If the rank of the matrix  \(\left( \begin{matrix} \lambda & -1 & 0 \\ 0 & \lambda & -1 \\ -1 & 0 & \lambda \end{matrix} \right) \)  is 2. Then \(\lambda \) is ________.

(a)

1

(b)

2

(c)

3

(d)

only real number

If \(\left| \begin{matrix} 2x & 5 \\ 8 & x \end{matrix} \right| =\left| \begin{matrix} 6 & -2 \\ 7 & 3 \end{matrix} \right| \) then x =

(a)

3

(b)

± 3

(c)

± 6

(d)

6

\(\int _{ 0 }^{ \infty }{ { x }^{ 4 }{ e }^{ -x } } \)dx is _______.

(a)

12

(b)

4

(c)

4!

(d)

64

If ∫ x sin x dx = - x cos x + α then α = __________ +c

(a)

sin x

(b)

cos x

(c)

C

(d)

none of these

The demand and supply function of a commodity are P(x) = (x − 5)² and S(x)= x² + x + 3 then the equilibrium quantity x₀ is ________.

(a)

5

(b)

2

(c)

3

(d)

19

The area of the region bounded by the curve y² = 2y - x and the y-axis _____ sq. units

(a)

\(\frac{4}{3}\)

(b)

\(\frac{2}{3}\)

(c)

4

(d)

\(\frac{16}{3}\)

The differential equation formed by eliminating a and b from \(y=a e^{x}+b e^{-x}\) is ______.

(a)

\(\frac{d^{2} y}{d x^{2}}-y=0\)

(b)

\(\frac{d^{2} y}{d x^{2}}-\frac{d y}{d x}=0\)

(c)

\(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } =0\)

(d)

\(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } -x=0\)

The I.F. of \(\frac { dy }{ dx } \)- y tan x = cos x is _____

(a)

sec x

(b)

cos x

(c)

e^tanx

(d)

cot x

Δf(x) = _______.

(a)

f(x+ h)

(b)

f(x) − f(x+h)

(c)

f(x + h) − f(x)

(d)

f (x) − f(x−h)

The knowledge of ______ is essential for the study of Numerical Analysis

(a)

Differences

(b)

finite differences

(c)

interpolation

(d)

extrapolation

\(\int _{ -\infty }^{ \infty }{ f(x)dx } \) is always equal to ________.

(a)

zero

(b)

one

(c)

E(X)

(d)

f(x)+1

If \(f(x)=\left\{\begin{array}{cc} \frac{A}{x}, & 1<x<e^{3} \\ 0, & \text { otherwise } \end{array}\right.\) is a p.d.f. of a continuous random variable. X then P(X≥e)

(a)

\(\frac{2}{3}\)

(b)

\(\frac{3}{2}\)

(c)

\(\frac{1}{6}\)

(d)

\(\frac{1}{8}\)

The average percentage of failure in a certain examination is 40. The probability that out of a group of 6 candidates atleast 4 passed in the examination are ________.

(a)

0.5443

(b)

0.4543

(c)

0.5543

(d)

0.4573

If Z is a standard normal variate, then p(0

(a)

0.5

(b)

1

(c)

0.25

(d)

0.75

An estimator is a sample statistic used to estimate a  ______.

(a)

population parameter

(b)

biased estimate

(c)

sample size

(d)

census

Out of 1000 T.V viewers, 320 watched a particular programme. Then the standard error is __________

(a)

-0.147

(b)

0.147

(c)

0.0147

(d)

-0.0147

A typical control charts consists of ________.

(a)

CL, UCL

(b)

CL, LCL

(c)

CL, LCL, UCL

(d)

UCL, LCL

Chance variation does not affect _____ of the product

(a)

price

(b)

value

(c)

quantity

(d)

quality

The Penalty in VAM represents difference between the first ________.

(a)

Two largest costs

(b)

Largest and Smallest costs

(c)

Smallest two costs

(d)

None of these

In least cost method if the minimum cost is not unique then the choice can be made as ___________

(a)

arbitrarily

(b)

unique

(c)

difference

(d)

summation

Two newspapers A and B are published in a city . Their market shares are 15% for A and 85% for B of those who bought A the previous year, 65% continue to buy it again while 35% switch over to B. Of those who bought B the previous year, 55% buy it again and 45% switch over to A. Find their market shares after one year

Integrate the following with respect to x.
(3 + x)(2 − 5x)

The marginal cost at a production level of x units is given by C '(x) = 85 +\(\frac{375}{x^2}\). Find the cost of producing 10 in elemental units after 15 units have been produced?

Solve the following differential equations
\(\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } -4\frac { dy }{ dx } +4y=0\)

Find the missing term from the following data

x	1	2	3	4
f(x)	100	-	126	157

Suppose, the life in hours of a radio tube has the following p.d.f
\(f(x)=\left\{\begin{array}{l} \frac{100}{x^{2}}, \text { when } x \geq 100 \\ 0, \text { when } x<100 \end{array}\right.\)
Find the distribution function.

The probability of the happening of an event X is 0.002 in an experiment. If an experiment is reported 1000 times, find the probability that the event X happens exactly twice? (e^-2 = 0.1353)

What is type I error.

Calculate the cost of living index by aggregate expenditure method

Commodity	Quantity	Price(Rs.)
	2000	2000	2003
A	100	8	12
B	25	6	7.50
C	10	5	5.25
D	20	48	52
E	65	15	16.50
F	30	19	27.00

What is the Assignment problem?

Consider the matrix of transition probabilities of a product available in the market in two brands A and B.
\(_{ B }^{ A }\left( \begin{matrix} \overset { A }{ 0.9 } & \overset { B }{ 0.1 } \\ 0.3 & 0.7 \end{matrix} \right) \)
Determine the market share of each brand in equilibrium position.

Evaluate \(\int { \frac { { sec }^{ 2 }x }{ 3+tanx } } dx\)

If the marginal cost (MC) of a production of the company is directly proportional to the number of units (x) produced, then find the total cost function, when the fixed cost is Rs. 5,000 and the cost of producing 50 units is Rs. 5,625.

Form the differential equation for y = (A + Bx)e^3x where A and B are constants.

Given y₃ = 2, y₄ = −6, y₅ = 8, y₆ = 9 and y₇ = 17 Calculate Δ⁴y₃

The probability distribution of a discrete random variable. X is given by

X	-2	2	5
P(X=x)	\(\frac{1}{4}\)	\(\frac{1}{4}\)	\(\frac{1}{2}\)

then find 4E(X²)- Var (2X)

If the chance of running a bus service according to schedule is 0.8, calculate the probability on a day schedule with 10 services :
(i) exactly one is late
(ii) atleast one is late

A company market car tyres. Their lives are normally distributed with a mean of 50,000 kms and standard derivation of 2000 kms. A test sample of 64 tyres has a mean life of 51250 km. Can you conclude that the sample mean differs significantly from the population mean? (Test at 5% level).

Calculate the cost of living index by aggregate expenditure method:

Commodity	Weights 2010	Price (Rs.)
		2010	2015
P	80	22	25
Q	30	30	45
R	25	42	50
S	40	25	35
T	50	36	52

For the given pay-off matrix, choose the best alternative for the given states of nature under
(i) Maximin (ii) Minimax princple

Alternative	States of Nature
	Good	Fair	Bad
A	100	60	+50
B	80	50	+10
C	40	20	+5

1. Calculate the Laspeyre’s, Paasche’s and Fisher’s price index number for the following data. Interpret on the data.

   Commodities	Base Year		Current Year
   	Price	Quantity	Price	Quantity
   A	170	562	72	632
   B	192	535	70	756
   C	195	639	95	926
   D	187	128	92	255
   E	185	542	92	632
   F	150	217	180	314
   7	12.6	12.7	12.5	12.8
   8	12.4	12.3	12.6	12.5
   9	12.6	12.5	12.3	12.6
   10	12.1	12.7	12.5	12.8

2. Obtain an initial basic feasible solution to the following transportation problem using Vogels' approximation method.
   

1. In a particular university 40% of the students are having news paper reading habit. Nine university students are selected to find their views on reading habit. Find the probability that
   (i) none of those selected have news paper reading habit
   (ii) all those selected have news paper reading habit
   (iii) atleast two third have news paper reading habit.

2. Assign four trucks 1, 2, 3 and 4 to vacant spaces A, B, C, D, E and F so that distance travelled is minimized. The matrix below shows the distance.
   

1. Marks in an aptitude test given to 800 students of a school was found to be normally distributed 10% of the students scored below 40 marks and 10% of the students scored above 90 marks. Find the number of students scored between 40 and 90?

2. Measurements of the weights of a random sample of 200 ball bearings made by certain machine during one week showed a mean of 0.824 newtons and a S.D. of 0.042 newton's. Find
   a) 95% and
   b) 99% confidence limits for the mean weight of all the ball bearings.

1. Suppose that the quantity needed Q_d = 42 -4p-4\(\frac { dp }{ dt } +\frac { { d }^{ 2 }p }{ { dt }^{ 2 } } \) and quantity supplied Q_s = -6 + 8p where p is the price. Find the s equilibrium price for market clearance.

2. The following data gives the melting point of a alloy of lead and zinc where ‘t’ is the temperature in degree c and P is the percentage of lead in the alloy

   P	40	50	60	70	80	90
   T	180	204	226	250	276	304

   Find the melting point of the alloy containing 84 percent lead.

1. Evaluate \(\int _{ 0 }^{ \frac { \pi }{ 2 } }\) x sin x dx

2. Evaluate \(\int { \frac { \left( { x }^{ 2 }+1 \right) dx }{ { \left( x-1 \right) }^{ 2 }\left( x+3 \right) } } \)

1. The price of 3 Business Mathematics books, 2 Accountancy books and one Commerce book is Rs. 840. The price of 2 Business Mathematics books, one Accountancy book and one Commerce book is Rs. 570. The price of one Business Mathematics book, one Accountancy book and 2 Commerce books is Rs. 630. Find the cost of each book by using Cramer’s rule.

2. A manufacturer of ball pens claims that a certain pen he manufactures has a mean writing life of 400 pages with a standard deviation of 20 pages. A purchasing agent selects a sample of 100 pens and puts them for test. The mean writing life for the sample was 390 pages. Should the purchasing agent reject the manufactures claim at 1% level?

1. The price of a machine is Rs. 5,00,000 with an estimated life of 12 years. The estimated salvage value is Rs. 30,000. The machine can be rented at Rs. 72,000 per year. The present value of the rental payment is calculated at 9% interest rate. Find out whether it is advisable to rent the machine.(e^−1.08 = 0.3396).

2. The probability function of a random variable X is given by
   \(p(x)=\left\{\begin{array}{l} \frac{1}{4}, \text { for } x=-2 \\ \frac{1}{4}, \text { for } x=0 \\ \frac{1}{2}, \text { for } x=10 \\ 0, \text { elsewhere } \end{array}\right.\)
   Evaluate the following probabilities.
   P(X<0)