??
??
? ?
45
A 2B '
16
???
??
??
??
OR
Given that
? ??
??
??
??
??
??
0 a 3
A 2 0 1
b 1 0
is skew symmetric matrix.
? ? ?
? ? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ? ??
? ? ? ?
? ? ? ? ?
? ? ? ?
? ? ?
? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ?
?
? ? ?
T
T
AA
0 a 3 0 2 b
2 0 1 a 0 1
b 1 0 3 1 0
0 2 b 0 a 3
a 0 1 2 0 1
3 1 0 b 1 0
By equality of Matrices,
a 2 and b 3
CBSE XII | Mathematics
Sample Paper – 2 Solution
3. Projection of a vector a on another vector b is
b
a.
|b|
=
? ?
? ?
ˆ ˆ
3i j 4k
3 12 15
ˆ ˆ
i 3k .
9 1 16 26 26
??
?
? ? ?
??
4. Equation of a line passing through (a, b, c) and with direction cosines (l, m, n) is given
by
x a y b z c
l m n
? ? ?
??
Here point is (0, 0, 0)
Direction cosines of the line parallel to the x-axis are 1, 0, 0
? required line is
x y z
1 0 0
??
Section B
5.
11
3
sin sin cos x 1
5
??
??
??
??
??
Operating both sides by sin
-1
we get,
? ?
1 1 1
11
11
11
3
sin cos x sin 1
52
3
cos x sin
25
3
cos x cos
5
3
x Since, sin x cos x
52
? ? ?
??
??
??
?
? ? ?
? ??
? ? ?
??
??
??
??
??
??
? ??
? ? ? ?
??
??
6.
??
?
5 x x 1
Given matrix is singular if 0
24
? ? ? ? ?
? ? ?
??
(20 4x) (2x 2) 0
18 6x 0
x3
Page 3
CBSE XII | Mathematics
Sample Paper – 2 Solution
CBSE Board
Class XII Mathematics
Sample Paper – 2 Solution
SECTION – A
1. A = {1, 2, 3, 4, 5}
Now 1 * 2 = L.C.M. (1, 2) = 2
1 * 3 = 3, 1 * 4 = 4, 1 * 5 = 5
2 * 3 = 6 ? A
So * is not a binary operation on A
2. A’ =
23
12
???
??
??
A =
21
32
???
??
??
B =
10
12
???
??
??
? ?
41
A 2B
56
???
??
??
??
? ?
45
A 2B '
16
???
??
??
??
OR
Given that
? ??
??
??
??
??
??
0 a 3
A 2 0 1
b 1 0
is skew symmetric matrix.
? ? ?
? ? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ? ??
? ? ? ?
? ? ? ? ?
? ? ? ?
? ? ?
? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ?
?
? ? ?
T
T
AA
0 a 3 0 2 b
2 0 1 a 0 1
b 1 0 3 1 0
0 2 b 0 a 3
a 0 1 2 0 1
3 1 0 b 1 0
By equality of Matrices,
a 2 and b 3
CBSE XII | Mathematics
Sample Paper – 2 Solution
3. Projection of a vector a on another vector b is
b
a.
|b|
=
? ?
? ?
ˆ ˆ
3i j 4k
3 12 15
ˆ ˆ
i 3k .
9 1 16 26 26
??
?
? ? ?
??
4. Equation of a line passing through (a, b, c) and with direction cosines (l, m, n) is given
by
x a y b z c
l m n
? ? ?
??
Here point is (0, 0, 0)
Direction cosines of the line parallel to the x-axis are 1, 0, 0
? required line is
x y z
1 0 0
??
Section B
5.
11
3
sin sin cos x 1
5
??
??
??
??
??
Operating both sides by sin
-1
we get,
? ?
1 1 1
11
11
11
3
sin cos x sin 1
52
3
cos x sin
25
3
cos x cos
5
3
x Since, sin x cos x
52
? ? ?
??
??
??
?
? ? ?
? ??
? ? ?
??
??
??
??
??
??
? ??
? ? ? ?
??
??
6.
??
?
5 x x 1
Given matrix is singular if 0
24
? ? ? ? ?
? ? ?
??
(20 4x) (2x 2) 0
18 6x 0
x3
CBSE XII | Mathematics
Sample Paper – 2 Solution
7. ??
32
We have, I= x 3x 5x
? ?
? ?
? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
?
2
2
2
2
dI
3x 6x 5
dx
3x 6x 3 2
3 x 2x 1 2
3 x 1 2 0
dI
Since 0 for all x, the income I of the doctor is increasing for all values of x.
dx
Thus, the insurance agent can ensure growth in the income of the doctor.
8. ? ? ? ? The distance of the plane 3x 4y 12z 3 0 from the origin (0,0,0) is
? ? ?
? ? ?
? ? ?
??
2 2 2
3(0) 4(0) 12(0) 3
=
(3) ( 4) (12)
0 0 0 3
=
9 16 144
?
?
?
?
3
=
169
3
13
3
13
Page 4
CBSE XII | Mathematics
Sample Paper – 2 Solution
CBSE Board
Class XII Mathematics
Sample Paper – 2 Solution
SECTION – A
1. A = {1, 2, 3, 4, 5}
Now 1 * 2 = L.C.M. (1, 2) = 2
1 * 3 = 3, 1 * 4 = 4, 1 * 5 = 5
2 * 3 = 6 ? A
So * is not a binary operation on A
2. A’ =
23
12
???
??
??
A =
21
32
???
??
??
B =
10
12
???
??
??
? ?
41
A 2B
56
???
??
??
??
? ?
45
A 2B '
16
???
??
??
??
OR
Given that
? ??
??
??
??
??
??
0 a 3
A 2 0 1
b 1 0
is skew symmetric matrix.
? ? ?
? ? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ? ??
? ? ? ?
? ? ? ? ?
? ? ? ?
? ? ?
? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ?
?
? ? ?
T
T
AA
0 a 3 0 2 b
2 0 1 a 0 1
b 1 0 3 1 0
0 2 b 0 a 3
a 0 1 2 0 1
3 1 0 b 1 0
By equality of Matrices,
a 2 and b 3
CBSE XII | Mathematics
Sample Paper – 2 Solution
3. Projection of a vector a on another vector b is
b
a.
|b|
=
? ?
? ?
ˆ ˆ
3i j 4k
3 12 15
ˆ ˆ
i 3k .
9 1 16 26 26
??
?
? ? ?
??
4. Equation of a line passing through (a, b, c) and with direction cosines (l, m, n) is given
by
x a y b z c
l m n
? ? ?
??
Here point is (0, 0, 0)
Direction cosines of the line parallel to the x-axis are 1, 0, 0
? required line is
x y z
1 0 0
??
Section B
5.
11
3
sin sin cos x 1
5
??
??
??
??
??
Operating both sides by sin
-1
we get,
? ?
1 1 1
11
11
11
3
sin cos x sin 1
52
3
cos x sin
25
3
cos x cos
5
3
x Since, sin x cos x
52
? ? ?
??
??
??
?
? ? ?
? ??
? ? ?
??
??
??
??
??
??
? ??
? ? ? ?
??
??
6.
??
?
5 x x 1
Given matrix is singular if 0
24
? ? ? ? ?
? ? ?
??
(20 4x) (2x 2) 0
18 6x 0
x3
CBSE XII | Mathematics
Sample Paper – 2 Solution
7. ??
32
We have, I= x 3x 5x
? ?
? ?
? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
?
2
2
2
2
dI
3x 6x 5
dx
3x 6x 3 2
3 x 2x 1 2
3 x 1 2 0
dI
Since 0 for all x, the income I of the doctor is increasing for all values of x.
dx
Thus, the insurance agent can ensure growth in the income of the doctor.
8. ? ? ? ? The distance of the plane 3x 4y 12z 3 0 from the origin (0,0,0) is
? ? ?
? ? ?
? ? ?
??
2 2 2
3(0) 4(0) 12(0) 3
=
(3) ( 4) (12)
0 0 0 3
=
9 16 144
?
?
?
?
3
=
169
3
13
3
13
CBSE XII | Mathematics
Sample Paper – 2 Solution
OR
The shortest distance is given by,
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
2 1 1 2
12
21
12
2 1 1 2
22
12
A A . B B
BB
ˆˆ ˆ ˆ ˆ ˆ ˆ
A A i j 2k 4i j 3i 2k
ˆ ˆˆ
i j k
ˆˆ
B B 1 2 3 2i j
2 4 5
ˆ ˆ ˆ ˆ
A A . B B 3i 2k . 2i j 6
B B 2 1 5
66
so shortest distance between two lines units
55
??
?
?
? ? ? ? ? ? ? ? ?
? ? ? ? ?
?
? ? ? ? ? ? ? ?
? ? ? ?
?
??
9. Let A
0 q r r s
r q 0 p q
s r q p 0
??
? ? ?
??
A is a skew symmetric matrix of order 3
?A’ = -A
?|A’| = |A| = -|A|
?|A| = -|A|
?2|A| = 0
?|A|= 0
10. Let I =
1 cot x
dx
x log sin x
?
?
?
Put x + log sin x = t
?(1 + cot x) dx = dt
so integral I becomes
?
??
? ? ?
dt
log| t | C
t
log| x logsin x| C
Page 5
CBSE XII | Mathematics
Sample Paper – 2 Solution
CBSE Board
Class XII Mathematics
Sample Paper – 2 Solution
SECTION – A
1. A = {1, 2, 3, 4, 5}
Now 1 * 2 = L.C.M. (1, 2) = 2
1 * 3 = 3, 1 * 4 = 4, 1 * 5 = 5
2 * 3 = 6 ? A
So * is not a binary operation on A
2. A’ =
23
12
???
??
??
A =
21
32
???
??
??
B =
10
12
???
??
??
? ?
41
A 2B
56
???
??
??
??
? ?
45
A 2B '
16
???
??
??
??
OR
Given that
? ??
??
??
??
??
??
0 a 3
A 2 0 1
b 1 0
is skew symmetric matrix.
? ? ?
? ? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ? ??
? ? ? ?
? ? ? ? ?
? ? ? ?
? ? ?
? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ?
?
? ? ?
T
T
AA
0 a 3 0 2 b
2 0 1 a 0 1
b 1 0 3 1 0
0 2 b 0 a 3
a 0 1 2 0 1
3 1 0 b 1 0
By equality of Matrices,
a 2 and b 3
CBSE XII | Mathematics
Sample Paper – 2 Solution
3. Projection of a vector a on another vector b is
b
a.
|b|
=
? ?
? ?
ˆ ˆ
3i j 4k
3 12 15
ˆ ˆ
i 3k .
9 1 16 26 26
??
?
? ? ?
??
4. Equation of a line passing through (a, b, c) and with direction cosines (l, m, n) is given
by
x a y b z c
l m n
? ? ?
??
Here point is (0, 0, 0)
Direction cosines of the line parallel to the x-axis are 1, 0, 0
? required line is
x y z
1 0 0
??
Section B
5.
11
3
sin sin cos x 1
5
??
??
??
??
??
Operating both sides by sin
-1
we get,
? ?
1 1 1
11
11
11
3
sin cos x sin 1
52
3
cos x sin
25
3
cos x cos
5
3
x Since, sin x cos x
52
? ? ?
??
??
??
?
? ? ?
? ??
? ? ?
??
??
??
??
??
??
? ??
? ? ? ?
??
??
6.
??
?
5 x x 1
Given matrix is singular if 0
24
? ? ? ? ?
? ? ?
??
(20 4x) (2x 2) 0
18 6x 0
x3
CBSE XII | Mathematics
Sample Paper – 2 Solution
7. ??
32
We have, I= x 3x 5x
? ?
? ?
? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
?
2
2
2
2
dI
3x 6x 5
dx
3x 6x 3 2
3 x 2x 1 2
3 x 1 2 0
dI
Since 0 for all x, the income I of the doctor is increasing for all values of x.
dx
Thus, the insurance agent can ensure growth in the income of the doctor.
8. ? ? ? ? The distance of the plane 3x 4y 12z 3 0 from the origin (0,0,0) is
? ? ?
? ? ?
? ? ?
??
2 2 2
3(0) 4(0) 12(0) 3
=
(3) ( 4) (12)
0 0 0 3
=
9 16 144
?
?
?
?
3
=
169
3
13
3
13
CBSE XII | Mathematics
Sample Paper – 2 Solution
OR
The shortest distance is given by,
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
2 1 1 2
12
21
12
2 1 1 2
22
12
A A . B B
BB
ˆˆ ˆ ˆ ˆ ˆ ˆ
A A i j 2k 4i j 3i 2k
ˆ ˆˆ
i j k
ˆˆ
B B 1 2 3 2i j
2 4 5
ˆ ˆ ˆ ˆ
A A . B B 3i 2k . 2i j 6
B B 2 1 5
66
so shortest distance between two lines units
55
??
?
?
? ? ? ? ? ? ? ? ?
? ? ? ? ?
?
? ? ? ? ? ? ? ?
? ? ? ?
?
??
9. Let A
0 q r r s
r q 0 p q
s r q p 0
??
? ? ?
??
A is a skew symmetric matrix of order 3
?A’ = -A
?|A’| = |A| = -|A|
?|A| = -|A|
?2|A| = 0
?|A|= 0
10. Let I =
1 cot x
dx
x log sin x
?
?
?
Put x + log sin x = t
?(1 + cot x) dx = dt
so integral I becomes
?
??
? ? ?
dt
log| t | C
t
log| x logsin x| C
CBSE XII | Mathematics
Sample Paper – 2 Solution
OR
Given
?
?
?
2
2
cos2x 2sin x
I dx
cos x
? ?
? ?
?
?
??
? ? ?
?
?
? ? ?
?
??
?
?
?
?
?
2
2
2 2 2
22
2
22
2
22
2
2
cos2x 2sin x
I dx
cos x
cos x sin x 2sin x
I dx ..... cos2x cos x sin x
cos x
cos x sin x
I dx
cos x
1
I dx .... cos x sin x 1
cos x
I sec x dx
I tan x c
11. ? ? ? ? Let a 2i j 5k, then
? ? ? ? ?
? ? ? ?
??
? ? ?
? ? ? ? ? ?
? ? ? ?
?
2 2 2
a ( 2) (1) ( 5)
a 4 1 25
a 30
a 2i j 5k 2 1 5
Now, a i j k
30 30 30 30 a
21
Thus, the direction cosines of the vector 2i j 5k are ,
30 30
5
and .
30
12. A =
a 0 0
0 a 0
0 0 a
??
??
??
??
??
then |A| = a³
Since, a is a nonzero real number so |A| ? 0
If A is an invertible matrix of order n, then |adj(A)| = |A|
n -1
So |adj A| = |A|² = (a³)
²
= a
6
Read More
|
204 videos|290 docs|139 tests
|
FAQs on Sample Solution Paper 2 - Math, Class 12 - Mathematics (Maths) Class 12 - JEE
| 1. What are the main topics covered in the Math Class 12 exam? |  |
Ans. The main topics covered in the Math Class 12 exam are algebra, calculus, coordinate geometry, mathematical reasoning, statistics, and probability.
| 2. How should I prepare for the Math Class 12 exam? | |
Ans. To prepare for the Math Class 12 exam, it is important to understand the concepts thoroughly, practice a variety of problems, solve previous year question papers, and seek help from teachers or tutors if needed.
| 3. Are calculators allowed in the Math Class 12 exam? | |
Ans. Yes, calculators are generally allowed in the Math Class 12 exam. However, it is important to check the specific guidelines provided by the exam board to ensure which types of calculators are permitted.
| 4. Can I use graph paper during the Math Class 12 exam? | |
Ans. Yes, in most cases, graph paper is allowed in the Math Class 12 exam. It is advisable to check the exam guidelines to confirm the specific requirements regarding the use of graph paper.
| 5. How can I improve my problem-solving skills for the Math Class 12 exam? | |
Ans. To improve problem-solving skills for the Math Class 12 exam, it is recommended to practice solving a wide range of problems, understand different problem-solving techniques, and analyze the solutions to identify patterns and strategies. Additionally, seeking guidance from teachers or attending problem-solving workshops can also be beneficial.
About this Document
|
4.93/5 Rating |
|
Dec 25, 2024 Last updated |
Document Description: Sample Solution Paper 2 - Math, Class 12 for JEE 2024 is part of Mathematics (Maths) Class 12 preparation.
The notes and questions for Sample Solution Paper 2 - Math, Class 12 have been prepared according to the JEE exam syllabus. Information about Sample Solution Paper 2 - Math, Class 12 covers topics
like and Sample Solution Paper 2 - Math, Class 12 Example, for JEE 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Sample Solution Paper 2 - Math, Class 12.
Introduction of Sample Solution Paper 2 - Math, Class 12 in English is available as part of our Mathematics (Maths) Class 12
for JEE & Sample Solution Paper 2 - Math, Class 12 in Hindi for Mathematics (Maths) Class 12 course.
Download more important topics related with notes, lectures and mock test series for JEE
Exam by signing up for free. JEE: Sample Solution Paper 2 - Math, Class 12 | Mathematics (Maths) Class 12 - JEE
Description
Full syllabus notes, lecture & questions for Sample Solution Paper 2 - Math, Class 12 | Mathematics (Maths) Class 12 - JEE - JEE | Plus excerises question with solution to help you revise complete syllabus for Mathematics (Maths) Class 12 | Best notes, free PDF download
Information about Sample Solution Paper 2 - Math, Class 12
In this doc you can find the meaning of Sample Solution Paper 2 - Math, Class 12 defined & explained in the simplest way possible. Besides explaining types of
Sample Solution Paper 2 - Math, Class 12 theory, EduRev gives you an ample number of questions to practice Sample Solution Paper 2 - Math, Class 12 tests, examples and also practice JEE
tests
|
Explore Courses for JEE exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
Related Searches
shortcuts and tricks
,Previous Year Questions with Solutions
,video lectures
,Sample Solution Paper 2 - Math
,Sample Solution Paper 2 - Math
,Class 12 | Mathematics (Maths) Class 12 - JEE
,Important questions
,practice quizzes
,Semester Notes
,Class 12 | Mathematics (Maths) Class 12 - JEE
,past year papers
,ppt
,Objective type Questions
,Summary
,Viva Questions
,study material
,Class 12 | Mathematics (Maths) Class 12 - JEE
,Exam
,Sample Paper
,Sample Solution Paper 2 - Math
,Extra Questions
,Free
,mock tests for examination
,MCQs
;
Additional Information about Sample Solution Paper 2 - Math, Class 12 for JEE Preparation
Sample Solution Paper 2 - Math, Class 12 Free PDF Download
The Sample Solution Paper 2 - Math, Class 12 is an invaluable resource that delves deep into the core of the JEE exam.
These study notes are curated by experts and cover all the essential topics and concepts, making your preparation more efficient and effective.
With the help of these notes, you can grasp complex subjects quickly, revise important points easily,
and reinforce your understanding of key concepts. The study notes are presented in a concise and easy-to-understand manner,
allowing you to optimize your learning process. Whether you're looking for best-recommended books, sample papers, study material,
or toppers' notes, this PDF has got you covered. Download the Sample Solution Paper 2 - Math, Class 12 now and kickstart your journey towards success in the JEE exam.
Importance of Sample Solution Paper 2 - Math, Class 12
The importance of Sample Solution Paper 2 - Math, Class 12 cannot be overstated, especially for JEE aspirants.
This document holds the key to success in the JEE exam.
It offers a detailed understanding of the concept, providing invaluable insights into the topic.
By knowing the concepts well in advance, students can plan their preparation effectively.
Utilize this indispensable guide for a well-rounded preparation and achieve your desired results.
Sample Solution Paper 2 - Math, Class 12 Notes
Sample Solution Paper 2 - Math, Class 12 Notes offer in-depth insights into the specific topic to help you master it with ease.
This comprehensive document covers all aspects related to Sample Solution Paper 2 - Math, Class 12.
It includes detailed information about the exam syllabus, recommended books, and study materials for a well-rounded preparation.
Practice papers and question papers enable you to assess your progress effectively.
Additionally, the paper analysis provides valuable tips for tackling the exam strategically.
Access to Toppers' notes gives you an edge in understanding complex concepts.
Whether you're a beginner or aiming for advanced proficiency, Sample Solution Paper 2 - Math, Class 12 Notes on EduRev are your ultimate resource for success.
Sample Solution Paper 2 - Math, Class 12 JEE Questions
The "Sample Solution Paper 2 - Math, Class 12 JEE Questions" guide is a valuable resource for all aspiring students preparing for the
JEE exam. It focuses on providing a wide range of practice questions to help students gauge
their understanding of the exam topics. These questions cover the entire syllabus, ensuring comprehensive preparation.
The guide includes previous years' question papers for students to familiarize themselves with the exam's format and difficulty level.
Additionally, it offers subject-specific question banks, allowing students to focus on weak areas and improve their performance.
Study Sample Solution Paper 2 - Math, Class 12 on the App
Students of JEE can study Sample Solution Paper 2 - Math, Class 12 alongwith tests & analysis from the EduRev app,
which will help them while preparing for their exam. Apart from the Sample Solution Paper 2 - Math, Class 12,
students can also utilize the EduRev App for other study materials such as previous year question papers, syllabus, important questions, etc.
The EduRev App will make your learning easier as you can access it from anywhere you want.
The content of Sample Solution Paper 2 - Math, Class 12 is prepared as per the latest JEE syllabus.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup