ICSE Mathematics Previous Year Paper with Solutions - 2024 | ICSE Class 12 Science Previous Year Papers PDF Download

 Page 1


Mathematics [Official] 
CISCE 
Academic Year: 2023-2024 
Date & Time: 20th February 2024, 2:00 pm 
 
Duration: 3h                                                                                                     Marks: 70 
SECTION A - 65 MARKS 
Q1. In subparts (i) to (x) choose the correct options and in subparts (xi) to (xv), 
answer’s the question’s as instructed. 
1.1. Let L be a set of all straight lines in a plane. The relation R on L defined as 
'perpendicular to' is ______. 
1. Symmetric and Transitive 
2. Transitive 
3. Symmetric 
4. Equivalence 
Solution 
Let L be a set of all straight lines in a plane. The relation R on L defined as 
'perpendicular to' is symmetric. 
Explanation: 
The relation is symmetric, meaning that if a line (I) is perpendicular to line (m), then 
line (m) is also perpendicular to line I. 
However, if line (l) is perpendicular to line (m) and line (m) is perpendicular to line (n). 
Then, lines 'I' and 'n' are parallel rather than perpendicular, but is parallel. 
As a result, the provided relation is only symmetric. 
1.2. The order and degree of the differential equation  
 
Page 2


Mathematics [Official] 
CISCE 
Academic Year: 2023-2024 
Date & Time: 20th February 2024, 2:00 pm 
 
Duration: 3h                                                                                                     Marks: 70 
SECTION A - 65 MARKS 
Q1. In subparts (i) to (x) choose the correct options and in subparts (xi) to (xv), 
answer’s the question’s as instructed. 
1.1. Let L be a set of all straight lines in a plane. The relation R on L defined as 
'perpendicular to' is ______. 
1. Symmetric and Transitive 
2. Transitive 
3. Symmetric 
4. Equivalence 
Solution 
Let L be a set of all straight lines in a plane. The relation R on L defined as 
'perpendicular to' is symmetric. 
Explanation: 
The relation is symmetric, meaning that if a line (I) is perpendicular to line (m), then 
line (m) is also perpendicular to line I. 
However, if line (l) is perpendicular to line (m) and line (m) is perpendicular to line (n). 
Then, lines 'I' and 'n' are parallel rather than perpendicular, but is parallel. 
As a result, the provided relation is only symmetric. 
1.2. The order and degree of the differential equation  
 
 
Solution 
The order and degree of the differential equation 
 
Explanation: 
The given differential equation is  
 
Here, the highest derivative is 2 
? Order = 2 and the power of the highest derivative is 1. 
? Degree = 1. 
1.3. Let A be a non-empty set. 
Statement 1: Identity relation on A is Reflexive. 
Statement 2: Every Reflexive relation on A is an Identity relation. 
1. Both the statements are true. 
2. Both the statements are false. 
3. Statement 1 is true and Statement 2 is false. 
4. Statement 1 is false and Statement 2 is true. 
Solution 
Statement 1 is true and Statement 2 is false. 
Page 3


Mathematics [Official] 
CISCE 
Academic Year: 2023-2024 
Date & Time: 20th February 2024, 2:00 pm 
 
Duration: 3h                                                                                                     Marks: 70 
SECTION A - 65 MARKS 
Q1. In subparts (i) to (x) choose the correct options and in subparts (xi) to (xv), 
answer’s the question’s as instructed. 
1.1. Let L be a set of all straight lines in a plane. The relation R on L defined as 
'perpendicular to' is ______. 
1. Symmetric and Transitive 
2. Transitive 
3. Symmetric 
4. Equivalence 
Solution 
Let L be a set of all straight lines in a plane. The relation R on L defined as 
'perpendicular to' is symmetric. 
Explanation: 
The relation is symmetric, meaning that if a line (I) is perpendicular to line (m), then 
line (m) is also perpendicular to line I. 
However, if line (l) is perpendicular to line (m) and line (m) is perpendicular to line (n). 
Then, lines 'I' and 'n' are parallel rather than perpendicular, but is parallel. 
As a result, the provided relation is only symmetric. 
1.2. The order and degree of the differential equation  
 
 
Solution 
The order and degree of the differential equation 
 
Explanation: 
The given differential equation is  
 
Here, the highest derivative is 2 
? Order = 2 and the power of the highest derivative is 1. 
? Degree = 1. 
1.3. Let A be a non-empty set. 
Statement 1: Identity relation on A is Reflexive. 
Statement 2: Every Reflexive relation on A is an Identity relation. 
1. Both the statements are true. 
2. Both the statements are false. 
3. Statement 1 is true and Statement 2 is false. 
4. Statement 1 is false and Statement 2 is true. 
Solution 
Statement 1 is true and Statement 2 is false. 
Explanation: 
Consider A = {a, b, c} and define a relation R as R = {(a, a), (b, b), (c, c), (a, b). 
Then R is a reflexive relation on A, but not an identity relation, because R contains the 
elements (a, b). 
1.4. The graph of the function f is shown below. 
 
Of the following options, at what values of x is the function f NOT differentiable? 
1. At x = 0 and x = 2 
2. At x = 1 and x = 3 
3. At x = – 1 and x = 1 
4. At x = – 1.5 and x = 1.5 
Solution 
At x = 0 and x = 2 
Explanation: 
 
Hence, x = 0 and x = 2, the function f is not differentiable. 
1.5.  
 
Page 4


Mathematics [Official] 
CISCE 
Academic Year: 2023-2024 
Date & Time: 20th February 2024, 2:00 pm 
 
Duration: 3h                                                                                                     Marks: 70 
SECTION A - 65 MARKS 
Q1. In subparts (i) to (x) choose the correct options and in subparts (xi) to (xv), 
answer’s the question’s as instructed. 
1.1. Let L be a set of all straight lines in a plane. The relation R on L defined as 
'perpendicular to' is ______. 
1. Symmetric and Transitive 
2. Transitive 
3. Symmetric 
4. Equivalence 
Solution 
Let L be a set of all straight lines in a plane. The relation R on L defined as 
'perpendicular to' is symmetric. 
Explanation: 
The relation is symmetric, meaning that if a line (I) is perpendicular to line (m), then 
line (m) is also perpendicular to line I. 
However, if line (l) is perpendicular to line (m) and line (m) is perpendicular to line (n). 
Then, lines 'I' and 'n' are parallel rather than perpendicular, but is parallel. 
As a result, the provided relation is only symmetric. 
1.2. The order and degree of the differential equation  
 
 
Solution 
The order and degree of the differential equation 
 
Explanation: 
The given differential equation is  
 
Here, the highest derivative is 2 
? Order = 2 and the power of the highest derivative is 1. 
? Degree = 1. 
1.3. Let A be a non-empty set. 
Statement 1: Identity relation on A is Reflexive. 
Statement 2: Every Reflexive relation on A is an Identity relation. 
1. Both the statements are true. 
2. Both the statements are false. 
3. Statement 1 is true and Statement 2 is false. 
4. Statement 1 is false and Statement 2 is true. 
Solution 
Statement 1 is true and Statement 2 is false. 
Explanation: 
Consider A = {a, b, c} and define a relation R as R = {(a, a), (b, b), (c, c), (a, b). 
Then R is a reflexive relation on A, but not an identity relation, because R contains the 
elements (a, b). 
1.4. The graph of the function f is shown below. 
 
Of the following options, at what values of x is the function f NOT differentiable? 
1. At x = 0 and x = 2 
2. At x = 1 and x = 3 
3. At x = – 1 and x = 1 
4. At x = – 1.5 and x = 1.5 
Solution 
At x = 0 and x = 2 
Explanation: 
 
Hence, x = 0 and x = 2, the function f is not differentiable. 
1.5.  
 
1. –4 
2. 0 
3. –1 
4. 4 
Solution 
 
Explanation: 
 
= – cosec 30° – sec 120° 
= – cosec 30° – sec [(90° + 30°)] 
= – 2 – [– cosec 30°] 
= – 2 + cosec 30° 
= – 2 + 2 
= 0 
1.6.  
 
Page 5


Mathematics [Official] 
CISCE 
Academic Year: 2023-2024 
Date & Time: 20th February 2024, 2:00 pm 
 
Duration: 3h                                                                                                     Marks: 70 
SECTION A - 65 MARKS 
Q1. In subparts (i) to (x) choose the correct options and in subparts (xi) to (xv), 
answer’s the question’s as instructed. 
1.1. Let L be a set of all straight lines in a plane. The relation R on L defined as 
'perpendicular to' is ______. 
1. Symmetric and Transitive 
2. Transitive 
3. Symmetric 
4. Equivalence 
Solution 
Let L be a set of all straight lines in a plane. The relation R on L defined as 
'perpendicular to' is symmetric. 
Explanation: 
The relation is symmetric, meaning that if a line (I) is perpendicular to line (m), then 
line (m) is also perpendicular to line I. 
However, if line (l) is perpendicular to line (m) and line (m) is perpendicular to line (n). 
Then, lines 'I' and 'n' are parallel rather than perpendicular, but is parallel. 
As a result, the provided relation is only symmetric. 
1.2. The order and degree of the differential equation  
 
 
Solution 
The order and degree of the differential equation 
 
Explanation: 
The given differential equation is  
 
Here, the highest derivative is 2 
? Order = 2 and the power of the highest derivative is 1. 
? Degree = 1. 
1.3. Let A be a non-empty set. 
Statement 1: Identity relation on A is Reflexive. 
Statement 2: Every Reflexive relation on A is an Identity relation. 
1. Both the statements are true. 
2. Both the statements are false. 
3. Statement 1 is true and Statement 2 is false. 
4. Statement 1 is false and Statement 2 is true. 
Solution 
Statement 1 is true and Statement 2 is false. 
Explanation: 
Consider A = {a, b, c} and define a relation R as R = {(a, a), (b, b), (c, c), (a, b). 
Then R is a reflexive relation on A, but not an identity relation, because R contains the 
elements (a, b). 
1.4. The graph of the function f is shown below. 
 
Of the following options, at what values of x is the function f NOT differentiable? 
1. At x = 0 and x = 2 
2. At x = 1 and x = 3 
3. At x = – 1 and x = 1 
4. At x = – 1.5 and x = 1.5 
Solution 
At x = 0 and x = 2 
Explanation: 
 
Hence, x = 0 and x = 2, the function f is not differentiable. 
1.5.  
 
1. –4 
2. 0 
3. –1 
4. 4 
Solution 
 
Explanation: 
 
= – cosec 30° – sec 120° 
= – cosec 30° – sec [(90° + 30°)] 
= – 2 – [– cosec 30°] 
= – 2 + cosec 30° 
= – 2 + 2 
= 0 
1.6.  
 
 
Solution 
 
Explanation: