Mathematics-III

The value of Chebyshev polynomials $ T_2(x) $ is

The maximum number of edges in a simple graph with n vertices is

If R is a relation on a finite set having n elements, then the number of relations on A is

Let R be a relation on a set A such that $ R = R^{-1} $, then R is

Let $ f(x) = \frac{ax+b}{cx+d} $, then $ fof(x) = x $ provided

SD is defined as

Let A and B are two possible outcomes of an experiment and suppose $ P(A) = 0.3 $, $ P(B) = K $, $ P(A \cup B) = 0.6 $. If A and B are mutually exclusive events then the value of K is

If the mean of poisson distribution is m, then SD of this distribution is

The median of the numbers 11, 10, 12, 13, 9 is

A hypothesis is false but is accepted, then there is an error of type _________.

Find the generating function of Chebyshev polynomials.

Show that $ T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x) $, where $ T_n(x) $ is Chebyshev polynomials.

What is the wavelet transform?

For any three sets A, B, C prove that $ A \times (B \cup C) = (A \times B) \cup (A \times C) $.

Prove that the relation R on the set $ N \times N $ defined by $ (a,b)R(c,d) \in R \Leftrightarrow a+d = b+c $ for all $ (a,b), (c,d) \in N \times N $ is an equivalence relation.

Show that the function $ f: Q \rightarrow Q $ given by $ f(x) = 2x - 3 $ for all $ x \in Q $ is a bijection.

Show that: $ \int_0^p x(\text{ber}^2 x + \text{bei}^2 x)dx = p(\text{ber } p \text{ bei}' p - \text{bei } p \text{ ber}' p) $.

Write the vertex set and the edge set, and give a table showing the edge endpoint function for the given graph.

Discuss Skewness and Kurtosis for the following frequency distribution:

In a partially destroyed laboratory record of an analysis of a correlation data, the following results only are eligible: Variance of $ x = 9 $. Regression equations: $ 40x - 18y = 214 $, $ 8x - 10y + 66 = 0 $. Find (i) mean values of x and y, (ii) coefficient of correlation between x and y, and (iii) the standard deviation of y and angle between the lines of regressions.

Find the mean and variance of binomial distribution.

The probability that a pen manufactured by a company will be defective is 1/10. If 12 such pens are manufactured, find the probability that (i) exactly two will be defective, (ii) at least two will be defective and (iii) none will be defective.

In a test 2000 electric bulbs, it was found that the life of a particular make was normally distributed with an average life of 2040 hours and SD of 60 hours. Estimate the number of bulbs likely to burn for (i) more than 2150 hours, (ii) less than 1950 hours and (iii) more than 1950 hours and less than 2160 hours.

Find the curve of best fit of the type $ y = ae^{bx} $ to the following data by method of least square:

The mean of a certain normal population is equal to the standard error of the mean of the samples of 100 from that distribution. Find the probability that the mean of the sample of 25 from the distribution will be negative.

An unbiased coin is thrown n times. It is desired that the relative frequency of the appearance of head should lie between 0.49 and 0.51. Find the smallest value of n that will ensure this result with 90% confidence.

The value of Chebyshev polynomials $ T_2(x) $ is

The maximum number of edges in a simple graph with n vertices is

If R is a relation on a finite set having n elements, then the number of relations on A is

Let R be a relation on a set A such that $ R = R^{-1} $, then R is

Let $ f(x) = \frac{ax+b}{cx+d} $, then $ fof(x) = x $ provided

SD is defined as

Let A and B are two possible outcomes of an experiment and suppose $ P(A) = 0.3 $, $ P(B) = K $, $ P(A \cup B) = 0.6 $. If A and B are mutually exclusive events then the value of K is

If the mean of poisson distribution is m, then SD of this distribution is

The median of the numbers 11, 10, 12, 13, 9 is

A hypothesis is false but is accepted, then there is an error of type _________.

Find the generating function of Chebyshev polynomials.

Show that $ T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x) $, where $ T_n(x) $ is Chebyshev polynomials.

What is the wavelet transform?

For any three sets A, B, C prove that $ A \times (B \cup C) = (A \times B) \cup (A \times C) $.

Prove that the relation R on the set $ N \times N $ defined by $ (a,b)R(c,d) \in R \Leftrightarrow a+d = b+c $ for all $ (a,b), (c,d) \in N \times N $ is an equivalence relation.

Show that the function $ f: Q \rightarrow Q $ given by $ f(x) = 2x - 3 $ for all $ x \in Q $ is a bijection.

Show that: $ \int_0^p x(\text{ber}^2 x + \text{bei}^2 x)dx = p(\text{ber } p \text{ bei}' p - \text{bei } p \text{ ber}' p) $.

Write the vertex set and the edge set, and give a table showing the edge endpoint function for the given graph.

Discuss Skewness and Kurtosis for the following frequency distribution:

In a partially destroyed laboratory record of an analysis of a correlation data, the following results only are eligible: Variance of $ x = 9 $. Regression equations: $ 40x - 18y = 214 $, $ 8x - 10y + 66 = 0 $. Find (i) mean values of x and y, (ii) coefficient of correlation between x and y, and (iii) the standard deviation of y and angle between the lines of regressions.

Find the mean and variance of binomial distribution.

The probability that a pen manufactured by a company will be defective is 1/10. If 12 such pens are manufactured, find the probability that (i) exactly two will be defective, (ii) at least two will be defective and (iii) none will be defective.

In a test 2000 electric bulbs, it was found that the life of a particular make was normally distributed with an average life of 2040 hours and SD of 60 hours. Estimate the number of bulbs likely to burn for (i) more than 2150 hours, (ii) less than 1950 hours and (iii) more than 1950 hours and less than 2160 hours.

Find the curve of best fit of the type $ y = ae^{bx} $ to the following data by method of least square:

The mean of a certain normal population is equal to the standard error of the mean of the samples of 100 from that distribution. Find the probability that the mean of the sample of 25 from the distribution will be negative.

An unbiased coin is thrown n times. It is desired that the relative frequency of the appearance of head should lie between 0.49 and 0.51. Find the smallest value of n that will ensure this result with 90% confidence.

The value of Chebyshev polynomials $ T_3(x) $ is

The value of $ \frac{d}{dx}(x \text{ bei}' x) $ is

Which of the following functions is inverse of itself?

Given $ f(x) = \log\left(\frac{1+x}{1-x}\right) $ and $ g(x) = \frac{3x+x^3}{1-3x^2} $. Then $ f \circ g(x) $ equals

If a function $ f: (2,\infty) \rightarrow B $ defined by $ f(x) = x^2 - 4x + 5 $ is a bijection, then B =

If $ r_1 $ and $ r_2 $ are two regression coefficients, then signs of $ r_1 $ and $ r_2 $ depend on _________.

Regression coefficient of y on x is 0.7 and that of x on y is 3.2. Is the correlation coefficient consistent? (State Yes or No)

The SD of the binomial distribution is

A hypothesis is true but is rejected, then there is an error of type _________.

A hypothesis is false but is accepted, then there is an error of type _________.

State and prove the orthogonality property of Chebyshev polynomials.

Show that $ (1-x^2)T_n'(x) = nT_{n-1}(x) - nxT_n(x) $, where $ T_n(x) $ is Chebyshev polynomials.

What are the applications of wavelets transform?

For any three sets A, B and C, prove that $ A \times (B \cup C) = (A \times B) \cap (A \times C) $. (Note: There is a typo in the original question paper which asks to prove intersection instead of union, please prove accordingly or state the standard relation).

Prove that the relation R on the set $ N \times N $ defined by $ (a,b)R(c,d) \in R \Leftrightarrow a+d = b+c $ for all $ (a,b), (c,d) \in N \times N $ is an equivalence relation.

Show that the function $ f: R \rightarrow R $ defined by $ f(x) = 3x^3 + 3 $ for all $ x \in R $ is a bijection.

Derive the expression of ber and bei functions.

Consider the given graph. Find all edges that are incident on $ v_1 $, all vertices that are adjacent to $ v_1 $, all edges that are adjacent to $ e_1 $, all loops, all parallel edges, all vertices that are adjacent to themselves, and all isolated vertices.

The first four moments about the working mean 28.5 of a distribution are 0.294, 7.144, 42.409 and 454.98. Calculate the moment about the mean. Also evaluate $ \beta_1 $, $ \beta_2 $ and comment upon the skewness and kurtosis of the distribution.

Compute the Karl Pearson's coefficient correlation using the data given below:

Find the moment-generating function of Poisson distribution.

In a certain factory turning out razor blades, there is a small chance of 0.002 for any blade to be defective. The blades are supplied in packets of 10. Calculate the approximate number of packets containing no defective, one defective and two defective blades respectively in a consignment of 10000 packets.

In a normal distribution, 31% of the items are under 45 and 8% are over 64. Find the mean and SD of the distribution.

Discuss the method of least square.

The mean of a certain normal population is equal to the standard error of the mean of the samples of 100 from that distribution. Find the probability that the mean of the sample of 25 from the distribution will be negative.

An unbiased coin is thrown n times. It is desired that the relative frequency of the appearance of head should lie between 0.49 and 0.51. Find the smallest value of n that will ensure this result with 90% confidence.

The value of Chebyshev polynomials $ T_3(x) $ is

The value of $ \frac{d}{dx}(x \text{ bei}' x) $ is

Which of the following functions is inverse of itself?

Given $ f(x) = \log\left(\frac{1+x}{1-x}\right) $ and $ g(x) = \frac{3x+x^3}{1-3x^2} $. Then $ f \circ g(x) $ equals

If a function $ f: (2,\infty) \rightarrow B $ defined by $ f(x) = x^2 - 4x + 5 $ is a bijection, then B =

If $ r_1 $ and $ r_2 $ are two regression coefficients, then signs of $ r_1 $ and $ r_2 $ depend on _________.

Regression coefficient of y on x is 0.7 and that of x on y is 3.2. Is the correlation coefficient consistent? (State Yes or No)

The SD of the binomial distribution is

A hypothesis is true but is rejected, then there is an error of type _________.

A hypothesis is false but is accepted, then there is an error of type _________.

State and prove the orthogonality property of Chebyshev polynomials.

Show that $ (1-x^2)T_n'(x) = nT_{n-1}(x) - nxT_n(x) $, where $ T_n(x) $ is Chebyshev polynomials.

What are the applications of wavelets transform?

For any three sets A, B and C, prove that $ A \times (B \cup C) = (A \times B) \cap (A \times C) $. (Note: There is a typo in the original question paper which asks to prove intersection instead of union, please prove accordingly or state the standard relation).

Prove that the relation R on the set $ N \times N $ defined by $ (a,b)R(c,d) \in R \Leftrightarrow a+d = b+c $ for all $ (a,b), (c,d) \in N \times N $ is an equivalence relation.

Show that the function $ f: R \rightarrow R $ defined by $ f(x) = 3x^3 + 3 $ for all $ x \in R $ is a bijection.

Derive the expression of ber and bei functions.

Consider the given graph. Find all edges that are incident on $ v_1 $, all vertices that are adjacent to $ v_1 $, all edges that are adjacent to $ e_1 $, all loops, all parallel edges, all vertices that are adjacent to themselves, and all isolated vertices.

The first four moments about the working mean 28.5 of a distribution are 0.294, 7.144, 42.409 and 454.98. Calculate the moment about the mean. Also evaluate $ \beta_1 $, $ \beta_2 $ and comment upon the skewness and kurtosis of the distribution.

Compute the Karl Pearson's coefficient correlation using the data given below:

Find the moment-generating function of Poisson distribution.

In a certain factory turning out razor blades, there is a small chance of 0.002 for any blade to be defective. The blades are supplied in packets of 10. Calculate the approximate number of packets containing no defective, one defective and two defective blades respectively in a consignment of 10000 packets.

In a normal distribution, 31% of the items are under 45 and 8% are over 64. Find the mean and SD of the distribution.

Discuss the method of least square.

The mean of a certain normal population is equal to the standard error of the mean of the samples of 100 from that distribution. Find the probability that the mean of the sample of 25 from the distribution will be negative.

An unbiased coin is thrown n times. It is desired that the relative frequency of the appearance of head should lie between 0.49 and 0.51. Find the smallest value of n that will ensure this result with 90% confidence.

If $ P_n $ is the Legendre polynomial of first kind, then the value of $ \int_{-1}^1 P_{n+1}^2 dx $ is

If $ J_n $ the Bessel's function of first kind, then the value of $ J_{3/2} $ is

The general solution of $ \frac{d^2y}{dx^2} + 9y = \sin^3 x $ is

The general solution of $ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 $ is

The radius of convergence of the series $ \sum_{n=0}^\infty (3+4i)^n z^n $ is

The value of the integral $ \oint_{|z|=2} \frac{e^{2z}}{(z+1)^4} dz $ is

Let A, B and C be any three independent events. Which one of the following is incorrect statement?

A random variable X has a Poisson distribution. If $ 4P(X=2) = P(X=1) + P(X=0) $ then the variance of X is

The moment-generating function of a continuous random variable X be given as, $ M_X(t) = (1-t)^{-9}, |t|<1 $. Then its mean and variance are

For the differential equation $ t(t-2)^2y'' + ty' + y = 0 $, $ t=0 $ is

Solve the following differential equation in series form: $ x(x-1)\frac{d^2y}{dx^2} + (3x-1)\frac{dy}{dx} + y = 0 $.

Show that $ P_{2n}(0) = (-1)^n \frac{(2n)!}{2^{2n}(n!)^2} $.

Show that $ P_n(x) $ is the coefficient of $ t^n $ in the expansion of $ (1-2xt+t^2)^{-1/2} $ in ascending power of t.

Prove that $ J_{5/2}(x) = \sqrt{\frac{2}{\pi x}}\left[\left(\frac{3-x^2}{x^2}\right)\sin x - \frac{3\cos x}{x}\right] $.

Solve the differential equation $ (p^2+q^2)y = qz $.

Solve the linear partial differential equation $ \frac{\partial^2 z}{\partial x^2} + \frac{\partial^2 z}{\partial x \partial y} - 6\frac{\partial^2 z}{\partial y^2} = y \cos x $.

A tightly stretched string with fixed end points $ x=0 $ and $ x=l $ is initially at rest in its equilibrium position. If it is set vibrating by giving to each of its points on initial velocity $ \lambda x(l-x) $, find the displacement of the string at any distance x from one end at any time t.

Prove that $ (1-x^2)P_n'(x) = (n+1)[xP_n(x) - P_{n+1}(x)] $.

If $ f(z) = u+iv $ is an analytic function and $ u-v = e^{-x}[(x-y)\sin y - (x+y)\cos y] $, then find u, v and analytic function $ f(z) $.

Show that if $ f(z) = u+iv $ is an analytic function and (i) Re $ f(z) $ = constant, (ii) Im $ f(z) $ = constant, then $ f(z) $ is a constant.

Find all possible Taylor and Laurent series expansions of the function $ f(z) = 1/[(z+1)(z+2)^2] $ about the point $ z=1 $.

Evaluate the integral $ \int_0^{2\pi} \frac{d\theta}{2+\sin\theta} $ by using the residue theorem.

The odds that person X speaks the truth are 3:2 and the odds that person Y speaks the truth are 5:3. In what percentage of cases are they likely to contradict each other on an identical point?

A random variable X has the following probability function:
P(x) values for x = 0, 1, 2, 3, 4, 5, 6 are respectively $ 0, k, 2k, 2k, 3k, k^2, 2k^2, 7k^2+k $.
(i) Find k.
(ii) Evaluate $ P(X<6) $, $ P(X\ge6) $ and $ P(0 (iii) If $ P(X\le\alpha)>1/2 $, find the minimum value of $ \alpha $.
(iv) Determine the distribution function of X.

Let X be a non-negative random variable such that $ \log X = Y $ (say) is normally distributed with mean $ \mu $ and variance $ \sigma^2 $.
(i) Write down the probability density function of X. Find $ E(X) $ and var(X).
(ii) Find the median and the mode of the distribution of X.

In one sample of 8 observations, the sum of the squares of deviations of the sample values from the sample mean was 84.4 and in other sample of 10 observations it was 102.6. Test whether this difference is significant at 5 per cent level, given that the 5 per cent point of F for $ n_1=7 $ and $ n_2=9 $ degrees of freedom is 3.29.

If $ P_n $ is the Legendre polynomial of first kind, then the value of $ \int_{-1}^1 P_{n+1}^2 dx $ is

If $ J_n $ the Bessel's function of first kind, then the value of $ J_{3/2} $ is

The general solution of $ \frac{d^2y}{dx^2} + 9y = \sin^3 x $ is

The general solution of $ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 $ is

The radius of convergence of the series $ \sum_{n=0}^\infty (3+4i)^n z^n $ is

The value of the integral $ \oint_{|z|=2} \frac{e^{2z}}{(z+1)^4} dz $ is

Let A, B and C be any three independent events. Which one of the following is incorrect statement?

A random variable X has a Poisson distribution. If $ 4P(X=2) = P(X=1) + P(X=0) $ then the variance of X is

The moment-generating function of a continuous random variable X be given as, $ M_X(t) = (1-t)^{-9}, |t|<1 $. Then its mean and variance are

For the differential equation $ t(t-2)^2y'' + ty' + y = 0 $, $ t=0 $ is

Solve the following differential equation in series form: $ x(x-1)\frac{d^2y}{dx^2} + (3x-1)\frac{dy}{dx} + y = 0 $.

Show that $ P_{2n}(0) = (-1)^n \frac{(2n)!}{2^{2n}(n!)^2} $.

Show that $ P_n(x) $ is the coefficient of $ t^n $ in the expansion of $ (1-2xt+t^2)^{-1/2} $ in ascending power of t.

Prove that $ J_{5/2}(x) = \sqrt{\frac{2}{\pi x}}\left[\left(\frac{3-x^2}{x^2}\right)\sin x - \frac{3\cos x}{x}\right] $.

Solve the differential equation $ (p^2+q^2)y = qz $.

Solve the linear partial differential equation $ \frac{\partial^2 z}{\partial x^2} + \frac{\partial^2 z}{\partial x \partial y} - 6\frac{\partial^2 z}{\partial y^2} = y \cos x $.

A tightly stretched string with fixed end points $ x=0 $ and $ x=l $ is initially at rest in its equilibrium position. If it is set vibrating by giving to each of its points on initial velocity $ \lambda x(l-x) $, find the displacement of the string at any distance x from one end at any time t.

Prove that $ (1-x^2)P_n'(x) = (n+1)[xP_n(x) - P_{n+1}(x)] $.

If $ f(z) = u+iv $ is an analytic function and $ u-v = e^{-x}[(x-y)\sin y - (x+y)\cos y] $, then find u, v and analytic function $ f(z) $.

Show that if $ f(z) = u+iv $ is an analytic function and (i) Re $ f(z) $ = constant, (ii) Im $ f(z) $ = constant, then $ f(z) $ is a constant.

Find all possible Taylor and Laurent series expansions of the function $ f(z) = 1/[(z+1)(z+2)^2] $ about the point $ z=1 $.

Evaluate the integral $ \int_0^{2\pi} \frac{d\theta}{2+\sin\theta} $ by using the residue theorem.

The odds that person X speaks the truth are 3:2 and the odds that person Y speaks the truth are 5:3. In what percentage of cases are they likely to contradict each other on an identical point?

A random variable X has the following probability function:
P(x) values for x = 0, 1, 2, 3, 4, 5, 6 are respectively $ 0, k, 2k, 2k, 3k, k^2, 2k^2, 7k^2+k $.
(i) Find k.
(ii) Evaluate $ P(X<6) $, $ P(X\ge6) $ and $ P(0 (iii) If $ P(X\le\alpha)>1/2 $, find the minimum value of $ \alpha $.
(iv) Determine the distribution function of X.

Let X be a non-negative random variable such that $ \log X = Y $ (say) is normally distributed with mean $ \mu $ and variance $ \sigma^2 $.
(i) Write down the probability density function of X. Find $ E(X) $ and var(X).
(ii) Find the median and the mode of the distribution of X.

In one sample of 8 observations, the sum of the squares of deviations of the sample values from the sample mean was 84.4 and in other sample of 10 observations it was 102.6. Test whether this difference is significant at 5 per cent level, given that the 5 per cent point of F for $ n_1=7 $ and $ n_2=9 $ degrees of freedom is 3.29.