Sample Solution Paper 9 - Math, Class 12

# Sample Solution Paper 9 - Math, Class 12 | Mathematics (Maths) Class 12 - JEE PDF Download

``` Page 1

CBSE XII | Mathematics
Sample Paper – 9 Solution

Mathematics
Class XII
Sample Paper – 9 Solution

SECTION – A

1. a42, means element at 3
rd
row and 2
nd
column
So,
a32 = 10

2. Differentiating w.r.t. x, we get,
? ? ? ?
? ? ? ?
? ?
d
sin cosx
dx
d
cos cosx cosx
dx
sin x cos cosx
?
??

3. DE:
2
22
2
2
dy dy
x1
dx dx
squaring
dy dy dy
x 2x 1
dx dx dx
dy
x 2x 1
dx
? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
??
??
??
??

It is linear, since x is independent variable.

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
a a b b c c
cos
a b c a b c
substituting we get
8
cos
53
??
??
? ? ? ?
??
??
??
??

Page 2

CBSE XII | Mathematics
Sample Paper – 9 Solution

Mathematics
Class XII
Sample Paper – 9 Solution

SECTION – A

1. a42, means element at 3
rd
row and 2
nd
column
So,
a32 = 10

2. Differentiating w.r.t. x, we get,
? ? ? ?
? ? ? ?
? ?
d
sin cosx
dx
d
cos cosx cosx
dx
sin x cos cosx
?
??

3. DE:
2
22
2
2
dy dy
x1
dx dx
squaring
dy dy dy
x 2x 1
dx dx dx
dy
x 2x 1
dx
? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
??
??
??
??

It is linear, since x is independent variable.

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
a a b b c c
cos
a b c a b c
substituting we get
8
cos
53
??
??
? ? ? ?
??
??
??
??

CBSE XII | Mathematics
Sample Paper – 9 Solution

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
a a b b c c
cos
a b c a b c
substituting we get
10
cos
9 22
??
??
? ? ? ?
??
??
??
??

SECTION – B

5. f: R+ ? [4, 8)  defined by f (x) = x² + 4
f is 1 – 1
Let x1, x2, ? R+ such that f(x1) = f(x2)

22
12
22
12
1 2 1 2
x 4 x 4
xx
x x x , x R
?
? ? ? ?
??
? ? ?

? f is 1 – 1
f is onto: Let y ? [4, 8)
f(x) = y ? x
2
+ 4 = y
? x = y4 ?
Since y ? [4, 8) ? x ? R+
For y ? [4, 8) there is a x ? R+ such that f (x) = y.
So f is onto.
So, f is bijective function and hence f is invertible.
The inverse of f is defined by
f: [4, 8) ? R+
f
-1
(y) = y4 ?
Page 3

CBSE XII | Mathematics
Sample Paper – 9 Solution

Mathematics
Class XII
Sample Paper – 9 Solution

SECTION – A

1. a42, means element at 3
rd
row and 2
nd
column
So,
a32 = 10

2. Differentiating w.r.t. x, we get,
? ? ? ?
? ? ? ?
? ?
d
sin cosx
dx
d
cos cosx cosx
dx
sin x cos cosx
?
??

3. DE:
2
22
2
2
dy dy
x1
dx dx
squaring
dy dy dy
x 2x 1
dx dx dx
dy
x 2x 1
dx
? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
??
??
??
??

It is linear, since x is independent variable.

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
a a b b c c
cos
a b c a b c
substituting we get
8
cos
53
??
??
? ? ? ?
??
??
??
??

CBSE XII | Mathematics
Sample Paper – 9 Solution

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
a a b b c c
cos
a b c a b c
substituting we get
10
cos
9 22
??
??
? ? ? ?
??
??
??
??

SECTION – B

5. f: R+ ? [4, 8)  defined by f (x) = x² + 4
f is 1 – 1
Let x1, x2, ? R+ such that f(x1) = f(x2)

22
12
22
12
1 2 1 2
x 4 x 4
xx
x x x , x R
?
? ? ? ?
??
? ? ?

? f is 1 – 1
f is onto: Let y ? [4, 8)
f(x) = y ? x
2
+ 4 = y
? x = y4 ?
Since y ? [4, 8) ? x ? R+
For y ? [4, 8) there is a x ? R+ such that f (x) = y.
So f is onto.
So, f is bijective function and hence f is invertible.
The inverse of f is defined by
f: [4, 8) ? R+
f
-1
(y) = y4 ?

CBSE XII | Mathematics
Sample Paper – 9 Solution

6.
We have,
? ?
1 0 0 2 0 0
A 0 1 0 and, B 0 3 0
0 0 2 0 0 1
1 0 0 2 0 0 3 0 0
A B 0 1 0 0 3 0 0 2 0 diag 3 2 1
0 0 2 0 0 1 0 0 1
and,
3 0 0 8 0 0 11 0 0
3A 4B 0 3 0 0 12 0 0 9 0 diag 1
0 0 6 0 0 4 0 0 2
? ? ? ?
? ? ? ?
? ? ?
? ? ? ?
? ? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ? ?
? ? ? ? ? ?
? ? 1 9 2

7.
x
x 2x
e
I dx
5 4e e
?
?
??

??
xx
Let e t e dx dt

Now integral I becomes,
? ?
2
2
2
dt
I
5 4t t
dt
I
5 4 4 4t t
dt
I
9 4 4t t
?
?
?
?
??
??
? ? ? ?
??
? ? ?

2
22
dt
I
9 (t 2)
dt
I
3 (t 2)
?
?
??
??
??
??

1
x
1
(t 2)
I sin C
3
(e 2)
I sin C
3
?
?
?
? ? ?
?
? ? ?

Page 4

CBSE XII | Mathematics
Sample Paper – 9 Solution

Mathematics
Class XII
Sample Paper – 9 Solution

SECTION – A

1. a42, means element at 3
rd
row and 2
nd
column
So,
a32 = 10

2. Differentiating w.r.t. x, we get,
? ? ? ?
? ? ? ?
? ?
d
sin cosx
dx
d
cos cosx cosx
dx
sin x cos cosx
?
??

3. DE:
2
22
2
2
dy dy
x1
dx dx
squaring
dy dy dy
x 2x 1
dx dx dx
dy
x 2x 1
dx
? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
??
??
??
??

It is linear, since x is independent variable.

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
a a b b c c
cos
a b c a b c
substituting we get
8
cos
53
??
??
? ? ? ?
??
??
??
??

CBSE XII | Mathematics
Sample Paper – 9 Solution

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
a a b b c c
cos
a b c a b c
substituting we get
10
cos
9 22
??
??
? ? ? ?
??
??
??
??

SECTION – B

5. f: R+ ? [4, 8)  defined by f (x) = x² + 4
f is 1 – 1
Let x1, x2, ? R+ such that f(x1) = f(x2)

22
12
22
12
1 2 1 2
x 4 x 4
xx
x x x , x R
?
? ? ? ?
??
? ? ?

? f is 1 – 1
f is onto: Let y ? [4, 8)
f(x) = y ? x
2
+ 4 = y
? x = y4 ?
Since y ? [4, 8) ? x ? R+
For y ? [4, 8) there is a x ? R+ such that f (x) = y.
So f is onto.
So, f is bijective function and hence f is invertible.
The inverse of f is defined by
f: [4, 8) ? R+
f
-1
(y) = y4 ?

CBSE XII | Mathematics
Sample Paper – 9 Solution

6.
We have,
? ?
1 0 0 2 0 0
A 0 1 0 and, B 0 3 0
0 0 2 0 0 1
1 0 0 2 0 0 3 0 0
A B 0 1 0 0 3 0 0 2 0 diag 3 2 1
0 0 2 0 0 1 0 0 1
and,
3 0 0 8 0 0 11 0 0
3A 4B 0 3 0 0 12 0 0 9 0 diag 1
0 0 6 0 0 4 0 0 2
? ? ? ?
? ? ? ?
? ? ?
? ? ? ?
? ? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ? ?
? ? ? ? ? ?
? ? 1 9 2

7.
x
x 2x
e
I dx
5 4e e
?
?
??

??
xx
Let e t e dx dt

Now integral I becomes,
? ?
2
2
2
dt
I
5 4t t
dt
I
5 4 4 4t t
dt
I
9 4 4t t
?
?
?
?
??
??
? ? ? ?
??
? ? ?

2
22
dt
I
9 (t 2)
dt
I
3 (t 2)
?
?
??
??
??
??

1
x
1
(t 2)
I sin C
3
(e 2)
I sin C
3
?
?
?
? ? ?
?
? ? ?

CBSE XII | Mathematics
Sample Paper – 9 Solution

8.
x
3
(x 4)e
I .dx
(x 2)
?
?
?
?

x
33
x
23
x'
23
xx
23
x
2
x 2 2
I e .dx
(x 2) (x 2)
12
I e .dx
(x 2) (x 2)
Thus the given integral is of the form,
12
I e f(x) f '(x) dx where, f(x) ; f (x)
(x 2) (x 2)
e 2e
I dx dx
(x 2) (x 2)
1 d 1
e
dx
(x 2) (x
?
?
?
??
??
?
??
??
??
??
??
??
??
??
??
??
??
?
? ? ? ?
??
??
??
??
??
2
x
2
dx
2)
e
So,I C
(x 2)
?
?? ??
?? ??
??
??
?? ??
??
?

OR
2
3
x
dx
1x
?
?

Let 1 + x
3
= t
? 0 + 3x
2
dx = dt
?
2
dt
x dx
3
?
2
3
3
dt
x
3
dx
t
1x
1 dt

3t
1
log t c
3
1
log 1 x c
3
??
?
??
??
??
??
?
??
?
??
? ? ?

Page 5

CBSE XII | Mathematics
Sample Paper – 9 Solution

Mathematics
Class XII
Sample Paper – 9 Solution

SECTION – A

1. a42, means element at 3
rd
row and 2
nd
column
So,
a32 = 10

2. Differentiating w.r.t. x, we get,
? ? ? ?
? ? ? ?
? ?
d
sin cosx
dx
d
cos cosx cosx
dx
sin x cos cosx
?
??

3. DE:
2
22
2
2
dy dy
x1
dx dx
squaring
dy dy dy
x 2x 1
dx dx dx
dy
x 2x 1
dx
? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
??
??
??
??

It is linear, since x is independent variable.

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
a a b b c c
cos
a b c a b c
substituting we get
8
cos
53
??
??
? ? ? ?
??
??
??
??

CBSE XII | Mathematics
Sample Paper – 9 Solution

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
a a b b c c
cos
a b c a b c
substituting we get
10
cos
9 22
??
??
? ? ? ?
??
??
??
??

SECTION – B

5. f: R+ ? [4, 8)  defined by f (x) = x² + 4
f is 1 – 1
Let x1, x2, ? R+ such that f(x1) = f(x2)

22
12
22
12
1 2 1 2
x 4 x 4
xx
x x x , x R
?
? ? ? ?
??
? ? ?

? f is 1 – 1
f is onto: Let y ? [4, 8)
f(x) = y ? x
2
+ 4 = y
? x = y4 ?
Since y ? [4, 8) ? x ? R+
For y ? [4, 8) there is a x ? R+ such that f (x) = y.
So f is onto.
So, f is bijective function and hence f is invertible.
The inverse of f is defined by
f: [4, 8) ? R+
f
-1
(y) = y4 ?

CBSE XII | Mathematics
Sample Paper – 9 Solution

6.
We have,
? ?
1 0 0 2 0 0
A 0 1 0 and, B 0 3 0
0 0 2 0 0 1
1 0 0 2 0 0 3 0 0
A B 0 1 0 0 3 0 0 2 0 diag 3 2 1
0 0 2 0 0 1 0 0 1
and,
3 0 0 8 0 0 11 0 0
3A 4B 0 3 0 0 12 0 0 9 0 diag 1
0 0 6 0 0 4 0 0 2
? ? ? ?
? ? ? ?
? ? ?
? ? ? ?
? ? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ? ?
? ? ? ? ? ?
? ? 1 9 2

7.
x
x 2x
e
I dx
5 4e e
?
?
??

??
xx
Let e t e dx dt

Now integral I becomes,
? ?
2
2
2
dt
I
5 4t t
dt
I
5 4 4 4t t
dt
I
9 4 4t t
?
?
?
?
??
??
? ? ? ?
??
? ? ?

2
22
dt
I
9 (t 2)
dt
I
3 (t 2)
?
?
??
??
??
??

1
x
1
(t 2)
I sin C
3
(e 2)
I sin C
3
?
?
?
? ? ?
?
? ? ?

CBSE XII | Mathematics
Sample Paper – 9 Solution

8.
x
3
(x 4)e
I .dx
(x 2)
?
?
?
?

x
33
x
23
x'
23
xx
23
x
2
x 2 2
I e .dx
(x 2) (x 2)
12
I e .dx
(x 2) (x 2)
Thus the given integral is of the form,
12
I e f(x) f '(x) dx where, f(x) ; f (x)
(x 2) (x 2)
e 2e
I dx dx
(x 2) (x 2)
1 d 1
e
dx
(x 2) (x
?
?
?
??
??
?
??
??
??
??
??
??
??
??
??
??
??
?
? ? ? ?
??
??
??
??
??
2
x
2
dx
2)
e
So,I C
(x 2)
?
?? ??
?? ??
??
??
?? ??
??
?

OR
2
3
x
dx
1x
?
?

Let 1 + x
3
= t
? 0 + 3x
2
dx = dt
?
2
dt
x dx
3
?
2
3
3
dt
x
3
dx
t
1x
1 dt

3t
1
log t c
3
1
log 1 x c
3
??
?
??
??
??
??
?
??
?
??
? ? ?

CBSE XII | Mathematics
Sample Paper – 9 Solution

9. We have to differentiate it w.r.t. x two times
? ?
? ?
? ?
2
2
2
2
2
2
Differentiating
dy
2y m 2x
dx
dy
y mx............... 1
dx
differentiating again
d y dy
ym
dx dx
from 1
d y dy y dy
y
dx dx x dx
which is the required differential equation
??
??
??
? ? ?
??
??
??
??
??
??

10.
r. 3i 4j 12k 13 0
22
2 2 2 2
2
22
3x 4y 12z 13 0
Distance of point (1, 1, p) from the plane , is given by
3 1 4 1 12 p 13 20 12p
3 4 12 3 4 12
Distance of point ( 3, 0, 1) from the plane , is given by
3 3 4 0 12 1 13
8
3 4 12 3
2
22
4 12

22
2 2 2 2
The two distances are equal
20 12p 8
3 4 12 3 4 12
20 12p 8
20 12p 8
7
p 1,
3

```

## Mathematics (Maths) Class 12

204 videos|288 docs|139 tests

## FAQs on Sample Solution Paper 9 - Math, Class 12 - Mathematics (Maths) Class 12 - JEE

 1. What are the different types of functions in mathematics?
Ans. In mathematics, there are several types of functions. Some commonly used types include linear functions, quadratic functions, exponential functions, logarithmic functions, and trigonometric functions.
 2. How can I determine if a function is one-to-one or onto?
Ans. To determine if a function is one-to-one, we need to check if different inputs produce different outputs. If for every pair of distinct inputs, the function produces distinct outputs, then it is one-to-one. On the other hand, to determine if a function is onto, we need to check if every element in the range of the function has a corresponding element in the domain.
 3. What is the difference between a function and an equation?
Ans. A function is a relationship between a set of inputs (domain) and a set of outputs (range), where each input corresponds to exactly one output. An equation, on the other hand, is a statement that shows the equality between two expressions. While a function can be represented by an equation, not all equations represent functions.
 4. How do I find the domain and range of a function?
Ans. To find the domain of a function, we need to determine the set of all possible inputs or x-values for which the function is defined. The range, on the other hand, is the set of all possible outputs or y-values that the function can produce. The domain and range can be found by analyzing the restrictions or limitations of the function, such as division by zero or taking the square root of a negative number.
 5. Can a function have more than one inverse?
Ans. No, a function cannot have more than one inverse. By definition, the inverse of a function is a function that "undoes" the original function, swapping the roles of the input and output variables. If a function has an inverse, it is unique. However, not all functions have inverses. A function must be one-to-one to have an inverse.

## Mathematics (Maths) Class 12

204 videos|288 docs|139 tests

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