MASTERS TUITION CENTER DEEPAK SIR QUADRATIC EQUATIONS

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Basil Gregory
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1 Type-30 1) Find two consecutive natural numbers whose square have the sum ) The sum of the squares of three consecutive natural numbers is 149. Find them. 3) Three consecutive natural numbers are such that the square jf the middle number exceeds the difference of the squares of the other two by 60. Find the three numbers. Type-31 4) Find two consecutive positive even integers whose squares have the sum ) Find two consecutive positive odd integers whose squares have the sum ) If the product of two positive consecutive even. integers is 288. Find the integers. Type-32 7) The product of two successive multiples of 5 is 300. Find them. 8) Find two consecutive multiples of 3 whose product is ) The product of two successive multiples of 4 is 28 more than the first multiple. Find them. Type-33 10) Divide 57 into two parts whose product is ) Divide 29 into two parts such that the sum of the squares of the parts is ) Divide 16 into two parts such that the twice the square of the larger part exceeds the square of smaller part by 164.The sum of two numbers is 8 and the sum of their squares is 34. Find the numbers. Type-34 13) A two digit number contains the smaller of two digits in the unit's place. The product of the digits is 24 and the difference between the digits is 5. Find the number. 14) A two digit number is such that the product of its digits is 8. When 18 is subtracted from the number, the digits interchange their place. Find the number. 15) A two digit number is such that the product of its digits is 12. When 36 is added to the number, the digits interchange their place. Find the number. Type-35 16) A person on tour has Rs. 360 for his expenses. If he extends his tour for 4 days, he has to cut down his daily expenses by Rs. 3. Taking the original duration of tour as x, form an equation in x and solve it. 17) A shopkeeper buys a number of books of Rs. 80, if he had bought 4 more books for the same amount, each book would have cost Rs. 1 less. Taking the original number of books as x, form an equation in x and solved it. 18) A trader bought a number of articles for Rs Ten were damaged and he sold each of the rest at Rs. 2 more than what he paid for it, thus, clearing a profit of Rs. 60 on the whole transaction. Taking the number of articles he bought as x, form an equation in x and solve it. 19) The hotel bill for a number of people for overnight stay is Rs. 4800, if there were 4 more, the bill each person had to pay would have reduced by Rs Find the number of people staying overnight. 20) A trader buys x articles for a total cost of Rs. 600.

2 (i) Write down the cost of one article in terms of A;. If the cost per article were Rs. 5 more, the number of articles that can be bought for Rs. 600 would be four less. (i i ) Write down the equation in x for the above situation and solve it to find x. Type-36 21) An aeroplane travelled a distance of 400 km at an average speed of x km/hr. On the return journey, the speed was increased by 40 km/hr. Write down an expression for the time taken for: (i) The onward journey, (i i ) The return journey. If the return journey took 30 minutes less than the onward journey, write down an equation in x and find its value. 22) The distance by road between two towns A and B, is 216 km, and by train it is 208 km. A car travels at a speed of km/hr, and the train travels at a speed which is 16 km/hr faster than the car. Calculate. (i) The time taken by the car to reach town B from A, in terms of x. (ii) The time taken by the train to reach town B from A, in terms of x. (iii) If the train takes 2 hours less than the car, to reach town B, obtain an equation in x, and solve it. (iv) Hence find the speed of the train. 23) Car A travels x km for every litre of petrol, while Car B travels (x + 5) km for every litre of petrol. a. Write down the number of litres of petrol, used by Car A and Car B in covering a distance of 400 km. (i f ) If Car A uses 4 litres of petrol more than Car B in covering the 400 km, write down an equation in x and solve it to determine the number of litres of petrol/used by Car B for the journey. 24) A train covers a distance of 600 km at x km/hr. Had the speed been (x + 20) km/hr, the time taken to cover the distance would have been reduced by 5 hours. Write down an equation in x and solve it to evaluate x. 25) A train travels a distance of 300 km at constant speed, if the speed of the train is increased by 5 km/hr, the journey would have been 2 hours less. Find the original speed of the train. 26) Prateek covers a distance of 1.5 km between home to school at a speed of 'x' km/hr and his sister Meenu covers the same distance at a speed of(x-4) km/hr. (i Find the time taken by Prateek and his sister Meenu in reaching the school. (i i ) If Meenu takes 6 minutes more than Prateek, find the value of 'x'. Type-37 27) The speed of a boat in still water is 15 km/hr. It can go 30 km upstream and return down stream to the original point in 4 hours 30 minutes. Find the speed of stream. 28) The speed of a boat in still water is 8 km/hr. It can go 15 km upstream and 22 km down stream in 5 hours. Find the speed of stream. 29) Shivika can row a boat at the rate of 4 km/hr in still water. She takes 8 hours in going 12 km upstream and 12 km downstream. Find the speed of stream.

3 Type-38 30) The hypotenuse of a right-angled triangle is 6 m more than twice the shortest side, if the third side is 2m less than the hypotenuse. Find all sidej of the triangle. 31) The length of hypotenuse of a right-angled triangle is one unit more than twice the length of the shortest side and other side is one unit less than twice the length of shortest side. Find all sides of the triangle. Also find the area. 32) The hypotenuse of a right-angled triangle is cm if the smaller leg is tripled and longer leg is doubled, the new hypotenuse will be 9 5 cm. Find the remaining two sides of the triangle. 33) The lengths (in metres) of the sides of a right triangle are 2x -1, 4x and 4x+ 1, x > 0. Find, (i) value of x and (ii) the area of the triangle. 34) A 112 cm long wire is bent to form a right- angled triangle with hypotenuse 50 cm. Find the area of the triangle so formed. Type-39 35) The product of Shubham's age 5 years ago with his age 9 years later is 15. Find his present age. 36) One year ago, the father was 8 times as old as his son. Now his age is the square of his son age. Find their present ages. 37) Radha is twice as old as Mohini. After 3 years, Radha's age will be 1 times the age of Mohini. Find their present ages. 38) Two years ago, a man's age was three times the square of his son's age. In three years' time, his age will be four times his son's age. Find their present ages. 39) When Mrs. Pratibha was asked her age, she replied, "If you subtract 21 times my age from the square of my age, the result is 100". Find her age. Type-40 40) A takes 6 days less than the time taken by B to finish a work, if both can finish the work in 4 days. Find the time of B to finish the work. 41) Two pipes running together can fill a cistern in 2 minutes, if one pipe takes one minute more than the other to fill the cistern. Find the time in which each pipe fill the cistern. 42) A tank can be filled by one pipe in 'x' minutes and emptied by another in (x + 5) minutes. Both the pipes when opened together, can fill an empty tank in 16.8 minutes. Find V. Type-41 43) A square lawn has a path 4 m wide around it. The area of the path is 196 m 2. Find the side of the lawn. 44) A rectangular garden 10 m by 16 m is to be surrounded by a concrete walk of uniform width. Given that the area of the walk is 120 sq. m, assuming the width of the walk to be x, form an equation in x and solve it to find the value of x. 45) A rectangle of ai 3a 105 cm 2 has its length equal to x cm. Write down its breadth in terms of x. Given that the perimeter is 44 cm. Write down an equation in x and solve it to find the dimension of the rectangle. 46) The length of a rectangle exceeds its breadth by 5m. If the breadth were doubled and the length reduced by 9 m, the area of the rectangle would have increased by 140 m 2. Find its dimensions. 47) The perimeter of a rectangular plot is 180 m and its area is 1800 m 2. Take the length of the plot as x m. Use the perimeter 180 m to write the value of the breadth in terms of x. Use the

4 values of length, breadth and the area to write an equation inx. Solve the equation to calculate the length and breadth of the plot. Type-42 48) One-fourth of a herd of Camels was seen in the forest, twice the square root of the herd had gone to a mountain and the remaining 15 Camels were seen on the bank of a river. Find total number of Camels. 49) 7/2 times the square root of the total number of Swans are playing in a pond and remaining two are swimming in the water. Find total number of Swans. 50) Out of a certain number of Saras birds, one- fourth are moving about a lotus plant, oneninth coupled with one-fourth as well as 7 times the square root of the total number of Saras birds move on a hill, 56 saras birds remain at the tree. Find the total num 1 """- "f Saras birds. 51) Angiy Arjun carried some arrows for fighting with Bheesham. With half that arrows he cut down the arrows thrown by Bheesham on him and with six other arrows he killed the rath driver of Bheesham, with one arrow each he knocked down respectively the rath, flag and the bow of the Bheesham. Finally, with one more than four times the square root of the arrows, he laid Bheesham unconscious on an arrow bed. Find the total number of arrows Arjun had. Type-43 52) Pankaj sold an article for Rs. 56 which cost him Rs. x. He finds that he has gained x % on his outlay. 53) Find 2x articles cost Rs. (5x + 54) and (x + 2) similar articles cost Rs. (lox - 4). Find x. 54) Reeta bougth a saree for Rs. 60x and sold it for Rs. ( x) at a loss x %. Find the cost price of saree. 55) A man sells a table for Rs. 96 and gains as much per cent as the table cost him. Find the cost of the table. Type-44 56) In a certain positive fraction, the denominator is greater than the numerator by 3. If 1 is subtracted from both the denominator and numerator, the fraction is decreased by 1/14. Find the fraction. 57) The sum of the numerator and denominator of a certain positive fraction is 8. If 2 is added to both the numerator and denominator, the fraction is increased by 4/35. Find the fraction. 58) The denominator of a fraction is 1 more than its numerator. The sum of the fraction and its reciprocal is 2 j Find the fraction. Type-45 59) The total surface area of a hollow metal cylinder, open at both ends, of external radius 10 cm and height 12 cm is 512 n cm 2. Taking x to be inner, radius, write down an equation in x and 60) The total surface area of a hollow metal cylinder open at both ends of external radius 18 cm, and height 20 cm is 1568 n cm 2. Takingx to be inner radius, write down an equation in x and 61) The total surface area of a hollow metal cylinder open at both ends, of external radius 8 cm, and height 10 cm is 338 n cm 2. Taking x to be inner radius, write down an equation in x and

5 Type-46 62) Two circles touch each other externally. The sum of their areas is 130 n cm 2 and the distance between their centres is 14 cm. Find the radius of each circle. 63) X and Y are the centres of circles of radius 9 cm and 2 cm and XY = 17 cm. Z is the centre of a circle of radius x cm, which touches the above circles externally. Given that ZXZY = 90. Find the radius x. 64) P and Q are centres of circles of radius 9 cm and 2 cm respectively. PQ = 17 cm. R is the centre of a circle of radius x cm, which touches the above circles externally. Given that ZPRQ = 90, write an equation in x and solve it. [2004] Type-47 65) In the morning assembly of a school, 480 students are arranged in rows and columns. If there are 4 more students in each row than the number of columns, find the number of students in each row. 66) In a picture hall, the number of rows was equal to the number of seats in each row. If the number of rows is doubled and the number of seats in each row is reduced by 5, then the total number of seats is increased by 375. How many rows were there? 67) In a mango groove the trees are planted in horizontal rows. There are 6 trees more in each horizontal row. Altogether there are 720 trees. Find the number of trees in horizontal rows. 68) In an auditorium, seats were arranged in rows and columns. The number of rows was equal to the number of seats in each row. When the number of rows was doubled and the number of seats in each row was reduced by 10, the total number of seats increased by 300. Find : i.the number of rows in the originalarrangement. (ii The number of seats in the auditorium after re-arrangement.

4. QUADRATIC EQUATIONS

4. QUADRATIC EQUATIONS . QUADRATIC EQUATIONS Important Terms, Definitions and Results The roots of a quadratic equation ax + bx + c = 0 An equation of the form ax + bx + c = 0, where a, b, c are real numbers and a 0, is called