Goa Board Class 12 Question Paper 2025 Mathematics

2025 II 24 0930 Seat No.

Time : 3 Hours MATHEMATICS
Subject Code

H 4 7 5 4

Total No. of Questions : 36 (Printed Pages : 8) Maximum Marks : 80

INSTRUCTIONS : (i) The question paper consists of 36 questions.

(ii) Question numbers 1 to 8 are Multiple Choice type of
questions of one mark each.

(iii) Question Numbers 9 to 16 are very short answer type
questions of one mark each.

(iv) Question Numbers 17 to 22 are short answer type-I
questions of two marks each.

(v) Question numbers 23 to 28 are short answer type-II
questions of three marks each.

(vi) Question numbers 29 to 34 are long answer type-I
questions of four marks each.

(vii) Question numbers 35 to 36 are long answer type-II
questions of five marks each.

(viii) There is no overall choice. However an internal choice
has been provided in two questions of 4 marks each
and 2 questions of 5 marks each.

(ix) Use of calculators is not permitted.

(x) Graph should be drawn on the answer paper only.

H-4754 1 P.T.O.

0 4
1. If A = x–5 0 is skew symmetric matrix, then the value of x is .........

• 1

• 2

• 3

• 4

2. The relation R = { (a, a), (b, b), (c, c) } on the set A = {a, b, c} is ...........

• Identity

• Reflexive only

• Symmetric only

• Equivalence

3
3. The value of cos–1 cos is ..................
2

•
2

3
•
2

5
•
2

7
•
2

H-4754 2

dy
4. The integrating factor of differential equation cos x + y sin x = 1 is .......
dx

• sec x + tan x

• Log(sec x + tan x)

• esec x x

• sec x

• x2x–1

• 2x2xlog x

• 2x2x(1 + log x)

• 2x2x (1 – 2 log x)

dy
6. The value of at (4, 1) of y3 – x = 5 is ...................... .
dx

5
•
4

1
•
12

1
•
24

1
•
3

H-4754 3 P.T.O.

7. 2 x 3 x dx equals to ................

3x
• +c
ln3

2x
• +c
ln2

2 x3x
• +c
ln2ln3

6x
• +c
ln6

8. The direction ratios of the line 6x – 2 = 3y + 1 = 2z – 2 is .................

• 3, 2, 1

1 1
• , ,–1
3 3

• 1, 2, 3

1 1
• , ,–1
6 3

9. If f : R R and g : R R are two functions defined by f(x) = 2x + 1 and

g (x) = x2 – 2 respectively, find (g o f (x).

10. Find the principal value of sec–1(–2).

dy
11. Find if y = sin (log x).
dx
2
12. Evaluate ( x3 + 1) dx .
–2

13. Find the vector equation of the line through the points A(3, 4, – 7) and

B (1, – 1, 6).

H-4754 4

1 1
14. If A and B are two independent events with P(A) = and P(B) = find
3 4

P B'|A .

15. Find the magnitude of a, where a = (i + 3 j – 2 k) × (– i + 3 k)

16. Find the interval in which the function f(x) = 2x2 + 12x + 1 is increasing.

17. Prove that tan–1 x + cot–1 x = , x R.
2

18. Using the First Principle of derivative, find the derivative of y = a2x+5 with

respect to x.

1
19. Show that the function f : R – {3} R – {0} defined by f(x) = is
x–3

bijective.

20. Find the general solution of the differential equation :

dy
(x + 2) = x2 + 4x – 9
dx

1 2 x
21. Find x, if [2x 3] = 0
–3 0 3

2
x
d2 y dy
22. If 2tdt + y = 1 , prove that y 2 +
2 + 1 = 0.
0
dx dx

dx
23. Using integration, prove that log x + x2 + a2 + c
x + a2
2

1 2 2
24. If A = 2 1 2 , prove that A2 = 4A + 5I
2 2 1

H-4754 5 P.T.O.

1
25. Find dx.
x[1 – 4 (log x)2 ]

26. Find the area of the parallelogram whose diagonals are 3i + j – 2k and

i – 3 j + 4 k . Also find its adjacent sides.

27. Solve the differential equation x2dy + (xy + y2)dx = 0
3
1 1 d2 y t
28. If x = t – and y = t + , prove that 2
=4 2 .
t t dx t +1

29. Solve the following linear programming problem by graphical method

Maximize Z = 6x + 5y subject to the constraints

2x + y 6

x + y 5

x + 3y 3,

x, y 0
1 –1 1
30. If A = 2 1 –3 , using adjoint method, find A–1. Hence solve the system
1 1 1
of equations :

x – y + z = 4

2x + y – 3z = 0

x + y + z = 2

H-4754 6

31. Discuss the continuity of the function f(x) at x = 0, where

(e4 x – 1) (1 – cos x)
if x 0
x3
f(x) = 4 cos x – 6 if x 0
log (1 + x)
if x 0
x 1 + x – 1 cosec x

If the function is discontinuous, state the type of discontinuity at x = 0.

1
32. Find dx
1 – 2sin x

Or

32. Find x 2 cos 2 x dx

33. Using integration, find the area enclosed between the parabola 4y = x2 and

the straight line x = 4y – 2.

34. Find the equation of the line passing through the point (1, 2, 3) and parallel

to the line of intersection of planes x – y + 2z = 6 and 3x + y – z = 5.

Or

34. Find the vector and Cartesian equation of the plane passing through three

non-collinear points (0, –1, – 1), (4, 5, 1) and (3, 9, 4) and determine whether

(–4, 4, 4) lie on the plane.

35. The radius of a solid cylindrical object is increasing at the rate of 2 cm/sec

and its altitude is decreasing at the rate of 3 cm/sec. Find the rate of change

of total surface area when the radius is 3 cm and altitude is 5 cm.

H-4754 7 P.T.O.

Or

35. A cricket field is of the shape of a rectangle with semicircle at each end. The

sports authority wants to construct a running track of 440 m enclosing the

cricket field. What is the maximum area of the rectangular cricket field ?

36. The members of a private company rent cars from three rental agencies, 50%

from agency X, 30% from agency Y and 20% from agency Z. From the past

experience, it is known that 90% of cars from agency X are AC cars, 70%

of cars from agency Y are AC cars, 20% of cars from agency Z are AC cars.

A rental car is selected and delivered to the company. It is found to be an

AC car. Find the probability that the rental car is from agency Z.

Or

36. In a group of 30 scientists working on an experiment, 20 never committed

error in their work and are reporting results elaborately. Two scientists are

selected at random from the group. Find the probability distribution of the

number of scientists who committed error in the work and reporting. Also,

find the mean of the distribution.

H-4754 8