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CBSE XII | Mathematics
Board Paper 2018 – Delhi Set 3 Solution
CBSE Board
Class XII Mathematics
Board Paper 2018 Solution
SECTION – A
1. Given that the magnitude of each of the two vectors a and b have same magnitude.
2
2
a b ..........(i)
the angle between a and b is 60
60 ........(ii)
9
and a b .........(iii)
2
Since,
a b a b cos
9
a a cos60 From (i),(ii) and (iii)
2
19
a
22
a9
a3
a b 3 ............... From (i)
??
?
? ? ? ?
??
? ? ?
? ? ?
? ? ?
??
??
? ? ?
2.
? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
??
? ? ? ?
??
??
??
??
??
? ? ? ? ? ? ? ?
??
? ? ? ?
? ? ? ?
?? ??
? ? ? ? ?
??
??
??
?
11
1 1 1 1
11
11
11
tan 3 cot 3
tan 3 cot 3 cot x cot x
tan 3 cot 3
tan 3 cot 3
tan x cot x
22
2
Page 2
CBSE XII | Mathematics
Board Paper 2018 – Delhi Set 3 Solution
CBSE Board
Class XII Mathematics
Board Paper 2018 Solution
SECTION – A
1. Given that the magnitude of each of the two vectors a and b have same magnitude.
2
2
a b ..........(i)
the angle between a and b is 60
60 ........(ii)
9
and a b .........(iii)
2
Since,
a b a b cos
9
a a cos60 From (i),(ii) and (iii)
2
19
a
22
a9
a3
a b 3 ............... From (i)
??
?
? ? ? ?
??
? ? ?
? ? ?
? ? ?
??
??
? ? ?
2.
? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
??
? ? ? ?
??
??
??
??
??
? ? ? ? ? ? ? ?
??
? ? ? ?
? ? ? ?
?? ??
? ? ? ? ?
??
??
??
?
11
1 1 1 1
11
11
11
tan 3 cot 3
tan 3 cot 3 cot x cot x
tan 3 cot 3
tan 3 cot 3
tan x cot x
22
2
CBSE XII | Mathematics
Board Paper 2018 – Delhi Set 3 Solution
3. Given that ? ab denotes the larger of ‘a’ and ‘b’.
? ?
? ? ? ? ? ?
? ? ?
? ? ? ? ? ? ?
and a b a b 3
5 10 5 10 3 10 3 13
4. 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 3
CBSE XII | Mathematics
Board Paper 2018 – Delhi Set 3 Solution
CBSE Board
Class XII Mathematics
Board Paper 2018 Solution
SECTION – A
1. Given that the magnitude of each of the two vectors a and b have same magnitude.
2
2
a b ..........(i)
the angle between a and b is 60
60 ........(ii)
9
and a b .........(iii)
2
Since,
a b a b cos
9
a a cos60 From (i),(ii) and (iii)
2
19
a
22
a9
a3
a b 3 ............... From (i)
??
?
? ? ? ?
??
? ? ?
? ? ?
? ? ?
??
??
? ? ?
2.
? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
??
? ? ? ?
??
??
??
??
??
? ? ? ? ? ? ? ?
??
? ? ? ?
? ? ? ?
?? ??
? ? ? ? ?
??
??
??
?
11
1 1 1 1
11
11
11
tan 3 cot 3
tan 3 cot 3 cot x cot x
tan 3 cot 3
tan 3 cot 3
tan x cot x
22
2
CBSE XII | Mathematics
Board Paper 2018 – Delhi Set 3 Solution
3. Given that ? ab denotes the larger of ‘a’ and ‘b’.
? ?
? ? ? ? ? ?
? ? ?
? ? ? ? ? ? ?
and a b a b 3
5 10 5 10 3 10 3 13
4. 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
Board Paper 2018 – Delhi Set 3 Solution
SECTION B
5. A black and a red die are rolled together.
? S =
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ?
1, 1 , 1, 2 , 1, 3 , 1, 4 , 1, 5 , 1, 6 ,
2, 1 , 2, 2 , 2, 3 , 2, 4 , 2, 5 , 2, 6 ,
3, 1 , 3, 2 , 3, 3 , 3, 4 , 3, 5 , 3, 6
4, 1 , 4, 2 , 4, 3 , 4, 4 , 4, 5 , 4, 6 ,
5, 1 , 5, 2 , 5, 3 , 5, 4 , 5, 5 , 5, 6 ,
6, 1 , 6
,
, 2 , 6
? ? ? ? ? ? ? ?
, 3 , 6, 4 , 6, 5 , 6, 6
? ?
? ?
? ?
? ?
? ?
??
??
??
??
??
? ?
We need to find the probability of obtaining the sum 8, given that red die resulted in a
number less than 4.
Let A be the event that a number on red die is less than 4.
Let B be the event that the sum of the numbers is 8.
A =
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
1, 1 , 1, 2 , 1, 3 , 1, 4 , 1, 5 , 1, 6 ,
2, 1 , 2, 2 , 2, 3 , 2, 4 , 2, 5 , 2, 6
3, 1 , 3, 2 , 3, 3 , 3, 4 , 3, 5 , 3, 6
??
??
??
??
??
??
??
and
B = {(2, 6), (3, 5), (4, 4), (5, 3), (6, 2)}
?A ? B = {(2, 6), (3, 5)}
n (A) = 18, n (B) = 5, n (A ? B ) = 2, and n(S) = 36
? ? ? ? ? ?
? ?
? ?
? ?
? ? ? ?
?
? ? ? ?
18 5 2
P A , P B and P A B
36 36 36
The conditional probability of obtaining the sum equal to 8, given that the red die
resulted in a number less than 4 is given by,
2
P A B
1
36
P B / A
18
P A 9
36
Page 4
CBSE XII | Mathematics
Board Paper 2018 – Delhi Set 3 Solution
CBSE Board
Class XII Mathematics
Board Paper 2018 Solution
SECTION – A
1. Given that the magnitude of each of the two vectors a and b have same magnitude.
2
2
a b ..........(i)
the angle between a and b is 60
60 ........(ii)
9
and a b .........(iii)
2
Since,
a b a b cos
9
a a cos60 From (i),(ii) and (iii)
2
19
a
22
a9
a3
a b 3 ............... From (i)
??
?
? ? ? ?
??
? ? ?
? ? ?
? ? ?
??
??
? ? ?
2.
? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
??
? ? ? ?
??
??
??
??
??
? ? ? ? ? ? ? ?
??
? ? ? ?
? ? ? ?
?? ??
? ? ? ? ?
??
??
??
?
11
1 1 1 1
11
11
11
tan 3 cot 3
tan 3 cot 3 cot x cot x
tan 3 cot 3
tan 3 cot 3
tan x cot x
22
2
CBSE XII | Mathematics
Board Paper 2018 – Delhi Set 3 Solution
3. Given that ? ab denotes the larger of ‘a’ and ‘b’.
? ?
? ? ? ? ? ?
? ? ?
? ? ? ? ? ? ?
and a b a b 3
5 10 5 10 3 10 3 13
4. 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
Board Paper 2018 – Delhi Set 3 Solution
SECTION B
5. A black and a red die are rolled together.
? S =
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ?
1, 1 , 1, 2 , 1, 3 , 1, 4 , 1, 5 , 1, 6 ,
2, 1 , 2, 2 , 2, 3 , 2, 4 , 2, 5 , 2, 6 ,
3, 1 , 3, 2 , 3, 3 , 3, 4 , 3, 5 , 3, 6
4, 1 , 4, 2 , 4, 3 , 4, 4 , 4, 5 , 4, 6 ,
5, 1 , 5, 2 , 5, 3 , 5, 4 , 5, 5 , 5, 6 ,
6, 1 , 6
,
, 2 , 6
? ? ? ? ? ? ? ?
, 3 , 6, 4 , 6, 5 , 6, 6
? ?
? ?
? ?
? ?
? ?
??
??
??
??
??
? ?
We need to find the probability of obtaining the sum 8, given that red die resulted in a
number less than 4.
Let A be the event that a number on red die is less than 4.
Let B be the event that the sum of the numbers is 8.
A =
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
1, 1 , 1, 2 , 1, 3 , 1, 4 , 1, 5 , 1, 6 ,
2, 1 , 2, 2 , 2, 3 , 2, 4 , 2, 5 , 2, 6
3, 1 , 3, 2 , 3, 3 , 3, 4 , 3, 5 , 3, 6
??
??
??
??
??
??
??
and
B = {(2, 6), (3, 5), (4, 4), (5, 3), (6, 2)}
?A ? B = {(2, 6), (3, 5)}
n (A) = 18, n (B) = 5, n (A ? B ) = 2, and n(S) = 36
? ? ? ? ? ?
? ?
? ?
? ?
? ? ? ?
?
? ? ? ?
18 5 2
P A , P B and P A B
36 36 36
The conditional probability of obtaining the sum equal to 8, given that the red die
resulted in a number less than 4 is given by,
2
P A B
1
36
P B / A
18
P A 9
36
CBSE XII | Mathematics
Board Paper 2018 – Delhi Set 3 Solution
6. Given two vectors are ?? i 2j 3k and ?? 3i 2j k .
? ? ? ? ? ?
?
? ? ?
?
? ? ?
?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ?
? ? ? ? ?
? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
2 2 2
2 2 2 2 2 2
ab
sin .........(i)
ab
To find a b
i j k
a b 1 2 3
3 2 1
a b 2 6 i 1 9 j 2 6 k
a b 4i 8j 4k
a b 4 8 4 96 4 6............(i)
a 1 ( 2) 3 14 ........(ii) and b 3 ( 2) 3 14......
?
??
? ? ?
? ? ? ?
.....(iii)
ab
Since sin
ab
46
sin ......From (i), (ii) and (iii)
14 14
4 6 2 6
sin
7
14 14
7. Given
?
?
bx 5
y ae
Differentiating with y with respect to x,
bx 5
bx 5
2
2
2
2
2
2
2
dy
ae b
dx
dy
by ....(i)( y ae )
dx
Diffrentiating (i) with respect to x,
d y dy
b
dx dx
d y 1 dy dy
.....from (i)
dx y dx dx
d y dy
y
dx dx
?
?
? ? ?
? ? ?
??
??
??
??
??
??
??
??
??
Page 5
CBSE XII | Mathematics
Board Paper 2018 – Delhi Set 3 Solution
CBSE Board
Class XII Mathematics
Board Paper 2018 Solution
SECTION – A
1. Given that the magnitude of each of the two vectors a and b have same magnitude.
2
2
a b ..........(i)
the angle between a and b is 60
60 ........(ii)
9
and a b .........(iii)
2
Since,
a b a b cos
9
a a cos60 From (i),(ii) and (iii)
2
19
a
22
a9
a3
a b 3 ............... From (i)
??
?
? ? ? ?
??
? ? ?
? ? ?
? ? ?
??
??
? ? ?
2.
? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
??
? ? ? ?
??
??
??
??
??
? ? ? ? ? ? ? ?
??
? ? ? ?
? ? ? ?
?? ??
? ? ? ? ?
??
??
??
?
11
1 1 1 1
11
11
11
tan 3 cot 3
tan 3 cot 3 cot x cot x
tan 3 cot 3
tan 3 cot 3
tan x cot x
22
2
CBSE XII | Mathematics
Board Paper 2018 – Delhi Set 3 Solution
3. Given that ? ab denotes the larger of ‘a’ and ‘b’.
? ?
? ? ? ? ? ?
? ? ?
? ? ? ? ? ? ?
and a b a b 3
5 10 5 10 3 10 3 13
4. 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
Board Paper 2018 – Delhi Set 3 Solution
SECTION B
5. A black and a red die are rolled together.
? S =
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ?
1, 1 , 1, 2 , 1, 3 , 1, 4 , 1, 5 , 1, 6 ,
2, 1 , 2, 2 , 2, 3 , 2, 4 , 2, 5 , 2, 6 ,
3, 1 , 3, 2 , 3, 3 , 3, 4 , 3, 5 , 3, 6
4, 1 , 4, 2 , 4, 3 , 4, 4 , 4, 5 , 4, 6 ,
5, 1 , 5, 2 , 5, 3 , 5, 4 , 5, 5 , 5, 6 ,
6, 1 , 6
,
, 2 , 6
? ? ? ? ? ? ? ?
, 3 , 6, 4 , 6, 5 , 6, 6
? ?
? ?
? ?
? ?
? ?
??
??
??
??
??
? ?
We need to find the probability of obtaining the sum 8, given that red die resulted in a
number less than 4.
Let A be the event that a number on red die is less than 4.
Let B be the event that the sum of the numbers is 8.
A =
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
1, 1 , 1, 2 , 1, 3 , 1, 4 , 1, 5 , 1, 6 ,
2, 1 , 2, 2 , 2, 3 , 2, 4 , 2, 5 , 2, 6
3, 1 , 3, 2 , 3, 3 , 3, 4 , 3, 5 , 3, 6
??
??
??
??
??
??
??
and
B = {(2, 6), (3, 5), (4, 4), (5, 3), (6, 2)}
?A ? B = {(2, 6), (3, 5)}
n (A) = 18, n (B) = 5, n (A ? B ) = 2, and n(S) = 36
? ? ? ? ? ?
? ?
? ?
? ?
? ? ? ?
?
? ? ? ?
18 5 2
P A , P B and P A B
36 36 36
The conditional probability of obtaining the sum equal to 8, given that the red die
resulted in a number less than 4 is given by,
2
P A B
1
36
P B / A
18
P A 9
36
CBSE XII | Mathematics
Board Paper 2018 – Delhi Set 3 Solution
6. Given two vectors are ?? i 2j 3k and ?? 3i 2j k .
? ? ? ? ? ?
?
? ? ?
?
? ? ?
?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ?
? ? ? ? ?
? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
2 2 2
2 2 2 2 2 2
ab
sin .........(i)
ab
To find a b
i j k
a b 1 2 3
3 2 1
a b 2 6 i 1 9 j 2 6 k
a b 4i 8j 4k
a b 4 8 4 96 4 6............(i)
a 1 ( 2) 3 14 ........(ii) and b 3 ( 2) 3 14......
?
??
? ? ?
? ? ? ?
.....(iii)
ab
Since sin
ab
46
sin ......From (i), (ii) and (iii)
14 14
4 6 2 6
sin
7
14 14
7. Given
?
?
bx 5
y ae
Differentiating with y with respect to x,
bx 5
bx 5
2
2
2
2
2
2
2
dy
ae b
dx
dy
by ....(i)( y ae )
dx
Diffrentiating (i) with respect to x,
d y dy
b
dx dx
d y 1 dy dy
.....from (i)
dx y dx dx
d y dy
y
dx dx
?
?
? ? ?
? ? ?
??
??
??
??
??
??
??
??
??
CBSE XII | Mathematics
Board Paper 2018 – Delhi Set 3 Solution
8. 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
9. Given
? ? ? ? ? ?
32
C x 0.005x 0.02x 30x 5000
Marginal cost is the rate of change of total cost with respect to output.
??
? ? ? ? ? ? ?
?
? ? ? ? ? ?
??
22
2
dC
Marg inal cost
dx
Marg inal cost 0.005(3x ) 0.02(2x) 30 0.015x 0.04x 30
Marg inal cost at x 3
Marg inal cost 0.015 3 0.04 3 30
Marg inal cost 30.015
Therefore Marg inal cost is Rs. 30.015.
10. Given
?
???
?
??
??
1
1 cosx
y tan
sinx
? ?
2
2
1
1
1
1
x
x
1 cosx 2cos
2cos
2
2
y tan
xx
xx
2sin .cos
and sin x 2sin cos
22
22
x
y tan cot
2
x
y tan tan
22
x
y tan tan x x
22
Differentiating with respect to x,
dy 1 1
0
dx 2 2
?
?
?
?
??
??
??
??
??
?? ??
??
??
??
?
??
??
??
??
??
??
??
?? ???
? ? ?
?? ??
?? ??
?
?? ? ? ?
??
? ? ? ? ?
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FAQs on CBSE Past Year Paper Session (2018) Solutions, Math Class 12 - Mathematics (Maths) Class 12 - JEE
| 1. What is CBSE Past Year Paper Session (2018)? |  |
Ans. CBSE Past Year Paper Session (2018) refers to the previous year's question papers for the Central Board of Secondary Education (CBSE) Class 12 exam that was conducted in the year 2018. These question papers are useful for students to practice and prepare for their upcoming exams.
| 2. Why are CBSE Past Year Papers important for Class 12 Math? | |
Ans. CBSE Past Year Papers for Class 12 Math are important as they provide students with an understanding of the exam pattern, marking scheme, and types of questions that can be asked in the actual exam. By solving these papers, students can assess their preparation level and identify their strengths and weaknesses in different topics of the subject.
| 3. How can CBSE Past Year Paper Session (2018) Solutions help in exam preparation? | |
Ans. CBSE Past Year Paper Session (2018) Solutions provide step-by-step explanations and solutions to the questions asked in the previous year's exam. These solutions help students in understanding the right approach and methodology to solve the questions. By practicing these solutions, students can improve their problem-solving skills and enhance their overall exam preparation.
| 4. Are the CBSE Past Year Paper Session (2018) Solutions available for free? | |
Ans. The availability of CBSE Past Year Paper Session (2018) Solutions for free depends on the source. Some educational websites or platforms may provide these solutions for free, while others may require a subscription or payment to access them. It is advisable to explore different platforms and resources to find the most suitable option as per individual preferences.
| 5. Can CBSE Past Year Paper Session (2018) Solutions guarantee success in the Class 12 Math exam? | |
Ans. CBSE Past Year Paper Session (2018) Solutions are a valuable resource for exam preparation, but they cannot guarantee success on their own. Success in the Class 12 Math exam depends on various factors, including consistent and dedicated study, understanding of concepts, and practice of a wide range of questions. While solving past year papers and referring to their solutions can enhance preparation, it is essential to adopt a comprehensive study approach to excel in the exam.
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,mock tests for examination
,Previous Year Questions with Solutions
,Viva Questions
,study material
,CBSE Past Year Paper Session (2018) Solutions
,Math Class 12 | Mathematics (Maths) Class 12 - JEE
,Math Class 12 | Mathematics (Maths) Class 12 - JEE
,video lectures
,Semester Notes
,shortcuts and tricks
,Summary
,Math Class 12 | Mathematics (Maths) Class 12 - JEE
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Additional Information about CBSE Past Year Paper Session (2018) Solutions, Math Class 12 for JEE Preparation
CBSE Past Year Paper Session (2018) Solutions, Math Class 12 Free PDF Download
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CBSE Past Year Paper Session (2018) Solutions, Math Class 12 Notes
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CBSE Past Year Paper Session (2018) Solutions, Math Class 12 JEE Questions
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The content of CBSE Past Year Paper Session (2018) Solutions, Math Class 12 is prepared as per the latest JEE syllabus.
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