Find k, if the equations x + y + z = 7,  x + 2y + 3z = 18,  y + kz = 6 are inconsistent

Evaluate \(\int\left[\frac{1}{\log x}-\frac{1}{(\log x)^{2}}\right] d x\)

Integrate \(\int{\sqrt{1-sin2x}dx }\)

A firm’s marginal revenue function is MR = 20e^-x/10 \(\left( 1-\frac { x }{ 10 } \right) \). Find the corresponding demand function.

A company receives a shipment of 200 cars every 30 days. From experience it is known that the inventory on hand is related to the number of days. Since the last shipment, I x( )= − 200 0 2. x . Find the daily holding cost for maintaining inventory for 30 days if the daily holding cost is ₹3.5

If A=\(\left( \begin{matrix} 1 & 1 & -1 \\ 2 & -3 & 4 \\ 3 & -2 & 3 \end{matrix} \right) \) and B=\(\left( \begin{matrix} 1 & -2 & 3 \\ -2 & 4 & -6 \\ 5 & 1 & -1 \end{matrix} \right) \), then find the rank of AB and the rank of BA.

At marina two types of games viz., Horse riding and Quad Bikes riding are available on hourly rent. Keren and Benita spent Rs. 780 and Rs. 560 during the month of May.

Find the hourly charges for the two games (rides). (Use determinant method).

Evaluate \(\int { \frac { 7x-1 }{ { x }^{ 2 }-5x+6 } dx } \)

Evaluate \(\int { \left( { x }^{ 2 }-2x+5 \right) } { e }^{ -x }dx\)

The cost of over haul of an engine is Rs. 10,000 The operating cost per hour is at the rate of 2x − 240 where the engine has run x km. Find out the total cost if the engine run for 300 hours after overhaul.

Find the rank of each of the following matrices.
\(\left( \begin{matrix} 5 & 6 \\ 7 & 8 \end{matrix} \right) \)

Find the rank of each of the following matrices.
\(\left( \begin{matrix} 1 & -1 \\ 3 & -6 \end{matrix} \right) \)

Evaluate \(\int { \left( { x }^{ 3 }+7 \right) \left( x-4 \right) dx } \)

Integrate the following with respect to x.
\(\sqrt { 1-\sin2x } \)

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

The rank of m x n matrix whose elements are unity is ________.

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 ________.

The rank of the matrix  \(\left( \begin{matrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 4 & 9 \end{matrix} \right) \) is ________.

If \(\rho (A)\) = r  then which of the following is correct?

ഽ2^xdx is _______.

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 ________.

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

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

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

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

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

Consider the problem of assigning five jobs to five persons. The assignment costs are given as follows. Determine the optimum assignment.

Solve the following assignment problem.

Solve the transportation problem for which the cost origin, availabilities and destination requirements are given below using
i) North west corner and
(i) Least cost method

Solve the following TPP through vogel's approximation method 

Find the initial basic feasible solution for the following transportation problem by VAM

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

Solve the following assignment problem. Cell values represent cost of assigning job A, B, C and D to the machines I, II, III and IV.

Consider the problem of assigning five jobs to five persons. The assignment costs are given as follows. Determine the optimum assignment schedule.

Obtain an initial basic feasible solution to the following transportation problem by north west corner method.

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

Determine an initial basic feasible solution to the following transportation problem using North West corner rule.

Determine how much quantity should be stepped from factory to various destinations for the following transportation problem using the least cost method.

Find the initial basic feasible solution for the following transportation problem by Vogel's approximation method.

Solve the following assignment problem. Cell values represent cost of assigning job A, B, C and D to the operators I, II, III and IV.

Obtain the initial solution for the following problem

Determine an initial basic feasible solution to the following transportation problem using North West corner rule.

Here O_i and D_j represent i^th origin and j^th destination.

Obtain an initial basic feasible solution to the following transportation problem using least cost method.

Here O_i and D_j denote i^th origin and j^th destination respectively.

Determine how much quantity should be stepped from factory to various destinations for the following transportation problem using the least cost method

Cost are expressed in terms of rupees per unit shipped.

Solve the following assignment problem.

Obtain the initial solution for the following problem using north-west corner rule.

Determine an initial basic feasible solution to the following transportation problem using feast cost method.

Consider the following pay-off (profit) matrix action, states

Determine the best action using maximin principle.

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

The following is the pay-off matrix (in rupees) for three strategies and three states of nature. Select a strategy using maximin principle.

What is transportation problem?

Write mathematical form of transportation problem.

What is feasible solution and non degenerate solution in transportation problem?

What do you mean by balanced transportation problem?

What is the Assignment problem?

A set of non-negative values that satisfies the constants in a transportation problem is a ___________

The total transportation cost is ___________

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

Vogel's approximation method yields an initial basic feasible solution which is very close to the solution.

To assign different jobs to the different machines to minimize the overall cost is ___________

The transportation problem is said to be unbalanced if _______.

In a non – degenerate solution number of allocations is _______.

In a degenerate solution number of allocations is _______.

The Penalty in VAM represents difference between the first ________.

Number of basic allocation in any row or column in an assignment problem can be _______.

Fit a straight line trend to the following data using the method of least square. Estimate the trend for 2007.

From the data given below, calculate seasonal indices.

Compute
(i) Laspeyre's
(ii) Paasche's 
(iii) Fisher's price index number for 2000 from the following data.

Calculate Fisher's ideal index from the following data and verify that it satisfies both time reversal and factor reversal test

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.

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

Given below are the data relating to the production of sugarcane in a district.
Fit a straight line trend by the method of least squares and tabulate the trend values.

Given below are the data relating to the sales of a product in a district.
Fit a straight line trend by the method of least squares and tabulate the trend values.

Calculate the seasonal index for the monthly sales of a product using the method of simple averages.

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

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

Fit a straight line trend for the following data using the method of least squares.

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

Find a trend line to the following data by the method of sami-averages.

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

Compute Fisher's price index number for the following data.

Fit a trend line by the method of semi-averages for the given data.

Calculate three-yearly moving averages of number of students studying in a higher secondary school in a particular village from the following data.

Calculate four-yearly moving averages of number of students studying in a higher secondary school in a particular city from the following data.

A machine drills hole in a pipe with a mean diameter of 0.532 cm and a standard deviation of 0.002 cm. Calculate the control limits for mean of samples 5.

The following data gives the readings for 8 samples of size 6 each in the production of a certain product. Find the control limits using mean chart.

Given for n = 6, A₂ = 0.483,

Using the method ofleast squares, fit a straight line trend for Σx = 10, Σy = 16.9, Σx² = 30, Σxy = 47.4 and n = 7.

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

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

Calculate the cost of living index by aggregate expenditure method

Construct the cost of living index for 2003 on the basis of 2000 from the following data using family budget method.

Fit a trend line by the method of freehand method for the given data

Fit a trend line by the method of semi-averages for the given data.

Define Time series.

What is the need for studying time series?

State the uses of time series.

The component of a time series which is attached to short term fluctuations is __________

Cyclic variations in a time series are caused by __________

The terms prosperity, recession, depression  and recovery are in particular attached to __________

A decline in the sales of ice cream during November to March is associated with __________

Index number is a __________

A time series is a set of data recorded ________.

A time series consists of ________.

The components of a time series which is attached to short term fluctuation is ________.

Factors responsible for seasonal variations are ________.

The additive model of the time series with the components T, S, C and I is ________.

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.

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.

The mean breaking strength of cables supplied by a manufactures is 1900 \(\frac{n}{m^{2}}\) with a standard deviation of \(\frac{n}{m^{2}}\). The manufacture introduced a new technique in the manufacturing process and claimed that the breaking strength of cables has increased. In order to test the claim, a sample of 60 cables is tested. It is found that the mean breaking strength of the samplcd cables is \(1960 \frac{n}{m^{2}}\). Can we support the claimn at 1% level of significance.

A motor vehicle company desires to introduce a new model vehicle. The company claims that the mean fuel consumption of its new model is lower than that of the existing one which is 27 kms/litre. A sample of 100 vehicles of the new model is sclected and their fuel consumption are observed as 30 kms/litre with a standard deviation off 3 kms/litre. Test the claim of the company at 5% level of significance

A company producing LED bulbs finds that the mean life spar of the population of the bulbs is 2000 hours with a standard deviation of 150 hours. A sample of 100 bulbs is found to have the mean life spar of 1950 hours. Test at 5% level of significance wheather the mean life spar of the bulbs is significantly different from 2000 hours.

Using the following random number table,

Draw a sample of 10 children with their height from the population of 8,585 children as classified here under.

A machine produces a component of a product with a standard deviation of 1.6 cm in length. A random sample of 64 componentsvwas selected from the output and this sample has a mean length of 90 cm. The customer will reject the part if it is either less than 88 cm or more than 92 cm. Does the 95% confidence interval for the true mean length of all the components produced ensure acceptance by the customer?

A sample of 100 measurements at breaking strength of cotton thread gave a mean of 7.4 and a standard deviation of 1.2 gms. Find 95% confidence limits for the mean breaking strength of cotton thread.

The mean life time of a sample of 169 light bulbs manufactured by a company is found to be 1350 hours with a standard deviation of 100 hours. Establish 90% confidence limits within which the mean life time of light bulbs is expected to lie.

An auto company decided to introduce a new six cylinder car whose mean petrol consumption is claimed to be lower than that of the existing auto engine. It was found that the mean petrol consumption for the 50 cars was 10 km per litre with a standard deviation of 3.5 km per litre. Test at 5% level of significance, whether the claim of the new car petrol consumption is 9.5 km per litre on the average is acceptable.

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.

A sample of five measurements of the diameter of a sphere were recorded by a scientist as 6.33, 6.37,6.36,6.32 and 6.37 mm. Determine the point estimate of
(a) mean
(b) variance.

A random sample of marks in mathematics secured by 50 students out of 200 students showed a mean of 75 and a standard deviation of 10. Find the 95% confidence limits for the estimate of their mean marks.

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).

The mean life time of 50 electric bulbs produced by a manufacturing company is estimated to be 825 hours with the S.D. of 110 hours. If II is the mean life time of all the bulbs produced by the company, test the hypothesis that μ = 900 hours at 5% level of significance.

Using the following random number table (Kendall-Babington Smith)

Draw a random sample of 10 four- figure numbers starting from 1550 to 8000.

From the following data, select 68 random samples from the population of heterogeneous group with size of 500 through stratified random sampling, considering the following categories as strata.
Category 1: Lower income class - 39%
Category 2: Middle income class - 38%
Category 3: Upper income class - 23%

Find the sample size for the given standard deviation 10 and the standard error with respect of sample mean is 3.

A die is thrown 9000 times and a throw of 3 or 4 is observed 3240 times. Find the standard error of the proportion for an unbiased die. .

The standard deviation of a sample of size 50 is 6.3. Determine the standard error whose population standard deviation is 6?

A random sample of size 50 with mean 67.9 is drawn from a normal population. If it is known that the standard error of the sample \(\sqrt { 0.7 } \) , find 95% confidence interval for the population mean.

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

Out of 1500 school students, a sample of 150 selected to test the accuracy of solving a problem in B.M. and of them 10 did a mistake. Calculate the standard error of sample proportion.

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)

The income distribution of the population of a village has a mean of Rs. 6000 and a variance of Rs. 32,400. Could a sample of 64 persons with a mean income of Rs. 5950 belong to this population. (Test at 1% level of significance).

Using the Kendall-Babington Smith - Random number table, Draw 5 random samples.

Using the following Tippett’s random number table,

Draw a sample of 15 houses from Cauvery Street which has 83 houses in total.

A server channel monitored for an hour was found to have an estimated mean of 20 transactions transmitted per minute. The variance is known to be 4. Find the standard error.

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?

What is population?

The central limit theorem states that the sampling distribution of the mean will approach normal distribution __________

Probability of rejecting null hypothesis. when it is true is _______

The number of ways in which one can select 2 customers out of 10 customers is __________

The standard error of the sample mean is __________

Which of the following statements is true?

A ________ may be finite or infinite according as the number of observations or items in it is finite or infinite.

A __________ of statistical individuals in a population is called a sample.

A finite subset of statistical individuals in a population is called ________.

Any statistical measure computed from sample data is known as _________.

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

Four coins are tossed simultaneously. What is the probability of getting
a) atleast 2 heads
b) atmost 2 heads.

20% of the bolts produced in a factory are found to be defective. Find the probability that in a sample of 10 bolts chosen at random exactly 2 will be defective using
(i) Binomial distribution
(ii) Poisson distribution (e^-2 = 0.1353)

The mean weight of 500 male students in a certain college is 151 pounds and the S.D is 15 pounds. Assuming the weights are normally distributed, find how many students weight
(i) between 120 and 155 pounds
(ii) more than 185 pounds.

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?

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?

If the average rain falls on 9 days in every thirty days, find the probability that rain will fall on atleast two days of a given week.

The sum and product of the mean and variance of a binomial distribution are 24 and 128. Find the distribution.

An insurance company has discovered that only about 0.1 per cent of the population is involved in a certain type of accident each year. If its 10,000 policy holders were randomly selected from the population, what is the probability that not more than 5 of its clients are involved in such an accident next year? (e⁻¹⁰=.000045)

One fifth percent of the the blades produced by a blade manufacturing factory turn out to be defective. The blades are supplied in packets of 10. Use Poisson distribution to calculate the approximate number of packets containing no defective, one defective and two defective blades respectively in a consignment of 1,00,000 packets (e^–0.2 =.9802)

If the probability that an individual suffers a bad reaction from injection of a given serum is 0.001, determines the probability that out of 2,000 individuals
(a) exactly 3, and
(b) more than 2 individuals will suffer a bad reaction.

The probability that an event A happens in one treat of an experiment is 0.4. Three independent treats of the experiment are performed. Find the p!probability that the event A happens at least once.

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

If a random variable X follows Poisson distribution such that P(X = 2) = 9. P(X = 4) + 90 P(X = 6) then find the mean and variance.

Find the value of K if X is a normal variate whose p.d.f is given by f(x)  = \(\frac { 1 }{ K } \)e^8x-4x2, -∞

Obtain K, μ and σ2 of of the normal distribution whose probability distribution function is f(x) = \(K{ e }^{ -2x^{ 2 }+4x-2 }\), -∞

A and B play a game in which their chance of winning are in the ratio 3 : 2 Find A’s chance of winning atleast three games out of five games played.

The probability that a student get the degree is 0.4 Determine the probability that out of 5 students
(i) one will be graduate
(ii) atleast one will be graduate

The mean of Binomials distribution is 20 and standard deviation is 4. Find the parameters of the distribution.

If x is a binomially distributed random variable with E(x) = 2 and van (x) = 4/3 Find P(x = 5)

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

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

If the mean of the binomial distribution with 9 trial is 6, then find the variance.

If the mean of the binomial distribution is 20 and standard deviation is 4, then find the number of events.

Suppose X is a binomial variate X ~ B (5, p) and P(X = 2) = P(X = 3), then find p.

If 10 coins are tossed, find the probability that exactly 5 heads appears.

A fair coin is tossed 6 times. Find the probability that exactly 2 heads occurs.

Verfy the following statement:
The mean of a Binomial distribution is 12 and its standard deviation is 4.

In tossing of a five fair coin, find the chance of getting exactly 3 heads.

In a Poisson distribution the first probability term is 0.2725. Find the next Probability term 

In a book of 520 pages, 390 typo-graphical errors occur. Assuming Poisson law for the number of errors per page, find the probability that a random sample of 5 pages will contain no error.

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

The standard normal distribution is represented by ___________

In 8 throws of a die, 1 or 3 is considered a success. Then the mean number of successes is ___________

In a poison distribution, mean is 16, then standard deviation is ___________

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

Normal distribution was invented by ________.

If X ~ N(9,81) the standard normal variate Z will be ________.

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

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

In a parametric distribution the mean is equal to variance is ________.

A discrete random variable X has the following probability distribution.

Find the value of e. Also, find the mean of the distribution.

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

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

The probability distribution of the discrete random variables X and Y are given below

Prove that E(Y²) = 2E(X).

The random variable X tan take only the values 0,1,2. Given that P(X = 0) = P(X = 1) = P and E(X²) = E(X), find the value of p.

The probability distribution of a random variable X is

Calculate
(i) A if E(X) = 2.94
(ii) V(X)

A random variable X has the following probability function

(i) Find a, Evaluate
(ii) P(X < 3),
(iii) P(X > 2) and
(iv) P(2 < X \(\leq\) 5).

Construct the distribution function for the discrete random variable X whose probability distribution is given below. Also draw a graph of p(x) and F(x).

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

The amount of bread (in hundreds of pounds) x that a certain bakery is able to sell in a day is found to be a numerical valued random phenomenon, with a probability function specified by the probability density function f(x) is given  by
\(f(x)=\left\{\begin{array}{l} Ax,for \ 0≤x10 \\ A(20−x),for \ 10 ≤x< 20 \\ 0,\quad \quad \quad otherwise \end{array}\right.\)
(a) Find the value of A.
(b) What is the probability that the number of pounds of bread that will be sold tomorrow is
(i) More than 10 pounds,
(ii) Less than 10 pounds, and
(iii) Between 5 and 15 pounds?

A continuous random variable X has the following probability function

(i) Find k
(ii) Ealuate p(x<6), p(x\(\ge \)6) and p(0)
(iii) If P(X\(\le\)x).\(\frac{1}{2}\), then find the minimum value of x.

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

then find 4E(X²)- Var (2X)

A random variable. X has following distribution

Find E(2X+3)²

If a continuous random variable. X has the p.d.f. f(x) = 4k(x-1)³, 1 ≤ x ≤ 3 then find p[-2 ≤ X ≤ 2]

A player tosses two unbiased coins. He wins Rs. 5 if two heads appear, Rs. 2 if one head appear and Rs.1 if no head appear. Find the expected amount to win.

If the probability density function of a random variable. X is given by f(x) = \(\frac{2x}{9}\),0

\(\text { If } \ p(x) \ = \begin{cases}\frac{x}{20}, & x=0,1,2,3,4,5 \\ 0, & \text { otherwise }\end{cases}\)
Find
(i) P(X<3) and 
(ii) P(2\(\leq\)4)

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.

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

A coin is tossed thrice. Let X be the number of observed heads. Find the cumulative distribution function of X.

A continuous random variable X has the following p.d.f f(x) = ax, 0\(\le\)x\(\le\)1
Determine the constant a and also find P\(\\ \left[ X\le \frac { 1 }{ 2 } \right] \)

Determine whether the following is a probability distribution of a random variable X.

An unbiased die is rolled. If the random variable X is defined as
X(w) = {1, the outcome w is an even number    
{0, if the outcome w is an odd number
Find the probability distribution of X.

Two eggs are drawn at random without replacement from a bag containing two bad eggs and eight good eggs. Find the probability of getting two bad eggs?

A random variable X has the probability mass function

then find k

A discrete random variable. X has the following probability distribution

Pind the value of a and P(X< 3)

The number of cars in a household is given below.

Estimate the probability mass function. Verify p(x_i ) is a probability mass function.

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.

Construct cumulative distribution function for the given probability distribution.

The discrete random variable X has the probability function

Show that k = 0.1.

Two coins are tossed simultaneously. Getting a head is termed as success. Find the probability distribution of the number of successes.

If \(f(x)=\left\{\begin{array}{lc} k x^{2} & 0  if a p.d.f. then the value of k is ___________

A random variable X has the following probability distribution

Then P(1≤X≤4) is

A random variable X has the following probability mass function

Then λ is

X is a discrete random variable. Which take values 0,1,2 and P(X = 0) = \(\frac{144}{169}\), P(X = 1) = \(\frac{1}{169}\), then the value of P(X = 2) is ________

Given E(X + c) = 8 and E(X - c) = 12, then the value of c is ___________

Value which is obtained by multiplying possible values of random variable with probability of occurrence and is equal to weighted average is called ________.

Demand of products per day for three days are 21, 19, 22 units and their respective probabilities are 0.29, 0.40, 0.35. Profit per unit is 0.50 paisa then expected profits for three days are ________.

Probability which explains x is equal to or less than particular value is classified as ________.

Given E(X)=5 and E(Y)=−2, then E(X−Y) is ________.

A variable that can assume any possible value between two points is called ________.

Estimate the production for 1962 and 1965 from the following data

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

From the data, find the number of students whose height is between 80 cm and 90 cm

From the following table, estimate the premium for a policy maturing at the age of 58.

Using Lagrange's formula find the value of y when x = 4 from the following table.

Using Newton’s formula for interpolation estimate the population for the year 1905 from the table:

The values of y = f(x) for x = 0,1,2, ...,6 are given by

Estimate the value of y (3.2) using forward interpolation formula by choosing the four values that will give the best approximation.

From the following table find the number of students who obtained marks less than 45.

Using appropriate interpolation formula find the number of students whose weight is between 60 and 70 from the data given below

The population of a certain town is as follows

Using appropriate interpolation formula, estimate the population during the period 1946.

From the following data, estimate the population for the year 1986 graphically.

Using graphic method, find the value of y when x=27.

Find y when x = 0.2 given that

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

Find the number of men getting wages between Rs. 30 and Rs. 35 from the following table.

Using graphic method, find the value of y when x = 38 from the following data:

Construct a forward difference table for the following data

Construct a forward difference table for y = f(x) = x³+2x+1 for x = 1,2,3,4,5

By constructing a difference table and using the second order differences as constant, find the sixth term of the series 8,12,19,29,42…

Prove that f(4) = f(3) + Δf(2) + Δ² f(1) + Δ³ f(1) taking ‘1’ as the interval of differencing.

Find the missing term from the following data.

If f(0) = 5, f(1) = 6, f(3) = 50, find f(2) by using Lagrange's formula.

Find the missing term from the following data

When h = 1, find Δ (x³).

Find the second order backward differences of f(x).

Find (i) Δe^ax
(ii) Δ²e^x
(iii) Δ log x

Evaluate \({ \Delta }^{ 2 }\left( \frac { 1 }{ x } \right) \) by taking ‘1’ as the interval of differencing.

If f (x) = e^ax then show that f(0), Δf(0), Δ²f(0) are in G.P

Prove that 
(1 + Δ)(1 - ∇) = 1

Prove that
∇Δ = Δ -  ∇

The differential equation \(\left( \frac { dx }{ dy } \right) ^{ 2 }+5y^{ \frac { 1 }{ 3 } }\)= x is _____________

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

The amount present in a radio active element disintegrates at a rate proportional to its amount. The differential equation corresponding to the above statement is _____________ (k is negative).

The differential equation satisfied by all the straight lines in xy plane is _____________

If y = k.e^λx then its differential equation where k is arbitrary constant is _____________

Δ²y₀ = _______.

Δf(x) = _______.

E ≡ _______.

If h = 1, then Δ(x²) = _______.

If c is a constant then Δc = _______.

Solve: \(\frac { dy }{ dx } \) = sin(x + y)

Solve: x²\(\frac { dy }{ dx } \) = y²+2xy given that y = 1, when x = 1

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

Solve: (D² + 14D + 49)y = e^-7x + 4.

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.

Find the differential equation of the family of straight lines y = mx + c when
(i) m is the arbitrary constant
(ii) c is the arbitrary constant
(iii) m and c both are arbitrary constants.

Solve 3e^xtan ydx +(1 + e^x)sec²ydy = 0 given y(0) = \(\frac { \pi }{ 4 } \)

Solve : x - y \(\frac { dx }{ dy } =a\left( { x }^{ 2 }+\frac { dx }{ dy } \right) \)

The normal lines to a given curve at each point(x,y) on the curve pass through the point (1, 0). The curve passes through the point (1, 2). Formulate the differential equation representing the problem and hence find the equation of the curve.

The sum of Rs. 2,000 is compounded continuously, the nominal rate of interest being 5% per annum. In how many years will the amount be double the original principal? (log_e2 = 0.6931)

Form the differential equation by eliminating α and β from (x − α)² + (y − β)² = r²

Find the differential equation of the family of all straight lines passing through the origin.

Form the differential equation that represents all parabolas each of which has a latus rectum 4a and whose axes are parallel to the x axis.

Find the differential equation of all circles passing through the origin and having their centers on the y axis.

Find the differential equation of the family of parabola with foci at the origin and axis along the x-axis.

Form the differential equation by eliminating α and β from (x − α)² + (y − β)² = r²

Find the differential equation of the family of all straight lines passing through the origin.

Form the differential equation that represents all parabolas each of which has a latus rectum 4a and whose axes are parallel to the x axis.

Find the differential equation of all circles passing through the origin and having their centers on the y axis.

Find the differential equation of the family of parabola with foci at the origin and axis along the x-axis.

Write down the order and degree of the following differential equations.
\(\left( \frac { dy }{ dx } \right) ^{ 3 }-4\left( \frac { dy }{ dx } \right) \)+y = 3e^x

Write down the order and degree of the following differential equations.
\(\left( \frac { dy }{ dx } \right) ^{ 2 }-7\frac { d^{ 3 }y }{ { dx }^{ 3 } } +y\frac { { d }^{ 2 }y }{ dx^{ 2 } } +4\frac { dy }{ dx } \)- log x = 0

Write down the order and degree of the following differential equations.
\(\sqrt { 1+\left( \frac { dy }{ dx } \right) ^{ 2 } } \)= 4x

Write down the order and degree of the following differential equations.
\(\left[ 1+\left( \frac { dy }{ dx } \right) ^{ 2 } \right] ^{ \frac { 2 }{ 3 } }=\frac { d^{ 2 }y }{ { dx }^{ 2 } } \)

Find the differential equation for y = mx + \(\frac { a }{ m } \) where m is arbitrary constant.

Find the order and degree of the following differential equations.
\(\frac { dy }{ dx } +2y={ x }^{ 3 }\)

Find the differential equation of the following
y = cx + c − c³

Find the order and degree of the following differential equations
\(\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } +3{ \left( \frac { dy }{ dx } \right) }^{ 2 }+4y=0\)

Solve: (x² + x + 1)dx + (y²− y + 3)dy = 0

Solve: ydx − xdy = 0

The differential equation \(\left( \frac { dx }{ dy } \right) ^{ 2 }+5y^{ \frac { 1 }{ 3 } }\)= x is _____________

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

The amount present in a radio active element disintegrates at a rate proportional to its amount. The differential equation corresponding to the above statement is _____________ (k is negative).

The differential equation satisfied by all the straight lines in xy plane is _____________

If y = k.e^λx then its differential equation where k is arbitrary constant is _____________

The degree of the differential equation \(\frac { { d }^{ 4 }y }{ { dx }^{ 4 } } { -\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) }^{ 4 }+\frac { dy }{ dx } =3\) ______.

The order and degree of the differential equation \(\sqrt { \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } } =\sqrt { \frac { dy }{ dx } +5 } \) are respectively ______.

The order and degree of the differential equation \(\left(\frac{d^{2} y}{d x^{2}}\right)^{\frac{3}{2}}-\sqrt{\left(\frac{d y}{d x}\right)}-4=0\) are respectively ______.

The differential equation \({ \left( \frac { dx }{ dy } \right) }^{ 3 }+2{ y }^{ \frac { 1 }{ 2 } }\) = x is ______.

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

Find the area of the region bounded by the line y = x - 5, the x-axis and between the ordinates x = 3and x = 7

The Marginal revenue for a commodity is MR=\(\frac { { e }^{ x } }{ 100 } +x+{ x }^{ 2 }\), find the revenue function.

The elasticity of demand with respect to price for a commodity-is a constant and is equal to 2. Find the demand function and hence the total revenue function, given that when the price is 1, the demand is 4.

A company determines that the marginal cost of producing x units is C'(x) = 10.6x. The fixed cost is Rs. 50. The selling price per unit is Rs.5. Find the profit function.

The demand and supply functions under pure competition are P_d = 16 - x² and p_s = 2x² + 4. Find the consumer's surplus and producer's surplus at the market equilibrium price.

Find the area bounded by y = 4x + 3 with x- axis between the lines x = 1 and x = 4

Find the area of the parabola \({ y }^{ 2 }=8x\) bounded by its latus rectum.

Sketch the graph \(y=\left| x+3 \right| \) and evaluate \(\int _{ -6 }^{ 0 }{ \left| x+3 \right| } \) dx.

Using integration find the area of the circle whose center is at the origin and the radius is a units.

Using integration find the area of the region bounded between the line x = 4 and the parabola y² = 16x.

Find the area contained between the x-axis and one arc of the curve y = cos x bounded between
\(x=-\frac { \pi }{ 2 } and\quad x=\frac { \pi }{ 2 } \)

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

Find the area bounded by one arc of the curve y = sin ax and the x-axis.

Find the area of the region bounded by the line x - y =1, x-axis and the lines x = -2 and x = 0.
x - y = 1

Determine the cost of producing 3000 units of commodity if the marginal cost in rupees per unit is C'(x) = \(\frac{x}{3000}+2.50\)

Find the area bounded by y = x between the lines x = −1 and x = 2 with x -axis.

Find the area bounded by the line y = x, the x-axis and the ordinates x = 1, x = 2

Using integration, find the area of the region bounded by the line y −1 = x, the x axis and the ordinates x = –2, x = 3.

Find the area of the region lying in the first quadrant bounded by the region y = 4x², x = 0, y = 0 and y = 4

The marginal cost function of manufacturing x shoes is 6 +10x − 6x². The cost producing a pair of shoes is Rs. 12. Find the total and average cost function.

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

Find the area under the curve y = 4x² - 8x + 6 bounded by the Y-axis, X-axis and the ordinate at x = 2.

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

The marginal cost function of manufacturing x units of a commodity is 3x² - 2x + 8. If there is no fixed cost, find the total cost function?

If the marginal revenue for a commodity is MR = 9 - 6x² + 2x, find the total revenue function.

Find the area of the region bounded by the line x − 2y − 12 = 0 , the y-axis and the lines y = 2, y = 5.

Find the area of the region bounded by the parabola \(y=4{ - }x^{ 2 }\) , x −axis and the lines x = 0, x = 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.

Find the area bounded by the lines y − 2x − 4 = 0, y = 1, y = 3 and the y-axis

If MR = 20 − 5x + 3x², find total revenue function.

The area bounded by y = 2x - x² and X-axis is _________ sq. units

The area of the region bounded by the ellipse __________

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

The area of the region bounded by the line 2y = -x + 8, X - axis and the lines x = 2 and x = 4 is ________ sq.units.

The area enclosed by the curve y = cos²x in [0,\(\pi\)] the lines x=0, x = \(\pi\) and the X-axis is ________sq.units.

Area bounded by the curve y = x (4 − x) between the limits 0 and 4 with x − axis is ________.

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

Area bounded by the curve y = \(\frac{1}{x}\) between the limits 1 and 2 is ________.

If the marginal revenue function of a firm is MR = \({ e }^{ \frac { -x }{ 10 } }\), then revenue is ________.

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

If f'(x) = a sin x + b cos x and f'(0) = 4, f(0) = 3, f\(\left( \frac { \pi }{ 2 } \right) \) = 5, find f(x).

Evaluate \(\int { \frac { { x }^{ 7 } }{ { x }^{ 5 }+1 } } dx\)

Evaluate ഽ x³ sin (x⁴) dx

Evaluate \(\int { \frac { 1 }{ { 3x }^{ 2 }+13x-10 } } dx\)

Evaluate ഽx. log (1 + x) dx

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

Evaluate \(\int { \frac { { 3x }^{ 2 }+6x+1 }{ \left( x+3 \right) \left( { x }^{ 2 }+1 \right) } } dx\)

Integrate the following with respect to x.
\(\frac { { 4x }^{ 2 }+2x+6 }{ { \left( x+1 \right) }^{ 2 }\left( x-3 \right) } \)

Integrate the following with respect to x. 
\(\frac { { 3x }^{ 2 }-2x+5 }{ { \left( x-1 \right) }\left( x^{ 2 }+5 \right) } \)

Evaluate \(\int { { \left( \log x \right) }^{ 2 } } dx\)

Evaluate  \(\int { \frac { { { x }^{ 4 }+{ x }^{ 4 }+1 } }{ { x }^{ 2 }-x+1 } } \)

Evaluate \(\int { \frac { cos2x-cos2\alpha }{ cosx-cos\alpha } } dx\)

Evaluate \(\int { \frac { { ({ a }^{ x }{ +b }^{ x }) }^{ 2 } }{ { a }^{ x }b^{ x } } dx } \)

If f' (x) = 3x² - \(\frac { 2 }{ { x }^{ 3 } } \) and f(1) = 0, find f(x)

Evaluate \(\int { \frac { { 8 }^{ 1+x }+{ 4 }^{ 1-x } }{ { 2 }^{ x } } } dx\)

 Evaluate \(\int \frac{a x^{2}+b x+c}{\sqrt{x}} d x\)

Evaluate \(\int { \frac { { 2x }^{ 2 }-14x+24 }{ x-3 } dx } \)

Evaluate \(\int { \frac { x+2 }{ \sqrt { 2x+3 } } } dx\)

Evaluate \(\int { \frac { 1 }{ \sqrt { x+2 } -\sqrt { x-2 } } } dx\)

 Integrate the following with respect to x.
\(\sqrt{x}\)(x³ − 2x + 3)

Evaluate \(\int { { a }^{ 3{ log }_{ a }x } } dx\)

Evaluate \(\int { \frac { 2{ cos }^{ 2 }x-cos2x }{ { sin }^{ 2 }x } } \)

Evaluate ∫ tan²x dx

Evaluate \(\int { \frac { 2+3cosx }{ { sin }^{ 2 }x } } dx\)

Evaluate \(\int { \frac { { 2 }^{ x }+{ 3 }^{ x } }{ { 5 }^{ x } } dx } \)

Evaluate \(\int \sqrt{2 x+1} \ d x\)

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

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

Evaluate \(\int { \left( { x }^{ 3 }+7 \right) \left( x-4 \right) dx } \)

Integrate the following with respect to x.
\(\sqrt { 3x+5 } \)

\(\int { \left( x-1 \right) } { e }^{ -x }\) dx = __________ +c

If \(\int { \frac { { 2 }^{ \frac { 1 }{ x } } }{ { x }^{ 2 } } } dx=k{ 2 }^{ \frac { 1 }{ x } }\) +c, then k is ___________

\(\int { { \left| x \right| }^{ 3 } } \)dx = ________________ +c

\(\int { \frac { 2 }{ { \left( { e }^{ x }+{ e }^{ -x } \right) }^{ 2 } } } \) dx = ____________ +c 

\(\int { { e }^{ x } } \) (1-cot x +cot² x) dx = _______________ +c

ഽ\(\frac { 1 }{ { x }^{ 3 } } \)dx is _______.

ഽ2^xdx is _______.

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

\(\int \frac{\sin 5 x-\sin x}{\cos 3 x} d x\) is _______.

ഽ\(\frac{logx}{x}\) dx , x > 0 is _______.

The sum of three numbers is 6. If we multiply the third number by 2 and add the first number to the result we get 7. By adding second and third numbers to three times the first number we get 12. Find the numbers using rank method

A mixture is to be made of three foods A, B, C. The three foods A, B, C contain nutrients P, Q, R as shown below

How to form a mixture which will have 8 ounces of P, 5 ounces of Q and 7 ounces of R? (Cramer's rule).

For what values of k, the system of equations kx+ y+z = 1, x+ ky+z= 1, x+ y+kz = 1 have
(I) Unique solution
(ii) More than one solution
(iii) no solution

Using determinants, find the quadratic defined by f(x) = ax² + bx + c if
f(1) = 0,
f(2) = - 2 and
f(3) = -6.

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?

Show that the equations 2x + y + z = 5, x + y + z = 4, x − y + 2z = 1 are consistent and hence solve them.

Show that the equations x + y + z = 6, x + 2y + 3z = 14, x + 4y + 7z = 30 are consistent and solve them.

Show that the equations are inconsistent x − 4y + 7z = 14, 3x + 8y − 2z = 13, 7x − 8y + 26z = 5 

Find k, if the equations x + 2y − 3z = −2, 3x − y − 2z = 1, 2x + 3y − 5z = k are consistent.

Find k, if the equations x + y + z = 7,  x + 2y + 3z = 18,  y + kz = 6 are inconsistent

Find the rank of the matrix
\(A=\left( \begin{matrix} 2 & 4 & 5 \\ 4 & 8 & 10 \\ -6 & -12 & -15 \end{matrix} \right) \)

Find the rank of the matrix \(A=\left( \begin{matrix} 1 & 2 & -4 \\ 2 & -1 & 3 \\ 8 & 1 & 9 \end{matrix}\begin{matrix} 5 \\ 6 \\ 7 \end{matrix} \right) \)

Show that the equations 2x - y + z = 7, 3x + y - 5z = 13, x + y + z = 5 are consistent and have a unique solution.

Show that the equations x + 2y = 3, y - z = 2, x + y + z = 1 are consistent and have infinite sets of solution.

Show that the equations x- 3y + 4z = 3, 2x - 5y + 7z = 6, 3x - 8y + 11z = 1 are inconsistent

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

Find the rank of the matrix \(\left( \begin{matrix} 5 & 3 & 0 \\ 1 & 2 & -4 \\ -2 & -4 & 8 \end{matrix} \right) \)

Find the rank of the matrix \(\left( \begin{matrix} 1 & 2 & -1 \\ 2 & 4 & 1 \\ 3 & 6 & 3 \end{matrix}\begin{matrix} 3 \\ -2 \\ -7 \end{matrix} \right) \)

Find the rank of the matrix A = \(\left( \begin{matrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 5 & 7 \end{matrix} \right) \)

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

Find the rank of the matrix \(\left[ \begin{matrix} 7 & -1 \\ 2 & 1 \end{matrix} \right] \)

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

Solve x + 2y = 3 and x +y = 2 using Cramer's rule.

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

Show that the equations x + y + z = 6, x + 2y + 3z = 14 and x + 4y + 7z = 30 are consistent

Find the rank of the matrix \(\begin{pmatrix} 1 & 5 \\ 3 & 9 \end{pmatrix}\)

Find the rank of the matrix \(\begin{pmatrix} -5 & -7 \\ 5 & 7 \end{pmatrix}\)

Find the rank of each of the following matrices.
\(\left( \begin{matrix} 5 & 6 \\ 7 & 8 \end{matrix} \right) \)

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

Find the rank of each of the following matrices.
\(\left( \begin{matrix} 1 & -1 \\ 3 & -6 \end{matrix} \right) \)

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

The rank of an n x n matrix each of whose elements is 2 is __________

The value of \(\left| \begin{matrix} { 5 }^{ 2 } & { 5 }^{ 3 } & { 5 }^{ 4 } \\ { 5 }^{ 3 } & { 5 }^{ 4 } & { 5^{ 5 } } \\ { 5 }^{ 4 } & { 5 }^{ 5 } & { 5 }^{ 6 } \end{matrix} \right| \)

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

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

If A = (1 2 3), then the rank of AA^T is ________.

The rank of m x n matrix whose elements are unity is ________.

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 ________.

If A = \(\begin{pmatrix} 2 & 0 \\ 0 & 8 \end{pmatrix}\),then \(\rho (A)\) is ________.

The rank of the matrix  \(\left( \begin{matrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 4 & 9 \end{matrix} \right) \) is ________.

Which of the following is not an elementary transformation?

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

\(\int _{ 0 }^{ 1 }{ (2x+1) } dx\) is _______.

\(\int { { \left| x \right| }^{ 3 } } \)dx = ________________ +c

Area bounded by the curve y = x (4 − x) between the limits 0 and 4 with x − axis is ________.

The rank of m x n matrix whose elements are unity is ________.

If A, B are two n x n non-singular matrices, then ___________

\(\int \frac{\sin 5 x-\sin x}{\cos 3 x} d x\) is _______.

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

For a demand function p, if \(\int \frac{d p}{p}=k \int \frac{d x}{x}\) then k is equal to ________.

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

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

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

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

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

Show that the equations are inconsistent x − 4y + 7z = 14, 3x + 8y − 2z = 13, 7x − 8y + 26z = 5 

Investigate for what values of ‘a’ and ‘b’ the following system of equations x + y + z = 6,x + 2y + 3z = 10, x + 2y + az = b have
(i) no solution
(ii) a unique solution
(iii) an infinite number of solutions.

Show that the following system of equations have unique solution:
x + y + z = 3, x + 2y + 3z = 4, x + 4y + 9z = 6 by rank method.

An amount of Rs. 5,000/- is to be deposited in three different bonds bearing 6%, 7% and 8% per year respectively. Total annual income is Rs. 358/-. If the income from first two investments is Rs. 70/- more than the income from the third, then find the amount of investment in each bond by rank method.

Two types of soaps A and B are in the market. Their present 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 and when is the equilibrium reached?

Show that the equations 2x - y + z = 7, 3x + y - 5z = 13, x + y + z = 5 are consistent and have a unique solution.

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

Two products A and B currently share the market with shares 60% and 40% each respectively. Each week some brand switching latees place. Of those who bought A the previous week 70% buy it again whereas 30% switch over to B. Of those who bought B the previous week, 80% buy it again whereas 20% switch over to A. Find their shares after one week and after two weeks.

If f' (x) = 3x² - \(\frac { 2 }{ { x }^{ 3 } } \) and f(1) = 0, find f(x)

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

Find the rank of the matrix \(\left[ \begin{matrix} 7 & -1 \\ 2 & 1 \end{matrix} \right] \)

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

For what value of x, the matrix
\(A=\left| \begin{matrix} 1 & -2 & 3 \\ 1 & 2 & 1 \\ x & 2 & -3 \end{matrix} \right| \) is singular?

Evaluate \(\int { { a }^{ 3{ log }_{ a }x } } dx\)

Evaluate ∫ tan²x dx

The value of \(\left| \begin{matrix} { 5 }^{ 2 } & { 5 }^{ 3 } & { 5 }^{ 4 } \\ { 5 }^{ 3 } & { 5 }^{ 4 } & { 5^{ 5 } } \\ { 5 }^{ 4 } & { 5 }^{ 5 } & { 5 }^{ 6 } \end{matrix} \right| \)

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

\(\int { \frac { 2 }{ { \left( { e }^{ x }+{ e }^{ -x } \right) }^{ 2 } } } \) dx = ____________ +c 

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

The value of the integral \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { { \sqrt { cosx } } }{ \sqrt { cosx } +\sqrt { sinx } } } dx= \)

Find k, if the equations x + y + z = 7,  x + 2y + 3z = 18,  y + kz = 6 are inconsistent

Investigate for what values of ‘a’ and ‘b’ the following system of equations x + y + z = 6,x + 2y + 3z = 10, x + 2y + az = b have
(i) no solution
(ii) a unique solution
(iii) an infinite number of solutions.

An automobile company uses three types of Steel S₁, S₂ and S₃ for providing three different types of Cars C₁, C₂ and C₃. Steel requirement R (in tonnes) for each type of car and total available steel of all the three types are summarized in the following table.

Determine the number of Cars of each type which can be produced by Cramer’s rule.

A new transit system has just gone into operation in Chennai. Of those who use the transit system this year, 30% will switch over to using metro train next year and 70% will continue to use the transit system. Of those who use metro train this year, 70% will continue to use metro train next year and 30% will switch over to the transit system. Suppose the population of Chennai city remains constant and that 60% of the commuters use the transit system and 40% of the commuters use metro train 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?

The subscription department of a magazine sends out a letter to a large mailing list inviting subscriptions for the magazine. Some of the people receiving this letter already subscribe to the magazine while others do not. From this mailing list, 60% of those who already subscribe will subscribe again while 25% of those who do not now subscribe will subscribe. On the last letter it was found that 40% of those receiving it ordered a subscription. What percent of those receiving the current letter can be expected to order a subscription?

In a transition probability matrix, all the entries are greater than or equal to _______.

The rank of an n x n matrix each of whose elements is 2 is __________

The value of \(\int _{ -\frac{\pi}{2}}^{ \frac{\pi}{2}}\) cos x dx is _______.

The value of \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ cosx } { e }^{ sinx }dx=\)

Area bounded by the curve y = \(\frac{1}{x}\) between the limits 1 and 2 is ________.

Show that the equations are inconsistent x − 4y + 7z = 14, 3x + 8y − 2z = 13, 7x − 8y + 26z = 5 

Investigate for what values of ‘a’ and ‘b’ the following system of equations x + y + z = 6,x + 2y + 3z = 10, x + 2y + az = b have
(i) no solution
(ii) a unique solution
(iii) an infinite number of solutions.

An amount of Rs. 5,000/- is to be deposited in three different bonds bearing 6%, 7% and 8% per year respectively. Total annual income is Rs. 358/-. If the income from first two investments is Rs. 70/- more than the income from the third, then find the amount of investment in each bond by rank method.

Solve by Cramer’s rule x + y + z = 4, 2x − y + 3z = 1, 3x + 2y − z = 1

A total of Rs. 8,500 was invested in three interest earning accounts. The interest rates were 2%, 3% and 6% if the total simple interest for one year was Rs. 380 and the amount, invested at 6% was equal to the sum of the amounts in the other two accounts, then how much was invested in each account? (use Cramer’s rule).

Show that the equations 2x + y = 5,4x + 2y = 10 are consistent and solve them.

Show that the equations 2x - y + z = 7, 3x + y - 5z = 13, x + y + z = 5 are consistent and have a unique solution.

Evaluate \(\int { \frac { { 8 }^{ 1+x }+{ 4 }^{ 1-x } }{ { 2 }^{ x } } } dx\)

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

The marginal cost function is MC = 300 \({ x }^{ \frac { 2 }{ 5 } }\) and fixed cost is zero. Find out the total cost and average cost functions.

Find the rank of the matrix A = \(\left( \begin{matrix} 1 & 1 & 1 \\ 3 & 4 & 5 \\ 2 & 3 & 4 \end{matrix}\begin{matrix} 1 \\ 2 \\ 0 \end{matrix} \right) \) 

Solve the equations 2x + 3y = 7, 3x + 5y = 9 by Cramer’s rule.

A total of Rs. 8,600 was invested in two accounts. One account earned \(4\frac { 3 }{ 4 } %\)% annual interest and the other earned \(6\frac { 1 }{ 2 } %\)% annual interest. If the total interest for one year was Rs. 431.25, how much was invested in each account? (Use determinant method).

Find the rank of each of the following matrices.
\(\left( \begin{matrix} 1 & 2 & -1 \\ 2 & 4 & 1 \\ 3 & 6 & 3 \end{matrix}\begin{matrix} 3 \\ -2 \\ -7 \end{matrix} \right) \)

Integrate the following with respect to x.
\(\frac { 1 }{ x{ \left( \log x \right) }^{ 2 } } \)

Find the rank of each of the following matrices.
\(\left( \begin{matrix} 5 & 6 \\ 7 & 8 \end{matrix} \right) \)

Evaluate \(\int \sqrt{2 x+1} \ d x\)

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

Integrate the following with respect to x.
(4x + 2) \(\sqrt { { x }^{ 2 }+x+1 } \)

Evaluate the following
\(\int _{ 0 }^{ \infty }{ { e }^{ -mx } } { x }^{ 6 }dx\)

Evaluate \(\int \sqrt{2 x+1} \ d x\)

Evaluate \(\int { \left( { x }^{ 3 }+7 \right) \left( x-4 \right) dx } \)

Integrate the following with respect to x.
\(\frac { 8x+13 }{ \sqrt { 4x+7 } } \)

Evaluate \(\int { \frac { \cos2x }{ { \sin }^{ 2 }{ x \cos }^{ 2 }x } dx } \) 

Evaluate ഽ\(\sqrt { { x }^{ 2 }-16 } \)dx

If A =\(\left( \begin{matrix} 1 \\ 2 \\ 3 \end{matrix} \right) \) then the rank of AA^T is ________.

If \(\rho (A)=\rho (A,B)\) the number of unknowns, then the system is______.

In a transition probability matrix, all the entries are greater than or equal to _______.

\(\int \frac{\sin 5 x-\sin x}{\cos 3 x} d x\) is _______.

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