Goa Board Class 12 Question Paper Mar 2019 Mathematics and Statistics

2019 III 13 1000 Seat No.

Time : 2½ Hours MATHEMATICS & STATISTICS
Subject Code

H 6 0 6
Total No. of Questions : 30 (Printed Pages : 8) Maximum Marks : 80

INSTRUCTIONS : (i) All questions are compulsory.

(ii) The question paper consists of 30 questions divided

into five sections A, B, C, D and E.

(iii) Section A contains 7 questions of 1 mark each,

which are multiple choice type questions, Section B

contains 7 questions of 2 marks each, Section C

contains 7 questions of 3 marks each, Section D

contains 7 questions of 4 marks each and Section

E contains 2 questions of 5 marks each.

(iv) There is no overall choice in the paper. However

internal choice is provided in 2 questions of 3

marks, 2 questions of 4 marks each and 2 ques-

tions of 5 marks. In questions with choices, only

one of the choices is to be attempted.

(v) Use of calculators is not permitted.

(vi) Logarithmic tables will be supplied on request.

(vii) Graphs should be drawn on the answer paper

only.

Section A

Question Nos. 1 to 7 carry 1 mark each. In each question, four options are

provided, out of which one is correct. Write the correct option.

3 2 1 4
1. If 2X = , then matrix X = ............
1 1 1 1

2 2

2 0

1 1

1 0

1 1
2 2
1
0
2
1 1

1 0

dy
2. If y = log (sec x), then = ............
dx
sec x tan x
sec x
tan x
sec x + tan x
3. Which statement among the following statements is false ?
Banker’s gain is interest on true discount
Banker’s discount is interest on face value
True discount is interest on present value
Banker’s gain is interest on Banker’s discount

4. The total time from the beginning of the first payment period to the end of
the last payment period of an annuity is called :

Payment interval
Interval of deferment
Term
Time period
5. In a partnership deed on admission of a new partner, the premium brought
by the new partner is shared among the old partners in ...............

Sacrificing ratio
Gaining ratio
New profit sharing ratio
Old profit sharing ratio

1
6. dx ............. C
x2 a2

1 x a
log
2a x a

1 x a
log
2a x a

log ( x x2 a2 )

log ( x x2 a2 )
/4

7. sec2 x dx ..............
0

0
1
2
–1
1

Section B

Question Nos. 8 to 14 carry 2 marks each.

8. If f : R+ (1, ) such that f (x) = x2 + 1, then show that the function ‘f ’
is bijective.

ab
9. If a * b = and (x * 5) × 3 = 15, then find x.
2

2 3
10. If A = , then show that A3 = A.
1 2

11. If the point (1, 1), (0, 2) and (3, y) are collinear points, then find y using

determinant.

12. Total cost function of a commodity is given by C(x) = 3x2 + 7x + 12. Find

x so that marginal cost is equal to average cost.

13. Write the linear constraints of a linear programming problem if the feasible

region of the problem is given by the shaded region in the following

figure :

2 3
14. If A and B are two independent events such that P(A) = and P(B) = ,
3 4

then find P(A B ).

Section C

Question Nos. 15 to 21 carry 3 marks each.

15. Find x, y and z if :

3 1 2 2 x
2
3 2 1 4 1 0 y .
1
0 1 1 1 z

16. If ex + ey = ex+y, then show that :

dy ey
.
dx ex

17. A monopolist’s demand function for one of its products is p(x) = ax + b,

where x is the number of units produced and p is the price per unit. If on

selling 10 units, the price is Rs. 1,200 per unit and on selling 7 units, the

price per unit is Rs. 1,500, then find total revenue function and marginal

revenue function of the product.

Or

Profit function of a commodity is given by P(x) = 600x – 2000 – 3x2, where

x is number of items. Then find x so that profit is maximum.

18. Find :

x 1
2
dx.
x 2x 3

19. Evaluate :

x sin x dx.
0

20. Solve the differential equation :

y( x 1) dx x( y 1) dy 0.

Also find the particular solution given that y = 1 when x = 1.

Or

Solve the differential equation :

dy
x2 x2 y2 xy.
dx

21. Form the differential equation by eliminating the arbitrary constants A and

B from the equation :

y Ax 2 Bx.

Section D

Question Nos. 22 to 28 carry 4 marks each.

22. By using properties of determinants prove that :

2a b a a

a 2a b a (4 a b) (a b)2 .

a a 2a b

23. If f is a function such that :
f (x) = 4x4 – 3x3, 0 < x < 1
= Ax + B, 1 < x < 2
= 5x2 – 8x, 2 < x < 3
and f is continuous on its domain, then find A and B.

log t dy
24. If x = t2 log t and y , then find when t = 1.
t dx
25. There are three boxes containing bulbs. In Box I out of 10 bulbs 4 are
defective. In Box II out of 9 bulbs 3 are defective. In Box III out of 8 bulbs
4 are defective. One of the boxes is selected at random and a bulb is selected
from the box at random. If the selected bulb is defective, then find the
probability that it is from box II.
Or
A die is tossed 3 times. If getting a number greater than 2 is a success
then find the probability distribution of number of successes. Also find mean
of the probability distribution.
26. Solve the following linear programming problem graphically.
Minimize :
Z = 8x + 9y
Subject to the constraints :
x + y < 8, 4x + 3y > 7,
x + 6y > 7, x > 0, and y > 0.
27. A bill of Rs. 60,000 drawn on 2nd January, 2017 for 9 months and discounted
on 12th May 2017 for Rs. 58,680. Find the rate of interest per annum.
28. Find :

sin x dx
.
cos2 x 5 cos x 7

Or
Find :
1
2
dx.
2 sin x 3 cos2 x 1

Section E

Question Nos. 29 and 30 carry 5 marks each.

29. A loan of Rs. 5,05,700 to purchase a car is re-paid in 18 half yearly equal

instalments, at 8% per annum compounded half yearly. The first

instalment being paid at the end of first half year. Find the amount of each

instalment.

Or

Find the amount of deferred annuity of Rs. 9,000 payable at the end of

every quarter year, the first instalment being made at the end of 2 years

and last instalment at the end of 10 years, if money is worth 12% per annum
compounded quarterly.
2
30. X and Y started a business in partnership. X invested of the capital and
3
rest by Y. After 9 months, X withdrew half of his investment. After n months,

they divided their profit in the ratio 3 : 2. Then find n.

Or

A, B and C are partners of a firm contributing Rs. 5,00,000, Rs. 3,00,000

and Rs. 2,00,000 respectively. The partnership deed provided an interest of
5% on the capital investment. B and C gets Rs. 30,000 and Rs. 20,000 per

month respectively, as salary for their services to the firm. At the end of
the year the firm gets a profit of Rs. 12,00,000. What will each receive if

the remaining profit is shared in ratio of capitals.