The number of Hawkins-Simon conditions for the viability of an input - output analysis is ________.

(a)

1

(b)

3

(c)

4

(d)

2

Which of the following matrix has no inverse.

(a)

\(\begin{pmatrix} -1 & 1 \\ 1 &-4 \end{pmatrix}\)

(b)

\(\begin{pmatrix} 2 & -1 \\ -4 &2 \end{pmatrix}\)

(c)

\(\begin{pmatrix} cos\ a & sin\ a \\ -sin\ a & cos\ a \end{pmatrix}\)

(d)

\(\begin{pmatrix} sin\ a & cos\ a \\ -cos\ a & sin\ a \end{pmatrix}\)

If nC₃ = nC₂, then the value of nC₄ is _______.

(a)

2

(b)

3

(c)

4

(d)

5

Number of words with or without meaning that can be formed using letters of the word "EQUATION" , with no repetition of letters is _____.

(a)

7!

(b)

3!

(c)

8!

(d)

5!

The slope of the line 7x + 5y - 8 = 0 is _______.

(a)

7/5

(b)

-7/5

(c)

5/7

(d)

-9/7

The locus of the point P which moves such that P is at equidistance from their coordinate axes is _______.

(a)

\(y={1\over x}\)

(b)

y = -x

(c)

y = x

(d)

\(y=-{1\over x}\)

The value of \(cosec^{-1}\left(\frac{2}{\sqrt{3}}\right)\) is ________.

(a)

\(\frac{\pi}{4}\)

(b)

\(\frac{\pi}{2}\)

(c)

\(\frac{\pi}{3}\)

(d)

\(\frac{\pi}{6}\)

If \(\tan A=\frac{1}{2}\) and \(\tan B=\frac{1}{3}\) then tan(2A + B) is equal to ______.

(a)

1

(b)

2

(c)

3

(d)

4

If the function f(x) is continuous at x = a if \(\lim _{ x\rightarrow a }{ f\left( x \right) } \) is equal to ________.

(a)

f(-a)

(b)

f\((\frac{1}{a})\)

(c)

2f(a)

(d)

f(a)

If y = x and z = \(\frac{1}{x}\) then \(\frac{dy}{dz}=\)________.

(a)

x²

(b)

1

(c)

-x²

(d)

\(-\frac{1}{x^2}\)

The elasticity of demand for the demand function x = \(\frac { 1 }{ p } \) is______.

(a)

0

(b)

1

(c)

\(-\frac { 1 }{ p } \)

(d)

\(\infty \)

If f(x,y) is a homogeneous function of degree n, then \(x\frac { \partial f }{ \partial x } +y\frac { \partial f }{ \partial y } \) is equal to ________.

(a)

(n–1)f

(b)

n(n–1)f

(c)

nf

(d)

f

Purchasing price of one share of face value 100 available at a discount of \(9\frac{1}{2}\%\) with brokerage \(\frac{1}{2}\%\) is ________.

(a)

Rs. 89

(b)

Rs. 90

(c)

Rs. 91

(d)

Rs. 95

The present value of the perpetual annuity of Rs. 2000 paid monthly at 10 % compound interest is _______.

(a)

Rs. 2,40,000

(b)

Rs. 6,00,000

(c)

Rs. 20,40,000

(d)

Rs. 2,00,400

Harmonic mean is better than other means if the data are for _________.

(a)

Speed or rates.

(b)

Heights or lengths.

(c)

Binary values like 0 and 1

(d)

Ratios or proportions.

The probability of obtaining an even prime number on each die, when a pair of dice is rolled is _________.

(a)

1/36

(b)

0

(c)

1/3

(d)

1/6

If r(X,Y) = 0 the variables X and Y are said to be ______.

(a)

Positive correlation

(b)

Negative correlation

(c)

No correlation

(d)

Perfect positive correlation

The correlation coefficient from the following data N = 25, ΣX = 125, ΣY = 100, ΣX² = 650, ΣY² = 436, ΣXY = 520 ________.

(a)

0.667

(b)

-0.006

(c)

-0.667

(d)

0.70

A solution which maximizes or minimizes the given LPP is called ______.

(a)

a solution

(b)

a feasible solution

(c)

an optimal solution

(d)

none of these

Which of the following is not correct?

(a)

Objective that we aim to maximize or minimize

(b)

Constraints that we need to specify

(c)

Decision variables that we need to determine

(d)

Decision variables are to be unrestricted

Evaluate\(\left| \begin{matrix} 1 & 3 & 4 \\ 102 & 18 & 36 \\ 17 & 3 & 6 \end{matrix} \right| \)

Evaluate the following using binomial theorem: (101)⁴

Find the acute angle between the lines 2x - y + 3 = 0 and x + y + 2 = 0.

Evaluate : \(\cos\left[\frac{\pi}{3}-\cos^{-1}\left(\frac{1}{2}\right)\right]\)

Find \(\frac{dy}{dx}\) if x = at², y = 2at

Find the maximum and minimum values of x³-6x²+7

Find the number of shares which will give an annual income of Rs. 3,600 from 12% stock of face value Rs. 100.

A die is thrown twice and the sum of the number appearing is observed to be 6. What is the conditional probability that the number 4 has appeared at least once?

From the following data calculate the correlation coefficient Σxy = 120, Σx² = 90, Σy² = 640

A retired person has Rs. 70,000 to invest and two types of bonds are available in the market for investment. First type of bond yields an annual income of 8% on the amount invested and the second type yields 10% per annum. As per norms, he has to invest a minimum of Rs. 10,000 in the first type and not more than Rs.30,000 in the second type. How should he plan his investment, so as to get maximum returns after one year of investment? Formulate the above as LPP.

The technology matrix of an economic system of two industries is \(\left[ \begin{matrix} 0.8 & 0.2 \\ 0.9 & 0.7 \end{matrix} \right] \) Test whether the system is viable as per Hawkins – Simon conditions.

 Find the middle terms in the expansion of \({ \left( { 2x }^{ 2 }-\frac { 3 }{ { x }^{ 3 } } \right) }^{ 10 }\)

Find the locus of a point such that the sum of its distances from the points (0, 2) and (0, -2) is 6.

Convert the following into the product of trigonometric functions cos55^o + sin 55^o

Verify the existence of the function \(f(x)=\left\{\begin{array}{l} 5 x-4 \text { if } 0< x \leq 1 \\ 4 x^3-3 x \text { if } 1< x< 2 \end{array} \text { at } x=1\right.\)

The production function for a commodity is P = 10L + 0.1 L² +15K-0.2K² +2KL  where L is labour and K is Capital.
(i) Calculate the marginal products of two inputs when 10 units of each of labour and Capital are used
(ii) If 10 units of capital are used, what is the upper limit for use of labour which a rational producer will  never exceed?

A man wishes to pay back his depts of Rs.3783 due after 3 years by 3 equal yearly instalments. Find the amount of each instalments,money being worth 5% p.a. compounded annually

Calculate Quartile deviation and Coefficient of Quartile deviation of the following data.

Marks:	0	10	20	30	40	50	60	70
No. of students:	150	142	130	120	72	30	12	4

Calculate the correlation co-efficient from the below data:

X	1	2	3	4	5	6	7	8	9
Y	9	8	10	12	11	13	14	16	5

Construct the network for the projects consisting of various activities and their precedence relationships are as given below:

Immediate Predecessor	A	B	C	D	E	F	G	H	I
Activity	B	C	D,E,F	G	I	H	J	K	L

1. Two commodities A and B are produced such that 0.4 tonne of A and 0.7 tonne of B are required to produce a tonne of A. Similarly 0.1 tonne of A and 0.7 tonne of B are needed to produce a tonne of B. Write down the technology matrix. If 68 tonnes of A and 10.2 tonnes of B are required, find the gross production of both of them.

2. Resolve into partial fractions for the following : \(\frac{x+2}{(x-1)(x+3)^2}\)

1. Using the principle of mathematical induction, prove that 1.3 + 2.3² + 3.3³ + ... + n.3ⁿ =\(\frac { (2n-1){ 3 }^{ n+1 }+3 }{ 4 } for\ all\ n\in N\)

2. If the equation ax² + 5xy - 6y² + 12x + 5y + c = 0 represents a pair of perpandicular straight lines, find a and c.

1. Prove that \(\frac{\cos4x+\cos3x+\cos2x}{\sin4x+\sin3x+\sin2x}=\cot3x\)

2. Prove that (sin 3x + sin x) sin x + (cos 3x - cos x) cos x = 0.

1. Draw the graph of the following function f(x) = e^-2x

2. Find the stationary values and stationary points for the function f(x) = 2x³ + 9x² + 12x + 1

1. The total revenue (TR) for commodity x is \(TR=12x+{x^2\over2}-{x^3\over 3}\)S.T. at the highest point of average revenue (AR), AR = MR

2. a bank pays 8% interest compounded quarterly. Determine the equal deposits to be made at the end of each quarter for 3 years so as to receive Rs.300 at the end of 3 years.

1. Compute upper Quartiles, lower Quartiles, D₄ and P₆₀, P₇₅ from the following data.

   CI	10-20	20-30	30-40	40-50	50-60	60-70	70-80
   Frequency	12	19	5	10	9	6	6

2. Calculate correlation coefficient for the following data.

   X	25	18	21	24	27	30	36	39	42	48
   Y	26	35	48	28	20	36	25	40	43	39

1. A project schedule has the following characteristics

   Activity	1-2	1-3	2-4	3-4	3-5	4-9	5-6	5-7	6-8	7-8	8-10	9-10
   Time	4	1	1	1	6	5	4	8	1	2	5	7

   Construct the network and calculate the earliest start time, earliest finish time, latest start time and latest finish time of each activity and determine the Critical path of the project and duration to complete the project.

2. Reshma wishes to mix two types of food P and Q in such a way that the Vitamin contents of the mixture contain at least 8 units of vitamin A and 11 units of vitamin B. Food P costs Rs.60/kg and Food Q costs Rs.80/kg. Food P contains 3 units 1 kg of vitamin A and 5 units 1 kg of vitamin B while food Q contains 4 units 1 kg of vitamin A and 2 units 1 kg of vitamin B. Determine the minimum cost of the mixture.