\(\left| { A }_{ n\times n } \right| \) = 3 \(\left| adjA \right| \) = 243 then the value n is _______.

For what value of k, the matrix \(A=\left( \begin{matrix} 2 & k \\ 3 & 5 \end{matrix} \right) \) has no inverse?

If \(\int _{ 0 }^{ 1 }{ f(x) } dx=1,\int _{ 0 }^{ 1 }{ xf(x) } dx=a\) and \(\int _{ 0 }^{ 1 }{ { x }^{ 2 }f(x) } dx={ a }^{ 2 }\), then \(\int _{ 0 }^{ 1 }{ { (a-x) }^{ 2 } } f(x)\) dx is _______.

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

If \(\int { \frac { 1 }{ \left( x+2 \right) \left( { x }^{ 2 }+1 \right) } } \) dx = a log \(\left| 1+{ x }^{ 2 } \right| \) +b tan^-1 x + \(\frac { 1 }{ 5 } log\left| x+2 \right| \) +c then ___________

The area lying above the X-axis and under the parabola y = 4x - x² is ______ sq. units

The solution of \(\frac { dy }{ dx } \) = e^x-y is _____________

Δ²y₀ = _______.

Lagrange’s interpolation formula can be used for _______.

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

If X is a discrete random variable and p(x) is the probability of X, then the expected value of this random variable is equal to ________.

If the p.d.f of a continuous random variable. X is \(f(x)=\left\{\begin{array}{l} \frac{x}{2}, 0 then E(3X²-2X) = _________

If F(x) is the probability distribution function, then F(- ∞) is_______.

If X ~ N(μ, σ²), the maximum probability at the point of inflexion of normal distribution is ________.

The point estimate mean of the following data is __________.
21.1, 25.0, 20.0, 16.0, 12.0, 10.0, 17.0, 18.0, 13.0,11.0

Any hypothesis which is complementary to the null hypothesis is _______ hypothesis.

Least square method of fitting a trend is ________.

Cyclic variations in a time series are caused by __________

Variation due to assignable causes in the product occur due to, _____

The methods of funding feasible solution to a transportation problem ___________

Solve: 2x + 3y = 4 and 4x + 6y = 8 using Cramer's rule.

Calculate the producer’s surplus at x = 5 for the supply function p = 7 + x.

Find the area of the region bounded by the parabola x² = 4y, y = 2, y = 4 and the y-axis.

Solve: x dy +y dx = 0

When h = 1, find Δ (x³).

Find the mean for the probability density function \(f(x)=\begin{cases} \frac { 1 }{ 24 } ,-12\le x\le 12 \\ 0,\quad otherwise \end{cases}\)

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

A sample of 400 students is found to have mean height of 171.38 cms, Can it reasonable 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)

Calculate the 3-yearlymoving averages of the production figures (in tonnes) for the following data.

For the given pay-off matrix, find the optimal decision under the minimax principle.

Solve: 2x - 3y - 1 = 0, 5x + 2y - 12 = 0 by Cramer's rule.

Find the area under the demand curve xy = 1 bounded by the ordinates x = 3, x = 9 and x-axis

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

If y₇₅ = 2459, y₅₀ = 2018, y₈₅ = 1180, and y₉₀ =402, find y₈₂

If you toss a fair coin three times, the outcome of an experiment consider as random variable which counts the number of heads on the upturned faces. Find out the probability mass function and check the properties of the probability mass function.

An urn contains 4 white and 6 red balls. Four balls are drawn at random from the urn. Find the probability distribution of the number of white balls.

The standard deviation of a binomial distribution (q +p)¹⁶ is 2. Find its mean.

A sample of 100 students is chosen from a large group of students. The average height of these students is 162 cm and standard deviation (S.D) is 8 cm. Obtain the standard error for the average height of large group of students of 160 cm?

Fit a trend line to the following data by graphic  method.

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

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?

Evaluate \(\int _{ 0 }^{ \infty }{ { e }^{ -\frac { x }{ 2 } } } dx\)

Evaluate ഽ x³ sin (x⁴) dx

Solve (x² + 1)\(\frac { dy }{ dx } \) + 2xy = 4x² 

The net profit p and quantity x satisfy the differential equation \(\frac { dp }{ dx } =\frac { 2{ p }^{ 3 }-{ x }^{ 3 } }{ 3x{ p }^{ 2 } } \). Find the relationship between the net profit and demand given that p = 20, when x = 10.

From the following data, calculate the value of e^1.75

The following information is the probability distribution of successes.

Determine the expected number of success.

The probability distribution of a random variation X is given below.

Find
(i) V(X)
ii) V\((\frac{X}{2})\)

Assuming one in 80 births is a case of twins, calculate the probability of 2 or more sets of twins on a day when 30 births occur.

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?

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?

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.

Calculate Fisher’s price index number and show that it satisfies both Time Reversal Test and Factor Reversal Test for data given below.

Solve the following assignment problem.