CBSE Class 12 Mathematics Standard Question Paper 2026 with
Solution Set (30-2-3)
General Instructions
1. The question paper consists of six sections — Section A to Section F.
2. Time allowed is 3 hours 15 minutes and the maximum marks are 90.
3. All questions are compulsory unless otherwise stated.
4. Section A contains Multiple Choice Questions (MCQs). Choose the correct answer
from the given options.
5. Section B includes:
• True/False questions
• Fill in the blanks
• Very short answer questions (one or two words)
6. Section C contains short answer type questions.
7. Section D contains long descriptive questions with internal choices.
8. Section E contains long answer questions. Attempt the required number as in-
structed.
9. Section F consists of Map Work. Mark and label the places correctly on the outline
map of India.
10. Figures to the right indicate full marks for each question.
11. Write neatly and draw diagrams wherever necessary.
12. Write answers only in the space provided or as instructed.
Section A
1. The distance of the point A(4a, 3a) from x-axis is :
(A) 3a
(B) − 3a
(C) 4a
(D) − 4a
Correct Answer: (A) 3a
Solution:
1
,Step 1: Understanding the Concept:
The distance of a point P (x, y) from the x-axis is defined as the absolute value of its y-
coordinate, represented as |y|.
This represents the vertical displacement of the point from the horizontal axis.
Step 2: Key Formula or Approach:
Distance from x-axis = |y-coordinate|.
Step 3: Detailed Explanation:
Given the point A(4a, 3a):
The x-coordinate is 4a.
The y-coordinate is 3a.
By applying the definition, the distance from the x-axis is the magnitude of the y-coordinate.
Distance = |3a|
Since distances are scalar magnitudes and in such geometric contexts a is typically considered
a positive parameter unless specified, the distance is 3a.
Step 4: Final Answer:
The distance of the point A from the x-axis is 3a.
Quick Tip
To avoid confusion: The distance from the x-axis is the y-value, and the distance from
the y-axis is the x-value.
2. The natural number 1 is :
(A) a prime number.
(B) a composite number.
(C) prime as well as composite.
(D) neither prime nor composite.
Correct Answer: (D) neither prime nor composite.
Solution:
Step 1: Understanding the Concept:
Natural numbers are classified based on the number of factors they possess.
A prime number is a natural number > 1 that has exactly two distinct factors: 1 and itself.
A composite number is a natural number > 1 that has more than two factors.
2
,Step 2: Detailed Explanation:
The number 1 has only one factor, which is 1 itself.
Because it does not have exactly two distinct factors, it fails the definition of a prime number.
Because it does not have more than two factors, it fails the definition of a composite number.
Historically and mathematically, 1 is treated as a unit, serving as the multiplicative identity,
and is excluded from both categories.
Step 3: Final Answer:
The number 1 is neither prime nor composite.
Quick Tip
The number 2 is the smallest prime number and the only even prime number. 1 is often
called a ”unit” in number theory.
3. Given cot θ = 3, the value of cos θ is :
1
(A) 3
(B) √1
10
(C) √3
√10
10
(D) 3
Correct Answer: (C) √3
10
Solution:
Step 1: Understanding the Concept:
In a right-angled triangle, the cotangent ratio is the ratio of the adjacent side to the opposite
side. The cosine ratio is the ratio of the adjacent side to the hypotenuse.
Step 2: Key Formula or Approach:
Base (B)
1. cot θ = Perpendicular (P)
√
2. Pythagoras Theorem: H = P 2 + B 2
B
3. cos θ = H
Step 3: Detailed Explanation:
Given cot θ = 3, we can write this as 31 .
Let Base (B) = 3k and Perpendicular (P ) = 1k.
Calculate Hypotenuse (H):
p p √ √
H= (3k)2 + (1k)2 = 9k 2 + k 2 = 10k 2 = k 10
Now, find cos θ:
3
, Base 3k 3
cos θ = = √ =√
Hypotenuse k 10 10
Step 4: Final Answer:
The value of cos θ is √310 .
Quick Tip
Alternatively, use the identity cosec2 θ = 1 + cot2 θ.
cosec2 θ = 1 + 32 = 10 =⇒ sin θ = √110 .
Then, cos θ = cot θ × sin θ = 3 × √1 = √3 .
10 10
4. For any natural number n, 5n ends with the digit :
(A) 0
(B) 5
(C) 3
(D) 2
Correct Answer: (B) 5
Solution:
Step 1: Understanding the Concept:
The last digit (unit digit) of the power of a number depends on the cyclic property of that base
number’s units digit.
Step 2: Detailed Explanation:
Consider the powers of 5 for natural numbers n:
For n = 1, 51 = 5
For n = 2, 52 = 25
For n = 3, 53 = 125
For n = 4, 54 = 625
Observation: Regardless of the power, the last digit is always 5.
This is because when we multiply a number ending in 5 by another 5, the result always ends
in 5 × 5 = 25, which has 5 as its unit digit.
Step 3: Final Answer:
For any natural number n, 5n ends with the digit 5.
4
Solution Set (30-2-3)
General Instructions
1. The question paper consists of six sections — Section A to Section F.
2. Time allowed is 3 hours 15 minutes and the maximum marks are 90.
3. All questions are compulsory unless otherwise stated.
4. Section A contains Multiple Choice Questions (MCQs). Choose the correct answer
from the given options.
5. Section B includes:
• True/False questions
• Fill in the blanks
• Very short answer questions (one or two words)
6. Section C contains short answer type questions.
7. Section D contains long descriptive questions with internal choices.
8. Section E contains long answer questions. Attempt the required number as in-
structed.
9. Section F consists of Map Work. Mark and label the places correctly on the outline
map of India.
10. Figures to the right indicate full marks for each question.
11. Write neatly and draw diagrams wherever necessary.
12. Write answers only in the space provided or as instructed.
Section A
1. The distance of the point A(4a, 3a) from x-axis is :
(A) 3a
(B) − 3a
(C) 4a
(D) − 4a
Correct Answer: (A) 3a
Solution:
1
,Step 1: Understanding the Concept:
The distance of a point P (x, y) from the x-axis is defined as the absolute value of its y-
coordinate, represented as |y|.
This represents the vertical displacement of the point from the horizontal axis.
Step 2: Key Formula or Approach:
Distance from x-axis = |y-coordinate|.
Step 3: Detailed Explanation:
Given the point A(4a, 3a):
The x-coordinate is 4a.
The y-coordinate is 3a.
By applying the definition, the distance from the x-axis is the magnitude of the y-coordinate.
Distance = |3a|
Since distances are scalar magnitudes and in such geometric contexts a is typically considered
a positive parameter unless specified, the distance is 3a.
Step 4: Final Answer:
The distance of the point A from the x-axis is 3a.
Quick Tip
To avoid confusion: The distance from the x-axis is the y-value, and the distance from
the y-axis is the x-value.
2. The natural number 1 is :
(A) a prime number.
(B) a composite number.
(C) prime as well as composite.
(D) neither prime nor composite.
Correct Answer: (D) neither prime nor composite.
Solution:
Step 1: Understanding the Concept:
Natural numbers are classified based on the number of factors they possess.
A prime number is a natural number > 1 that has exactly two distinct factors: 1 and itself.
A composite number is a natural number > 1 that has more than two factors.
2
,Step 2: Detailed Explanation:
The number 1 has only one factor, which is 1 itself.
Because it does not have exactly two distinct factors, it fails the definition of a prime number.
Because it does not have more than two factors, it fails the definition of a composite number.
Historically and mathematically, 1 is treated as a unit, serving as the multiplicative identity,
and is excluded from both categories.
Step 3: Final Answer:
The number 1 is neither prime nor composite.
Quick Tip
The number 2 is the smallest prime number and the only even prime number. 1 is often
called a ”unit” in number theory.
3. Given cot θ = 3, the value of cos θ is :
1
(A) 3
(B) √1
10
(C) √3
√10
10
(D) 3
Correct Answer: (C) √3
10
Solution:
Step 1: Understanding the Concept:
In a right-angled triangle, the cotangent ratio is the ratio of the adjacent side to the opposite
side. The cosine ratio is the ratio of the adjacent side to the hypotenuse.
Step 2: Key Formula or Approach:
Base (B)
1. cot θ = Perpendicular (P)
√
2. Pythagoras Theorem: H = P 2 + B 2
B
3. cos θ = H
Step 3: Detailed Explanation:
Given cot θ = 3, we can write this as 31 .
Let Base (B) = 3k and Perpendicular (P ) = 1k.
Calculate Hypotenuse (H):
p p √ √
H= (3k)2 + (1k)2 = 9k 2 + k 2 = 10k 2 = k 10
Now, find cos θ:
3
, Base 3k 3
cos θ = = √ =√
Hypotenuse k 10 10
Step 4: Final Answer:
The value of cos θ is √310 .
Quick Tip
Alternatively, use the identity cosec2 θ = 1 + cot2 θ.
cosec2 θ = 1 + 32 = 10 =⇒ sin θ = √110 .
Then, cos θ = cot θ × sin θ = 3 × √1 = √3 .
10 10
4. For any natural number n, 5n ends with the digit :
(A) 0
(B) 5
(C) 3
(D) 2
Correct Answer: (B) 5
Solution:
Step 1: Understanding the Concept:
The last digit (unit digit) of the power of a number depends on the cyclic property of that base
number’s units digit.
Step 2: Detailed Explanation:
Consider the powers of 5 for natural numbers n:
For n = 1, 51 = 5
For n = 2, 52 = 25
For n = 3, 53 = 125
For n = 4, 54 = 625
Observation: Regardless of the power, the last digit is always 5.
This is because when we multiply a number ending in 5 by another 5, the result always ends
in 5 × 5 = 25, which has 5 as its unit digit.
Step 3: Final Answer:
For any natural number n, 5n ends with the digit 5.
4
