Class 12 PT II Question Paper 2025-26 Maths

CBSE CLASS 12
PT II Question Paper
Session 2025-26

MATHS

Exam CBSE Class 12
Subject Maths
Session 2025-26
Document Type PT II Question Paper

Notes · Sample Papers · Previous Year Papers · Mock Tests

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SECOND UNIT TEST
SESSION: 2025-2026
CLASS: XII MAXIMUM MARKS: 40
SUBJECT: MATHEMATICS TIME: 1 Hour 30 Minutes
General Instructions:

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1. The question paper consists of 2 printed pages and 17 questions in all and 5sections A, B, C, D and E

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2. Section - A comprises of 4 MCQs and 2 Assertion – Reason based questions of 1 mark each.

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3. Section - B comprises of 4 VSA type questions of 2 marks each.

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4. Section - C comprises of 4 SA type questions of 3 marks each.
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5. Section - D comprises of 2 LA type questions of 5 marks each.
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6. In Section E, Questions no.17 is Case study-based question, carrying 4 marks.
7. There is no overall choice. However, an internal choice has been provided in 2 questions in Section B, 2
questions in Section C, 01 question in Section D and in one subpart of question of Section E.
SECTION: A
This section comprises of multiple-choice questions (MCQs) (Q1.to Q4.) of 1 mark each.
Q.1 1
Evaluate: ∫ 𝑑𝑥
(A) 0 ( B) 𝜋 (C) (D)

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Q.2 Area of the region bounded by the curve x = 2y+3, the y-axis and the line y = -1, y=1 is 1

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(A)6 sq. units (B) 3/2 sq. units (C) 4 sq. units (D) 8 sq. units

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Q.3 1
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⃗ ⃗
If 𝑎⃗ = 2𝚤̂ − 𝚥̂ + 𝑘 and 𝑏 = 𝚤̂ + 𝚥̂ − 𝑘 , then 𝑎⃗ and 𝑏 are

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(A) collinear vectors which are not parallel (B) parallel vectors

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(C) perpendicular vectors (D) unit vectors
Q.4 If the line makes an angle of 𝜋/4 with positive directions each of Y and Z axis, then the 1
angle which it makes with positive direction of X axis is
(A)0 (B) 𝜋 (C) 𝜋/2 (D) 𝜋/4
Assertion - Reason Based Questions
Direction: Question numbers 5 and 6 are Assertion (A) and Reason (R) based questions
carrying 1 mark each. Two statements are given, one Assertion (A) and the other Reason (R).
Select the correct answer from the options (A), (B), (C) and (D) as given below:
(A) Both (A) and (R) are true and (R) is the correct explanation of (A).
(B) Both (A) and (R) are true but (R) is not the correct explanation of (A).

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(C) (A) is true but (R) is false.

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(D) (A) is false but (R) is true.

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Q.5 Assertion(A): Given two non-zero vectors 𝑎⃗ and 𝑏⃗ . If 𝑟⃗ is another non-zero vector such that 1

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𝑟⃗× (𝑎⃗+𝑏⃗)= 0⃗ . Then 𝑟⃗ is perpendicular to 𝑎⃗ × 𝑏⃗ .
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Reason (R): The vector 𝑎⃗ + 𝑏⃗ is perpendicular to the plane of 𝑎⃗and 𝑏⃗
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ag Assertion (A): If a line makes angles 𝛼, 𝛽 and 𝛾 with the positive direction of the coordinate
axes, then sin 𝛼 + sin 𝛽 + sin 𝛾 = 2
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Reason (R): The sum of the squares of the direction ratios of a line is always 1.
SECTION: B
This section comprises of 4 very short answer (VSA) type questions of 2 marks each.
Q.7 Find the value of ∫ 𝑥 (1 − 𝑥 ) 𝑑𝑥. 2
OR
If ∫ = , then find the value of 𝑎.

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Q.8 Find the area bounded by the curve y = cos x between x = 0 and x = 2𝜋 2

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Q.9
If the vectors 𝑎⃗ and 𝑏⃗ are such that |𝑎⃗| = 3, 𝑏⃗ = and 𝑎⃗ × 𝑏⃗ is a unit vector, then find the 2
angle between 𝑎⃗ and 𝑏⃗.
OR
Find the magnitude of each of the two vectors 𝑎⃗ and 𝑏⃗ , having the same magnitude such that
the angle between them is 600 and their scalar product is .
Q.10 Find he direction cosines of the line 6x-2 = 3y+1 =2z-2. 2
SECTION: C
This section comprises of 4 short answer (SA) type questions of 3 marks each.
Q.11 Using method of integration, find the area bounded by the curve |𝑥| + |𝑦| = 1. 3
Q.12 3
Evaluate ∫
√
OR
Evaluate ∫ 𝑑𝑥
Q.13 Find a vector of magnitude 6 units, perpendicular to each of the vectors 𝑎⃗ + 𝑏⃗ and 𝑎⃗ - 𝑏⃗ , 3
where 𝑎⃗ = 𝚤̂ + 𝚥̂ + 𝑘 and 𝑏⃗ = 𝚤̂ + 2𝚥 + 3𝑘 .
Q.14 Find the shortest distance between the lines: Line 1: 𝑟⃗ = 𝚤̂ + 𝚥̂ + 𝜆 2𝚤̂ − 𝚥̂ + 𝑘 3
Line 2: 𝑟⃗ = 2𝚤̂ + 𝚥̂ − 𝑘 + 𝜇 3𝚤̂ − 5𝚥̂ + 2𝑘
OR
Find the value of 𝜆, so that the lines = = 𝑎𝑛𝑑 = = are
perpendicular to each other.
SECTION: D
This section comprises of 2 long answer (LA) type questions of 5 marks each
Q.15 Find the equation of the line passing through the point of intersection of 5
the lines = = 𝑎𝑛𝑑 = = and perpendicular to these given lines.
Q.16 Using the method of integration find the area of the ∆ABC, coordinates of whose vertices are 5
𝐴(2,0), 𝐵(4,5)𝑎𝑛𝑑 𝐶 (6,3).
OR
Using integration, find the area of the region enclosed between the circle 𝑥 + 𝑦 = 16 and
the lines 𝑥 = −2 𝑎𝑛𝑑 𝑥 = 2.
SECTION: E
This section comprises of 01 case-study of 4 marks with subparts of marks 1, 1, 2 respectively.
Q.17 An engineer is designing a new metro rail network in a city. Initially, two metro lines, Line 𝐴
and Line 𝐵, each consisting of multiple stations are designed. The track for Line 𝐴 is
represented by 𝑙 ∶ = = , while the track for Line 𝐵 is represented by
𝑙 ∶ = = .
Based on the above information, answer the following questions: 1
(i) Find whether the two metro tracks are parallel.
(ii) Solar panels are to be installed on the rooftop of the metro stations. Determine the
equation of the line representing the placement of solar panels on the rooftop of 1
Line 𝐴’s stations, given that the panels are to be positioned parallel to Line 𝐴’s
track (𝑙 ) and pass through the point (1, −2, −3).
(iii) (a) To connect the stations, a pedestrian pathway perpendicular to the two metro
lines are to be constructed which passes through the point (3, 2, 1). Determine
the equation of the pedestrian pathway.
OR 2
(b) Find the shortest distance between Line 𝐴 and Line 𝐵.

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