PSEB 12th Sample Paper 2026 Maths

Punjab Board

Sample Academic Year
2026

Paper

Sample Question Paper
Mathematics
Class – 10+2
Time Allowed : 3 Hours Maximum Marks : 80
Instructions :
1) All questions are compulsory.
2) Question paper consists of 18 questions divided into 4 sections A, B, C and D.
3) Section A comprises of 1 question of 20 multiple choice type questions of 1 mark each.
4) Section B comprises of 7 questions of 2 marks each.
5) Section C comprises of 7 questions of 4 marks each.
6) Section D comprises of 3 questions of 6 marks each.
7) An internal choice is provided in 3 questions of Section C and D each. You have to attempt only one of
the alternatives in all such cases.
8) Use of calculator is not allowed.

Section – A
Q1 Choose the correct options in the following questions :
(i) Relation 𝑹 = {(𝒙, 𝒚) ∶ 𝒙 < 𝒚𝟐 𝐰𝐡𝐞𝐫𝐞 𝒙, 𝒚 ∈ ℝ } is
(a)Reflexive but not symmetric (b)Symmetric and transitive but not Reflexive 1
(c)Reflexive and Symmetric (d)Neither reflexive nor symmetric nor transitive
(ii) Range of function 𝐜𝐨𝐬 𝟏 is :
𝝅 𝝅 𝝅 1
(a)[𝟎, 𝝅] − 𝟐
(b)(𝟎, 𝝅) (c) − , − {𝟎} 𝟐 𝟐
(d)[𝟎, 𝝅]
(iii) 𝟏
Principal value of 𝐬𝐢𝐧 𝟏 𝟐 is
𝝅 𝝅 𝝅 𝝅 1
(a)𝟐 (b) 𝟑 (c) 𝟒 (d) 𝟔
(iv) If 𝑨 is a square matrix of order 𝟐 × 𝟐 and |𝑨| = 𝟓 then |𝑨𝒅𝒋. (𝑨)| is 1
(a)𝟐𝟓 (b)𝟏𝟐𝟓 (c)𝟓 (d)𝟏𝟎
(v) If
𝒙 − 𝟐𝒚 𝟎
=
−𝟓 𝟎
, then 𝒚 is equal to:-
𝟓 𝒙 𝟓 𝟑 1
(a)1 (b)3 (c)2 (d)4
(vi) If the order of the matrix 𝑨 is 𝟑 × 𝟐 then the order of the matrix (𝑨 ) is : 1
(a)𝟐 × 𝟑 (b) 𝟑 × 𝟐 (c) 𝟐 × 𝟐 (d) 𝟑 × 𝟑
(vii) 𝐬𝐢𝐧 𝟖𝒙
, 𝒙 ≠ 𝟎
If 𝒇(𝒙) = 𝟓𝒙 is continuous at 𝒙 = 𝟎 then value of 𝒎 is
𝒎+𝟏 , 𝒙=𝟎
𝟓 𝟖 𝟑 𝟓 1
(a)𝟖 (b)𝟓 (c)𝟓 (d) 𝟑
(viii) 𝒅𝒚
If 𝒚 = 𝒆𝐥𝐨𝐠 𝒙 then 𝒅𝒙 is 1
(a)𝐥𝐨𝐠 𝒙 − 𝒙 (b)𝒙𝒆𝐥𝐨𝐠 𝒙 (c)𝟏 (d)𝒆𝐥𝐨𝐠 𝒙 𝐥𝐨𝐠 𝒙
(ix) If 𝒚 = 𝐭𝐚𝐧 𝒙 then, at 𝒙 = 𝝅 , 𝒅𝒚 is equal to :
𝟒 𝒅𝒙 1
(a)𝟏 (b)√𝟐 (c)𝟐 (d)𝟒
(x) Radius of a circle is increasing at the rate of 𝟐 𝒎/𝒔. Rate of change of its circumference is : 1
(a)𝟒𝝅 𝒎/𝒔 (b)𝟐 𝒎/𝒔 (c) 𝟐𝝅 𝒎/𝒔 (d)𝟒 𝒎/𝒔
(xi) 𝝅/𝟑 √𝐜𝐨𝐬 𝒙
∫𝝅/𝟔 √𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬 𝒙 𝒅𝒙 is equal to
𝝅
√
𝝅 𝝅 𝝅 1
(a)𝟒 (b) 𝟔 (c)𝟏𝟐 (d) 𝟐
(xii) 𝟏 𝒅𝒙
∫𝟎 𝟏 𝒙𝟐 is equal to :
𝝅 𝝅 𝝅 𝝅 1
(a) (b) (c) (d)
𝟐 𝟒 𝟑 𝟔
(xiii)Order of differential equation
𝒅𝟐 𝒚 𝒅𝒚
− 𝟐 𝒅𝒙 + 𝟑𝒚 = 𝟎 is:
𝒅𝒙𝟐 1
(a)𝟑 (b) 𝟐 (c)𝟏 (d)𝟎
(xiv) If 𝒂⃗ is a non-zero vector then |𝒂⃗ × 𝒂⃗| is equal to 1
(a)|𝒂⃗| (b)|𝒂⃗|𝟐 (c) 𝟏 (d)𝟎

(xv) Name of the inequality 𝒂⃗. 𝒃⃗ ≤ |𝒂⃗||𝒃⃗| is : 1
(a)Cauchy-Schwartz Inequality (b)Triangle Inequality
(c)Rolle’s Inequality (d)Lagrange’s Inequality
(xvi) Direction ratios of the straight line 𝒓⃗ = 𝟐 ̂ − 𝟑 ̂ + 𝒌 + 𝒎 𝟗 ̂ − 𝟐 ̂ + 𝟓𝒌 are 1
(a)< 2, −3,1 > (b)< 9,2,5 > (c)< −2,3, −1 > (d)< 9, −2,5 >
(xvii) Vector equation of the line
𝒙 𝟓
=
𝒚 𝟑
=
𝒛 𝟑
is 1
𝟒 𝟓 𝟖
(a)𝒓⃗ = 𝟒 ̂ − 𝟓 ̂ − 𝟖𝒌 + 𝝁 𝟓 ̂ + 𝟑 ̂ − 𝟑𝒌 (b)𝒓⃗ = −𝟒 ̂ + 𝟓 ̂ + 𝟖𝒌 + 𝝁 𝟓 ̂ + 𝟑 ̂ − 𝟑𝒌
(c)𝒓⃗ = 𝟓 ̂ + 𝟑 ̂ − 𝟑𝒌 + 𝝁 𝟒 ̂ − 𝟓 ̂ − 𝟖𝒌 (d)𝒓⃗ = 𝟓 ̂ + 𝟑 ̂ − 𝟑𝒌 + 𝝁 −𝟒 ̂ + 𝟓 ̂ − 𝟖𝒌
(xviii) Objective function of a linear programming problem is : 1
(a)Always quadratic (b)Always linear
(c)May be linear or quadratic depending on the problem (d)May be cubic some times
(xix) 𝟏 𝟑 𝟏
If𝑷(𝑨) = 𝟐 , 𝑷(𝑩) = 𝟖 and 𝑷(𝑨 ∩ 𝑩) = 𝟓 then𝑷(𝑨|𝑩)is equal to : 1
𝟐 𝟖 𝟐 𝟓
(a)𝟓 (b) 𝟏𝟓 (c) 𝟑 (d) 𝟖
(xx) Ram and Rahim are contesting for two vacancies in a company. Probability of selection of Ram is 𝟗 and
𝟕 1
𝟒
that of Rahim is . What is the probability that both will be selected ?
𝟕
𝟔𝟏 𝟒 𝟖 𝟏𝟏
(a)𝟔𝟑 (b)𝟗 (c)𝟗 (d)𝟏𝟔

Section – B
𝟐 𝟏 𝟑
𝟏 −𝟏 2
Q2 If 𝑨 = , 𝑩= 𝟎 𝟐 then verify that (𝑨𝑩) = 𝑩 𝑨
𝟒 𝟏 𝟎
𝟓 𝟎
Q3 𝒅
Find 𝒅𝒙 (𝒙𝒙 ) and evaluate whether this result is true ∀ 𝒙 ∈ ℝ . 2

Q4 𝟐𝒙 𝟑 2
Evaluate ∫ 𝒙𝟐 𝟏 𝒅𝒙 .
Q5 Using integration, find the area bounded by the circle whose centre is at origin and radius is 4 units . 2
Q6 The volume of spherical balloon is increasing at the rate of 25 c.c./s . Find the rate of change of its 2
surface area at the instant when its radius is 5cm.
Q7 Find the points on the curve 𝒚 = 𝟐𝒙𝟑 − 𝟑𝒙𝟐 − 𝟏𝟐𝒙 + 𝟏𝟓 where rate of change of dependent variable 2
with respect to the independent variable vanishes. What do we call these type of points ?
Q8 Find the value of 𝒑 if the vectors 𝒑 ̂ − 𝟖 ̂ + 𝟓𝒌 and 𝟓 ̂ + 𝟐 ̂ − 𝟑𝒌 are perpendicular to each other. 2

Section – C
Q9 𝟓𝒙 𝟑
Prove that the function, 𝒇: ℝ → ℝ, 𝒇(𝒙) = 𝟒 is one-one and onto. 4
Q10(a) Using determinants, find the value of 𝒌 if the area of the triangle formed by the points 2
(−𝟑, 𝟔), (−𝟒, 𝟒) 𝐚𝐧𝐝 (𝒌, −𝟐) is 12 sq. units.
(b) If 𝑿 = 𝟑 𝟒
and 𝟐𝑿 − 𝒀 =
𝟓 𝟏𝟎
then find the matrix 𝒀. 2
𝟐 −𝟏 𝟑 −𝟓
Q11 If 𝒖 = 𝒙𝒚 , 𝒗 = 𝒚𝒙 and quantity 𝒚 remains 𝟑 times the quantity 𝒙 then find that amongst quantities 𝒖 and 𝒗, 4
which changes more rapidly with respect to quantity 𝒙 when 𝒙 = 𝟏. (Take 𝐥𝐨𝐠 𝒆 𝟑 = 𝟏. 𝟎𝟗 )

OR
𝟏 𝒕𝟐 𝟐𝒕 𝒅𝒚 𝒙
If 𝒙 = 𝟏 𝒕𝟐 , 𝒚 = 𝟏 𝒕𝟐 then prove that 𝒅𝒙 + 𝒚 = 𝟎.

Q12 𝟑𝒙 𝟐
Evaluate ∫ (𝒙𝟐 𝟏)(𝒙 𝟐) 𝒅𝒙 4
Q13 Solve the following linear programming problem graphically : 4
Maximize and minimize 𝒁 = 𝟒𝒙 + 𝟐𝒚 − 𝟕 subject to the constraints
𝒙 + 𝟑𝒚 ≤ 𝟔𝟎, 𝒙 + 𝒚 ≥ 𝟏𝟎, 𝒙 − 𝒚 ≤ 𝟎, 𝒙 ≥ 𝟎, 𝒚 ≥ 𝟎

Q14 𝒅𝒚 𝟐
Solve : 𝒙 𝐥𝐨𝐠 𝒙 𝒅𝒙 + 𝒚 = 𝒙 𝐥𝐨𝐠 𝒙. 4
OR
Solve : 𝒙 𝒅𝒚 − 𝟑𝒙 + 𝒙𝒚 + 𝒚 𝒅𝒙 = 𝟎 ; given that 𝒚 = 𝟏 when 𝒙 = 𝟏 .
𝟐 𝟐 𝟐 4
Q15 Bag I contains 7 red and 5 white balls. Bag II contains 3 red and 4 white balls. A bag is selected at random 4
and a ball is drawn from it. Detect that which ball has more chances of being drawn red or white ?
OR
A laboratory blood test is 𝟗𝟗% effective in detecting a certain disease when it is in fact present. 4
However, the test also yields a false positive result for 𝟎. 𝟓% of the healthy person tested (i.e. if a
healthy person is tested, then, with the probability 𝟎. 𝟎𝟎𝟓 , the test will imply he has the disease). If
𝟎. 𝟏% of the population actually has the disease, what is the probability that a person has the disease
given that his test result is positive ?
Section – D
Q16 Ajay, Sameer and Meenal have Rs.20/- each and some footballs, basketballs and volleyballs in their 6
shops. In a week, Ajay sold 3 footballs and a volleyball but he bought 2 basketballs for his shop and he
has Rs.35/- now. In same duration, Sameer sold 2 basketballs and 2 volleyballs but he bought a football
for his shop and he has Rs. 95/- now. Similarly, Meenal sold 2 footballs and a basketball but she bought 3
volleyballs for her shop and she has Rs.15/- now. Find the cost of a football, a basketball and a volleyball
by the help of matrices.
OR
(a) Express
𝟓 𝟏
as the sum of a symmetric matrix and a skew-symmetric matrix. 3
𝟑 𝟕
(b) If 𝑨 =
𝟑 𝟐
and 𝒇(𝒙) = 𝒙𝟐 − 𝟏𝟎𝒙 + 𝟏𝟑 then show that 𝒇(𝑨) = 𝑶 and using this result find 𝑨 𝟏 . 3
𝟒 𝟕
Q17(a) Prove that for any two non-zero vectors 𝒂⃗ and 𝒃⃗ , 𝒂⃗ + 𝒃⃗ ≤ |𝒂⃗| + |𝒃⃗| . Also write the name of this 4
inequality.
(b) Adjacent sides of a parallelogram are given by 𝟔 ̂ − ̂ + 𝟓𝒌 and ̂ + 𝟓 ̂ − 𝟐𝒌 . Find the area of 2
parallelogram.
OR
(a) Find the shortest distance between the following pairs of lines : 4
∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧
𝒓⃗ = 𝒊 − 𝟒 𝒋 + 𝟓𝒌 + 𝝁 𝟓 𝒊 + 𝟗 𝒋 + 𝒌 & 𝒓⃗ = 𝟐 𝒊 + 𝟖 𝒋 − 𝟔𝒌 + 𝝀 𝟑 𝒊 − 𝟐 𝒋 + 𝒌

(b) Find the angle between the lines 2
∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧
𝒓⃗ = 𝟑 𝒊 + 𝟖 𝒋 + 𝟑𝒌 + 𝝁 𝒊 + 𝟐 𝒋 − 𝒌 & 𝒓⃗ = −𝟑 𝒊 + 𝟗 𝒋 − 𝒌 + 𝝀 𝟓 𝒊 + 𝟑 𝒋 + 𝟒𝒌

Q18 Find the height of the right circular cone of maximum volume, which is inscribed in a sphere of radius 6
12cm.
OR
𝒙𝟐
Evaluate ∫ 𝒙𝟒 𝟏 𝒅𝒙 6

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