Goa Board Class 12 Question Paper June 2018 Mathematics and Statistics (New Pattern)

*H605*
2018 VI 12 1430 Seat No. :

Time : 2½ Hours MATHEMATICS & STATISTICS (New Pattern)

Subject Code

H 6 0 6

Total No. of Questions : 30 (Printed Pages : 7) Maximum Marks : 80

INSTRUCTIONS : 1) All questions are compulsory.
2) The question paper contains 30 questions divided into
five Sections A, B, C, D and E.
3) 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, Section E contains 2
questions of 5 marks each.
4) 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
choices only one of the choices to be attempted.
5) Use of calculators is not permitted.
6) Logarithm- tables will be provided on request.
7) Graph should be drawn on answer paper only.

SECTION – A

Question numbers 1 to 7 carry 1 mark each.
In each question, four options are provided, out of which one is correct. Select and
write the correct option.

1. A square matrix in which all the non-diagonal entries are zero and diagonal
entries are same numbers is called __________ matrix.
• Symmetric
• Scalar
• Skew-symmetric
• Singular

H-606 -1- P.T.O.

2. If R is a relation defined on set of integers z, such that R = { (a, b) : |a – b| is a
multiple of 3 } then R is ________
• Reflexive but not symmetric
• Symmetric but not reflexive
• Symmetric and reflexive
• Neither reflexive nor symmetric

3. If a * b = a + b – 7, where a, b
R, then identity element of the binary operation
‘ * ’ is ________
• 0 • 7 • 6 • –6
dy
4. If y = sin25x then
______
dx
• cos2 5x • 5 cos25x
• 5 sin10x • 10 sin5x

5.
tan xdx  ___________ +C.
• sec2 x • tanx secx • log |sinx| • log |secx|

6. In Bill of exchange, the difference between Banker’s discount and true discount
is called
• Banker’s gain • Face value
• Discounted value • Present value

7. In an Annuity if the first periodic payment is done after lapse of some periods
then the annuity is called
• Ordinary Annuity • Annuity due
• Deferred Annuity • Sinking fund

SECTION – B

Question numbers 8 to 14 carry 2 marks each.


1 0
2 3 
    2 
8. If A   1 1 , B  4 1 and C  
then find (2A – B) C.
    3 
 2 1 5 6 

H-606 -2-

9. If f and g are two real valued functions such that f (x) = x2 + 2 and g (x) = x
1
then find f
g and g
f .

10. If P = (3, 4 ), Q = (2, 2) and R = (1, 6) , then find area of
PQR using determinants.

11. Find
x e dx .
! N

12. A furniture dealer deals in sales of only tables and chairs. He has Rs. 5,00,000
to invest and a space to store at most 300 pieces. A table costs Rs. 2,500 and a
chair costs Rs. 500. He can sell a table at a profit of Rs. 500 and a chair at a
profit of Rs. 150. Formulate the constraints and objective function of this linear
programming problem to maximise his profit.

3
13. Define ‘Independent Events’. If A and B are two independent events, P(A)

2 4
and P(B)
, then find P(A
B) .
3

14. Variable cost of x units of a commodity is 2x2 + 3x, fixed cost is Rs. 1,800 and
total revenue of x units is 9x + 1808. Find the break even point of the commodity.

SECTION – C

Question numbers 15 to 21 carry 3 marks each.

15. Find the matrices X and Y such that

 2 1
X+Y= 

  1 1

 1  2
X – 2y = 
.
 2  2

H-606 -3- P.T.O.

(3 x
1)(1  2x )
16. If Y  , find dy .
(2x  3)(5 x
2) dx

17. Form the differential equation by eliminating the arbitrary constants A and B
from Ax2 + By2 = 1.

18. P, Q and R invested in a partnership deed Rs. 2,00,000, Rs. 4,00,000 and
Rs. 3,00,000 respectively. After 4 months P invested additional Rs. 2,00,000
and R invested Rs. 1,00,000. The profit for the year is Rs. 6,60,000. Find the
share of each partner in the profit.

19. Solve the differential equation.

(x2 + y2) dx – xydy = 0.

OR

Find the particular solution of the differential equation


2xtanydx – (x2 + 1) sec2ydy = 0 when x = 0 and y  .
4
20. In a partnership deed, Akbar and Birbal contributed Rs. 9,000 and Rs. 8,000
respectively. They decided to share their profit in the ratio of their capital. At the
end of the year Akbar gets Rs. 2,000 more than Birbal as a share of profit . Find
the total profit and share of each partner.

21. The cost function of the firm is given by C (x) = 900 – 10x + x2. Find the output
x, at which the cost is minimum.

OR
27
The demand function for a product is given by P  34
3x
. Find total
x
revenue function and marginal revenue function. Also find x so that the marginal
revenue is equal to average revenue.

H-606 -4-

SECTION – D

Question numbers from 22 to 28 carry 4 marks each.

e
1N

22. If f(x)  log(1 x) if
1 x  0

 Ax  B if 0  x  2

x
4
 if 2  x  3
x x
6

and f is continuous at x = 0 and x = 2, find the values of A and B.

23. If x = tsint + cost and y = tcost – sint, where t is the parameter, then find
dy

when t  .
dx 4

x x y
z

24. Prove that y y z
x  (x  y)(y  z) (z  x)(x
y
z)

z z x
y

using properties of determinant.
OR

x 1 3 5
Find the value of x if 2 x2 5
0 using properties of determinants.
2 3 x4

H-606 -5- P.T.O.

16
x
%

25. Evaluate 
!

dx .
!
!

16
x  6  x
!

26. Solve the following linear programming problem graphically

Maximize Z = 10x + 15y

Subject to the constraints

y – 2x
2

x+y
2

x
1

x, y
0

4 1
27. For X, Y, Z the probabilities of being selected as the manager of a firm are ,
7 7
2
and respectively. The respective probabilities for them to introduce a radical
7
change in marketing strategy are 0.3, 0.8 and 0.5. If change in marketing strategy
took place, then find the probability that it is due to the appointment of Y.

OR

An Urn contains 4 white balls, 3 red balls and 5 green balls. If four balls are
drawn from the urn one by one with replacement, then find the probability that all
the balls are white. Also find the probability that at least one is white ball in the
drawn of 4 balls.

28. A bill of Rs. 8,000 for 4 months was discounted for Rs. 7,904 on August 10th 2015. If
the rate of interest is 6% per annum, then find the date on which the bill was
drawn.

H-606 -6-

SECTION – E

Question numbers 29 to 30 carry 5 marks each.

(sin x
1)cos x
29. Find  dx .
(sinx  1)(sin x
2)(sin x
3)
OR

5x  2
Find
dx .
3 x  2x  1
30. Find present value of a sequence of annual payments of Rs. 3,000 each if first
payment being made at the end of 4 years, the last at the end of
1 2 years and money worth 7 % per annum compounded annually. [Given
that (1.07) –3 = 0.8164, (1.07) –4 = 0.7629, (1.07) –8 = 0.5820 and
(1.07)–12 = 0.5326]
OR

At the beginning of each quarter year Rs. 5,000 is deposited in a bank account
that pays interest of 8% per annum compounded quarterly. Find the amount in
the account at the end of 3 years (Use log tables).

_______________

H-606 -7- P.T.O.