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Page 1
CBSE XII | Mathematics
Board Paper 2014 – Delhi Set 3 Solution
CBSE Board
Class XII Mathematics
Board Paper 2014 Solution
Delhi
SECTION – A
1. Given that
2
AA ? .
We need to find the value of
? ?
3
7A I A , where I is the identity matrix. ??
Thus,
? ? ? ?
? ? ? ?
? ? ? ?
? ?
? ?
? ?
3
3 2 2 3
3
3 2 2 3 2 2 2
3
2
3
3
3
7A I A 7A I 3I A 3IA A
7A I A 7A I 3A 3A A A I I,I A A,IA A
7A I A 7A I 3A 3A A A A
7A I A 7A I 3A 3A A
7A I A 7A I 7A
7A I A I
? ? ? ? ? ? ?
?? ? ? ? ? ? ? ? ? ? ? ? ?
??
?? ? ? ? ? ? ? ? ? ?
??
? ? ? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ?
2. Given that
x y z 1 4
2x y w 0 5
?? ? ? ? ?
?
? ? ? ?
?
? ? ? ?
We need to find the value of x + y.
?? ? ? ? ?
?
? ? ? ?
?
? ? ? ?
?
? ? ?
??
ij ij
x y z 1 4
2x y w 0 5
Two matrices A and B are equal to each other, if they have the same dimensions
and the same elements a b , for i = 1,2,...,n and j = 1,2,...,m.
x y 1...(1)
2x y 0...(2)
Equa ?
? ? ?
??
tion (2) (1) is x = 1
Substituting the value of x = 1 in equation (1), we have
1 y 1
y2
Therefore, x + y = 1 + 2 = 3
Page 2
CBSE XII | Mathematics
Board Paper 2014 – Delhi Set 3 Solution
CBSE Board
Class XII Mathematics
Board Paper 2014 Solution
Delhi
SECTION – A
1. Given that
2
AA ? .
We need to find the value of
? ?
3
7A I A , where I is the identity matrix. ??
Thus,
? ? ? ?
? ? ? ?
? ? ? ?
? ?
? ?
? ?
3
3 2 2 3
3
3 2 2 3 2 2 2
3
2
3
3
3
7A I A 7A I 3I A 3IA A
7A I A 7A I 3A 3A A A I I,I A A,IA A
7A I A 7A I 3A 3A A A A
7A I A 7A I 3A 3A A
7A I A 7A I 7A
7A I A I
? ? ? ? ? ? ?
?? ? ? ? ? ? ? ? ? ? ? ? ?
??
?? ? ? ? ? ? ? ? ? ?
??
? ? ? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ?
2. Given that
x y z 1 4
2x y w 0 5
?? ? ? ? ?
?
? ? ? ?
?
? ? ? ?
We need to find the value of x + y.
?? ? ? ? ?
?
? ? ? ?
?
? ? ? ?
?
? ? ?
??
ij ij
x y z 1 4
2x y w 0 5
Two matrices A and B are equal to each other, if they have the same dimensions
and the same elements a b , for i = 1,2,...,n and j = 1,2,...,m.
x y 1...(1)
2x y 0...(2)
Equa ?
? ? ?
??
tion (2) (1) is x = 1
Substituting the value of x = 1 in equation (1), we have
1 y 1
y2
Therefore, x + y = 1 + 2 = 3
CBSE XII | Mathematics
Board Paper 2014 – Delhi Set 3 Solution
3.
11
Given that tan x tan y and xy<1.
4
??
?
??
? ?
11
1
1
We need to find the value of x+y+xy.
tan x tan y
4
xy
tan xy 1
1 xy 4
xy
tan tan tan
1 xy 4
xy
1
1 xy
x y 1 xy
x y xy 1
??
?
?
?
??
?? ??
? ? ?
??
?
??
?? ???? ??
??
?? ?? ??
?
?? ?? ??
?
??
?
? ? ? ?
? ? ? ?
4. Given that
3x 7 8 7
2 4 6 4
?
?
.
We need to find the value of x
? ?
3x 7 8 7
2 4 6 4
12x 14 32 42
12x 14 10
12x 10 14
12x 24
x2
?
?
? ? ? ? ?
? ? ? ?
? ? ? ?
? ? ?
? ? ?
Page 3
CBSE XII | Mathematics
Board Paper 2014 – Delhi Set 3 Solution
CBSE Board
Class XII Mathematics
Board Paper 2014 Solution
Delhi
SECTION – A
1. Given that
2
AA ? .
We need to find the value of
? ?
3
7A I A , where I is the identity matrix. ??
Thus,
? ? ? ?
? ? ? ?
? ? ? ?
? ?
? ?
? ?
3
3 2 2 3
3
3 2 2 3 2 2 2
3
2
3
3
3
7A I A 7A I 3I A 3IA A
7A I A 7A I 3A 3A A A I I,I A A,IA A
7A I A 7A I 3A 3A A A A
7A I A 7A I 3A 3A A
7A I A 7A I 7A
7A I A I
? ? ? ? ? ? ?
?? ? ? ? ? ? ? ? ? ? ? ? ?
??
?? ? ? ? ? ? ? ? ? ?
??
? ? ? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ?
2. Given that
x y z 1 4
2x y w 0 5
?? ? ? ? ?
?
? ? ? ?
?
? ? ? ?
We need to find the value of x + y.
?? ? ? ? ?
?
? ? ? ?
?
? ? ? ?
?
? ? ?
??
ij ij
x y z 1 4
2x y w 0 5
Two matrices A and B are equal to each other, if they have the same dimensions
and the same elements a b , for i = 1,2,...,n and j = 1,2,...,m.
x y 1...(1)
2x y 0...(2)
Equa ?
? ? ?
??
tion (2) (1) is x = 1
Substituting the value of x = 1 in equation (1), we have
1 y 1
y2
Therefore, x + y = 1 + 2 = 3
CBSE XII | Mathematics
Board Paper 2014 – Delhi Set 3 Solution
3.
11
Given that tan x tan y and xy<1.
4
??
?
??
? ?
11
1
1
We need to find the value of x+y+xy.
tan x tan y
4
xy
tan xy 1
1 xy 4
xy
tan tan tan
1 xy 4
xy
1
1 xy
x y 1 xy
x y xy 1
??
?
?
?
??
?? ??
? ? ?
??
?
??
?? ???? ??
??
?? ?? ??
?
?? ?? ??
?
??
?
? ? ? ?
? ? ? ?
4. Given that
3x 7 8 7
2 4 6 4
?
?
.
We need to find the value of x
? ?
3x 7 8 7
2 4 6 4
12x 14 32 42
12x 14 10
12x 10 14
12x 24
x2
?
?
? ? ? ? ?
? ? ? ?
? ? ? ?
? ? ?
? ? ?
CBSE XII | Mathematics
Board Paper 2014 – Delhi Set 3 Solution
5. Since differentiation operation is the inverse operation of integration, we have
? ? sin ? ? f x x x
Let ? ?
0
sin ?
?
x
f x t tdt
Let us do this by integration by parts.
Therefore assume u = t; du = dt
sin
cos
?
??
??
tdt dv
tv
? ? ? ? ? ?
? ?
? ? ? ?
0
0
Therefore,
= t cos cos
cos sin
Differentiating the above function with respect to x,
f x sin cos cos sin
?? ? ? ?
??
? ? ? ?
??? ? ? ? ? ? ?
??
?
x
x
f x t t dt
f x x x x C
x x x x x x
6. Since the vectors are parallel, we have
? ?
ab
3i 2j 9k i 2pj 3k
3i 2j 9k i 2 pj 3 k
Comparing the respective coefficients, we have
3;
2 p 2
2 3 p 2
1
p
3
??
? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ?
? ? ?
? ? ?
? ? ? ? ?
?
??
7. ? ? The set of natural numbers, N = 1, 2, 3, 4, 5, 6....
? ? ? ?
? ? ? ? ? ? ? ?
? ?
? ?
The relation is given as
R = x, y : 2 8
Thus, R = 6, 1 , 4, 2 , 2, 3
Domain = 6, 4, 2
Range = 1, 2, 3
?? xy
Page 4
CBSE XII | Mathematics
Board Paper 2014 – Delhi Set 3 Solution
CBSE Board
Class XII Mathematics
Board Paper 2014 Solution
Delhi
SECTION – A
1. Given that
2
AA ? .
We need to find the value of
? ?
3
7A I A , where I is the identity matrix. ??
Thus,
? ? ? ?
? ? ? ?
? ? ? ?
? ?
? ?
? ?
3
3 2 2 3
3
3 2 2 3 2 2 2
3
2
3
3
3
7A I A 7A I 3I A 3IA A
7A I A 7A I 3A 3A A A I I,I A A,IA A
7A I A 7A I 3A 3A A A A
7A I A 7A I 3A 3A A
7A I A 7A I 7A
7A I A I
? ? ? ? ? ? ?
?? ? ? ? ? ? ? ? ? ? ? ? ?
??
?? ? ? ? ? ? ? ? ? ?
??
? ? ? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ?
2. Given that
x y z 1 4
2x y w 0 5
?? ? ? ? ?
?
? ? ? ?
?
? ? ? ?
We need to find the value of x + y.
?? ? ? ? ?
?
? ? ? ?
?
? ? ? ?
?
? ? ?
??
ij ij
x y z 1 4
2x y w 0 5
Two matrices A and B are equal to each other, if they have the same dimensions
and the same elements a b , for i = 1,2,...,n and j = 1,2,...,m.
x y 1...(1)
2x y 0...(2)
Equa ?
? ? ?
??
tion (2) (1) is x = 1
Substituting the value of x = 1 in equation (1), we have
1 y 1
y2
Therefore, x + y = 1 + 2 = 3
CBSE XII | Mathematics
Board Paper 2014 – Delhi Set 3 Solution
3.
11
Given that tan x tan y and xy<1.
4
??
?
??
? ?
11
1
1
We need to find the value of x+y+xy.
tan x tan y
4
xy
tan xy 1
1 xy 4
xy
tan tan tan
1 xy 4
xy
1
1 xy
x y 1 xy
x y xy 1
??
?
?
?
??
?? ??
? ? ?
??
?
??
?? ???? ??
??
?? ?? ??
?
?? ?? ??
?
??
?
? ? ? ?
? ? ? ?
4. Given that
3x 7 8 7
2 4 6 4
?
?
.
We need to find the value of x
? ?
3x 7 8 7
2 4 6 4
12x 14 32 42
12x 14 10
12x 10 14
12x 24
x2
?
?
? ? ? ? ?
? ? ? ?
? ? ? ?
? ? ?
? ? ?
CBSE XII | Mathematics
Board Paper 2014 – Delhi Set 3 Solution
5. Since differentiation operation is the inverse operation of integration, we have
? ? sin ? ? f x x x
Let ? ?
0
sin ?
?
x
f x t tdt
Let us do this by integration by parts.
Therefore assume u = t; du = dt
sin
cos
?
??
??
tdt dv
tv
? ? ? ? ? ?
? ?
? ? ? ?
0
0
Therefore,
= t cos cos
cos sin
Differentiating the above function with respect to x,
f x sin cos cos sin
?? ? ? ?
??
? ? ? ?
??? ? ? ? ? ? ?
??
?
x
x
f x t t dt
f x x x x C
x x x x x x
6. Since the vectors are parallel, we have
? ?
ab
3i 2j 9k i 2pj 3k
3i 2j 9k i 2 pj 3 k
Comparing the respective coefficients, we have
3;
2 p 2
2 3 p 2
1
p
3
??
? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ?
? ? ?
? ? ?
? ? ? ? ?
?
??
7. ? ? The set of natural numbers, N = 1, 2, 3, 4, 5, 6....
? ? ? ?
? ? ? ? ? ? ? ?
? ?
? ?
The relation is given as
R = x, y : 2 8
Thus, R = 6, 1 , 4, 2 , 2, 3
Domain = 6, 4, 2
Range = 1, 2, 3
?? xy
CBSE XII | Mathematics
Board Paper 2014 – Delhi Set 3 Solution
8. Given that the cartesian equation of the line as
? ? ? ? ? ?
? ?
3 4 2 6
5 7 4
That is,
3 4 2 3
5 7 4
4
33
5 7 2
? ? ?
??
? ? ? ? ?
??
??
??
? ? ? ?
?
?
x y z
x y z
y
xz
Any point on the line is of the form:
5 3,7 4,2 3
Thus, the vector equation is of the form:
r , where is the position vector of any
point on the line and b is the vector parallel to the lin
? ? ? ?
?? a b a
? ? ?
?
? ? ? ? ? ?
? ?
e.
Therefore, the vector equation is
r 5 3 7 4 2 3
r 5 7 2 3 4 3
r 3 4 3 5 7 2
? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
i j k
i j k i j k
i j k i j k
? ? ?
? ? ?
?
9.
a
2
0
dx
Given that
4+x 8
?
?
?
a
2
0
-1
0
We need to find the value of a.
dx
Let I=
4+x 8
1
Thus, I= tan
2 2 8
?
??
?
??
??
?
?
?
a
x
1
1
1
1
tan
2 2 8
tan 2
28
tan
24
1
2
2
?
?
?
??
? ? ?
??
??
??
?
?
?
a
a
a
a
a
Page 5
CBSE XII | Mathematics
Board Paper 2014 – Delhi Set 3 Solution
CBSE Board
Class XII Mathematics
Board Paper 2014 Solution
Delhi
SECTION – A
1. Given that
2
AA ? .
We need to find the value of
? ?
3
7A I A , where I is the identity matrix. ??
Thus,
? ? ? ?
? ? ? ?
? ? ? ?
? ?
? ?
? ?
3
3 2 2 3
3
3 2 2 3 2 2 2
3
2
3
3
3
7A I A 7A I 3I A 3IA A
7A I A 7A I 3A 3A A A I I,I A A,IA A
7A I A 7A I 3A 3A A A A
7A I A 7A I 3A 3A A
7A I A 7A I 7A
7A I A I
? ? ? ? ? ? ?
?? ? ? ? ? ? ? ? ? ? ? ? ?
??
?? ? ? ? ? ? ? ? ? ?
??
? ? ? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ?
2. Given that
x y z 1 4
2x y w 0 5
?? ? ? ? ?
?
? ? ? ?
?
? ? ? ?
We need to find the value of x + y.
?? ? ? ? ?
?
? ? ? ?
?
? ? ? ?
?
? ? ?
??
ij ij
x y z 1 4
2x y w 0 5
Two matrices A and B are equal to each other, if they have the same dimensions
and the same elements a b , for i = 1,2,...,n and j = 1,2,...,m.
x y 1...(1)
2x y 0...(2)
Equa ?
? ? ?
??
tion (2) (1) is x = 1
Substituting the value of x = 1 in equation (1), we have
1 y 1
y2
Therefore, x + y = 1 + 2 = 3
CBSE XII | Mathematics
Board Paper 2014 – Delhi Set 3 Solution
3.
11
Given that tan x tan y and xy<1.
4
??
?
??
? ?
11
1
1
We need to find the value of x+y+xy.
tan x tan y
4
xy
tan xy 1
1 xy 4
xy
tan tan tan
1 xy 4
xy
1
1 xy
x y 1 xy
x y xy 1
??
?
?
?
??
?? ??
? ? ?
??
?
??
?? ???? ??
??
?? ?? ??
?
?? ?? ??
?
??
?
? ? ? ?
? ? ? ?
4. Given that
3x 7 8 7
2 4 6 4
?
?
.
We need to find the value of x
? ?
3x 7 8 7
2 4 6 4
12x 14 32 42
12x 14 10
12x 10 14
12x 24
x2
?
?
? ? ? ? ?
? ? ? ?
? ? ? ?
? ? ?
? ? ?
CBSE XII | Mathematics
Board Paper 2014 – Delhi Set 3 Solution
5. Since differentiation operation is the inverse operation of integration, we have
? ? sin ? ? f x x x
Let ? ?
0
sin ?
?
x
f x t tdt
Let us do this by integration by parts.
Therefore assume u = t; du = dt
sin
cos
?
??
??
tdt dv
tv
? ? ? ? ? ?
? ?
? ? ? ?
0
0
Therefore,
= t cos cos
cos sin
Differentiating the above function with respect to x,
f x sin cos cos sin
?? ? ? ?
??
? ? ? ?
??? ? ? ? ? ? ?
??
?
x
x
f x t t dt
f x x x x C
x x x x x x
6. Since the vectors are parallel, we have
? ?
ab
3i 2j 9k i 2pj 3k
3i 2j 9k i 2 pj 3 k
Comparing the respective coefficients, we have
3;
2 p 2
2 3 p 2
1
p
3
??
? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ?
? ? ?
? ? ?
? ? ? ? ?
?
??
7. ? ? The set of natural numbers, N = 1, 2, 3, 4, 5, 6....
? ? ? ?
? ? ? ? ? ? ? ?
? ?
? ?
The relation is given as
R = x, y : 2 8
Thus, R = 6, 1 , 4, 2 , 2, 3
Domain = 6, 4, 2
Range = 1, 2, 3
?? xy
CBSE XII | Mathematics
Board Paper 2014 – Delhi Set 3 Solution
8. Given that the cartesian equation of the line as
? ? ? ? ? ?
? ?
3 4 2 6
5 7 4
That is,
3 4 2 3
5 7 4
4
33
5 7 2
? ? ?
??
? ? ? ? ?
??
??
??
? ? ? ?
?
?
x y z
x y z
y
xz
Any point on the line is of the form:
5 3,7 4,2 3
Thus, the vector equation is of the form:
r , where is the position vector of any
point on the line and b is the vector parallel to the lin
? ? ? ?
?? a b a
? ? ?
?
? ? ? ? ? ?
? ?
e.
Therefore, the vector equation is
r 5 3 7 4 2 3
r 5 7 2 3 4 3
r 3 4 3 5 7 2
? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
i j k
i j k i j k
i j k i j k
? ? ?
? ? ?
?
9.
a
2
0
dx
Given that
4+x 8
?
?
?
a
2
0
-1
0
We need to find the value of a.
dx
Let I=
4+x 8
1
Thus, I= tan
2 2 8
?
??
?
??
??
?
?
?
a
x
1
1
1
1
tan
2 2 8
tan 2
28
tan
24
1
2
2
?
?
?
??
? ? ?
??
??
??
?
?
?
a
a
a
a
a
CBSE XII | Mathematics
Board Paper 2014 – Delhi Set 3 Solution
10. Given that a and b are two perpendicular vectors.
Thus, a b 0
Also given that, a b 13 and a =5.
We need to find the value of b.
??
??
2
2 2 2
2
22
2
2
2
Consider a b :
a b = a 2 a b b
13 5 2 0 b
169 25 b
b 169 25
b 144
b 12
?
? ? ? ?
? ? ? ?
??
??
?
?
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FAQs on CBSE Past Year Paper Session (2014) Solutions, Math Class 12 - Mathematics (Maths) Class 12 - JEE
| 1. What is the CBSE Past Year Paper Session? |  |
Ans. The CBSE Past Year Paper Session refers to a study session where students can practice and solve previous year question papers of the Central Board of Secondary Education (CBSE) exams. It helps students in familiarizing themselves with the exam pattern and marking scheme, and also allows them to assess their preparation level.
| 2. How can solving past year papers help in preparing for the Math Class 12 CBSE exam? | |
Ans. Solving past year papers can be extremely beneficial in preparing for the Math Class 12 CBSE exam. It gives students an idea about the type of questions asked in the exam, helps them understand the weightage given to different topics, and enables them to practice time management. Additionally, analyzing their performance in these papers can highlight areas of weakness that need more attention.
| 3. Where can I find the solutions for the CBSE Past Year Paper Session (2014) for Math Class 12? | |
Ans. You can find the solutions for the CBSE Past Year Paper Session (2014) for Math Class 12 in various sources. You can check online educational platforms, official CBSE websites, or refer to specialized study materials or guidebooks that provide solutions for past year papers. Additionally, you can seek help from your teachers or academic mentors for guidance in finding the solutions.
| 4. How can solving CBSE past year papers improve my exam performance? | |
Ans. Solving CBSE past year papers can significantly improve your exam performance in several ways. It helps you in gaining familiarity with the exam pattern, allows you to practice and refine your problem-solving skills, boosts your confidence, and reduces exam anxiety. By solving these papers, you can also identify common mistakes and rectify them, ultimately enhancing your overall understanding of the subject and improving your chances of scoring better in the actual exam.
| 5. Are the solutions provided for the CBSE Past Year Paper Session (2014) accurate and reliable? | |
Ans. The accuracy and reliability of the solutions for the CBSE Past Year Paper Session (2014) may vary depending on the source you refer to. It is advisable to cross-verify the solutions with trusted reference materials or consult with subject experts to ensure their correctness. Additionally, you can compare your solutions with those provided by reputable sources to gain further confidence in their accuracy.
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CBSE Past Year Paper Session (2014) Solutions, Math Class 12 Free PDF Download
The CBSE Past Year Paper Session (2014) Solutions, 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.
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CBSE Past Year Paper Session (2014) Solutions, Math Class 12 Notes
CBSE Past Year Paper Session (2014) Solutions, 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 CBSE Past Year Paper Session (2014) Solutions, Math Class 12.
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CBSE Past Year Paper Session (2014) Solutions, Math Class 12 JEE Questions
The "CBSE Past Year Paper Session (2014) Solutions, 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.
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The content of CBSE Past Year Paper Session (2014) Solutions, Math Class 12 is prepared as per the latest JEE syllabus.
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