if T = \(_{ B }^{ A }\left( \begin{matrix} \overset { A }{ 0.4 } & \overset { B }{ 0.6 } \\ 0.2 & 0.8 \end{matrix} \right) \) is a transition probability matrix, then at equilibrium A is equal to ________.

(a)

\(\frac { 1 }{ 4 } \)

(b)

\(\frac { 1 }{ 5 } \)

(c)

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

(d)

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

If the number of variables in a non-homogeneous system AX = B is n, then the system possesses a unique solution only when _______.

(a)

\(\rho (A)=\rho (A,B)>n\)

(b)

\(\rho(A)=\rho(A, B)=n\)

(c)

\(\rho (A)=\rho (A,B) < n\)

(d)

none of these

ഽ\(\frac { { e }^{ x } }{ { e }^{ x }+1 } \) dx is _______.

(a)

log\(\left| \frac { { e }^{ x } }{ { e }^{ x }+1 } \right| +c\)

(b)

log\(\left| \frac { { e }^{ x }+1 }{ { e }^{ x } } \right| +c\)

(c)

log\(\left| { e }^{ x } \right| +c\)

(d)

log\(\left| { e }^{ x }+1 \right| +c\)

If MR and MC denotes the marginal revenue and marginal cost functions, then the profit functions is ________.

(a)

P = ഽ(MR − MC) dx + k

(b)

P = ഽ(MR + MC) dx + k

(c)

P = ഽ(MR)(MC)dx + k

(d)

P = ഽ(R −C)dx + k

The integrating factor of the differential equation \(\frac{dx}{dy}+Px=Q\) is ______.

(a)

e^ഽPdx

(b)

^{\(\int P d x\)}

(c)

ഽPdy

(d)

e^ഽPdy

Which of the following is the homogeneous differential equation?

(a)

(3x−5)dx = (4y−1)dy

(b)

xy dx−(x³+y³)dy = 0

(c)

y²dx+(x² − xy  − y²)dy = 0

(d)

(x²+y)dx = (y²+x)dy

A formula or equation used to represent the probability distribution of a continuous random variable is called ________.

(a)

probability distribution

(b)

distribution function

(c)

probability density function

(d)

mathematical expectation

If Z is a standard normal variate, the proportion of items lying between Z = –0.5 and Z = –3.0 is ________.

(a)

0.4987

(b)

0.1915

(c)

0.3072

(d)

0.3098

An estimate of a population parameter given by two numbers between which the parameter would be expected to lie is called an………..interval estimate of the parameter.

(a)

point estimate

(b)

interval estimation

(c)

standard error

(d)

confidence

A typical control charts consists of ________.

(a)

CL, UCL

(b)

CL, LCL

(c)

CL, LCL, UCL

(d)

UCL, LCL

If number of sources is not equal to number of destinations, the assignment problem is called______.

(a)

balanced

(b)

unsymmetric

(c)

symmetric

(d)

unbalanced

\(\int { { 3 }^{ x+2 } } \) dx = ______________ +c

(a)

\(\frac { { 3 }^{ x } }{ log3 } \)

(b)

\(\frac { 9\left( { 3 }^{ x } \right) }{ log3 } \)

(c)

\(\frac { 3.{ 3 }^{ x } }{ log3 } \)

(d)

\(\frac { { 3 }^{ x } }{ 9log3 } \)

Choose the odd one out

(a)

\(\int_{0}^{\infty} e^{-t} d t\)

(b)

\(\int_{0}^{\infty} e^{-t} d t\)

(c)

\(\int_{0}^{\infty} x^{n-1} e^{-x} d x\)

(d)

\( \int_{0}^{\infty} e^{-4 x} x^{4} d x\)

The area bounded by the curve y = e^x, the X-axis and the lines x = 0 and x = 2 is__________.

(a)

e²-1

(b)

e²+1

(c)

e²

(d)

e²-2

The solution of the equation of the type \(\frac{d x}{d y}+P x=\mathbf{Q}\) (P and Q are function of y) is  _______________

(a)

\(\mathbf{y}=\int Q e^{\int p d x} d y+c\)

(b)

\(y e^{\int p d x}=\int Q e^{\int P d x} d x+c\)

(c)

\(x e^{\int_{P d} y}=\int Q e^{\int P d y} d y+c\)

(d)

\(x e^{\int p d y}=\int Q e^{\int P d x} d x+c\)

The degree of the differential equation \(\left(\frac{d^{2} y}{d x^{3}}\right)^{3}+\left(\frac{d y}{d x}\right)^{2}+\sin \left(\frac{d y}{d x}\right)+1=0 \text { is }\)________________

(a)

3

(b)

2

(c)

1

(d)

not defined

 The nationality of the mathematician Joseph Louis Laguange is _________

(a)

German

(b)

Spain

(c)

Italian

(d)

French

If f(x) = kx(1-x), 0< x < 1 is a p.d.f. then the value of k is ________.

(a)

\(\frac{1}{5}\)

(b)

\(\frac{2}{5}\)

(c)

\(\frac{3}{5}\)

(d)

6

If a random variable X has the following probability distribution
\(\begin{array}{|c|c|c|c|c|} \hline x & -1 & -2 & 1 & 2 \\ \hline p(x) & 1 / 3 & 1 / 6 & 1 / 6 & 1 / 3 \\ \hline \end{array}\)Then the expected value of X-is _________

(a)

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

(b)

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

(c)

\(\frac{1}{2}\)

(d)

\(\frac{1}{3}\)

The point estimate variance of 6.33, 6.37, 6.36, 6.32, 6.37 is

(a)

0.0022

(b)

0.00055

(c)

0.0055

(d)

0.055

Find the rank of the matrix A =\(\left( \begin{matrix} 1 & -3 \\ 9 & 1 \end{matrix}\begin{matrix} 4 & 7 \\ 2 & 0 \end{matrix} \right) \)

Integrate the following with respect to x.
\(\frac { { e }^{ 2x } }{ { e }^{ 2x }-2 } \)

Using Integration, find the area of the region bounded the line 2y + x = 8, the x axis and the lines x = 2, x = 4.

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

Evaluate Δ (log ax).

Prove that, V(X+b) = V(X)

Define critical value.

Construct a forward difference table for \(\mathrm{y}=\mathrm{f}(x)=x^{2}+2 x+2 \text { for } x=1,2,3,4\)

Find the probability distribution of X, the number of heads in 2 tosses of a coin

The following data shows the value of sample mean (\(\bar{X}\)) and the range R for 10 samples of size 5 each. Calculate the control limits for : mean chart and range chart.

Sample No.	1	2	3	4	5	6	7	8	9	10
Mean \(\bar{X}\)	11.2	11.8	10.8	11.6	11.0	9.6	10.4	9.6	10.6	10.0
Range	7	4	8	5	7	4	8	4	7	9

(Given for n = 5, A₂ = .577, D₃ = 0, D₄ = 2.115)

Find the rank of the matrix \(\left( \begin{matrix} 0 & -1 & 5 \\ 2 & 4 & -6 \\ 1 & 1 & 5 \end{matrix} \right) \)

Akash bats according to the following traits. If he makes a hit (S), there is a 25% chance that he will make a hit his next time at bat. If he fails to hit (F), there is a 35% chance that he will make a hit his next time at bat. Find the transition probability matrix for the data and determine Akash’s long- range batting average.

Integrate the following with respect to x.
x log x

Evaluate the following integrals:
ഽ\(\frac { dx }{ { { 2-3x-2x }^{ 2 } } } \)

The marginal cost function MC = 2 + 5e^x Find AC.

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

Two unbiased dice are thrown simultaneously and sum of the upturned faces considered as random variable. Construct a probability mass function.

What is the probability of guessing correctly atleast six of the ten answers in a TRUE/FALSE objective test?

Solve: \(\left(x^{2}-y x^{2}\right) d y+\left(y^{2}+x^{2} y^{2}\right) d x=0\)

A random variable X has the folowing probability distribution 

x	0	1	2	3	4	5	6	7	8
p(x)	a	3a	5a	7a	9a	11a	13a	15a	17a

(i) Find a
\(\text {(ii) } \mathrm{P}(x<3), \mathrm{P}(x>3) \text { and } \mathrm{P}(0<x<5)\)

1. The total number of units produced (P) is a linear function of amount of over times in labour (in hours) (l), amount of additional machine time (m) and fixed finishing time (a)
   i.e, P = a + bl + cm
   From the data given below, find the values of constants a, b and c

   Day	Production (in Units P)	Labour (in Hrs l)	Additional Machine Time (in Hrs m)
   Monday Tuesday Wednesday	6,950 6,725 7,100	40 35 40	10 9 12

   Estimate the production when overtime in labour is 50 hrs and additional machine time is 15 hrs.

2. Integrate the following with respect to x.
   \(e^{x}\left[\frac{x-1}{(x+1)^{3}}\right]\)

1. The marginal cost of production of a firm is given by C'(x) = 20 + \(\frac { x }{ 20 } \) the marginal revenue is given by R'(x) = 30 and the fixed cost is Rs. 100. Find the profit function

2. Solve x²ydx-(x³+y³)dy = 0

1. Find the missing entries from the following

   x	0	1	2	3	4	5
   y = f(x)	0	-	8	15	-	35

2. The marks obtained in a certain exam follow normal distribution with mean 45 and SD 10. If 1,300 students appeared at the examination, calculate the number of students scoring
   (i) less than 35 marks and
   (ii) more than 65 marks.

1. Calculate the seasonal index for the quarterly production of a product using the method of simple averages.

   Year	I Quarter	II Quarter	III Quarter	IV Quarter
   2005	255	351	425	400
   2006	269	310	396	410
   2007	291	332	358	395
   2008	198	289	310	357
   2009	200	290	331	359
   2010	250	300	350	400

2. Obtain an initial basic feasible solution to the following transportation problem using Vogel’s approximation method.
   

1. A new transit system has just gone into operation in a city. Of those who use the transit system this year, 10% will switch over to using their own car next year and 90% will continue to use the transit system. Of those who use their cars this year, 80% will continue to use their cars next year and 20% will switch over to the transit system. Suppose the population of the city remains constant and that 50% of the commuters use the transit system and 50% of the commuters use their own car this year,
   (i) What percent of commuters will be using the transit system after one year?
   (ii) What percent of commuters will be using the transit system in the long run?

2. The total cost of production y and the level of output x are related to the marginal cost of production by the equation (6x² + 2y²) dx - (x²+ 4xy) dy = 0. What is the relation between total cost and output if y = 2, when x = 1

1. A continuous random variable has the following p.d.f, \(f(x)= \begin{cases}k x^{2}, & 0 \leq x \leq 10 \\ 0, & \text { otherwise }\end{cases}\) find k and evaluate
   \((i)\ \mathrm{P}(0.2 \leq x \leq 0.5) \)
   (ii) \( \mathrm{P}(x \leq 3)\)

2. A sample of 400 students is found to have a mean height of 171.38 cms. Can it reasonably be regarded as a sample from a large population with mean height of 171.17 cms and standard deviation of 3.3 cms (Test at 5% level)

1. The followingdata relateto the life(inhours) of 10 samples of 6 electricbulbs each drawn at an intervalof one hour from a production process.Draw the controlchart for \(\overline { X } \) and \(\overline { R } \) and comment.

   Sample No	Lifetime (inhour)
   	1	2	3	4	5	6
   1	620	687	666	689	738	686
   2	501	585	524	585	653	668
   3	673	701	686	567	619	660
   4	646	626	572	628	631	743
   5	494	984	659	643	660	640
   6	634	755	625	582	683	555
   7	619	710	664	693	770	534
   8	630	723	614	535	550	570
   9	482	791	533	612	497	499
   10	706	524	626	503	661	754

   (For n = 6,A₂= 0.483,D₃ = 0,D₄ = 2.004)

2. Solve the transportation problem using
   (i) North west corner Rule
   (ii) Least cost method
   (iii) Vogel Approximation method