Goa Board Class 12 Question Paper June 2019 Mathematics and Statistics

2019 VI 12 1430 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 section A, B, C, D and E.

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

are multiple choice type questions. Section B contain

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 and 2 questions of 5 marks.

In questions with choice, 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.

1. If A is a square matrix of order 3 × 3, then (kA)T, k 0 is equal
to .............

kA

kAT

1 T
A
k

k3AT

2. A binary operation * defined on Q, the set of rational numbers, as
a * b = ab + 3, then * is .................

Associative but not commutative

commutative but not associative

Both commutative and associative

Neither commutative nor associative

3. If the Banker’s discount on a bill of Rs. 25,000 due in 6 months is Rs. 500
then the rate of interest is ...................... per annum.

2%

3%

4%

5%

1
dx ..............
4.
x2 a2

log| x x2 a2 | c

log| x x2 a2 | c

log| x x2 a2 | c

log| x x2 a2 | c

1 2 2 3
5. If A = and B = then 2A – 3B = ............
3 0 1 4

1 3

0 1

4 5

9 12

4 12

9 3

2 1

5 12

6. Present value of an annuity of Rs. 600 payable at the end of each 6 months
for 5 years, if money is worth 6% per annum compounded semi-annually
is ...................
(a10/ 0.03 8.5302, a10/0.06 7.3601)

5223.6

4312.1

4416.06

5118.12

7. In a partnership deed, three partners invested Rs. 11,000, Rs. 12,000 and

Rs. 13,000. then the sum of their profit sharing ratio is ..................

16

26

36

52

Section B

Question Nos. 8 to 14 carry 2 marks each.

8. Using determinants, find the equation of the line joining the points (2, 3)

and (4, 6).

9. Find the derivative of the function f (x) = log (x + 8). with respect to x, using

the first principle.

10. A company produces two types of foods F1 and F2. They cost Rs. 50 and

Rs. 30 per unit. Each type of food contains two types of nutrients N1 and

N2 in different quantities. Each unit of F1 contains 2 units of N1 and 4 units

of N2. Each unit of F2 contains 3 units of N1 and 2 units of N2. The minimum

daily requirements of N1 and N2 is 6 units and 8 units respectively. Write

the objective function and the constraints of the linear programming problem

to meet the minimum requirements.

11. Define Sacrificing ratio and gaining ratio in a partnership deed.

12. Evaluate :

2
2 x2 3x 1
dx.
1
x

13. A coin is tossed 3 times. If getting a head is success, find the probability

of getting atleast one success.

14. Find :

5
dx.
x(3 2 log x)

Section C

Question Nos. 15 to 21 carry 3 marks each.

15. Siya and Riya entered into a partnership investing Rs. 42,000 and Rs. 70,000

respectively and agreed to share their profits in the ratio of their capitals.

Siya is a working partner and gets Rs. 1,000 per month as working allowance.

Find the total earnings of each partner in a profit of Rs. 33,700 after one

year.

16. Check whether the relation R defined on the set of real numbers as

R = {(a, b) : a < b} is Reflexive, symmetric and transitive.

Or

If f : R R is defined by f (x) = 4x + 5, show that the function f (x) is one-

one and onto. Hence find inverse of the function.

dy y
17. If x8 y2 (x y)10 , then prove that .
dx x

Or

d2 y
If y = A cos (3x + 2) + B sin (3x + 2), then prove that 9y 0.
dx2

18. Evaluate :

/2
sin 2 x
dx.
0 (1 cos x)2

dy
19. If x = 3(t – sin t), y = 3 (1 – cos t) find .
dx

20. Solve the differential equation x log x dy – y dx = 0.

21. Show that the following differential equation is homogeneous and hence

solve it :

dy
x x y.
dx

Section D

Question Nos. 22 to 28 carry 4 marks each.

22. By using properties of determinants show that :

a2 2a 2a 1 1

2a 1 a 2 1 (a 1)3 .

3 3 1

23. Solve the following system of equations, using matrix method :

x + 2y + z = 7

x + 3z = 11

2x – 3y = 1.

24. Define removable type of discontinuity of a function at a point. Determine
whether the function f (x) defined below is continuous at x = 0, where :

x 1 1
f ( x) ; x 0
log(1 x)

2 x2 3 x
; x 0
6x
sin 3 x
; x < 0.
tan 4 x
25. Find :
1
dx.
3 sin x 4 cos x 5
Or
Find :
e x dx
(e x 1) ( e x 3) (2e x 1)
26. Solve the following linear programming problem graphically.
Maximise :
Z = 5x + 10y
Subject to the constraints :
x + 2y < 120
x + y > 60
x – 2y > 0, x, y, > 0
27. Two rotten apples are accidentally mixed with eight good ones. Two apples
are drawn one after another with replacement from this lot. Find the
probability distribution of the number of rotten apples.
Or
Of the students in a college, it is known that 60% reside in city and 40%
reside in village. Previous year results report that 30% of all students who
reside in the city attain A grade and 20% of the village students attain A
grade in their annual examination. At the end of the year, one student is
chosen at random from the college and he has an A grade, what is the
probability that the student is from city.

28. A bill of exchange for Rs. 10,000 drawn on 19 June for 5 months was

discounted for Rs. 9900 at the rate of 5% p.a. Find the date on which bill

was discounted.

Section E

Question Nos. 29 to 30 carry 5 marks each.

29. A machine costs a company Rs. 63,000 and its effective life is estimated to

be 12 years. A sinking fund is created in order to replace the machine by

a new model at the end of its lifetime. The scrap of the old machine would

yield Rs. 3,000. Find what amount should be set aside at the end of each

year to accumulate at compound interest of 15% per year. (Use logarithmic
table).

Or

Find the present value of annuity of Rs. 600 payable at the beginning of

each 3 months for a period of 15 years. If the interest rate is 6% p.a.

compounded quarterly. (Use logarithmic table).

30. The demand equation for a manufacturer’s product is p = 8000 – 40x – x2,

where p is the price per unit and x is the number of units sold. At what

level of output, x, the total revenue will be maximum ? Also find the price

per unit

Or

x4
If C = 5x3 – 24x2 – – 20x is a cost function, find the average cost
3
function. At what level of production x, is there a minimum average

cost ?