21.

The following table shows the mean and standard deviation of the marks of Mathematics and Physics scored by the students in a school:

	Mathematics	Physics
Mean	84	81
Standard Deviation	7	4

The correlation coefficient between the given marks is 0.86. Estimate the likely marks in physics if the marks in Mathematics are 92.

Bag A contains three red and four white balls; bag B contains two red and three white balls. If one ball is drawn from bag A and two balls from bag b, find the probability that:

(i) One ball is red and two balls are white.

(ii) All the three balls are of the same colour.

Three persons Aman, Bipin and Mohan attempt a mathematics problem independently. The odds in favour of Aman and Mohan solving the problem are 3:2 and 4:1 respectively and the odds against Bipin solving the problem are 2:1. Find:

(i) The probability that all the three will solve the problem.

(ii) The probability that problem will be solved.

If the sum and the product of the mean and variance of a Binomial Distribution are 1.8 and 0.8 respectively, find the probability distribution the probability of at least one success.

The price index for the following data for the year 2011 taking 2001 as the base year was 127. The simple average price relatives method was used. Find the value of x:

The profits of a paper bag manufacturing company (in laks of rupees) during each month of a year are :

Plot the given data on a graph sheet. Calculate the four monthly moving averages and plot these on the same graph sheet.

Find A^{- 1}, where A = $\left[\begin{array}{ccc}4& 2& 3\\ 1& 1& 1\\ 3& 1& - 2\end{array}\right]$

Hence, solve the following system of linear equations:

4x + 2y + 3z = 2

x + y + z =1

3x + y - 2z = 5

Find the locus of the complex number z = x + iy, satisfying relations age (z - 1) = $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$, and $\left|\mathrm{z} - 2 - 3\mathrm{i}\right| = 2$. Illustrate the locus on the Argand plane.

Solve the following differential equation:

ye^ydx = (y³ + 2xe^y)dy, given that x = 0, y = 1.

Find the value of $\mathrm{\lambda }$ for which the four points A, B, C, D with position vectors $- \hat{\mathrm{j}} - \hat{\mathrm{k}}; 4\hat{\mathrm{i}} + 5\hat{\mathrm{j}} + \mathrm{\lambda }\hat{\mathrm{k}}; 3\hat{\mathrm{i}} +\hspace{0.17em}9\hat{\mathrm{j}} + 4\hat{\mathrm{k}} \mathrm{and} - 4\hat{\mathrm{i}} + 4\hat{\mathrm{j}} + 4\hat{\mathrm{k}}$ are coplanar.