PSEB 12th Sample Paper 2027 Mathematics

FOR PUNJAB CLASS 12 PREPARATION

PSEB 12th Sample
Paper 2027
Mathematics
EXAM YEAR TYPE SUBJECT

Punjab Class 12 2027 Sample Paper Maths

Notes · Sample Papers · Previous Year Papers · Mock Tests

m
m .co

m .co s e m
se l a
Sample Question Paper ag
Mathematics
Time Allowed: 3 Hours Class XII 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.

m
.co
5) Section C comprises of 7 questions of 4 marks each.
6)
m
Section D comprises of 3 questions of 6 marks each.

.co e m
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.
em l as
l as
8) Use of calculator is not allowed.
Part-A ag Marks
g
Q.1 Choose theacorrect options in the following questions:
(i) A relation 𝑅 on the set 𝐴 = {1, 2, 3} defined as 𝑅 = {(1, 1), (2, 2), (1, 2), (2,1)} is: 1
(a) Reflexive and symmetric but not transitive
(b) Symmetric and transitive but not reflexive
(c) Reflexive and transitive but not symmetric
(d) An equivalence relation
(ii) Let 𝑓: [0, 𝜋] → [−1,1] be defined by 𝑓(𝑥) = sin 𝑥 . Then 𝑓 is: 1
(a) One-one but not onto
(b) Many-one onto
m
(c) Bijective
.co
em
(d) Neither one-one nor onto
(iii)
s
la (c) 𝑠𝑒𝑐
Which one of the following inverse trigonometric functions has principal value range [− , ] − {0}?
𝜋 𝜋
2 2
1
(a) 𝑡𝑎𝑛 −1
(b) 𝑠𝑖𝑛
2 a
g −1 −1
(d) 𝑐𝑜𝑠𝑒𝑐 −1
(iv) If 𝐴 = [
2 −3 0
] and 𝐵 = [ ] then the order of 𝐴 𝐵 is: ′
1
−5 4 −1 3
(a) 3 × 1 (b) 1 × 3 (c) 2 × 2 (d) 3 × 2
(v) If 𝐴 is a square matrix of order 3 × 3 such that ∣ 𝐴 ∣= 4, then the value of |2𝐴| is: 1
(a) 8 (b) 16 (c) 32 (d) 64
(vi) Let 𝐴 = [𝑎𝑖𝑗 ]𝑛×𝑛 be a square matrix. Which of the following statements correctly describes the conditions
1
for 𝐴 to be classified as a diagonal matrix?
(a) 𝑎𝑖𝑗 = 0 for all 𝑖 = 𝑗
m
m (b) 𝑎𝑖𝑗 = 0 for all 𝑖 ≠ 𝑗
.co
.co (c) 𝑎𝑖𝑗 = 1 for all 𝑖 = 𝑗, and 𝑎𝑖𝑗 = 0 for all 𝑖 ≠ 𝑗
s em
s em(vii) (d) 𝑎𝑖𝑗 = 𝑘 (𝑘 ≠ 0) for all 𝑖 = 𝑗, and 𝑎𝑖𝑗 = 0 for all 𝑖 ≠ 𝑗
g la |𝜆|
la
𝑥 2 −49
is continuous ata𝑥 = −7, then
, 𝑥 ≠ −7 1
ag If the function defined by: 𝑓(𝑥) = { 𝑥+7
𝜆 − 2, 𝑥 = −7
equals:

(a) −12 (b)0 (c)12 (d) 9
(viii) Which of the following standard indefinite integral formulas is incorrect? 1
𝑑𝑥 𝑥
(a) ∫ √𝑎2 −𝑥2 = sin−1 (𝑎) + 𝐶
𝑑𝑥 1 𝑥−𝑎
(b) ∫ 𝑥 2 −𝑎2 = 2𝑎 ln |𝑥+𝑎| + 𝐶
𝑑𝑥 1 𝑥
(c) ∫ 𝑎2 +𝑥2 = 𝑎 tan−1 (𝑎) + 𝐶
𝑑𝑥
(d) ∫ 2 2
= ln|𝑥 − √𝑥 − 𝑎 | + 𝐶
√𝑥 2 −𝑎2
m .
.co 1
s e m
s em l a
l a ag
ag For more Question Papers, Sample Papers, Notes & Syllabus visit Page 1 of 4

(ix) 𝑓′ (𝑥)
If ∫ [𝑓(𝑥)]2 𝑑𝑥 =
1
+ 𝐶, where 𝑓(𝑥) is a differentiable non-zero function, then 𝑔(𝑥) is equal to:
𝑔(𝑥) 1

(a) 𝑓(𝑥) (b) −𝑓(𝑥) (c) [𝑓(𝑥)]2 (d) − 𝑓(𝑥)
1

(x) The interval in which 𝑦 = 3 − 4𝑥 − 𝑥 2 is decreasing is: 1
(a) (−∞, −2) (b) (−∞, 2) (c) (−2, ∞) (d) (2, ∞)
(xi) The area enclosed by the curve 𝑦 = |𝑥|, the 𝑥-axis and the ordinates 𝑥 = −3 and 𝑥 = 0 is: 1
(a)
9
(b)
3 (c) 9 (d) 3
2 2
(xii) If 𝑓(𝑥) is an integrable function on [0, 𝜋], which of the following conditions is sufficient to conclude that
𝜋
1
∫ 𝑓 (𝑥) 𝑑𝑥 = 0?
0
(a) 𝑓(𝜋 − 𝑥) = 𝑓(𝑥)
(b) 𝑓(𝜋 − 𝑥) = −𝑓(𝑥)
(c) 𝑓(𝜋) = 𝑓(0)
(d) 𝑓(𝑥) is continuous on [0, 𝜋]
(xiii) Which of the following conditions is necessary to obtain a particular solution from a general solution? 1
(a) Order of the differential equation
(b) Degree of the differential equation
(c) An initial condition
(d) Homogeneity
(xiv) If 𝑎⃗ × 𝑏⃗⃗ = ⃗0⃗, where 𝑎⃗ ≠ ⃗0⃗ and 𝑏⃗⃗ ≠ ⃗0⃗, then: 1
(a) the vectors are perpendicular
(b) the vectors are parallel
(c) the vectors are equal
(d) the vectors are unit vectors
(xv) If the adjacent sides of a parallelogram are represented by 3𝑖̂ and 4𝑗̂, then area of the parallelogram is: 1
(a) −12 sq. units (b) 12𝑘̂ sq. units (c) 12 sq. units (d) 7 sq. units
(xvi) If vectors 𝑎⃗ = 3𝑖̂ + 5𝑗̂ − 𝑚𝑘̂ and 𝑏⃗⃗ = 𝑖̂ − 3𝑗̂ + 2𝑘̂ are perpendicular to each other, then 𝑚 + 1
1
equals:
(a) −6 (b) −5 (c) 5 (d) 6
(xvii) The vector equation of the line which passes through the point (3, −2, − 4) and parallel to the line
𝑥+4 y−1 z+7 1
5
= −2 = 7 is:
(a) 𝑟⃗ = 5𝑖̂ − 2𝑗̂ + 7𝑘̂ + 𝜆(3 𝑖̂ − 2𝑗̂ − 4𝑘̂ )
(b) 𝑟⃗ = 3 𝑖̂ − 2𝑗̂ − 4𝑘̂ + 𝜆(−4𝑖̂ + 𝑗̂ − 7𝑘̂ )
(c) 𝑟⃗ = −3 𝑖̂ + 2𝑗̂ + 4𝑘̂ + 𝜆(5𝑖̂ − 2𝑗̂ + 7𝑘̂ )
(d) 𝑟⃗ = 3 𝑖̂ − 2𝑗̂ − 4𝑘̂ + 𝜆(5𝑖̂ − 2𝑗̂ + 7𝑘̂ )
(xviii) Feasible solutions of a Linear Programming Problem (LPP) are: 1
(a) Points within the boundary of the feasible region
(b) Points within and on the boundary of the feasible region
(c) Points on the boundary of the feasible region
(d) Points outside the boundary of the feasible region
(xix) 𝐴 3 1
If 𝑃 (𝐵) = 5 , 𝑃(𝐵) = 2 , then 𝑃(𝐴 ∩ 𝐵) is equal to:
1
3 3 5 1
(a) (b) (c) (d)
10 5 6 5
(xx) Which of the following is not true? 1

2

For more Question Papers, Sample Papers, Notes & Syllabus visit Page 2 of 4

m
m .co

m .co s em
se g a
l𝑃(𝐴)
a
(a) If 𝐴 and 𝐵 are independent events, then 𝑃(𝐵/𝐴) = 𝑃(𝐵) provided
(b) If 𝐴 and 𝐵 are independent events, then 𝑃(𝐴 ∩ 𝐵 ) = 𝑃(𝐴)(1 − 𝑃(𝐵))′
≠0

(c) If 𝐴 and 𝐵′ are independent events, then 𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴) 𝑃(𝐵′ )
(d) If 𝐴′ and 𝐵′ are independent events, then 𝑃(𝐴′ ∩ 𝐵) = (1 − 𝑃(𝐴))𝑃(𝐵)

Part-B

Q.2 Show that all the diagonal elements of a skew-symmetric matrix are zero. 2
Q.3 2 m
.co
𝑑𝑦
Find 𝑑𝑥 , if 𝑥 = 𝑎(𝜃 + sin 𝜃) , 𝑦 = 𝑎(1 − cos 𝜃)
m
.co
Q.4 Satnam is pouring water into a cylindrical flask at the rate of 400 cm /s, while the flask is leaking at the 2
m
3

rate of 4 cm3 /s. Determine whether the water level in the flask is increasing or decreasing. Also, find the
s e
s em
rate of change of the water level if the radius of the flask is 3cm. l a
Q.5
g
Evaluate ∫la 1
𝑑𝑥 ag 2
Q.6 a area of the region bounded by the line 𝑦 = 𝑥, the x-axis, and the ordinates 𝑥 = −1 and 𝑥 = 1.
Find the
𝑠𝑖𝑛2 𝑥(1−cot 𝑥)2
2
Q.7 𝑑𝑦
Solve the differential equation 𝑑𝑥 = (1 + 𝑥 2 )(1 + 𝑦 2 ). 2
Q.8 An engineer wants to install a sensor perpendicular to two supporting rods represented by 𝑎⃗ = 2𝑖̂ − 𝑗̂ + 2
𝑘̂ , 𝑏⃗⃗ = 𝑖̂ + 𝑗̂ + 2𝑘̂. Find a vector along the direction in which the sensor should be installed.

Part-C

m
c. o
Q.9 (i) Find the value of sin−1 (sin
5𝜋
). 2
6

(ii)
1−cos 𝑥

s em
Express tan−1 (√1+cos 𝑥) , 0 < 𝑥 < 𝜋 in the simplest form. 2

Q.10
g la to the vertices of a triangle as −18. Without evaluating
A student obtains the determinant corresponding 4
a
the determinant again, answer the following:
(i) What is the area of the triangle?
(ii) Why the area is not negative?
(iii) Are the three points collinear? Give reason.
(iv) State the determinant-based condition for three points to be collinear.
Q.11 𝑑𝑦 2
3
3
𝑑𝑦 2 2
4
[1+(𝑑𝑥) ] [1+(𝑑𝑥) ]
If (𝑥 − 3)2 + (𝑦 − 4)2 = 25, prove that 2 + 𝑑2 𝑦
= 20.
𝑑2 𝑦

m
( 2) 𝑑𝑥2
𝑑𝑥

.co
OR
m
.co m
If 𝑓(𝑥) = (tan 𝑥) , 𝑔(𝑥) = 𝑥
𝑥 tan 𝑥
, then find that amongst 𝑓(𝑥) and 𝑔(𝑥) , which function changes
s e 4
em
𝜋
less rapidly with respect to the independent variable 𝑥, when 𝑥 = . Also find the difference between
s g l a 4

a ′ 𝜋 ′ 𝜋 𝜋

l 𝑓 ( ) and 𝑔 ( ) . (𝑇𝑎𝑘𝑒 𝜋 = 3.14 𝑎𝑛𝑑 𝑙𝑜𝑔 ( ) = −0.2 ).
a 𝑒

ag Q.12
4 4 4

𝜋 𝜋 𝜋
Let 𝐼 = ∫0 𝑥 𝑔(𝑥) 𝑑𝑥, where 𝑔(𝜋 − 𝑥) = 𝑔(𝑥). Show that 𝐼 = ∫0 𝑔(𝑥) 𝑑𝑥. Hence evaluate 4
2
𝜋 𝑥 sin 𝑥
∫0 1+ 𝑐𝑜𝑠2 𝑥 𝑑𝑥.
OR
𝑎 𝑎
Show that for any continuous function 𝑓(𝑥), ∫0 𝑓 (𝑥) 𝑑𝑥 = ∫0 𝑓 (𝑎 − 𝑥) 𝑑𝑥. Hence evaluate 4
𝜋/4
∫0 ln (1 + tan 𝑥) 𝑑𝑥.
Q.13 𝑑𝑦 2 4
Solve the differential equation 𝑥 log 𝑥
m + 𝑦 = log 𝑥
.
.co
𝑑𝑥 𝑥

e m
em
3
l as
l as ag
ag For more Question Papers, Sample Papers, Notes & Syllabus visit Page 3 of 4

Q.14 Solve the following linear programming problem graphically: 4
Maximise and minimise 𝑍 = 3𝑥 + 2𝑦 subject to the constraints
4𝑥 + 𝑦 ≥ 8, 𝑥 + 𝑦 ≤ 8, 𝑥 − 𝑦 ≥ 0, 𝑥 ≥ 0, 𝑦≥0
Q.15 Bag I contains 4 red balls and 7 white balls. Bag II contains 5 red balls and 6 white balls. One of the bags 4
is chosen at random and a ball is drawn from it. Find the probability of drawing:
(i) a red ball?
(ii) a white ball?
Which ball is more likely to be drawn?
OR
In a population, 1% of people have a specific disease. A diagnostic test for this disease gives a positive
result in 99% of people who actually have the disease, and in 2% of healthy people (false positive). A 4
person is selected at random. Find the probability that:
(i) the test result is positive.
(ii) the test result is negative.
Which test outcome (positive or negative) is more likely to occur for a randomly selected person?

Part-D

Q.16 If twice Ravneet's present age is subtracted from the sum of twice Manisha's present age and Navdeep's 6
present age, the result is 4. Five years ago, subtracting the sum of Manisha's age and Navdeep's age from
twice Ravneet's age gave 8. Five years hence, subtracting twice Manisha's age from the sum of Ravneet's
and Navdeep's ages will give 1. Using matrices, find the present ages of Manisha, Ravneet and Navdeep.
OR
𝑥
0 tan
2
If 𝐴′ = [ 𝑥 ] and 𝐼 is the identity matrix of order 2, show that 6
− tan 0
2
cos 𝑥 − sin 𝑥
𝐼 + 𝐴 = (𝐼 − 𝐴) [ ]
sin 𝑥 cos 𝑥

Q.17 Write the appropriate property to be used in evaluating the integral given by 6
𝜋
𝑥 𝑑𝑥
∫
25 𝑐𝑜𝑠 2 𝑥 + 49 𝑠𝑖𝑛2 𝑥
0
and hence evaluate.
OR
1
Prove that the volume of the largest right circular cylinder that can be inscribed in a sphere of radius R is 6
√3
of the volume of the sphere. A student claimed that "the cylinder of maximum volume have same radius as
that of half the radius of the sphere". Examine this claim using your result and justify?
Q.18 (a) The scalar product of the vector 𝑖̂ + 2𝑗̂ − 𝑘̂ with a unit vector along the sum of −𝑖̂ + 5𝑗̂ − 6𝑘̂ and 4
𝑝𝑖̂ − 2𝑗̂ + 3𝑘̂ is equal to one. Find the value of 𝑝.
(b) Find the area of the parallelogram whose adjacent sides are determined by the vectors 2
𝑎⃗ = 3𝑖̂ + 5𝑗̂ − 𝑘̂ and 𝑏⃗⃗ = 𝑖̂ − 3𝑗̂ + 2𝑘̂.

OR
(a) Find the shortest distance between the lines
4
𝑟⃗ = −2𝑖̂ + 𝑗̂ + 3𝑘̂ + 𝜇(3𝑖̂ − 2𝑗̂ − 𝑘̂ ) and 𝑟⃗ = −3 𝑖̂ − 𝑗̂ + 2𝑘̂ + 𝜆(𝑖̂ − 2𝑗̂ + 𝑘̂ )

𝑥−3 𝑦−1 𝑧+2 2−𝑥 𝑦+1 𝑧+56 2
(b) Find the angle between the pair of lines = = and = =
2 −1 −2 3 −5 −4

4

For more Question Papers, Sample Papers, Notes & Syllabus visit Page 4 of 4