Class 12 Exam > Class 12 Notes > ICSE Class 12 Science Previous Year Papers > ICSE Mathematics Previous Year Paper with Solutions - 2023
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Page 1
Mathematics
ISC Boards 2023
1. In subparts (i) to (x) choose the correct options and in subparts (xi) to (xv), answer the questions as instructed.
Answer (A)
Sol.
Let ?? = { 1, 2, 3}
?? = { ( 1, 1) , ( 2, 2) , ( 1, 2) , ( 3,3) , ( 2, 3) }.
For all ?? ? ?? , we have ( ?? , ?? )? ??
i.e., 1,2,3 ? ?? , we have ( 1, 1) , ( 2, 2) , ( 3, 3)? ??
So, ?? is reflexive.
Since ( 1, 2)? ?? but ( 2, 1)? ??
Hence, ?? is not symmetric.
Similarly, ( 1, 2) , ( 2,3)? ?? but ( 1, 3)? ??
Here, ?? is not transitive.
ii. If ?? is a square matrix of order 3, then |2?? | is equal to:
A. 2|?? |
B. 4|?? |
C. 8|?? |
D. 6|?? |
Answer (C)
Sol.
We know that,
|???? | = ?? ?? |?? |, where ?? is order of matrix.
? |2?? | = 2
3
|?? |
= 8|?? |
iii. If the following function is continuous at ?? = 2 then the value of k will be:
?? ( ?? )= {
2?? + 1, ???? ?? < 2
?? , ???? ?? = 2
3?? - 1, ???? ?? > 2
A. 2
B. 3
C. 5
D. -1
Answer (C)
i. A relation ?? on {1, 2, 3} is given by ?? = {(1, 1), (2, 2), (1, 2), (3,3), (2, 3)}. Then the relation ?? is:
A. Reflexive
B. Symmetric
C. Transitive
D. Symmetric and transitive.
Page 2
Mathematics
ISC Boards 2023
1. In subparts (i) to (x) choose the correct options and in subparts (xi) to (xv), answer the questions as instructed.
Answer (A)
Sol.
Let ?? = { 1, 2, 3}
?? = { ( 1, 1) , ( 2, 2) , ( 1, 2) , ( 3,3) , ( 2, 3) }.
For all ?? ? ?? , we have ( ?? , ?? )? ??
i.e., 1,2,3 ? ?? , we have ( 1, 1) , ( 2, 2) , ( 3, 3)? ??
So, ?? is reflexive.
Since ( 1, 2)? ?? but ( 2, 1)? ??
Hence, ?? is not symmetric.
Similarly, ( 1, 2) , ( 2,3)? ?? but ( 1, 3)? ??
Here, ?? is not transitive.
ii. If ?? is a square matrix of order 3, then |2?? | is equal to:
A. 2|?? |
B. 4|?? |
C. 8|?? |
D. 6|?? |
Answer (C)
Sol.
We know that,
|???? | = ?? ?? |?? |, where ?? is order of matrix.
? |2?? | = 2
3
|?? |
= 8|?? |
iii. If the following function is continuous at ?? = 2 then the value of k will be:
?? ( ?? )= {
2?? + 1, ???? ?? < 2
?? , ???? ?? = 2
3?? - 1, ???? ?? > 2
A. 2
B. 3
C. 5
D. -1
Answer (C)
i. A relation ?? on {1, 2, 3} is given by ?? = {(1, 1), (2, 2), (1, 2), (3,3), (2, 3)}. Then the relation ?? is:
A. Reflexive
B. Symmetric
C. Transitive
D. Symmetric and transitive.
Sol.
?? ( ?? )= {
2?? + 1, ???? ?? < 2
?? , ???? ?? = 2
3?? - 1, ???? ?? > 2
?? ( ?? ) is continuous at ?? = 2
? lim
?? ?2
-
?? ( ?? )= lim
?? ?2
+
?? ( ?? )= ?? ( 2)
? 2( 2)+ 1 = 3( 2)- 1 = ??
? ?? = 5
iv. An edge of a variable cube is increasing at the rate of 10cm/sec. How fast will the volume of the cube
increase if the edge is 5 cm long?
A. 75 ????
3
/??????
B. 750 ????
3
/??????
C. 7500 ????
3
/??????
D. 1250 ????
3
/??????
Answer (B)
Sol.
Let side of the cube is ??
Volume = ?? 3
Given
????
????
= 10 ???? /??????
?? = ?? 3
Differentiate w.r.t ??
????
????
= 3?? 2
.
????
????
= 3( 5)
2
× 10
= 750 ????
3
/??????
v. Let ?? ( ?? )= ?? 3
be a function with domain {0, 1, 2, 3}. Then domain of ?? -1
is:
A. { 3, 2, 1, 0}
B. { 0, -1, -2, -3}
C. { 0, 1, 8, 27}
D. { 0, -1, -8, -27}
Answer (C)
Sol.
?? ( ?? )= ?? 3
Domain = { 0, 1, 2, 3}
? ?? ( ?? )= { ( 0, 0) , ( 1, 1) , ( 2, 8) , ( 3, 27) }
? ?? -1
( ?? )= { ( 0, 0) , ( 1, 1) , ( 8, 2) , ( 27, 3) }
? Domain of ?? -1
= { 0, 1, 8, 27}
vi. For the curve ?? 2
= 2?? 3
- 7, the slope of the normal at (2, 3) is:
A. 4
B.
1
4
C. -4
D.
-1
4
Answer (D)
Sol.
?? 2
= 2?? 3
- 7
Differentiate w.r.t ??
Page 3
Mathematics
ISC Boards 2023
1. In subparts (i) to (x) choose the correct options and in subparts (xi) to (xv), answer the questions as instructed.
Answer (A)
Sol.
Let ?? = { 1, 2, 3}
?? = { ( 1, 1) , ( 2, 2) , ( 1, 2) , ( 3,3) , ( 2, 3) }.
For all ?? ? ?? , we have ( ?? , ?? )? ??
i.e., 1,2,3 ? ?? , we have ( 1, 1) , ( 2, 2) , ( 3, 3)? ??
So, ?? is reflexive.
Since ( 1, 2)? ?? but ( 2, 1)? ??
Hence, ?? is not symmetric.
Similarly, ( 1, 2) , ( 2,3)? ?? but ( 1, 3)? ??
Here, ?? is not transitive.
ii. If ?? is a square matrix of order 3, then |2?? | is equal to:
A. 2|?? |
B. 4|?? |
C. 8|?? |
D. 6|?? |
Answer (C)
Sol.
We know that,
|???? | = ?? ?? |?? |, where ?? is order of matrix.
? |2?? | = 2
3
|?? |
= 8|?? |
iii. If the following function is continuous at ?? = 2 then the value of k will be:
?? ( ?? )= {
2?? + 1, ???? ?? < 2
?? , ???? ?? = 2
3?? - 1, ???? ?? > 2
A. 2
B. 3
C. 5
D. -1
Answer (C)
i. A relation ?? on {1, 2, 3} is given by ?? = {(1, 1), (2, 2), (1, 2), (3,3), (2, 3)}. Then the relation ?? is:
A. Reflexive
B. Symmetric
C. Transitive
D. Symmetric and transitive.
Sol.
?? ( ?? )= {
2?? + 1, ???? ?? < 2
?? , ???? ?? = 2
3?? - 1, ???? ?? > 2
?? ( ?? ) is continuous at ?? = 2
? lim
?? ?2
-
?? ( ?? )= lim
?? ?2
+
?? ( ?? )= ?? ( 2)
? 2( 2)+ 1 = 3( 2)- 1 = ??
? ?? = 5
iv. An edge of a variable cube is increasing at the rate of 10cm/sec. How fast will the volume of the cube
increase if the edge is 5 cm long?
A. 75 ????
3
/??????
B. 750 ????
3
/??????
C. 7500 ????
3
/??????
D. 1250 ????
3
/??????
Answer (B)
Sol.
Let side of the cube is ??
Volume = ?? 3
Given
????
????
= 10 ???? /??????
?? = ?? 3
Differentiate w.r.t ??
????
????
= 3?? 2
.
????
????
= 3( 5)
2
× 10
= 750 ????
3
/??????
v. Let ?? ( ?? )= ?? 3
be a function with domain {0, 1, 2, 3}. Then domain of ?? -1
is:
A. { 3, 2, 1, 0}
B. { 0, -1, -2, -3}
C. { 0, 1, 8, 27}
D. { 0, -1, -8, -27}
Answer (C)
Sol.
?? ( ?? )= ?? 3
Domain = { 0, 1, 2, 3}
? ?? ( ?? )= { ( 0, 0) , ( 1, 1) , ( 2, 8) , ( 3, 27) }
? ?? -1
( ?? )= { ( 0, 0) , ( 1, 1) , ( 8, 2) , ( 27, 3) }
? Domain of ?? -1
= { 0, 1, 8, 27}
vi. For the curve ?? 2
= 2?? 3
- 7, the slope of the normal at (2, 3) is:
A. 4
B.
1
4
C. -4
D.
-1
4
Answer (D)
Sol.
?? 2
= 2?? 3
- 7
Differentiate w.r.t ??
2?? ????
????
= 6?? 2
?
????
????
=
3?? 2
??
At (2, 3),
????
????
|
( 2.3)
=
3( 2)
2
3
= 4
? Slope of Normal =
-1
4
vii. Evaluate: ?
?? ?? 2
+1
????
A. 2 log( ?? 2
+ 1)+ ??
B.
1
2
log( ?? 2
+ 1)+ ??
C. ?? ?? 2
+1
+ ??
D. log?? +
?? 2
2
+ ??
Answer (B)
Sol.
?? = ?
?? ?? 2
+ 1
????
Put ?? 2
+ 1 = ??
? 2?????? = ????
? ?? =
1
2
?
????
??
=
1
2
ln?? + ??
=
1
2
ln( ?? 2
+ 1)+ ??
viii. The derivative of log?? with respect to
1
?? is:
A.
1
??
B.
-1
?? 3
C. -
1
??
D. -??
Answer (D)
Sol.
Let
1
?? = ??
? -
1
?? 2
=
????
????
?
????
????
= -?? 2
Now, ?? = log??
????
????
= (
????
????
)(
????
????
)
= (
1
?? )( -?? 2
)
= -??
ix. The interval in which the function ?? ( ?? )= 5 + 36?? - 3?? 2
increases will be:
A. ( -8, 6)
B. ( 6, 8)
C. ( -6, 6)
D. ( 0, -6)
Answer (B)
Page 4
Mathematics
ISC Boards 2023
1. In subparts (i) to (x) choose the correct options and in subparts (xi) to (xv), answer the questions as instructed.
Answer (A)
Sol.
Let ?? = { 1, 2, 3}
?? = { ( 1, 1) , ( 2, 2) , ( 1, 2) , ( 3,3) , ( 2, 3) }.
For all ?? ? ?? , we have ( ?? , ?? )? ??
i.e., 1,2,3 ? ?? , we have ( 1, 1) , ( 2, 2) , ( 3, 3)? ??
So, ?? is reflexive.
Since ( 1, 2)? ?? but ( 2, 1)? ??
Hence, ?? is not symmetric.
Similarly, ( 1, 2) , ( 2,3)? ?? but ( 1, 3)? ??
Here, ?? is not transitive.
ii. If ?? is a square matrix of order 3, then |2?? | is equal to:
A. 2|?? |
B. 4|?? |
C. 8|?? |
D. 6|?? |
Answer (C)
Sol.
We know that,
|???? | = ?? ?? |?? |, where ?? is order of matrix.
? |2?? | = 2
3
|?? |
= 8|?? |
iii. If the following function is continuous at ?? = 2 then the value of k will be:
?? ( ?? )= {
2?? + 1, ???? ?? < 2
?? , ???? ?? = 2
3?? - 1, ???? ?? > 2
A. 2
B. 3
C. 5
D. -1
Answer (C)
i. A relation ?? on {1, 2, 3} is given by ?? = {(1, 1), (2, 2), (1, 2), (3,3), (2, 3)}. Then the relation ?? is:
A. Reflexive
B. Symmetric
C. Transitive
D. Symmetric and transitive.
Sol.
?? ( ?? )= {
2?? + 1, ???? ?? < 2
?? , ???? ?? = 2
3?? - 1, ???? ?? > 2
?? ( ?? ) is continuous at ?? = 2
? lim
?? ?2
-
?? ( ?? )= lim
?? ?2
+
?? ( ?? )= ?? ( 2)
? 2( 2)+ 1 = 3( 2)- 1 = ??
? ?? = 5
iv. An edge of a variable cube is increasing at the rate of 10cm/sec. How fast will the volume of the cube
increase if the edge is 5 cm long?
A. 75 ????
3
/??????
B. 750 ????
3
/??????
C. 7500 ????
3
/??????
D. 1250 ????
3
/??????
Answer (B)
Sol.
Let side of the cube is ??
Volume = ?? 3
Given
????
????
= 10 ???? /??????
?? = ?? 3
Differentiate w.r.t ??
????
????
= 3?? 2
.
????
????
= 3( 5)
2
× 10
= 750 ????
3
/??????
v. Let ?? ( ?? )= ?? 3
be a function with domain {0, 1, 2, 3}. Then domain of ?? -1
is:
A. { 3, 2, 1, 0}
B. { 0, -1, -2, -3}
C. { 0, 1, 8, 27}
D. { 0, -1, -8, -27}
Answer (C)
Sol.
?? ( ?? )= ?? 3
Domain = { 0, 1, 2, 3}
? ?? ( ?? )= { ( 0, 0) , ( 1, 1) , ( 2, 8) , ( 3, 27) }
? ?? -1
( ?? )= { ( 0, 0) , ( 1, 1) , ( 8, 2) , ( 27, 3) }
? Domain of ?? -1
= { 0, 1, 8, 27}
vi. For the curve ?? 2
= 2?? 3
- 7, the slope of the normal at (2, 3) is:
A. 4
B.
1
4
C. -4
D.
-1
4
Answer (D)
Sol.
?? 2
= 2?? 3
- 7
Differentiate w.r.t ??
2?? ????
????
= 6?? 2
?
????
????
=
3?? 2
??
At (2, 3),
????
????
|
( 2.3)
=
3( 2)
2
3
= 4
? Slope of Normal =
-1
4
vii. Evaluate: ?
?? ?? 2
+1
????
A. 2 log( ?? 2
+ 1)+ ??
B.
1
2
log( ?? 2
+ 1)+ ??
C. ?? ?? 2
+1
+ ??
D. log?? +
?? 2
2
+ ??
Answer (B)
Sol.
?? = ?
?? ?? 2
+ 1
????
Put ?? 2
+ 1 = ??
? 2?????? = ????
? ?? =
1
2
?
????
??
=
1
2
ln?? + ??
=
1
2
ln( ?? 2
+ 1)+ ??
viii. The derivative of log?? with respect to
1
?? is:
A.
1
??
B.
-1
?? 3
C. -
1
??
D. -??
Answer (D)
Sol.
Let
1
?? = ??
? -
1
?? 2
=
????
????
?
????
????
= -?? 2
Now, ?? = log??
????
????
= (
????
????
)(
????
????
)
= (
1
?? )( -?? 2
)
= -??
ix. The interval in which the function ?? ( ?? )= 5 + 36?? - 3?? 2
increases will be:
A. ( -8, 6)
B. ( 6, 8)
C. ( -6, 6)
D. ( 0, -6)
Answer (B)
Sol.
?? ( ?? )= ?? + ?????? - ?? ?? ??
? ?? '
( ?? )= ???? - ???? = -?? ( ?? - ?? )
? ?? '
( ?? )> ?? for ?? < ??
? function is increasing in ( -8, ?? )
x. Evaluate ? ?? 17
cos
4
?? ????
1
-1
A. 8
B. 1
C. -1
D. 0
Answer (D)
Sol.
?? = ? ?? 17
cos
4
?? ????
1
-1
Since, ?? 17
cos
4
?? is an odd function.
And we know that ? ?? ( ?? ) ???? = 0
?? -?? if ?? ( ?? ) is an odd function.
? ?? = 0
xi. Solve the differential equation:
????
????
= cosec??
Sol.
????
????
= cosec?? ?
????
cosec?? = ????
? sin?? ???? = ????
Integrating both sides
? ? sin?? ???? = ? ????
? - cos?? = ?? + ??
xii. For what value of ?? the matrix [
0 ?? -6 0
] is a skew symmetric matrix?
Sol.
?? = [
0 ?? -6 0
]
For skew symmetric matrix,
?? = -?? ??
? [
0 ?? -6 0
] = - [
0 -6]
?? 0
]
? [
0 ?? -6 0
] = [
0 6]
-?? 0
]
? ?? = 6
xiii. Evaluate ? |2?? + 1|????
1
0
Sol.
?? = ? |2?? + 1|????
1
0
?? = ? ( 2?? + 1) ????
1
0
[? 2?? + 1 > 0 ? ?? ? ( 0, 1) ]
= [?? 2
+ ?? ]
0
1
= 2
xiv. Evaluate ?
1+cos ?? sin
2
?? ????
Sol.
Page 5
Mathematics
ISC Boards 2023
1. In subparts (i) to (x) choose the correct options and in subparts (xi) to (xv), answer the questions as instructed.
Answer (A)
Sol.
Let ?? = { 1, 2, 3}
?? = { ( 1, 1) , ( 2, 2) , ( 1, 2) , ( 3,3) , ( 2, 3) }.
For all ?? ? ?? , we have ( ?? , ?? )? ??
i.e., 1,2,3 ? ?? , we have ( 1, 1) , ( 2, 2) , ( 3, 3)? ??
So, ?? is reflexive.
Since ( 1, 2)? ?? but ( 2, 1)? ??
Hence, ?? is not symmetric.
Similarly, ( 1, 2) , ( 2,3)? ?? but ( 1, 3)? ??
Here, ?? is not transitive.
ii. If ?? is a square matrix of order 3, then |2?? | is equal to:
A. 2|?? |
B. 4|?? |
C. 8|?? |
D. 6|?? |
Answer (C)
Sol.
We know that,
|???? | = ?? ?? |?? |, where ?? is order of matrix.
? |2?? | = 2
3
|?? |
= 8|?? |
iii. If the following function is continuous at ?? = 2 then the value of k will be:
?? ( ?? )= {
2?? + 1, ???? ?? < 2
?? , ???? ?? = 2
3?? - 1, ???? ?? > 2
A. 2
B. 3
C. 5
D. -1
Answer (C)
i. A relation ?? on {1, 2, 3} is given by ?? = {(1, 1), (2, 2), (1, 2), (3,3), (2, 3)}. Then the relation ?? is:
A. Reflexive
B. Symmetric
C. Transitive
D. Symmetric and transitive.
Sol.
?? ( ?? )= {
2?? + 1, ???? ?? < 2
?? , ???? ?? = 2
3?? - 1, ???? ?? > 2
?? ( ?? ) is continuous at ?? = 2
? lim
?? ?2
-
?? ( ?? )= lim
?? ?2
+
?? ( ?? )= ?? ( 2)
? 2( 2)+ 1 = 3( 2)- 1 = ??
? ?? = 5
iv. An edge of a variable cube is increasing at the rate of 10cm/sec. How fast will the volume of the cube
increase if the edge is 5 cm long?
A. 75 ????
3
/??????
B. 750 ????
3
/??????
C. 7500 ????
3
/??????
D. 1250 ????
3
/??????
Answer (B)
Sol.
Let side of the cube is ??
Volume = ?? 3
Given
????
????
= 10 ???? /??????
?? = ?? 3
Differentiate w.r.t ??
????
????
= 3?? 2
.
????
????
= 3( 5)
2
× 10
= 750 ????
3
/??????
v. Let ?? ( ?? )= ?? 3
be a function with domain {0, 1, 2, 3}. Then domain of ?? -1
is:
A. { 3, 2, 1, 0}
B. { 0, -1, -2, -3}
C. { 0, 1, 8, 27}
D. { 0, -1, -8, -27}
Answer (C)
Sol.
?? ( ?? )= ?? 3
Domain = { 0, 1, 2, 3}
? ?? ( ?? )= { ( 0, 0) , ( 1, 1) , ( 2, 8) , ( 3, 27) }
? ?? -1
( ?? )= { ( 0, 0) , ( 1, 1) , ( 8, 2) , ( 27, 3) }
? Domain of ?? -1
= { 0, 1, 8, 27}
vi. For the curve ?? 2
= 2?? 3
- 7, the slope of the normal at (2, 3) is:
A. 4
B.
1
4
C. -4
D.
-1
4
Answer (D)
Sol.
?? 2
= 2?? 3
- 7
Differentiate w.r.t ??
2?? ????
????
= 6?? 2
?
????
????
=
3?? 2
??
At (2, 3),
????
????
|
( 2.3)
=
3( 2)
2
3
= 4
? Slope of Normal =
-1
4
vii. Evaluate: ?
?? ?? 2
+1
????
A. 2 log( ?? 2
+ 1)+ ??
B.
1
2
log( ?? 2
+ 1)+ ??
C. ?? ?? 2
+1
+ ??
D. log?? +
?? 2
2
+ ??
Answer (B)
Sol.
?? = ?
?? ?? 2
+ 1
????
Put ?? 2
+ 1 = ??
? 2?????? = ????
? ?? =
1
2
?
????
??
=
1
2
ln?? + ??
=
1
2
ln( ?? 2
+ 1)+ ??
viii. The derivative of log?? with respect to
1
?? is:
A.
1
??
B.
-1
?? 3
C. -
1
??
D. -??
Answer (D)
Sol.
Let
1
?? = ??
? -
1
?? 2
=
????
????
?
????
????
= -?? 2
Now, ?? = log??
????
????
= (
????
????
)(
????
????
)
= (
1
?? )( -?? 2
)
= -??
ix. The interval in which the function ?? ( ?? )= 5 + 36?? - 3?? 2
increases will be:
A. ( -8, 6)
B. ( 6, 8)
C. ( -6, 6)
D. ( 0, -6)
Answer (B)
Sol.
?? ( ?? )= ?? + ?????? - ?? ?? ??
? ?? '
( ?? )= ???? - ???? = -?? ( ?? - ?? )
? ?? '
( ?? )> ?? for ?? < ??
? function is increasing in ( -8, ?? )
x. Evaluate ? ?? 17
cos
4
?? ????
1
-1
A. 8
B. 1
C. -1
D. 0
Answer (D)
Sol.
?? = ? ?? 17
cos
4
?? ????
1
-1
Since, ?? 17
cos
4
?? is an odd function.
And we know that ? ?? ( ?? ) ???? = 0
?? -?? if ?? ( ?? ) is an odd function.
? ?? = 0
xi. Solve the differential equation:
????
????
= cosec??
Sol.
????
????
= cosec?? ?
????
cosec?? = ????
? sin?? ???? = ????
Integrating both sides
? ? sin?? ???? = ? ????
? - cos?? = ?? + ??
xii. For what value of ?? the matrix [
0 ?? -6 0
] is a skew symmetric matrix?
Sol.
?? = [
0 ?? -6 0
]
For skew symmetric matrix,
?? = -?? ??
? [
0 ?? -6 0
] = - [
0 -6]
?? 0
]
? [
0 ?? -6 0
] = [
0 6]
-?? 0
]
? ?? = 6
xiii. Evaluate ? |2?? + 1|????
1
0
Sol.
?? = ? |2?? + 1|????
1
0
?? = ? ( 2?? + 1) ????
1
0
[? 2?? + 1 > 0 ? ?? ? ( 0, 1) ]
= [?? 2
+ ?? ]
0
1
= 2
xiv. Evaluate ?
1+cos ?? sin
2
?? ????
Sol.
?? = ?
1 + cos?? sin
2
??
= ? (
1
sin
2
?? +
cos?? sin
2
?? )????
= ?( cosec
2
?? + cosec?? · cot?? ) ????
= - cot?? - cosec?? + ??
xv. A bag contains 19 tickets, numbered from 1 to 19. Two tickets are drawn randomly in succession with
replacement. Find the probability that both the tickets drawn are even numbers.
Sol.
No. of even numbers from 1 to 19 is 9.
? Required probability is,
?? = (
9
19
)× (
9
19
)
=
81
361
2.
i. If ?? ( ?? )= [4 - ( ?? - 7)
3
]
1
5
is a real invertible function, then find ?? -1
( ?? )
Sol.
?? ( ?? )= [4 - ( ?? - 7)
3
]
1
5
Let ?? = [4 - ( ?? - 7) )
3
]
1
5
? ?? 5
= 4 - ( ?? - 7) )
3
? ( ?? - 7)
3
= 4 - ?? 5
? ?? - 7 = ( 4 - ?? 5
)
1
3
? ?? = 7 + ( 4 - ?? 5
)
1
3
Now, replace ?? by ??
? ?? = 7 + ( 4 - ?? 5
)
1
3
? ?? -1
( ?? )= 7 + ( 4 - ?? 5
)
1
3
OR
ii. Let ?? = R - { 2} and ?? = R - { 1}. If ?? ; ?? ? ?? is a function defined by ?? ( ?? )=
?? -1
?? -2
then show that ?? is a one-one and
an onto function.
Sol.
Given, ?? = R - { ?? }, ?? = R - { ?? }
?? ( ?? )=
?? - ?? ?? - ??
For one-one function
?? ( ?? ?? )= ?? ( ?? ?? )
?
?? ?? - ?? ?? ?? - ?? =
?? ?? - ?? ?? ?? - ??
? ?? ?? ?? ?? - ?? ?? ?? - ?? ?? + ?? = ?? ?? ?? ?? - ?? ?? - ?? ?? ?? + ??
? ?? ?? = ?? ??
? ?? ( ?? ) is one – one function.
For onto function ?? ? ??
? ?? =
?? - ?? ?? - ??
? ???? - ???? = ?? - ??
? ?? ( ?? - ?? )= ???? - ??
? ?? =
???? - ?? ?? - ??
? ?? ? ?? ? ?? ? ??
? ?? ( ?? ) is onto function.
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Oct 15, 2025
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