PSEB 12th Sample Paper 2024 Maths

SAMPLE PAPER
Punjab Board
2024

Time Allowed : 3 Hours MATHEMATICS-10+2 Max. Marks :80

Instructions:
1. All the questions are compulsory.
2. The question paper consists of 16 questions divided into 4 sections A,B,C and D.
3. Section A comprises of 3 questions :
(i) Q.No.1 consists of 16 Multiple Choice Questions carrying 1 mark each.
(ii) Q.No.2 consists of 8 Fill in the Blank type questions carrying 1 mark each.
(iii) Q.No.3 consists of 8 True/False type questions carrying 1 mark each.
4. Section B comprises of 5 questions of 2 marks each.
5. Section C comprises of 5 questions of 4 marks each.
6. Section D comprises of 3 questions of 6 marks each.
7. Internal choice has been provided in three questions of 2 marks, three questions of 4 marks and three
questions of 6 marks. You have to attempt only one of the alternatives in all such questions.
8. Use of calculator is not permitted.

Section – A

Q1 Choose the correct options in the following questions :
(i) Function : → , ( ) = − is :
1
(a)one-one only (b)onto only (c)one-one and onto (d)none of these
(ii) Relation given by = {( , ), ( , ), ( , ), ( , )} is
1
(a)reflexive only (b)symmetric only (c)transitive only (d) equivalence relation
(iii) is equal to :
1
(a) (b) (c) (d)
(iv) −
If = then value of is:
− − 1
(a) (b)− (c) (d)−
(v) If order of matrix is × and order of matrix is × then order of matrix is : 1
(a) × (b) × (c) × (d) ×
(vi) + , ≤
If ( ) = is continuous then value of is :
− , >5 1
(a) (b) (c) (d)
(vii) { ( )}is equal to :
1
(a) (b) (c) (d)
(viii) Critical point of the function ( ) = − + is :
1
(a) = (b) = (c) = (d) =
(ix) ∫ is equal to : 1
(a) + (b) + (c) + (d) +
/ /
(x) is equal to :
∫ / /
1
(a) (b) (c) (d)
(xi)
Degree of differential equation − + = is: 1
(a) (b) (c) (d)
(xii) If ⃗. ⃗ = | ⃗ × ⃗| then angle between vector ⃗ and vector ⃗ is :
1

(a) (b) (c) (d)
(xiii) If ⃗. ⃗ = then angle between vectors ⃗ and ⃗ is : 1
(a) (b) (c) (d)
(xiv) Direction ratios of line given by = = are : 1
(a) < 3,12, − > (b) < 3, − , > (c) < 3,6,7 > (d)< 3,6, − >
(xv) Common area for each constraint is called :
1
(a)infeasible region (b)feasible region (c)useless region (d)main region

(xvi) If ( ) = , ( )= and ( ∩ ) = then ( / ) is equal to :
1
(a) (b) (c) (d)

Q2 Fill in the blanks from the given options
(i) Value of ( )is ___________ 1
(ii) If =[ ] × such that = + then =__________ 1
(iii) If = then = _________ 1
(iv) If = then at = , = _____________ 1
(v) ∫ =__________ 1
(vi) Order of the differential equation − + = is _________ 1
(vii) Direction ratios of −axis are ______________ 1
(viii) Probability of occurrence of impossible event = ______________ 1

Q3 State true or false for the following statements :
(i) If is a square matrix then ( + ) is a skew-symmetric matrix. 1
(ii) If = then = . 1
(iii) If = then = 1
(iv) = +
1
(v) − = is a variable separable type of differential equation. 1
(vi) Scalar product of two perpendicular vectors is zero. 1
(vii) The point ( , − , ) lies on the − axis. 1
(viii) If ( ) = . then ( )= . 1

Section – B

Q4 Given a matrix = ,show that matrix = + is a symmetric matrix. 2
Q5 Find the interval in which function ( ) = + − is increasing. 2
OR
If = , then find 2
/
Q6 Evaluate ∫ / . 2
√
OR
√
Evaluate ∫ 2
√
Q7 Using integration find the area bounded by the parabola = straight lines = , = in the first
quadrant. 2
Q8 Find the value of if the vectors ⃗ = ̂ − ̂ − ⃗ = ̂ + ̂ − are perpendicular to each other. 2
OR
Find the angle between the lines : = = and = = 2

Section – C

Q9 Show that function : ℝ → ℝ , ( ) = is one-one and onto. 4
Q10 If = +( ) then find . 4
OR
Find the general solution of the differential equation ( − ) =( + ) . 4

Q11 / 4
Evaluate ∫ .
OR
Evaluate ∫ [ ( )+ ] 4
( )

Q12 Solve the following linear programming problem graphically:
Maximize and minimize = + subject to the constraints
4
+ ≤ , + ≥ , − ≥ , ≥ , ≥

Q13 Probability of solving a specific problem independently by A and B are 1/2 and 1/3 respectively. If both
try to solve the problem independently, find the probability that : 4
(i)the problem is solved (ii)exactly one of them solves the problem
OR
In an examination , 20 questions of true-false type are asked. Suppose a student tosses a fair coin to
determine his answer to each question. If coin falls heads, he answers true and if it falls tails, he answers 4
false. Find the probability that he answers at least 12 questions correctly.

Section – D

Q14 (a) Express the matrix = as a sum of a symmetric matrix and a skew-symmetric matrix. 4

− −
(b) If = and = then show that ( ) = 2
OR
Solve the following system of linear equations by matrix method :
6
− + = + − = + − =
Q15 Show that height of the cylinder of maximum volume that can be inscribed in a sphere of 30 cm is cm. 6
√
OR
Solve ∫ 6
Q16(a) Find the projection of the vector ⃗ = ̂ − ̂ + ̂ on the vector ⃗ = ̂ + ̂ − 2
(b) Find any daigonal of the parallelogram whose adjacent sides are given by the vectors ⃗ = ̂+ ̂+
4
and ⃗ = ̂ + ̂ + .Also find the area of the parallelogram.
OR 6
A line makes angles , , and with the diagonals of a cube, prove that
6
+ + + =

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