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Express the following statement in symbolic form and write its truth value.

"If 4 is an odd number, then 6 is divisible by 3 "

Find the values of x and y, if

`2[[1,3],[0,x]]+[[y,0],[1,2]]=[[5,6],[1,8]]`

Find the value of 'k' if the function

`f(X)=(tan7x)/(2x) , `

` =k, for

`=k, for x=0`

is continuos at x=0

price P for demand D is given as P = 183 +120D - 3D^{2} Find D for which the price is increasing

Prove that the following statement pattern is equivalent :

(p ∨ q) r and (p → r) ∧ (q → r)

Find the inverse of the following matrix by elementary row transformations if it exists. `A=[[1,2,-2],[0,-2,1],[-1,3,0]]`

Express the truth of each of the following statements by Venn diagram:

(a) Some hardworking students are obedient.

(b) No circles are polygons.

(c) All teachers are scholars and scholars are teachers.

If 'f' is continuous at x = 0, then find f(0).

`f(x)=(15^x-3^x-5^x+1)/(xtanx) , x!=0`

A firm wants to maximize its profit. The total cost function is C = 370Q + 550 and revenue is R = 730Q-3Q^{2}. Find the output for which profit is maximum and also find the profit amount at this output.

The ratio of number of boys and girls in a school is 3 : 2. If 20% of the boys and 30% of the girls are scholarship holders, find the percentage of students who are not scholarship holders.

Coefficient of rank correlation between x and y is 0.5 and `sumd_i^2`= 42. Assuming that no ranks are repeated, find the number of pairs of observations.

An agent charges 12% commission on the sales. What does he earn if the total sale amounts to Rs. 36,000? What does the seller get?

The present worth of the sum of Rs. 5,830, due 9 months hence, is Rs. 5,500. Find the rate of interest.

For a binomial distribution mean is 6 and variance is 2. Find n and p.

From a lot of 25 bulbs of which 5 are defective a sample of 5 bulbs was drawn at random with replacement. Find the probability that the sample will contain -

(a) exactly 1 defective bulb.

(b) at least 1 defective bulb.

Given the following table which relates to the number of parrots at age x, complete the life table for parrots.

X | 0 | 1 | 2 | 3 | 4 | 5 |

l | 1000 | 940 | 780 | 590 | 25 | 0 |

The equations given of the two regression lines are :

2x + 3y-6 = 0 and 5x + 7y-12 = 0.

Find :

(a) Correlation coefficient

(b) `sigma_x/sigma_y`

The number of complaints which a bank manager receives per day is a Poisson random variable with parameter m = 4. Find the probability that the manager will receive -

(a) only two complaints on any given day.

(b) at most two complaints on any given day

[Use e^{-4} =0.0183]

In the following data, one of the values of Y is missing. Arithmetic means of X and Y series are 6 and 8

X | 6 | 2 | 10 | 4 | 8 |

Y | 9 | 11 | ? | 8 | 7 |

(a) Estimate the missing observation.

(b) Calculate correlation coefficient.