Goa Board Class 12 Question Paper June 2018 Mathematics and Statistics _Old Pattern_

2018 VI 12 1430 Seat No. :

Time : 2½ Hours mathematics and statistics (Old Pattern)

Subject Code

H 6 0 6

Total No. of Questions : 7 (Printed Pages : 4) Maximum Marks : 80

Instructions : i) This question paper contains seven main questions.
ii) All seven questions are compulsory.
iii) Answer each main question on a fresh page.
iv) Use of calculator is not allowed.
v) Log tables will be supplied on request.
vi) Graphs should be drawn on the answer paper only.
vii) For each main question, the sub-questions carry the
following marks.
[A = 1 mark, B = 2 marks, C = 3 marks, D = 4 marks,
E = 5 marks]

1. A) Choose and write the correct alternative from those given below :
A matrix obtained by interchanging rows and columns in a given matrix is
called the _________ of the matrix.
• Adjoint • Multiplicative Inverse
• Additive Inverse • Transpose

3 2  1 0
B) Find X, if Y =   and 2X + Y =  .
1 4 −3 2

3 −2 1 0
C) If A =   and I =   , find K, so that A = KA – 2I.
2

4 − 2 0 1

1 0 0 
D) If A = 3 3 0  , find A–1, using the adjoint method.
5 2 −1

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

2. A) Determine the order and degree if defined, of the following differential equation.
2
 d2 y   dy 
 2  + cos   = 0 .
 dx   dx 

B) Using the properties of determinants, as far as possible, prove that
− a2 ab ac
ba −b2 bc = 4a2b2 c2 .
ca cb −c2

C) If x = cosθ + cos2θ and y = sinθ + sin2θ, find dy , where θ is the parameter.
dx
D) Solve graphically the following Linear programming problem.
Maximise Z = 20 x + 10y
Subject to the constraints
x + 2y ≤ 28
3x + y ≤ 24
x ≥ 0, y ≥ 0.

3. A) Given that E and F are events such that P(E) = 0.6, P(F) = 0.3 and P(E∩F) = 0.2,
find P(E/F).
B) Write the constraints for the following Linear Programming problem.
A factory manufactures two types of screws, A and B. Each type of screw
requires the use of two machines, an automatic and a hand operated. It takes
4 minutes on the automatic and 6 minutes on the hand operated machines
to manufacture a package of screws A, while it takes 6 minutes on automatic
and 3 minutes on the hand operated machines to manufacture a package of
screws B. Each machine is available for at the most 4 hours on any day.
dy
C) Find , if xy2 – x2y – 4 = 0.
dx

D) If a function f : R → R, is continuous on its domain, where
f(x) = 5b – 3ax –4 ≤ x <–2
= 4x + 1 –2 ≤ x ≤ 2
= ax2 + 17b 2<x≤4
Find the values of a and b.

H-606 -2-

4. A) Choose and write the correct alternative from those given below.
The difference between the resources and liabilities of a firm is called its
• Present worth • Future value • Profit • Loss
B) Using the first principle, find the derivative of function f : R → R where
f(x) = log5x.
7
C) Evaluate ∫ x dx.
2 x + 9−x
D) A bill drawn on April 14, 2016 for 8 months after date, was discounted on
July 24, 2016 at 5% per annum. If the bankers gain on the basis of simple
interest is Rs. 20, what is the face value of the bill ?

5. A) Choose and write the correct alternative from those given below.
The instantaneous rate of change of revenue at any level of output is called
• Average revenue • Cost • Profit • Marginal revenue
B) Find ∫x.cosxdx.
ex
C) Find ∫ dx .
(e + 4e +13)
2x x

D) Attempt any one of the following :
1) Given the cost function is C(x) = x3 – 57x2 + 315x + 20.
Find the level of output for which the cost is minimum.
2) The total revenue function is given by R(x) = 400 + 3x + 2x2.
Find : 1) The average revenue
2) Marginal revenue
3) Marginal revenue when x = 15 units
4) Average revenue when x = 10 units.

6. A) Choose and write the correct alternative from those given below :
If R = {(a, a), (b, b), (c, c), (b, c), (a, b)} is a relation on set A = {a, b, c} then
• R is reflexive • R is symmetric
• R is transitive • R is an equivalence relation.
B) A and B are partners sharing profits and losses in the ratio 2 : 5. They
1
surrender rd of each one’s share in favour of C, a new partner. Calculate
3
the new profit sharing ratio.
H-606 -3- P.T.O.

C) On set R of real numbers, an operation * is defined by a*b = a + b + ab.
Is *a binary operation ? Justify your answer.
If it is a binary operation, find whether * is commutative and associative.
D) X, Y and Z formed a partnership and contributed Rs. 30,000, Rs. 40,000 and
Rs. 50,000 respectively towards the capital. They agree to divide the annual
profit in proportion to the capital employed and to the time it is in use. After
6 months, Z withdrew Rs. 5,000 and Y added Rs. 5,000. At the end of the
year, the profit was Rs. 1,22,400. How will they share this profit ?
E) Attempt any one of the following :
1) Find the future amount of an annuity of Rs. 1,00,000 payable at the end of
each quarter for 6 years at 8% p.a. compounded quarterly. Use log tables.
2) Find the present value of an annuity due of Rs. 700 per annum payable
at the beginning of each year for 2 years allowing interest at 6% per
annum compounded annually. Use log tables.
7. A) Choose and write the correct alternative from those given below.
An annuity in which payments start after a lapse of a specified number of
payment periods is called __________.
• Deferred annuity • Ordinary annuity
• Sinking fund • Annuity due
B) Form a differential equation by eliminating the arbitrary constants A and B,
in the following equation.
Y = Ae2x + Be–2x
C) Solve the differential equation
sec2x.tany dx + sec2y tanx dy = 0.
3x −1
D) Find ∫ dx .
(x −1) (x − 2) (x − 3)

E) Attempt any one of the following :
1) Find the mean number of heads in three tosses of a fair coin.
2) A manufacturer has three machine operators A, B and C. The first operator
A produces 1% defective items, where as the other two operators B and
C produce 5% and 7% defective items respectively. A is on the job for
50% of the time, B is on the job for 30% of the time and C is on the job
for 20% of the time. A defective item is produced. What is the probability
that it was produced by A ?
__________
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