Sample Solution Paper 3 - Math, Class 12

# Sample Solution Paper 3 - Math, Class 12 | Mathematics (Maths) Class 12 - JEE PDF Download

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CBSE XII | Mathematics
Sample Paper 3 – Solution

Mathematics
Class XII
Sample Paper – 3 Solution
Time: 3 hour                            Total Marks: 100

SECTION A

1. If a vertical line intersects the graph of a relation in two or more points, then the relation
is not a function.
Graph should have no vertical lines

2. Magnitude of ?(î + j + k )= 1, for it to be a unit vector.

2 2 2 2
.(i+j+k) 3 3 1
1
3

3.

3
2
2
x0
x0
y x x 1
dy
3x 1
dx
At a point where the curve cuts y-axis, x = 0
dy
3x 1 1
dx ?
?
? ? ?
??
?
?
? ? ? ?
?
?
?

Page 2

CBSE XII | Mathematics
Sample Paper 3 – Solution

Mathematics
Class XII
Sample Paper – 3 Solution
Time: 3 hour                            Total Marks: 100

SECTION A

1. If a vertical line intersects the graph of a relation in two or more points, then the relation
is not a function.
Graph should have no vertical lines

2. Magnitude of ?(î + j + k )= 1, for it to be a unit vector.

2 2 2 2
.(i+j+k) 3 3 1
1
3

3.

3
2
2
x0
x0
y x x 1
dy
3x 1
dx
At a point where the curve cuts y-axis, x = 0
dy
3x 1 1
dx ?
?
? ? ?
??
?
?
? ? ? ?
?
?
?

CBSE XII | Mathematics
Sample Paper 3 – Solution

4.  64 women lost their jobs in factory III in the last two months.

OR
? A(adjA) | A|I
? ? ?
? ? ?
8 0 1 0
| A|I 8 8I
0 8 0 1
det(A) | A| 8

SECTION B

5.
?
?? ? ??
??
?? ??
?? ??
1
1
sin sin
32

?? ?? ??
? ? ?
?? ??
?? ??
sin
36

?? ??
??
??
??
?
??
sin
36
sin 1
2

6.
?
?
? ??
?
??
?
??
1
1
2x
I log dx ...(1)
2x

?
?
? ??
?
??
?
??
?? ? ? ? ?
? ? ? ? ? ?
? ? ? ?
??
? ? ? ?
?
? ??
?
??
?
??
1
1
2x
Let f(x) log and
2x
2 x 2 x
f( x) log log f(x).
2 x 2 x
f(x) is an odd function.
2x
Thus, log dx 0
2x

Page 3

CBSE XII | Mathematics
Sample Paper 3 – Solution

Mathematics
Class XII
Sample Paper – 3 Solution
Time: 3 hour                            Total Marks: 100

SECTION A

1. If a vertical line intersects the graph of a relation in two or more points, then the relation
is not a function.
Graph should have no vertical lines

2. Magnitude of ?(î + j + k )= 1, for it to be a unit vector.

2 2 2 2
.(i+j+k) 3 3 1
1
3

3.

3
2
2
x0
x0
y x x 1
dy
3x 1
dx
At a point where the curve cuts y-axis, x = 0
dy
3x 1 1
dx ?
?
? ? ?
??
?
?
? ? ? ?
?
?
?

CBSE XII | Mathematics
Sample Paper 3 – Solution

4.  64 women lost their jobs in factory III in the last two months.

OR
? A(adjA) | A|I
? ? ?
? ? ?
8 0 1 0
| A|I 8 8I
0 8 0 1
det(A) | A| 8

SECTION B

5.
?
?? ? ??
??
?? ??
?? ??
1
1
sin sin
32

?? ?? ??
? ? ?
?? ??
?? ??
sin
36

?? ??
??
??
??
?
??
sin
36
sin 1
2

6.
?
?
? ??
?
??
?
??
1
1
2x
I log dx ...(1)
2x

?
?
? ??
?
??
?
??
?? ? ? ? ?
? ? ? ? ? ?
? ? ? ?
??
? ? ? ?
?
? ??
?
??
?
??
1
1
2x
Let f(x) log and
2x
2 x 2 x
f( x) log log f(x).
2 x 2 x
f(x) is an odd function.
2x
Thus, log dx 0
2x

CBSE XII | Mathematics
Sample Paper 3 – Solution

7.
? ? ? ? Let p =2i 3j 4k  and  q = ai 6j 8k

? ? ? ? ?
?
? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ? ?
? ? ?
Two vectors  p and q will be collinear if,
i j k
p  q 0 2 3 4 0
a 6 8
i(24 24) j ( 16 4a) k(12 3a) 0
0i (16 4a)j (12 3a)k 0i 0j 0k
16 4a 0 and 12+3a = 0
a4

8.
?
??
??
??
? ? ?
? ??
??
?
??
?
? ??
??
? ? ??
??
? ? ? ?
?
?
??
1
25
Given A = , then
13
A 6 5 0
35
and adjA = , obtained from A by interchanging the main diagonal
12
elements and multiplying by ( 1) the non - diagonal elements.
35
35 12 adjA
A
12 A1
?

Page 4

CBSE XII | Mathematics
Sample Paper 3 – Solution

Mathematics
Class XII
Sample Paper – 3 Solution
Time: 3 hour                            Total Marks: 100

SECTION A

1. If a vertical line intersects the graph of a relation in two or more points, then the relation
is not a function.
Graph should have no vertical lines

2. Magnitude of ?(î + j + k )= 1, for it to be a unit vector.

2 2 2 2
.(i+j+k) 3 3 1
1
3

3.

3
2
2
x0
x0
y x x 1
dy
3x 1
dx
At a point where the curve cuts y-axis, x = 0
dy
3x 1 1
dx ?
?
? ? ?
??
?
?
? ? ? ?
?
?
?

CBSE XII | Mathematics
Sample Paper 3 – Solution

4.  64 women lost their jobs in factory III in the last two months.

OR
? A(adjA) | A|I
? ? ?
? ? ?
8 0 1 0
| A|I 8 8I
0 8 0 1
det(A) | A| 8

SECTION B

5.
?
?? ? ??
??
?? ??
?? ??
1
1
sin sin
32

?? ?? ??
? ? ?
?? ??
?? ??
sin
36

?? ??
??
??
??
?
??
sin
36
sin 1
2

6.
?
?
? ??
?
??
?
??
1
1
2x
I log dx ...(1)
2x

?
?
? ??
?
??
?
??
?? ? ? ? ?
? ? ? ? ? ?
? ? ? ?
??
? ? ? ?
?
? ??
?
??
?
??
1
1
2x
Let f(x) log and
2x
2 x 2 x
f( x) log log f(x).
2 x 2 x
f(x) is an odd function.
2x
Thus, log dx 0
2x

CBSE XII | Mathematics
Sample Paper 3 – Solution

7.
? ? ? ? Let p =2i 3j 4k  and  q = ai 6j 8k

? ? ? ? ?
?
? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ? ?
? ? ?
Two vectors  p and q will be collinear if,
i j k
p  q 0 2 3 4 0
a 6 8
i(24 24) j ( 16 4a) k(12 3a) 0
0i (16 4a)j (12 3a)k 0i 0j 0k
16 4a 0 and 12+3a = 0
a4

8.
?
??
??
??
? ? ?
? ??
??
?
??
?
? ??
??
? ? ??
??
? ? ? ?
?
?
??
1
25
Given A = , then
13
A 6 5 0
35
and adjA = , obtained from A by interchanging the main diagonal
12
elements and multiplying by ( 1) the non - diagonal elements.
35
35 12 adjA
A
12 A1
?

CBSE XII | Mathematics
Sample Paper 3 – Solution

OR

? Since A is skew symmetric matrix.
??
? ? ?
? ? ?
? ? ?
? ? ? ? ?
??
??
T
T
T3
T
T
Therefore,A A
AA
A ( 1) A
AA
A A ......(Since A A )
2 A 0
A0

9.
?
?
?
?
??
?
? ? ? ? ? ? ?
?
? ? ? ? ? ?
? ? ? ?
?
? ? ? ? ? ? ? ?
1
1
1
1
Let sec x .
Then,x sec and for x < 1,
2
Given expression = cot ( cot ),
2
cot (cot( ))
sec as 0 <
2

10.
?
??
?
??
? ? ?
??
1
2
11
22
5x
y = tan
1 6x
y = tan 3x tan 2x
dy 3 2
dx
1 9x 1 4x

Page 5

CBSE XII | Mathematics
Sample Paper 3 – Solution

Mathematics
Class XII
Sample Paper – 3 Solution
Time: 3 hour                            Total Marks: 100

SECTION A

1. If a vertical line intersects the graph of a relation in two or more points, then the relation
is not a function.
Graph should have no vertical lines

2. Magnitude of ?(î + j + k )= 1, for it to be a unit vector.

2 2 2 2
.(i+j+k) 3 3 1
1
3

3.

3
2
2
x0
x0
y x x 1
dy
3x 1
dx
At a point where the curve cuts y-axis, x = 0
dy
3x 1 1
dx ?
?
? ? ?
??
?
?
? ? ? ?
?
?
?

CBSE XII | Mathematics
Sample Paper 3 – Solution

4.  64 women lost their jobs in factory III in the last two months.

OR
? A(adjA) | A|I
? ? ?
? ? ?
8 0 1 0
| A|I 8 8I
0 8 0 1
det(A) | A| 8

SECTION B

5.
?
?? ? ??
??
?? ??
?? ??
1
1
sin sin
32

?? ?? ??
? ? ?
?? ??
?? ??
sin
36

?? ??
??
??
??
?
??
sin
36
sin 1
2

6.
?
?
? ??
?
??
?
??
1
1
2x
I log dx ...(1)
2x

?
?
? ??
?
??
?
??
?? ? ? ? ?
? ? ? ? ? ?
? ? ? ?
??
? ? ? ?
?
? ??
?
??
?
??
1
1
2x
Let f(x) log and
2x
2 x 2 x
f( x) log log f(x).
2 x 2 x
f(x) is an odd function.
2x
Thus, log dx 0
2x

CBSE XII | Mathematics
Sample Paper 3 – Solution

7.
? ? ? ? Let p =2i 3j 4k  and  q = ai 6j 8k

? ? ? ? ?
?
? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ? ?
? ? ?
Two vectors  p and q will be collinear if,
i j k
p  q 0 2 3 4 0
a 6 8
i(24 24) j ( 16 4a) k(12 3a) 0
0i (16 4a)j (12 3a)k 0i 0j 0k
16 4a 0 and 12+3a = 0
a4

8.
?
??
??
??
? ? ?
? ??
??
?
??
?
? ??
??
? ? ??
??
? ? ? ?
?
?
??
1
25
Given A = , then
13
A 6 5 0
35
and adjA = , obtained from A by interchanging the main diagonal
12
elements and multiplying by ( 1) the non - diagonal elements.
35
35 12 adjA
A
12 A1
?

CBSE XII | Mathematics
Sample Paper 3 – Solution

OR

? Since A is skew symmetric matrix.
??
? ? ?
? ? ?
? ? ?
? ? ? ? ?
??
??
T
T
T3
T
T
Therefore,A A
AA
A ( 1) A
AA
A A ......(Since A A )
2 A 0
A0

9.
?
?
?
?
??
?
? ? ? ? ? ? ?
?
? ? ? ? ? ?
? ? ? ?
?
? ? ? ? ? ? ? ?
1
1
1
1
Let sec x .
Then,x sec and for x < 1,
2
Given expression = cot ( cot ),
2
cot (cot( ))
sec as 0 <
2

10.
?
??
?
??
? ? ?
??
1
2
11
22
5x
y = tan
1 6x
y = tan 3x tan 2x
dy 3 2
dx
1 9x 1 4x

CBSE XII | Mathematics
Sample Paper 3 – Solution

OR

Let x be the length of an edge of the cube, V be the volume and S be the surface area
at any time t.

?
??
?
??
??
??
??
??
??
??
??
??
??
? ? ?
??
??
32
3
3
2
2
2
2
2
x 10
Then,V x and S 6x .
It is given that,
dV
9 cm / sec
dt
d
(x ) 9
dt
dx
3x 9
dt
dx 3

dt
x
Now, S = 6x
dS dx
12x
dt dt
dS 3
12x
dt
x
dS 36
dt x
dS 36
3.6 cm / sec
dt 10

11.
? ? ? ? ? ? ?
2 2 2 2 2 2 2
x (y b) a b    or x y 2by a .......(1)

?
? ? ? ? ?
? ? ? ?
2 2 2
dy
2x 2y
dy dy
dx
2x 2y 2b 0 2b .......(2)
dy
dx dx
dx
Substituting in (1),
dy
(x y a ) 2xy 0
dx

```

## Mathematics (Maths) Class 12

204 videos|288 docs|139 tests

## FAQs on Sample Solution Paper 3 - Math, Class 12 - Mathematics (Maths) Class 12 - JEE

 1. How can I calculate the area of a triangle?
Ans. To calculate the area of a triangle, you can use the formula A = (base * height) / 2. Simply multiply the length of the base of the triangle by its corresponding height, and then divide the product by 2.
 2. What is the difference between a line segment and a line?
Ans. A line segment is a part of a line that has two endpoints. It has a definite length and can be measured. On the other hand, a line extends infinitely in both directions and has no endpoints. It has no specific length and cannot be measured.
 3. How do I find the domain and range of a function?
Ans. To find the domain of a function, you need to determine all possible values for the independent variable (x) that the function can accept. The range of a function, on the other hand, refers to all the possible values of the dependent variable (y) that the function can produce.
 4. What is the difference between discrete and continuous data?
Ans. Discrete data consists of distinct, separate values that can only take on specific values within a certain range. It often involves counting and can be represented by whole numbers. Continuous data, on the other hand, can take any value within a given range. It is often measured and can be represented by decimal numbers.
 5. How do I solve a system of linear equations using substitution?
Ans. To solve a system of linear equations using substitution, follow these steps: 1. Solve one of the equations for one variable in terms of the other variable. 2. Substitute the expression obtained in step 1 into the other equation, replacing the corresponding variable. 3. Simplify the resulting equation and solve for the remaining variable. 4. Substitute the value found in step 3 back into one of the original equations to find the value of the other variable. 5. Check the solution by substituting the values of the variables back into both original equations.

## Mathematics (Maths) Class 12

204 videos|288 docs|139 tests

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