PSEB CLASS 12 MATHEMATICS GUESS PAPER MARCH 2026

Pre-Board Examination 2025-26
Class 12 – Mathematics
Section A (MCQs)

Time: 3 Hours    Maximum Marks: 80

Section A: Choose the correct option. (20 × 1 = 20)

1. Consider the non-empty set consisting of children in a family and a relation R defined as a R b if a is brother of b. Then R is:

(a) Transitive but not symmetric

(b) Both symmetric and transitive

(c) Symmetric but not transitive

(d) Neither symmetric nor transitive

2. Principal value of tan⁻¹(−1) is:

(a)

(b)

(c)

(d)

3. Domain of function f(x) = sin⁻¹x is equal to:

(a) [0, 1]

(b) R

(c) [−1, 1]

(d) None of these

4. If A and B are invertible matrices, then the inverse of AB is equal to:

(a) AB

(b) BA

(c) A⁻¹B⁻¹

(d) B⁻¹A⁻¹

5. If order of matrix A is 2 × 3 and order of matrix B is 3 × 5, then order of matrix B′A′ is:

(a) 5 × 2

(b) 2 × 5

(c) 5 × 3

(d) 3 × 2

6. If A is an invertible square matrix of order 4, then |adj A| is equal to:

(a) |A|⁴

(b) |A|⁵

(c) |A|³

(d) |A|²

7. If a function is continuous at x = 4, then the value of k is:

(a) 4

(b) 16

(c)

(d)

8. If a given function is , then its value is:

(a)

(b)

(c)

(d)

9. The rate of change of the area of a circle with respect to its radius r at r = 6 cm/sec is:

(a) 10π cm²/sec

(b) 12π cm²/sec

(c) 8π cm²/sec

(d) 11π cm²/sec

10. ∫ √( ) / √( ) dx is equal to:

(a)

(b)

(c)

(d)

11. ∫ ( ) is equal to:

(a)

(b)

(c)

(d)

12. Degree of equation is:

(a) 3

(b) 2

(c) 1

(d) 0

13. The integrating factor of the given differential equation is:

(a)

(b)

(c)

(d)

14. If →a is any vector then →a · →a is:

(a) 1

(b) 0

(c) |a|²

(d) None of these

15. If vectors →a and →b are perpendicular, then the value is:

(a) 6

(b) 9

(c) 3

(d) −3

16. Direction ratios of the line are:

(a) <3, 10, −7>

(b) <3, −5, 7>

(c) <3, 5, 7>

(d) <3, 5, −7>

17. Direction cosines of x-axis are:

(a) <1, 0, 0>

(b) <0, 1, 0>

(c) <0, 1, 1>

(d) <0, 1, 1>

18. The maximum value of Z subject to the constraints is:

(a) 0

(b) 12

(c) 20

(d) 32

19. If E is any event then P(E) belongs to:

(a) [0, 1]

(b) (0, 1)

(c) [−1, 1]

(d) (−1, 1)

20. If P(A) = ___ , P(B) = 0 then P(A | B) is:

(a) 0

(b)

(c) Not defined

(d) 1

Pre-Board 2025-26

Subject: Mathematics (Class: 12th)

Time: 3 hrs | M.M. 80

Section: B (7 × 2 = 14)

2. Using determinants, find the equation of line passing from the points (1, 4) and (-1, 2).

3. If y = x^sin-1x then, Find dy/dx.

4. Find the intervals in which the function f(x) = 6 - 9x - 2x² is strictly decreasing.

5. Find ∫ (sin³x + cos³x) / (sin²x . cos²x) dx.

6. Find the area of the region bounded by y² = 4x, x = 1, x = 4 and the x-axis in the First quadrant.

7. Find the general solution of the differential equation dy/dx = [y(x² - 1)] / [x(y² - 1)].

8. Find area of parallelogram whose adjacent sides are given by the vector a = i + j - k and b = 2i - j + 3k.

Section: C (7 × 4 = 28)

9. Show that the function f: R → R defined by f(x) = (2x - 1) / 3, x ∈ R is one-one and onto. Also find the inverse of f.

10. If A =

6 2
-1 3

and I =

1 0
0 1

then prove that (A - 4I)(A - 5I) = 0.

11. If f(x) = |x| sin(1/x) if x ≠ 0, and f(x) = 0 if x = 0, then discuss continuity of f(x) at x = 0.

(Or)

If y = e^ax Sin bx prove that d²y/dx² - 2a(dy/dx) + (a² + b²)y = 0.

12. Evaluate ∫ dx / [(x + 1)(x + 2)(x + 3)]

(Or)

Evaluate ∫ e^4x Cos 7x dx.

13. Solve the differential equation: (3xy + y²)dx = (x² + xy)dy

(Or)

Solve the differential equation: dy/dx - 2y = 3x.

14. Minimize z = 3x + 2y subject to the constraints x + y ≥ 8, 3x + 5y ≤ 15, x ≥ 0, y ≥ 0.

15. Bag I contains 5 red and 6 black balls, bag II contains 4 red and 7 black balls. One bag is chosen at random and a ball is drawn which is found to be red. Find the probability that it was drawn from the bag II.

Section: D (3 × 6 = 18)

16. Solve the following system of equations by matrix method:
2x - 3y + 5z = 11
3x + 2y - 4z = -5
x + y - 2z = -3

(Or)

a) Express the matrix

5 1 -1
2 -3 4
7 1 -3

as sum of symmetric and skew-symmetric matrices.
b) If A =

-1 3 0
-7 2 8

and B =

-5 0
0 3
1 -8

. Verify that (AB)' = B'A'.

17. Prove that the volume of the largest cone that can be inscribed in a sphere of radius r is 8/27 of the volume of the sphere.

(Or)

Evaluate ∫₀^π/2 log Sinx dx.

18. Find the shortest distance between the lines: (x - 1)/2 = (y - 2)/3 = (z - 3)/4 and (x - 2)/3 = (y - 3)/4 = (z - 5)/5.

(Or)

a) Find scalar projection of a = 2i - j - 3k on b = 3i - 5j + k.
b) Find the angle between vectors a + b and a - b if a = 2i - j + 3k and b = 3i - j - 2k.