If \(\rho(A) \neq \rho(A, B)\), then the system is _______.

(a)

Consistent and has infinitely many solutions

(b)

Consistent and has a unique solution

(c)

inconsistent

(d)

consistent

If A is a singular matrix, then Adj A is ___________

(a)

non-singular

(b)

singular

(c)

symmetric

(d)

not defined

ഽ\(\frac { sin2x }{ 2sinx } dx\) is _______.

(a)

sin x + c

(b)

\(\frac12\)sin x + c

(c)

cos x + c

(d)

\(\frac12\)cos x + c

The anti-derivative of f(x) = \(\sqrt { x } +\frac { 1 }{ \sqrt { x } } \) is ___________ +c

(a)

\(\frac { { 2 } }{ 3 } { x }^{ \frac { 3 }{ 2 } }+\frac { 2 }{ { x }^{ \frac { 1 }{ 2 } } } \)

(b)

\(\frac { { 3 } }{ 2 } { x }^{ \frac { 3 }{ 2 } }+2{ x }^{ \frac { 1 }{ 2 } }\)

(c)

\(\frac { { 2 } }{ 3 } { x }^{ \frac { 3 }{ 2 } }+2{ x }^{ \frac { 1 }{ 2 } }\)

(d)

none

Area bounded by the curve y = e^−2x between the limits 0 ≤ x ≤ ∞ is ________.

(a)

1 sq.units

(b)

\(\frac{1}{2}\) sq.unit

(c)

5 sq.units

(d)

2 sq.units

The area unded by the curves y = 2^x, x = 0 and x = 2 is________sq.units.

(a)

log_e2

(b)

3log_e2

(c)

\(\frac{3}{log_e2}\)

(d)

2log_e3

Solution of \(\frac { dy }{ dx } \) + Px = 0 ______.

(a)

x = ce^py

(b)

x = ce^−py

(c)

x = py + c

(d)

x = cy

The differential equation of all circles with centre at the origin is _____________

(a)

xdy +ydx = 0

(b)

xdy - ydx = 0

(c)

xdx + ydy = 0

(d)

xdx - ydy = 0

If ‘n’ is a positive integer Δⁿ[ Δ^-n​ ​​​​​​f(x)] _______.

(a)

f(2x)

(b)

f(x+ h)

(c)

f (x)

(d)

Δf(x)

Newton's forward interpolation formula is used when the value of y is required near the ______ of the table

(a)

end

(b)

beginning

(c)

left

(d)

right

E[X-E(X)]² is ________.

(a)

E(X)

(b)

E(X²)

(c)

V(X)

(d)

S.D(X)

Which of the following are correct?
(i) E(aX+b) = a E(X) + b
(ii) μ₂= μ₂¹ - (μ₁¹)²
(iii) μ²= variance
(iv) V (a X + b) = a² V(x)

(a)

all

(b)

i, ii and iii

(c)

ii and iii

(d)

i and iv

If P(Z > z) = 0.8508 what is the value of z (z has a standard normal distribution)?

(a)

–0.48

(b)

0.48

(c)

-1.04

(d)

1.04

The area under the standard normal curve between Z=-∞ and z=∞ is

(a)

0

(b)

0.5

(c)

1

(d)

0.75

A _______is one where each item in the universe has an equal chance of known opportunity of being selected.

(a)

Parameter

(b)

random sample

(c)

statistic

(d)

entire data

A sample of 100 students are drawn from 1550 student of a school. The mean weight and variance of the sample are 67.45 kg and 9 kg. Then the standard error is _________

(a)

.3

(b)

.9

(c)

.6745

(d)

6.745

Another name of consumer’s price index number is: ________.

(a)

Whole-sale price index number

(b)

Cost of living index

(c)

Sensitive

(d)

Composite

Choose the odd one out

(a)

Price index number

(b)

Quantity index number

(c)

cost of living index number

(d)

Ideal index number 

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

If the number of rows is____tothenumber of columns, then the assignment problem is said to be balanced.

(a)

equal

(b)

less

(c)

more

(d)

not equal

Solve: x + 2y = 3 and 2x + 4y = 6 using rank method.

Evaluate \(\int \sqrt{2 x+1} \ d x\)

Find the area under the curve y = 4x - x² included between x = 0, x = 3 and the X-axis.

Find the order and degree of the following differential equations.
(2 - y'')² = y''² + 2y'

When h = 1, find Δ (x³).

Find the expected value for the random variable of an unbiased die

In a Poisson distribution 3 P(X = 2) = P(X = 4), then find the parameter of the distribution.

Mention two branches of statistical inference?

Calculate the seasonal indices by the method of simple average for the following data.

Year	I quarter	II quarter	III quarter	IV quarter
1985	68	62	61	63
1986	65	58	66	61
1987	68	63	63	67

What is the difference between Assignment Problem and Transportation Problem?

Show that the equations 3x − 2y = 6, 6x − 4y = 10 are inconsistent

Evaluate \(\int _{ 1 }^{ 2 }{ \frac { log\quad x }{ { x }^{ 2 } } } dx\)

The price elasticity of demand for a commodity is \(\frac { p }{ { x }^{ 3 } } \). Find the demand function if the quantity of demand is 3, when the price is Rs. 2

Solve: (x²-yx²)dy + (y²+xy²)dx = 0

If h = 1 then prove that (E⁻¹Δ)x³ = 3x² − 3x + 1.

A random variable X can take all nonnegative integral values and the probabilities that X takes the value r is proportional to aT (0 < ∝ < 1). Find P(X = 0)

Weights of fish caught by a traveler are approximately normally distributed with a mean weight of 2.25 kg and a standard deviation of 0.25 kg. What percentage of fish weigh less than 2 kg?

A random sample of 500 apples was taken from large consignment and 45 of them were found to be bad. Find the limits at which the bad apples lie at 99% confidence level.

Write a brief note on seasonal variations

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. Solve the following equation by using Cramer’s rule
   2x + y −z = 3, x + y + z =1, x− 2y− 3z = 4

2. If \(\int _{ a }^{ b }{ dx } =1\) and \(\int _{ a }^{ b }{ xdx } =1\), then find a and b

1. Evaluate ഽ x³ sin (x⁴) dx

2. The marginal cost of production of a firm is given by C'(x) = 5 + 0.13x, the marginal revenue is given by R'(x) = 18 and the fixed cost is Rs. 120. Find the profit function.

1. Solve: (y-x)\(\frac { dy }{ dx } \) = a²

2. From the following table obtain a polynomial of degree y in x

   x	1	2	3	4	5
   y	1	-1	1	-1	1

1. A continuous random variable X has p.d.f
   f(x) = 5x⁴, 0\(\le\)x\(\le\)1 
   Find a₁ and a₂ such that
   i) P[X\(\le\)a₁] = P[X>a₁]   
   ii) P[X>a₂] = 0.05

2. A car hiring firm has two cars. The demand for cars on each day is distributed as a Poisson variate, with mean 1.5. Calculate the proportion of days on which
   (i) Neither car is used
   (ii) Some demand is refused

1. If the height of 300 students are normally distributed with mean 64.5 inches and standard deviation 3.3 inches find the height below which 99% of the student lie?

2. Explain in detail about the test of significance for single mean.

1. A car hire company has one car at each of five depots a,b,c,d and e. A customer in each of the fine towers A,B,C,D and E requires a car. The distance (in miles) between the depots (origins) and the towers(destinations) where the customers are given in the following distance matrix.
   
   How should the cars be assigned to the customers so as to minimize the distance travelled?

2. Solve the following assignment problem.
   

1. A sample poll of 100 voters chosen at random from all voters in a given district indicated that 55% of them were in favour of a particular candidate. Find
   (a) 95% confidence limits
   (b) 99% confidence limits for the proportion to all voters in favour of this candidate.

2. A quality control inspector has taken ten samples of size four packets each from a potato chips company. The contents of the sample are given below, Calculate the control limits for mean and range chart.

   Sample Number	Observations
   	1	2	3	4
   1	12.5	12.3	12.6	12.7
   2	12.8	12.4	12.4	12.8
   3	12.1	12.6	12.5	12.4
   4	12.2	12.6	12.5	12.3
   5	12.4	12.5	12.5	12.5
   6	12.3	12.4	12.6	12.6
   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

   (Given for n = 5,  A₂ = 0.58, D₃  = 0 and D₄ = 2.115)