PROBLEM ON QUADRATIC EQUATIONS

1) The product of two consecutive integers is 56. Find the integers.

2) The sum of the squares of two consecutive natural numbers is 41. Find the numbers.

3) Find the natural numbers which differ by 5 and the sum of whose squares is 97.

4) The sum of a number and its reciprocal is 4.25. find the number.

5) Two natural numbers differ by 3. Find the numbers, if sum of their reciprocals is 7/10.

6) Divide 15 into two parts such that the sum of their reciprocals is 3/10.

7) A can do a piece of work in x days and B can do the same work in x+16 days. If both working together can do it in 15 days; find x

8) The square of a number added to one-fifth of it, is equal to 26. Find the number.

9) The sum of the squares of two consecutive positive even Numbers is 52. Find the numbers.

10) Find two consecutive positive odd numbers, the sum of whose squares is 74.

11) Two numbers are in the ratio 3:5. Find the numbers; if the difference between their squares is 144.

12) Three positive numbers are in the ratio 1/2:1/3:1/4. Find the numbers; if the sum of their squares is 244.

13) Divide 20 into two parts such that three times the square of one part exceeds the other part by 10.

14) The sum of two numbers is 32 and their product is 175. Find the numbers.

15) Three consecutive natural numbers are such that the square of the middle number exceeds the difference of the squares of the other two by 60.

16) The ages of two sisters are 11 years and 14 years. In how many years time will the product of their ages be 304 ?

17) A stone is thrown vertically downwards and the formula

d= 16t²+ 4t gives the distance, d metres, that it falls in t seconds. How long does it take to fall 420 metres?

18) One pipe can fill a cistern in 3 hours less than the other. The two pipes together can fill the cistern in 6 hours 40 minutes. Find the time that each pipe will take to fill the cistern.

20) Out of the three consecutive positive integers, the middle number is p. If three times the square of the largest is greater than the sum of the squares of the other two numbers by 67; Calculate the value of p.

21) The sides of a right angled triangle containing the right angle are 4x cm and (2x - 1)cm. If the area of the triangle is 30cm²; find the length of its sides.

22) The hypotenuse of a right angled triangle is 26 cm and the sum of other two sides is 34 cm. Find the lengths of its sides.

23) The hypotenuse of a right angled triangle exceeds one side by 1 cm and other side by 18 cm; find the length of the sides of the triangle.

24) The length of of a rectangle plot exceeds its breadth by 12 m and the area is 1260m². Find its dimensions.

25) The perimeter of a rectangle plot is 104m and its area is 640 m². Find its dimensions.

26) A footpath of uniform width runs round the inside of a rectangular field 32m long and 24m wide. If the path occupies 208m², find the width of the footpath.

27) Two squares have sides x cm and (x+4) cm. The sum of their areas is 656cm². Find the side.

28) The length of a rectangle board exceeds its breadth by 8cm. If the length were decreased by 4cm and the breadth doubled, the area of the board would be increased by 256cm². Find the length of the board.

29) An area is paved with square tiles of a certain size and the number required is 600. If the tiles had been 1cm smaller each way, 864 would have been needed. Find the size of the larger tiles.

30) A farmer has 70m of fencing, with which he encloses three sides of a rectangular sheep pen; the fourth side being a wall. If the area of the pen is 600m², find the length of its shorter side.

31) A square lawn is bound on three sides by a path 4m wide. If the area of the path is 7/8 that of the lawn, find the dimensions of the lawn.

32) The area of a big room is 300 m². If the length were decreased by 5m and the breadth increased by 5m; the area would be unaltered. Find the length of the room.

33) The speed of an ordinary train is x km/hr and that of an express train is (x+25)km/hr

i) Find the time taken by each train to cover 300 km.

ii) If the ordinary train takes 2hrs more than the express train: Calculate speed of the train.

34) If the speed of a car is increased by 10km/hr, it takes 18 minutes less to cover a distance of 36km. Find the speed of the car.

35) If the speed of an aeroplane is reduced by 40km/hr, it takes 20 minutes more to cover 1200km. Find the speed of the aeroplane.

36) A train covers a distance of 300 km/hr. Another train covers the same distance at a speed of (x-5) km/hr.

I) Find the time which each train takes to cover the distance between the stations.

ii) If the second train takes 3 hours more than the first train, find the speed of each train.

37) A girl goes to her friend's house, which is at a distance of 12 km. She covers half of the distance at a speed of X km/hr. and the remaining at a speed of (x+2) km/hr. If she takes 2hrs 30 minutes to cover the whole distances; find x

38) A car made a run of 390 km in 'x' hours. If the speed had been 4 km per hour more, it would have taken 2 hours less for the journey. Find x

39) Rs 250 is divided equally among a certain number of children; if there were 25 children more, each would have received 50 paise less. Find the number of children.

40) A trader bought a number of articles for Rs1200. Ten were damaged and he sold each of the rest at Rs2 more than what he paid for it, thus getting a profit of Rs69 on the whole transaction.

41) Mr. Mehra sends his servant to the market to buy oranges worth Rs15. The servant having eaten three oranges on the way, Mr. Mehra pays 25 paise per orange more than the market price. Find number of oranges.

42) A man bought an article for Rs x and sold it for Rs 16. If his loss was x% find the cost price of the article.

43) A trader bought an article for Rs and sold it for Rs52, thereby making profit of (x-10)% on his outlay. Calculate the cost price, for trader.

44) By selling chair for Rs 75, Mohan gained as much % as its cost. Find the cost of chair.

45) A goods train leaves a station at 6 p.m., followed by an express train which leaves at 8pm. and travel 39 km per hour faster than the goods train. The express train arrives at a station, 180 km away, 36 minutes before the goods train. Find their speed.

46) The product of the digits of a two digit number is 24. If its units digit exceeds twice its ten's digit by 2; find the number.