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Selina solutions Selina ICSE Concise Mathematics for Class 10 chapter 5 Quadratic Equations

Chapters

Selina Selina ICSE Concise Mathematics Class 10

Selina ICSE Concise Mathematics for Class 10

Chapter 5 - Quadratic Equations

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

Without solving, comment upon the nature of roots of the following equations

7x2 – 9x +2 =0

Solve the following quadratic equations by factorization:

`sqrt6x^2 -   4x - 2sqrt6 = 0`

Without solving, comment upon the nature of roots of the following equations

6x2 – 13x +4 =0

Without solving, comment upon the nature of roots of the following equations

25x2 – 10x +1=0

Without solving, comment upon the nature of roots of the following equations :

`x^2 + 2sqrt3x - 9 = 0`

Without solving, comment upon the nature of roots of the following equations

x2 – ax – b2 =0

Without solving, comment upon the nature of roots of the following equations :

2x2 +8x +9=0

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

Find the value of p, if the following quadratic equation has equal roots: 4x2 – (p – 2)x + 1 = 0

Find the value of ‘p’, if the following quadratic equations have equal roots : x2 + (p – 3)x + p = 0

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

The equation 3x2 – 12x + (n – 5)=0 has equal roots. Find the value of n.

Find the value of m, if the following equation has equal roots : (m – 2)x2 – (5+m)x +16 =0

The sum of a number and its reciprocal is 4.25. Find the number.

Find the value of p for which the equation 3x2– 6x + k = 0 has distinct and real roots.

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

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

The sum of the squares of two positive integers is 208. If the square of the large number is 18 times the smaller. Find the numbers.

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

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

The denominator of a fraction is one more than twice the numerator. If the sum of the fraction and its reciprocal is 2.9; find the fraction.

The denominator of a fraction is one more than twice the numerator. If the sum of the fraction and its reciprocal is 2.9; find the fraction.

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

The monthly pocket money of Ravi and Sanjeev are in the ratio 5:7. Their expenditures are in the ratio 3:5. If each saves Rs. 80 every month, find their monthly pocket money.

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.
Assume the middle number to be x and form a quadratic equation satisfying the above statement. Hence; find the three numbers.

Out of 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.

Out of 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.

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; calculate ‘x’.

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.

A positive number is divided into two parts such that the sum of the squares of the two parts is 20. The square of the larger part is 8 times the smaller part. Taking x as the smaller part of the two parts, find the number.

A positive number is divided into two parts such that the sum of the squares of the two parts is 20. The square of the larger part is 8 times the smaller part. Taking x as the smaller part of the two parts, find the number.

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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 30 cm²; calculate the lengths of its sides.

Solve : x² – 10x – 24 = 0

Solve : x² – 16 = 0

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.

Solve `2x^2 - 1/2 x = 0`

The sides of a right-angled triangle are (x – 1) cm, 3x cm and (3x + 1) cm. Find:

(1) the value of x,
(2) the lengths of its sides,
(3) its area.

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

Solve : x(x – 5) = 24

The diagonal of a rectangle is 60 m more than its shorter side and the larger side is 30 m more than the shorter side. Find the sides of the rectangle.

Solve `9/2 x = 5 + x^2`

Solve `6/x = 1 + x`

The perimeter of a rectangle is 104 m and its area is 640 m². Find its length and breadth.

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

Solve `x = (3x + 1)/(4x)`

Two squares have sides x cm and (x + 4) cm. The sum of their area is 656 sq. cm. Express this as an algebraic equation in x and solve the equation to find the sides of the squares.

Solve `x + 1/x = 2.5`

Solve : (2x – 3)² = 49

The dimensions of a rectangular field are 50 m and 40 m. A flower bed is prepared inside this field leaving a gravel path of uniform width all around the flower bed. The total cost of laying the flower bed and gravelling the path at Rs 30 and Rs 20 per square metre, respectively, is Rs 52,000. Find the width of the gravel path.

Solve : 2(x² – 6) = 3(x – 4)

An area is paved with square tiles of a certain size and the number required is 128. If the tiles had been 2 cm smaller each way, 200 tiles would have been needed to pave the same area. Find the size of the larger tiles.

Solve : (x + 1)(2x + 8) = (x + 7)(x + 3)

A farmer has 70 m 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 600 sq. m, find the length of its shorter side.

Solve : x² – (a + b)x + ab = 0

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

(x + 3)² – 4(x + 3) – 5 = 0

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

4(2x – 3)² – (2x – 3) – 14 = 0

Solve `(3x - 2)/(2x - 3) = (3x - 8)/(x + 4)`

2x2 – 9x + 10 = 0, When
(1) x∈ N
(2) x∈ Q

Solve `(x - 3)/(x + 3) + (x + 3)/(x - 3) = 2 1/2`

Solve `4/(x + 2) - 1/(x + 3) = 4/(2x + 1)`

Solve `5/(x - 2) -  3/(x + 6) = 4/x`

Solve `(1 + 1/(x + 1))(1-1/(x - 1)) = 7/8`

Find the quadratic equation, whose solution set is: {3,5}

Find the quadratic equation, whose solution set is: {-2, 3}

Solve `x/3 + 3/(6 - x) = (2(6 + x))/15; (x ≠ 6)`

Solve the equation `9x^2 + (3x)/4 + 2 = 0` if possible for real values of x

Find the value of x, if a + 1=0 and x2 + ax – 6 =0.

Find the value of x, if a + 7=0; b + 10=0 and 12x2 = ax – b.

Use the substitution y= 2x +3 to solve for x, if 4(2x+3)2 – (2x+3) – 14 =0.

Without solving the quadratic equation 6x2 – x – 2=0, find whether x = 2/3 is a solution of this equation or not.

Determine whether x = -1 is a root of the equation x2 – 3x +2=0 or not.

If x = 2/3 is a solution of the quadratic equation 7x2+mx – 3=0; Find the value of m.

If x = -3 and x = 2/3 are solutions of quadratic equation mx+ 7x + n = 0, find the values of m and n

If quadratic equation x2 – (m + 1) x + 6=0 has one root as x =3; find the value of m and the root of the equation

Given that 2 is a root of the equation 3x² – p(x + 1) = 0 and that the equation px² – qx + 9 = 0 has equal roots, find the values of p and q.

Solve `x/a - (a + b)/x = (b(a +b))/(ax)`

Solve `(1200/x + 2)(x - 10) - 1200 = 60`

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The speed of an ordinary train is x km per hr and that of an express train is (x + 25) km per hr.
(1) Find the time taken by each train to cover 300 km.

(2) If the ordinary train takes 2 hrs more than the express train;calculate speed of the express train.

Solve the following quadratic equations by factorization:

`4/x - 3 = 5/(2x + 3)`

Solve the following equation using the formula

`x^2 + 10x + 21 = 0`

Solve of the following equation using the formula

`x^2 + 6x - 10 = 0`

Solve the following equation using the formula

`x^2 + 2x - 6 = 0`

Solve the following equation using the formula

`3x^2 + 2x - 1 = 0`

Solve the following equations using the formula

`2x^2 + 7x + 5 = 0`

Solve the following quadratic equations by factorization:

`2/3 x = - 1/6 x^2  - 1/3`

Solve the following quadratic equations by factorization:

`1/15 x^2 + 5/3 = 2/3 x`

Solve the following quadratic equations by factorization:

`x^2 - 6 = 2sqrt2`

Solve the following equation using the formula

`x^2 - 6x = 27`

Solve the following quadratic equations by factorization:

`(2x + 3)/(x + 3)= (x + 4)/(x + 2)`

Solve the following quadratic equations by factorization:

`(2x)/(x - 4) +  (2x- 5)/(x - 3) = 8 1/3`

Solve the following quadratic equations by factorization:

(x - 1)/(x - 2) +   (x - 3)/(x - 4)  =  3 1/3

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

Solve the following equations for x and give, in each case, your answer correct to one decimal place :

x2 – 8x+5=0

Solve the following equations for x and give, in given case, your answer correct to one decimal place :

5x2 +10x – 3 =0

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

Solve each of the following equations for x and give, in each case, your answer correct to two decimal places :

2x2 – 10x +5=0

Solve each of the following equations for x and give, in each case, your answer correct to two decimal places :
4x + 6/x + 13 = 0

Solve each of the following equations for x and give, in each case, your answer correct to two decimal places :

x2 – 3x – 9 =0

Solve each of the following equations for x and give, in each case, your answer correct to two decimal places :

`x^2 - 5x - 10 = 0`

A car covers a distance of 400 km at a certain speed. Had the speed been 12 km/h more, the time taken for the journey would have been 1 hour 40 minutes less. Find the original speed of the car.

Solve the following equations for x and give, in each case, your answer correct to 3 decimal places

3x2 – 12x – 1 =0

Solve the following equations for x and give, in each case, your answer correct to two decimal places :

`x^2 - 16x + 6 = 0`

Solve the following equations for x and give, in each case, your answer correct to 3 decimal places 

2x2 + 11x + 4= 0

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 distance at a speed of (x + 2) km/hr. If she takes 2 hrs 30 minutes to cover the whole distance, find ‘x’.

Solve x4 – 10x2 +9 =0

Solve:
x4 – 2x2 – 3 =0

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

Solve : (x2 – x)2 + 5(x2 – x)+ 4=0

Solve (x2 – 3x)2 – 16(x2 – 3x) – 36 =0

A goods train leaves a station at 6 p.m., followed by an express train which leaved at 8 p.m. and travels 20 km/hour faster than the goods train. The express train arrives at a station, 1040 km away, 36 minutes before the goods train. Assuming that the speeds of both the train remain constant between the two stations; calculate their speeds.

Solve `sqrt(x/(x- 3)) + sqrt((x - 3)/x) = 5/2`

Solve `((2x - 3)/(x -1)) - 4((x - 1)/(2x - 3)) = 3`

Solve `((3x + 1)/(x + 1)) + ((x + 1)/(3x + 1)) = 5/2`

A man bought an article for Rs x and sold it for Rs 16. If his loss was x per cent, find the cost price of the article.

Solve the equation `2x - 1/x = 7`.Write your answer correct to two decimal places.

Solve the following equation and give your answer correct to 3 significant figures: 5x² – 3x – 4 = 0

A trader bought an article for Rs x and sold it for Rs 52, thereby making a profit of (x – 10) per cent on his outlay. Calculate the cost price.

By selling a chair for Rs 75, Mohan gained as much per cent as its cost. Calculate the cost of the chair.

Solve for x using the quadratic formula. Write your answer correct to two significant figures.
(x – 1)2 – 3x + 4 = 0

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Solve `(2x)/(x - 3) + 1/(2x + 3) + (3x + 9)/(x - 3)(2x +3) = 0; x != 3, x != - 3/2`

The sum S of n successive odd numbers starting from 3 is given by the relation n(n + 2). Determine n, if the sum is 168.

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?

Solve: (2x+3)= 81

Solve `a^2x^2 - b^2 = 0`

The product of the digits of a two digit number is 24. If its unit’s digit exceeds twice its ten’s digit by 2; find the number.

Solve `x^2 - 11/4 x + 15/8 = 0`

One year ago, a man was 8 times as old as his son. Now his age is equal to the square of his son’s age. Find their present ages.

Solve `x + 4/x = -4; x != 0`

Solve: 2x4 – 5x² + 3 = 0

The age of the father is twice the square of the age of his son. Eight years hence, the age of the father will be 4 years more than three times the age of the son. Find their present ages.

Solve: x4 – 2x² – 3 = 0.

The speed of a boat in still water is 15 km/hr. It can go 30 km upstream and return downstream to the original point in 4 hours 30 minutes. Find the speed of the stream.

Solve `9(x^2 + 1/x^2) - 9(x - 1/x) - 52 = 0`

Mr. Mehra sends his servant to the market to buy oranges worth Rs 15. The servant having eaten three oranges on the way. Mr. Mehra pays Rs 25 paise per orange more than the market price.
Taking x to be the number of oranges which Mr. Mehra receives, form a quadratic equation in x. Hence, find the value of x.

Solve `2(x^2 + 1/x^2) - (x + 1/x) = 11`

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

Solve `(x^2 + 1/x^2) - 3(x - 1/x) - 2 = 0`

An employer finds that if he increased the weekly wages of each worker by Rs 5 and employs five workers less, he increases his weekly wage bill from Rs 3,150 to Rs 3,250. Taking the original weekly wage of each worker as Rs x; obtain an equation in x and then solve it to find the weekly wages of each worker.

A trader bought a number of articles for Rs 1,200. Ten were damaged and he sold each of the remaining articles at Rs 2 more than what he paid for it, thus getting 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.

Solve : (x² + 5x + 4)(x² + 5x + 6) = 120

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

The total cost price of a certain number of identical articles is Rs 4800. By selling the articles at Rs 100 each, a profit equal to the cost price of 15 articles is made. Find the number of articles bought.

Solve each of the following equations for x and give, in each case, your answer correct to two decimal places :

`x^2 - 5x - 10 = 0`

Solve of the following equations, giving answer upto two decimal places.

3x2 – x – 7 =0

Solve `(x/(x + 2))^2 - 7(x/(x + 2)) + 12 = 0; x != -2`

Solve : x2 – 11x – 12 =0; when x ∈ N

Solve x2 – 4x – 12 =0; when x ∈ I

Solve  2x2 – 9x + 10 =0; when x ∈ Q

Solve : (a + b)²x² – (a + b)x – 6 = 0; a + b ≠ 0

Solve `1/p + 1/q + 1/x = 1/(x + p + q)`

Solve x(x + 1) + (x + 2)(x + 3) = 42

Solve `1/(x - 1) - 2/(x + 2) = 3/(x + 3) - 4/(x + 4)`

For each equation given below find the values of m so that the equation has equal roots. Also find the solution of equation

`(m - 3)x^2 - 4x + 1 = 0` 

For each equation given below find the values of m so that the equation has equal roots. Also find the solution of equation

`3x^2 + 12x + (m + 7) = 0`

For each equation given below find the values of m so that the equation has equal roots. Also find the solution of equation

`x^2 - (m + 2)x + (m + 5) = 0`

Without solving the following quadratic equation Find the value of p for which the roots are equal

`px^2 - 4x + 3 = 0`

Without solving the following quadratic equation, find the value of m for which the given equation has equation has real and equal roots.

`x^2 + 2(m - 1)x + (m + 5) = 0`

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The distance by road between two towns A and B is 216 km, and by rail, it is 208 km. A car travels at a speed of x km/hr and the train travels at a speed which is 16 km/hr faster than the car. Calculate:
(1) the time is taken by the car to reach town B from A, in terms of x;
(2) the time is taken by the train to reach town B from A, in terms of x.
(3) If the train takes 2 hours less than the car, to reach town B, obtain an equation in x and solve it.
(4) Hence, find the speed of the train.

A trader buys x articles for a total cost of Rs 600.
(i) Write down the cost of one article in terms of x.
If the cost per article were Rs 5 more, the number of articles that can be bought for Rs 600 would be four less.
(ii) Write down the equation in x for the above situation and solve it for x.

A hotel bill for a number of people for the overnight stay is Rs 4800. If there were 4 people more, the bill each person had to pay, would have reduced by Rs 200. Find the number of people staying overnight.

An Aero plane 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:
(1) the onward journey;
(2) 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.

Rs 6500 was divided equally among a certain number of persons. Had there been 15 persons more, each would have got Rs 30 less. Find the original number of persons.

A plane left 30 minutes later than the scheduled time and in order to reach its destination 1500 km away in time, it has to increase its speed by 250 km/hr from its usual speed. Find its usual speed.

Two trains leave a railway station at the same time. The first train travels due west and the second train due north. The first train travels 5 km/hr faster than the second train. If after 2 hours, they are 50 km apart, find the speed of each train.

The sum S of first n even natural numbers is given by the relation S = n(n + 1). Find n, if the sum is 420.

The sum of the ages of a father and his son is 45 years. Five years ago, the product of their ages (in years) was 124. Determine their present ages.

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:
(1) the number of rows in the original arrangement.
(2) the number of seats in the auditorium after re-arrangement.

Mohan takes 16 days less than Manoj to do a piece of work. If both working together can do it in 15 days, in how many days will Mohan alone complete the work?

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.

In a certain positive fraction, the denominator is greater than the numerator by 3. If 1 is subtracted from the numerator and the denominator both, the fraction reduces by. Find the fraction.

In a two-digit number, the ten’s digit is bigger. The product of the digits is 27 and the difference between two digits is 6. Find the number.

Some school children went on an excursion by bus to a picnic spot at a distance of 300 km. While returning, it was raining and the bus had to reduce its speed by 5 km/hr and it took two hours longer for returning. Find the time taken to return.

Selina Selina ICSE Concise Mathematics Class 10

Selina ICSE Concise Mathematics for Class 10
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