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CBSE XII | Mathematics
Sample Paper – 10 Solution
Mathematics
Class XII
Sample Paper – 10 Solution
SECTION – A
1. By observation we find that
2 + x = 10
x = 8.
2.
? ?
d
cos x
dx
? ? ? ?
d
sin x x
dx
dy sin x
dx
2x
??
?
?
3. DE:
32
32
d y d y dy
y siny 0
dx dx dx
? ? ? ?
It is linear, since y siny ? is product of two different functions, and their individual
power is one.
Page 2
CBSE XII | Mathematics
Sample Paper – 10 Solution
Mathematics
Class XII
Sample Paper – 10 Solution
SECTION – A
1. By observation we find that
2 + x = 10
x = 8.
2.
? ?
d
cos x
dx
? ? ? ?
d
sin x x
dx
dy sin x
dx
2x
??
?
?
3. DE:
32
32
d y d y dy
y siny 0
dx dx dx
? ? ? ?
It is linear, since y siny ? is product of two different functions, and their individual
power is one.
CBSE XII | Mathematics
Sample Paper – 10 Solution
4. Let ? be the angles between, the given two lines
So, the angle between them given their direction cosines is given by
1 2 1 2 1 2
2 2 2 2 2 2
1 1 1 2 2 2
1
2
1
2
1
2
1
a a b b c c
cos
a b c a b c
substituting we get
a2
a3
b1
b2
c3
c1
11
cos
14
??
??
? ? ? ?
?
?
?
?
??
??
??
??
??
??
OR
Let ? be the angles between, the given two lines
So, the angle between them given their direction cosines is given by
? ? ? ? ? ? ? ? ? ? ? ?
? ?
1 2 1 2 1 2
2 2 2 2 2 2
1 1 1 2 2 2
1 2 1 2 1 2
1
a a b b c c
cos
a b c a b c
substituting we get
a a b b c c 2 1 7 2 3 4 0
cos 0
2
??
??
? ? ? ?
? ? ? ? ? ? ? ?
?
? ? ?
Page 3
CBSE XII | Mathematics
Sample Paper – 10 Solution
Mathematics
Class XII
Sample Paper – 10 Solution
SECTION – A
1. By observation we find that
2 + x = 10
x = 8.
2.
? ?
d
cos x
dx
? ? ? ?
d
sin x x
dx
dy sin x
dx
2x
??
?
?
3. DE:
32
32
d y d y dy
y siny 0
dx dx dx
? ? ? ?
It is linear, since y siny ? is product of two different functions, and their individual
power is one.
CBSE XII | Mathematics
Sample Paper – 10 Solution
4. Let ? be the angles between, the given two lines
So, the angle between them given their direction cosines is given by
1 2 1 2 1 2
2 2 2 2 2 2
1 1 1 2 2 2
1
2
1
2
1
2
1
a a b b c c
cos
a b c a b c
substituting we get
a2
a3
b1
b2
c3
c1
11
cos
14
??
??
? ? ? ?
?
?
?
?
??
??
??
??
??
??
OR
Let ? be the angles between, the given two lines
So, the angle between them given their direction cosines is given by
? ? ? ? ? ? ? ? ? ? ? ?
? ?
1 2 1 2 1 2
2 2 2 2 2 2
1 1 1 2 2 2
1 2 1 2 1 2
1
a a b b c c
cos
a b c a b c
substituting we get
a a b b c c 2 1 7 2 3 4 0
cos 0
2
??
??
? ? ? ?
? ? ? ? ? ? ? ?
?
? ? ?
CBSE XII | Mathematics
Sample Paper – 10 Solution
SECTION – B
5. Let X be the non-empty set for which P(X) is the power set.
ARB ? A ? B
i. ARA ? A ? A, every set is a subset of itself. R is reflexive
ii. If A, B, C ? P(X)
ARB ? A ? B, BRC ? B ? C
A ? B and B ? C ?A ? C
So ARC; Hence R is transitive.
iii. ARB ? A ? B does not imply B ? A
So B R A
R is not symmetric
R is reflexive, transitive but not symmetric ?R is not an equivalence relation
6. We have,
2A – 3B + 5C = O
2A = 3B – 5C
2 2 0 2 0 2
2A 3 5
3 1 4 7 1 6
6 6 0 10 0 10
2A
9 3 12 35 5 30
16 6 10
2A
26 2 18
?? ? ? ? ?
??
? ? ? ?
? ? ? ?
?? ? ? ? ?
??
? ? ? ?
? ? ? ?
???
?
??
? ? ?
??
16 6 10
1
A
2 26 2 18
???
?
??
? ? ?
??
8 3 5
A
13 1 9
???
?
??
? ? ?
??
Page 4
CBSE XII | Mathematics
Sample Paper – 10 Solution
Mathematics
Class XII
Sample Paper – 10 Solution
SECTION – A
1. By observation we find that
2 + x = 10
x = 8.
2.
? ?
d
cos x
dx
? ? ? ?
d
sin x x
dx
dy sin x
dx
2x
??
?
?
3. DE:
32
32
d y d y dy
y siny 0
dx dx dx
? ? ? ?
It is linear, since y siny ? is product of two different functions, and their individual
power is one.
CBSE XII | Mathematics
Sample Paper – 10 Solution
4. Let ? be the angles between, the given two lines
So, the angle between them given their direction cosines is given by
1 2 1 2 1 2
2 2 2 2 2 2
1 1 1 2 2 2
1
2
1
2
1
2
1
a a b b c c
cos
a b c a b c
substituting we get
a2
a3
b1
b2
c3
c1
11
cos
14
??
??
? ? ? ?
?
?
?
?
??
??
??
??
??
??
OR
Let ? be the angles between, the given two lines
So, the angle between them given their direction cosines is given by
? ? ? ? ? ? ? ? ? ? ? ?
? ?
1 2 1 2 1 2
2 2 2 2 2 2
1 1 1 2 2 2
1 2 1 2 1 2
1
a a b b c c
cos
a b c a b c
substituting we get
a a b b c c 2 1 7 2 3 4 0
cos 0
2
??
??
? ? ? ?
? ? ? ? ? ? ? ?
?
? ? ?
CBSE XII | Mathematics
Sample Paper – 10 Solution
SECTION – B
5. Let X be the non-empty set for which P(X) is the power set.
ARB ? A ? B
i. ARA ? A ? A, every set is a subset of itself. R is reflexive
ii. If A, B, C ? P(X)
ARB ? A ? B, BRC ? B ? C
A ? B and B ? C ?A ? C
So ARC; Hence R is transitive.
iii. ARB ? A ? B does not imply B ? A
So B R A
R is not symmetric
R is reflexive, transitive but not symmetric ?R is not an equivalence relation
6. We have,
2A – 3B + 5C = O
2A = 3B – 5C
2 2 0 2 0 2
2A 3 5
3 1 4 7 1 6
6 6 0 10 0 10
2A
9 3 12 35 5 30
16 6 10
2A
26 2 18
?? ? ? ? ?
??
? ? ? ?
? ? ? ?
?? ? ? ? ?
??
? ? ? ?
? ? ? ?
???
?
??
? ? ?
??
16 6 10
1
A
2 26 2 18
???
?
??
? ? ?
??
8 3 5
A
13 1 9
???
?
??
? ? ?
??
CBSE XII | Mathematics
Sample Paper – 10 Solution
7.
?
?
22
sin x cos x
dx
sin xcosx
?
?
?
?
?
?
??
?
??
??
? ? ? ? ?
22
22
sin x cos x
dx
sin xcosx
(cos x sin x)
2 dx
2sin xcosx
cos2x
2 dx
sin2x
2cos2x
dx
sin2x
f '(x)
log |sin2x| C .......( dx log f(x) c)
f(x)
8.
?
??
2
dx
5 8x x
?
?
?
?
?
? ? ?
?
? ? ? ? ?
?
? ? ?
?
??
??
??
??
??
??
??
2
2
2
22
dx
(x 8x 5)
dx
(x 8x 16 16 5)
dx
[(x 4) 21]
dx
( 21) (x 4)
1 21 (x 4)
log C
2 21 (x 4) 21
1 x 4 21
log C
2 21 x 4 21
Page 5
CBSE XII | Mathematics
Sample Paper – 10 Solution
Mathematics
Class XII
Sample Paper – 10 Solution
SECTION – A
1. By observation we find that
2 + x = 10
x = 8.
2.
? ?
d
cos x
dx
? ? ? ?
d
sin x x
dx
dy sin x
dx
2x
??
?
?
3. DE:
32
32
d y d y dy
y siny 0
dx dx dx
? ? ? ?
It is linear, since y siny ? is product of two different functions, and their individual
power is one.
CBSE XII | Mathematics
Sample Paper – 10 Solution
4. Let ? be the angles between, the given two lines
So, the angle between them given their direction cosines is given by
1 2 1 2 1 2
2 2 2 2 2 2
1 1 1 2 2 2
1
2
1
2
1
2
1
a a b b c c
cos
a b c a b c
substituting we get
a2
a3
b1
b2
c3
c1
11
cos
14
??
??
? ? ? ?
?
?
?
?
??
??
??
??
??
??
OR
Let ? be the angles between, the given two lines
So, the angle between them given their direction cosines is given by
? ? ? ? ? ? ? ? ? ? ? ?
? ?
1 2 1 2 1 2
2 2 2 2 2 2
1 1 1 2 2 2
1 2 1 2 1 2
1
a a b b c c
cos
a b c a b c
substituting we get
a a b b c c 2 1 7 2 3 4 0
cos 0
2
??
??
? ? ? ?
? ? ? ? ? ? ? ?
?
? ? ?
CBSE XII | Mathematics
Sample Paper – 10 Solution
SECTION – B
5. Let X be the non-empty set for which P(X) is the power set.
ARB ? A ? B
i. ARA ? A ? A, every set is a subset of itself. R is reflexive
ii. If A, B, C ? P(X)
ARB ? A ? B, BRC ? B ? C
A ? B and B ? C ?A ? C
So ARC; Hence R is transitive.
iii. ARB ? A ? B does not imply B ? A
So B R A
R is not symmetric
R is reflexive, transitive but not symmetric ?R is not an equivalence relation
6. We have,
2A – 3B + 5C = O
2A = 3B – 5C
2 2 0 2 0 2
2A 3 5
3 1 4 7 1 6
6 6 0 10 0 10
2A
9 3 12 35 5 30
16 6 10
2A
26 2 18
?? ? ? ? ?
??
? ? ? ?
? ? ? ?
?? ? ? ? ?
??
? ? ? ?
? ? ? ?
???
?
??
? ? ?
??
16 6 10
1
A
2 26 2 18
???
?
??
? ? ?
??
8 3 5
A
13 1 9
???
?
??
? ? ?
??
CBSE XII | Mathematics
Sample Paper – 10 Solution
7.
?
?
22
sin x cos x
dx
sin xcosx
?
?
?
?
?
?
??
?
??
??
? ? ? ? ?
22
22
sin x cos x
dx
sin xcosx
(cos x sin x)
2 dx
2sin xcosx
cos2x
2 dx
sin2x
2cos2x
dx
sin2x
f '(x)
log |sin2x| C .......( dx log f(x) c)
f(x)
8.
?
??
2
dx
5 8x x
?
?
?
?
?
? ? ?
?
? ? ? ? ?
?
? ? ?
?
??
??
??
??
??
??
??
2
2
2
22
dx
(x 8x 5)
dx
(x 8x 16 16 5)
dx
[(x 4) 21]
dx
( 21) (x 4)
1 21 (x 4)
log C
2 21 (x 4) 21
1 x 4 21
log C
2 21 x 4 21
CBSE XII | Mathematics
Sample Paper – 10 Solution
OR
Let I =
2
2
4
2
2
1
1
x1
x
dx dx
1
x1
x
x
??
?
?
?
?
?
(Dividing numerator and denominator by x
2
)
I
2
2
1
1
x
dx
1
x2
x
?
?
?
??
??
??
??
Substituting x -
1
x
= t,
2
1
1 dx dt
x
??
? ? ?
??
??
we get,
? ?
22
2
1
dt dt
I
t2
t2
1t
tan
22
?
??
??
?
?
?
1
2
1
1
x
1
x
tan
22
1 x 1
tan c
2 2x
?
?
??
?
??
?
??
??
??
??
?
??
??
??
??
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FAQs on Sample Solution Paper 10 - Math, Class 12 - Mathematics (Maths) Class 12 - JEE
| 1. What are the different types of matrices? |  |
Ans. Matrices can be classified into different types based on their properties. Some common types of matrices are:
1. Square Matrix: A matrix is said to be square if it has an equal number of rows and columns.
2. Diagonal Matrix: A matrix is said to be diagonal if all its non-diagonal elements are zero.
3. Identity Matrix: An identity matrix is a square matrix in which all the diagonal elements are one and all the non-diagonal elements are zero.
4. Symmetric Matrix: A matrix is said to be symmetric if it is equal to its transpose.
5. Skew-Symmetric Matrix: A matrix is said to be skew-symmetric if it is equal to the negative of its transpose.
| 2. What is the determinant of a matrix? | |
Ans. The determinant of a square matrix is a scalar value that can be computed from the elements of the matrix. It is denoted by |A| or det(A), where A is the matrix. The determinant provides important information about the matrix, such as whether it is invertible or singular.
To calculate the determinant of a 2x2 matrix [a b; c d], the formula is ad - bc. For larger matrices, there are various methods like cofactor expansion, row reduction, or using properties of determinants.
| 3. How do you solve a system of linear equations using matrices? | |
Ans. A system of linear equations can be solved using matrices by representing the equations in matrix form and applying matrix operations. Here's a step-by-step process:
1. Write the system of equations in matrix form, where the coefficients of variables form a matrix A, the variables form a column matrix X, and the constants form a column matrix B. The equation becomes AX = B.
2. Calculate the inverse of matrix A, denoted by A^(-1), if it exists. If A^(-1) does not exist, the system of equations either has no solution or infinitely many solutions.
3. Multiply both sides of the equation AX = B by A^(-1) to isolate X. The equation becomes X = A^(-1)B.
4. Compute the product A^(-1)B to obtain the values of variables in X.
5. If the system of equations has infinitely many solutions, the values of variables will be expressed in terms of parameters.
| 4. How can matrices be used in solving optimization problems? | |
Ans. Matrices can be used to solve optimization problems, particularly in linear programming. In linear programming, the goal is to maximize or minimize a linear objective function subject to linear constraints.
1. Define decision variables: Identify the variables that need to be optimized and represent them as a column matrix X.
2. Formulate the objective function: Express the objective function in terms of the decision variables and coefficients. This forms a linear combination that needs to be maximized or minimized.
3. Set up the constraints: Determine the constraints that the decision variables must satisfy. Represent the constraints as a system of linear equations or inequalities.
4. Represent the problem in matrix form: Create a coefficient matrix A for the constraints, a column matrix B for the constants, and a column matrix C for the coefficients of the objective function. The problem can now be written as AX ≤ B and CX as the objective function.
5. Apply linear programming techniques: Use matrix operations and optimization algorithms to solve the problem and find the optimal values for the decision variables.
| 5. What are eigenvalues and eigenvectors of a matrix? | |
Ans. Eigenvalues and eigenvectors are important concepts in linear algebra associated with square matrices.
Eigenvalues:
- Eigenvalues are scalar values that represent the scaling factor of eigenvectors.
- To find eigenvalues, we solve the characteristic equation det(A - λI) = 0, where A is the matrix, λ is the eigenvalue, and I is the identity matrix.
- The eigenvalues can be real or complex numbers.
Eigenvectors:
- Eigenvectors are non-zero vectors that remain in the same direction after a linear transformation by the matrix.
- To find eigenvectors, we substitute each eigenvalue into the equation (A - λI)X = 0 and solve for X.
- Eigenvectors are not unique and can be multiplied by any scalar value.
Eigenvalues and eigenvectors have various applications, such as solving systems of differential equations, image compression, and principal component analysis.
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Dec 22, 2024 Last updated |
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