Goa Board Class 12 Question Paper 2025 Mathematics & Statistics

2025 II 24 0930 Seat No.

Time : 3 Hours MATHEMATICS & STATISTICS
Subject Code

H 4 6 0 6
Total No. of Questions : 36 (Printed Pages : 8) Maximum Marks : 80

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

(ii) Question numbers 1–8 are multiple choice type questions
of 1 mark each.

(iii) Question numbers 9-16 are very short answer type
questions of 1 mark each.

(iv) Question numbers 17-22 are short answer type-I
questions of 2 marks each.

(v) Question numbers 23-28 are short answer type-II
questions of 3 marks each.

(vi) Question numbers 29-34 are long answer type-I
questions of 4 marks each.

(vii) Question numbers 35-36 are long answer type-II
questions of 5 marks each.

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

(ix) Use of calculator is not permitted.

(x) Log tables will be supplied on request.

(xi) Graph should be drawn on the answer paper itself.

H-4606 1 P.T.O.

x + 10 3 6 3
1. If =
0 –4 0 4 y , tthen x + y = ......................

• 5

• 3

• –3

• –5

3x 6
2. If A = 4 2 is a singular matrix, then the value of ‘x’ is ..................

• –4
• –2
• 2
• 4

3. The rate at which the revenue changes with respect to the number of items
sold at an instant is known as ................

• Demand

• Average revenue

• Marginal revenue

• Marginal average revenue
dy
4. If y = log(x + 1), then at x = 1 is ...................
dx
1
•
2
• 2

• log2

1
• log 2

H-4606 2

• Ordinary annuity

• Annuity due

• Deferred annuity

• Sinking fund

6. The sum of the order and degree of the differential equation
3 2
d2 y dy
+ – x2 y5 = 0 is ..................
dx2 dx

• 2

• 3

• 4

• 5

7. If ‘A’ and ‘B’ are mutually exclusive events of a sample space ‘S’, such that
1 3
P(A) = , P(A B) = , then P(B) = .....................
2 5

1
•
10

2
•
10

3
•
10

5
•
10

H-4606 3 P.T.O.

8. Let “L” be the set of all lines in a plane and “R” be a relation defined on

the set L by {(L1, L2) : L1 is perpendicular to L2}. Then “R” is ................

• Reflexive

• Symmetric

• Transitive

• Equivalence

9. Using determinants, find the area of triangle ABC whose vertices are

A (1, 0), B (0, 2) and C (–3, 2).

2 dy
10. If y = 2 x , find
dx
0 –5 8
11. Identify the type of matrix A = 5 0 12
–8 –12 0

13. Define : Independent events.

1
14. Evaluate : e x dx
0

3
15. If x 2 4 = 2, find the value of ’x’.

16. Differentiate ‘4xy + 5 = 0’ with respect to ‘x’.

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17. Write the constraints for the following shaded region.

18. Define the following terms :

(1) Bill of exchange

(2) Bankers discount

dy
19. If x = (t2 + 1)3 and y = (2t2 – 5)2, where ‘t’ is a parameter, find .
dx

2
d 2 y 1 dy
20. Verify that y = 2e 3x
is the solution of the differential equation =
dx 2 y dx

21. “X”, “Y” and “Z” were partners sharing profit in the ratio 4 : 3 : 2. “Y” retires

from the firm and “X” and “Z” decide to share profits in the ratio 3 : 2.

Calculate the gaining ratio.

cos x
22. Find dx
sin2 x – 9

2x – 3
23. Show that the function f : R R defined by f(x) = , x R is one-
4

one and onto.

H-4606 5 P.T.O.

4 –2
2 4 0
24. If AT = –1 0 and 2A – 3B = 5 – 3 1 , then find (4A – B)T, where “T”
3 5

denotes the transpose of a matrix.

3
x x 4 2
25. Find the derivative of y = 4 , with respect to ‘x’.
(4 x – 3) 3

26. A bill of ` 4500 drawn on 7th May 2024 for 6 months was discounted on
29th August 2024 for a cash payment of ` 4419. Find the rate of interest
charged by the bank.

27. Find the particular solution of the differential equation

(x2 – yx2) dy – (y2 + x2 y2)dx = 0 ; x = y = 2
1 1
28. Prove that log –1 dx = 0
0 x

29. Solve the following linear programming problem graphically.

Minimize Z = 20x + 10y

Subject to

x + 2y 40

3x + y 30

4x + 3y 60

x, y 0

30. Solve the following system of linear equations using the matrix method.

x + y + z = 3

2x – y + z = 2

x – 2y + 3z = 2

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31. Find the values of the constants ‘A’ and ‘B’ if the function f(x) defined below

is continuous at x = 0.
12 x. log(1 + Ax)
f (x) = x < 0
x2
= 3B + 5 cosx x = 0
1 – cos 4 x
= x(e4 x – 1) x > 0
5 – 2x
32. Find : dx
(1 – x) (2 + x) (4 + x)

Or
3 2
x – 4 x – 27 x – 20
Find : x 2 – 6 x – 15
dx

33. Three boxes are given each containing red and black marbles as indicated
below :
Box I : 6 red and 4 black marbles
Box II : 4 red and 6 black marbles
Box III : 2 red and 8 black marbles.

A box is chosen at random and a marble is drawn from the box. The marble
drawn is red. Find the probability that the marble drawn is from box I.

34. Amit, Rohan and Snehal started a partnership and invested ` 1,00,000,
` 80,000 and ` 1,20,000 respectively. Snehal took a loan of ` 70,000 and paid
9% interest to the firm. The firm earned a profit of ` 1,43,700, in addition
to the interest from the loan. Find each partner’s total earnings if the profit
is distributed in the ratio of their capital investment.

Or

‘A’, ‘B’ and ‘C’ invested ` 20,000, ` 18,000 and ` 12,000 in a business. ‘A’
and ‘B’ receives 12% and 8% of annual profit for services. The remaining
profit is divided among ‘A’, ‘B’ and ‘C’ in proportion to their capitals. At the
end of the year, ‘A’ receives altogether ` 648 more than ‘B’. Find the annual
profit.

H-4606 7 P.T.O.

35. A company is planning to market a new model of a product. From the

market survey it is found that the demand function is x = – 1500p + 30,000.

The fixed cost to the company for the product is ` 28,000 and the cost per

unit is ` 8. What price should the company charge to obtain maximum

profit.

Or

Given the total cost function for ‘x’ units of a commodity as
x5
C(x) = + 7 x 4 + 4 x2 – 3 x + 15.
5

Find (1) Marginal Cost

(2) Average Cost.
d x. MC – C( x)
Also show that (AC) = , where MC is Marginal Cost and AC
dx x2
is Average Cost.

36. What equal payments made at the beginning of each month for 8 years will

pay for a piece of land, priced at ` 6,19,600, if the rate of interest is 12%

per annum compounded monthly. (Use log tables).

Or

Ms. Pearl wants to collect ` 6,10,000 to purchase a car after 16 years. How

much money she should deposit at the end of each year in a company paying

compound interest at the rate of 16% p.a. (Use log tables).

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