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COMPLEX NUMBERS
1. What is iota (i)?
The equations of the form x2 + 1 = 0 or x2 + 9 = 0 are not solvable in real numbers (R) i.e. there is no real number whose square is a negative real number.
EULER was the first mathematician to introduce the symbol i (iota) for the square root of - 1, which is a solution of x2 + 1 = 0 i.e. i2 = -- 1. He also called this symbol as imaginary unit.
So, i = sq.root of - 1 i.e. / - 1
i2 = - 1
i3 = - i
i4 = 1
2. What is the probability of getting 53 sundays in a leap year?
Solution
There are 366 days in a leap year i.e. 52 complete weeks and 2 more days.
Therefore, a leap year can have 53 sundays only when one of the extra day is sunday. Those two days can be:
Total possible outcomes = {(Sunday, Monday), (Mon, Tues.), (Tues., Wed.), (Wed., Thurs), (Thurs, Fri), (Fri, sat), (Sat, Sun.)}
No. of total possible outcomes = 7
Favourable outcomes = {(Sun,Mon), (Sat,Sun)}
Favourable outcomes 2
P(E) = ------------------------------------- = --- Ans
Total possible outcomes 7
3.
4. The total expenditure of a family on a certain quantity of seven household consumables was found to be Rs.10,000 in the year 1990. if the cost of living index number for the year 1995 taking 1990 as base yearis 142.5. find the total expenditure of the family on the same quantity of the same seven household
consumables in the year 1995?
Solution Expenditure in current year
Cost of living index = ------------------------------------- x 100
Expenditure in base year
Expenditure in 1995
Therefore, 142.5 = ---------------------------- x 100
10,000
Expenditure in 1995 = Rs.14,250 Ans.
5. A man travels 600km partly by train and partly by car. if he covers 400km by train and the rest by rar. it takes him 6hr and 30mts. but if he travels 200km by train and the rest by car, he takes an hour longer. find the speed of the train and that of the car?
Solution Let the speed of train be x km/hr and that of car be y km/hr.
Time taken by train + Time taken by car = 6 1/2 hrs = 13/2 hrs.
400 / x + 200 / y = 13/2 Using Time = distance/speed
& 200 / x + 400 / y = 15/2
Solve the two equations to get the values of x and y as 1200/11 km/hr and 1200/17 km/hr
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