In the study learning material (SLM), you have seen many situations where Poisson distribution is suitable and discussed some examples of such situations. Create your own example for a situation other than which are discussed in SLM. If you denote your created random variable by X then find the probability that X is less than 2.

Suppose two friends Anjali and Prabhat trying to meet for a date to have lunch say between 2 pm to 3 pm. Suppose they follow the following rules for this meeting:

• Each of them will reach either on time or 10 minutes late or 20 minutes late or 30 minutes late or 40 minutes late or 50 minutes late or 1 hour late. All these arrival times are equally likely for both of them.
• Whoever of them reaches first will wait for the other to meet only for 10 minutes. If within 10 minutes the other does not reach, he/she leaves the place and they will not meet.

Find the probability of their meeting.

A spherical balloon is being Inflated at the rate of 900 cm³/sec. How fast is the Radius of the balloon Increasing when the Radius is15 cm.

Suriti wants to Invest at most 12000 in saving certificates and National Saving Bonds. She has to invest at least 2000 in Saving certificates and at least 4000 in National Saving Bonds. If Rate of Interest on saving certificates is 8% per annum and rate of interest on national saving bond is 10% per annum. How much money should she invest to earn maximum yearly income? Find also the maximum yearly income.

Evaluate the integral using beta and gamma functions.

Show that

  i)   

ii) 

Evaluate the integral   ,where D = {(x, y): x ≥ 0, y ≥ 0, x+y≤1}, by D considering D as a region of Type I and then as a region of Type II.

If m times the m^th term of an an A.P. is n^th times its nth term, show that (m + n)^th term of the A.P. is zero.

If f: [0, 5]→ R be a function defined by f(x) = x²+2x+1, x ∈ [0, 5]. Show that f is Riemann integrable using both definitions. Also, verify that the results of both definition match.

Find the inverse of the matrix A = , if it exists, -1 2 5

Test the convergence of the series

Find the area bounded by the curves x² = y and y=x.

Give an example of a set which is convex but not affine. Justify your claim with a proper explanation.

If a positive number exceeds its positive square root by 12, then find the number.

Solve the inequality  and graph the solution set.

If a, ẞ are roots of equation 2x²-3x-5=0, them find a Quadratic equation whose roots are α², β2

If 1, w, w² are Cube Roots of unity show that (1+ w)² - (1+w)³ + w² =0.

Find the equation of a line passing through points A(2, 3, 5) and B(5, 8, 9). Also, find the coordinates of a point on this line which is at a distance of 10 units from point A opposite to the side of point B.

Integrate function f(x) = x/[(x+1) (2x-1)] w.r.t x

In R we have a built-in data set "trees". A screenshot of the first four rows together with the R code to obtain it is given as follows. To get more detail about this data set you can run? trees command on R console.

           > head (tree, 4)

          Girth    Height   Volume 

1       8.3          70         10.3

2       8.6          65          10.3

3        8.8          63           10.2

4        10.5       72           16.4

Note that all the three variables of this data set are numeric. So, assuming each row of this data set is a point in 3-dimension. Find the distances between the points corresponding to the first and the third rows using the Manhattan and Chebyshev distance formula.