Nootan solutions for मैथमैटिक्स [अंग्रेजी] कक्षा १० आईसीएसई chapter 5 - Quadratic equations [Latest edition]

In the following, show that the given numbers are the solution of the given equation:

x² − 7x + 12 = 0; x = 4

In the following, show that the given numbers are the solution of the given equation:

`6x^2 + x − 7 = 0; x = 7/6`

In the following, show that the given numbers are the solution of the given equation:

2x² + 5x − 3 = 0; x = −3

In the following, show that the given numbers are the solution of the given equation:

`x^2 + 3sqrt2x - 8=0; x = sqrt2`

In the following, determine whether the given number are the solution of the given equation or not:

`x^2 - sqrt2 x -4=0; x= -sqrt2`

In the following, show that the given numbers are the solution of the given equation:

4x² − 4x + 1 = 0; x = −`1/2`

In the following, show that the given numbers are the solution of the given equation:

7x² + 11x − 6 = 0; x = 2

In the following, show that the given numbers are the solution of the given equation:

`2x^2 - 2sqrt6x + 3 =0; x = sqrt3`

If x = 3 is a root of the quadratic equation x² + mx – 9 = 0 then find the value of m.

If x = −4 is a root of the quadratic equation 2x² − 7x + q = 0, then find the value of q.

If x = 3 and x = –1 are the roots of x² + mx + n = 0, then find the values of m and n.

If x = `1/2` and x = –2 are the roots of 2x² + px + q = 0, find the values of p and q.

If x = 5 is a root of both the equations x² + mx – 15 = 0 and 2x² – 7x + n = 0, then find the value of mn + 2m – n.

Solve the following equation by factorisation method:

5x² − 3x − 2 = 0

Solve the following equation by factorisation method:

9x² − 64 = 0

Solve the following equation by factorisation method:

42 − x² = 3x − 2x

Solve the following equation by factorisation method:

8x² − 22x − 21 = 0

Solve the following equation by factorisation method:

21x² + 8x − 4 = 0

Solve the following equation by factorisation method:

10x² + 11x − 6 = 0

Solve the following equation by factorisation method:

12x² − 11x + 2 = 0

Solve the following equation by factorisation method:

x² + ax − bx − ab = 0

Solve the following quadratic equations by factorization:

ax² + (4a² − 3b)x − 12ab = 0

Solve the following equation by factorisation method:

a²b²x² − (a² + b²) x + 1 = 0

Solve the following equation by factorisation method:

abx² − (a² + b²)x + ab = 0

Solve the following equation by factorisation method:

`sqrt3x^2 + 10x + 7sqrt3 = 0`

Solve the following equation by factorisation method:

`5sqrt3x^2 + 7x-2sqrt3 = 0`

Solve the following equation by factorisation method:

`2/3 x^2-1/3 x-1 = 0, x\cancel=0`

Solve the following equation by factorisation method:

`16/x^2 - 6/x - 1 = 0, x\cancel=0`

Solve the following equation by factorisation method:

`(x+1)/x + x/(x+1) = 4 1/4, x\cancel=0, -1`

Solve the following equation by factorisation method:

`(x-1)/(x+1) + (x+1)/(x-1) = 29/10, x\cancel= 1, -1`

Solve the following equation by factorisation method:

`x+1/x = 10/3, x\cancel=0`

Solve the following equation by factorisation method:

`(x+4)/(x+5) - x/(x+1) = 1/8, x \cancel= -5,-1`

Solve the following equation by factorisation method:

`1/x + ((x-1))/2 = x, x\cancel=0`

Solve the following equation by factorisation method:

`x = (4x+1)/5x, x = 0`

Solve the following equation by factorisation method:

`a/(ax-1) + b/(bx-1) = a+b, a+b \cancel= 0, ab \cancel= 0`

Solve the following equation by factorisation method:

`1/(2a+b+2x) = 1/(2a) + 1/b + 1/(2x), x \cancel= 0, a \cancel= 0, b \cancel= 0, x \cancel= -((2a+b))/2`

Solve the following equation by factorisation method:

`sqrt(6x+7) = x`

Solve the following equation by factorisation method:

`sqrt(x(x-7)) = 3sqrt2`

Solve the following equation by factorisation method:

`2((2x+3)/(x-3))-25((x-3)/(2x+3)) = 5, x\cancel=3, x\cancel= -3/2`

Solve the following equation by factorisation method:

`((4x-3)/(2x+1))-10((2x+1)/(4x-3))=3,x≠-1/2,3/4` 

If one root of the equation 5x² + kx − 2 = 0 is 1, find the value of k and also find other root.

If one root of `sqrt2 x^2-kx-2sqrt2 = 0  "is"  2sqrt2`, find the value of k and also find other root.

If x² + kx – 12 = 0 and k + 1 = 0, find the value of x.

Solve: 2x − 3 = `sqrt(2x^2-2x+21)`

Solve for ‘x’ if 3(2x + 3)² + 7 (2x + 3) − 40 = 0 by using the substitution y = 2x + 3.

Solve for ‘x’ if (x + 2)² − 6 (x + 2) – 16 = 0 by using the substitution y = x + 2.

If x = k is a solution of the equation x(3x – 7) – 10 = 0, then find the value of k.

Solve by completing the square:

y² − 5y − 6 = 0

Solve by completing the square:

x² − x − 20 = 0

Solve by completing the square:

4x² − 3 = 2x

Solve by completing the square:

2x² + 5x − 6 = 0

Solve by completing the square:

z² − 3z + 1 = 0

Solve by completing the square:

x² + 3x − 8 = 0

Solve by completing the square:

3x² − 5x − 1 = 0

Solve by completing the square:

4x² + 4√3x + 3 = 0

Solve using quadratic formula:

3x² + x − 4 = 0

Solve by completing the square:

7x + 2 = −3x²

Solve by completing the square:

2x² + 2x − 12 = 0

Solve by completing the square:

`1/2 x^2 - sqrt11 x + 1 = 0`

Solve by completing the square:

x² − 2ax + 3x − 6a = 0

Solve the quadratic equation: x² – 2ax + (a² – b²) = 0 for x.

Solve the following equation and give your answer up to two decimal places:
x² − 5x − 10 = 0

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

Solve the following quadratic equation: x² + 4x − 8 = 0

Give your answer correct to one decimal place.

Solve 5x² − 3x − 4 = 0 and give your answer correct to two significant figures.

Solve the following equation:

`x - 18/x = 6` Give your answer correct to two significant figures.

Solve:

`(x+2)/(x-2) + (x-2)/(x+2) - 4 = 0`

Solve the following quadratic equation for x and give your answer correct to three significant figures: 2x² − 10x + 5 = 0

Determine the discriminant of the following:

x² + 2x + 4 = 0

Determine the discriminant of the following:

`3sqrt3x^2 + 10x + sqrt3 = 0`

Determine the nature of root of the following:

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

Determine the nature of root of the following:

2x² − 3x + 4 = 0

Determine the nature of root of the following:

x² + x + 1 = 0

Determine the nature of root of the following:

x² − 4x + 2 = 0

Determine whether the following equation has real roots or not. If yes, find them:

3x² − 2x + 2 = 0

Determine whether the following equation has real roots or not. If yes, find them:

`3x^2 + 2 - sqrt5x - 5 = 0`

Determine whether the following equation has real roots or not. If yes, find them:

3x² + 9x + 4 = 0

Determine whether the following equation has real roots or not. If yes, find them:

7x² + 8x + 2 = 0

`x^2-6x+4=0`

Determine whether the following equation has real roots or not. If yes, find them:

`3x^2 + 3sqrt5x - 5 = 0`

Show that the following equation has repeated roots:

4x² + 20x + 25 = 0

Show that the following equation has repeated roots:

9x² − 6x + 1 = 0

Find the value of ‘p’ for which the roots of the following equation are real and equal:

4x² + px + 9 = 0

Find the value of ‘p’ for which the roots of the following equation are real and equal:

9x² − 24x + p = 0

Find the value of ‘p’ for which the roots of the following equation are real and equal:

(3p + 1) x² + 2(p + 1) x + p = 0

Find the value of ‘p’ for which the roots of the following equation are real and equal:

(p + 1) x² − 2(p − 1) x + 1 = 0

Find the value of ‘p’ for which the roots of the following equation are real and equal:

x² − 2(p + 1) x + p² = 0

Find the value of k for which the given quadratic equation has real and distinct roots: 

x² − kx + 9 = 0

Find the value of ‘k’ for which the following quadratic equation has real roots:

x² + kx + 4 = 0

Find the values of k for which the given quadratic equation has real and distinct roots:

kx² + 6x + 1 = 0

In the following determine the set of values of k for which the given quadratic equation has real roots:

2x² − 5x − k = 0

Find the value of ‘k’ for which the following quadratic equation has real roots:

2x² + kx − 4 = 0

If the equation (1 + m²) x² + 2mcx + c² – a² = 0 has equal roots then show that c² = a² (1 + m²)

The product of two consecutive positive integers is 156. Find the integers.

Find two consecutive positive integers such that the sum of their squares is 181.

The product of two consecutive even natural numbers is 440. Find the numbers.

The sum of two numbers is 8 and 15 times the sum of their reciprocals is also 8. Find the numbers.

Find three consecutive positive numbers such that the square of the middle number exceeds the difference of the squares of the other two by 60.

Divide 12 into two parts such that their product is 32.

If a number is added to three times its reciprocal, the result is `5 3/5`. Find the number.

The sum of two numbers is 40 and the difference of their squares is 320. Find the numbers.

Find three consecutive positive integers such that sum of square of first and product of other two is 277.

Find three consecutive positive integers such that sum of their square is 149.

Sum of two natural numbers is 8 and the difference of their reciprocal is `2/15`. Find the numbers.

The denominator of a fraction is 2 more than its numerator. If 1 is subtracted from both the numerator and denominator, the fraction is decreased by `1/21.` Find the fraction.

The difference of denominator and numerator of a fraction is 4. If 5 is added to both numerator and denominator, the fraction is increased by `1/15`. Find the fraction.

The sum of a fraction and its reciprocal is `13/6`. Find the fraction if its numerator is 1 less than the denominator.

A two digit number is such that the product of its digits is 16. If 54 is subtracted from the number, the digits are interchanged. Find the number.

A two digit number is seven times the sum of its digits and also equal to 12 less than three times the product of its digits. Find the numbers.

A rectangular field is 16 m long and 10 m wide. There is a path of uniform width all around it, having an area of 120m . Find the width of the path. 

The side of a square exceeds the side of another square by 4 cm and the sum of the area of the two squares is 400 sq cm. Find the dimensions of the two squares.

The perimeter of a rectangular field is 82 m and its area is 400 m². Find the breadth of the rectangle.

The length of a rectangle is 3 cm more than its width. The area is 40 cm². Find the dimensions of the rectangle.

The area of a triangle is 18 cm² and the sum of its base and altitude is 12 cm. Find the base and altitude.

A farmer wishes to grow a 100 m² rectangular vegetable garden. Since he has with him only 30 m barbed wire, he fences three sides of the rectangular garden letting compound wall of his house act as the fourth side fence. Find the dimensions of his garden.

The length of the hypotenuse of a right-angled triangle exceeds the length of the base by 2 cm and exceeds twice the length of the altitude by 1 cm. Find the length of each side of the triangle. 

A bus covers a distance of 240 km at a uniform speed. Due to heavy rain its speed gets reduced by 10 km/h and as such it takes two hrs longer to cover the total distance. Assuming the uniform speed to be ‘x’ km/h, form an equation and solve it to evaluate ‘x’.

In a flight of 6000 km, an aircraft was slowed down due to bad weather. Its average speed for the trip was reduced by 400 km/h and time increased by 30 minutes. Find the original duration of flight.

A man covers a distance of 100 km, travelling with a uniform speed of x km/hr. Had the speed been 5 km/hr more it would have taken 1 hour less. Find x the original speed.

The time taken by a person to cover 150 km was 2.5 hrs more than the time taken in the return journey. If he returned at a speed of 10 km/h more than the speed when going, find his speed per hour in each direction. 

A plane left 40 min late due to bad weather and in order to reach its destination, 1600 km away in time, it has to increase its speed by 400 km/h from its usual speed. Find the usual speed of the plane.

A cyclist cycles non-stop from A to B, a distance of 14 km at a certain average speed. If his average speed reduced by 1 km per hour, he takes `1/3` h more to cover the same distance. Find his original average speed.

Car A travels ‘x’ km for every litre of petrol, while car B travels (x + 5) km for every litre of petrol.

1. Write down the number of litres of petrol used by car A and car B in covering a distance of 400 km.
2. If car A uses 4 litres of petrol more than car B in covering 400 km. write down an equation, in A and solve it to determine the number of litres of petrol used by car B for the journey.

By selling an article for ₹ 24 a trader loses as much percent as the cost price. Find the cost price of the article.

By selling an article for ₹ 144, a shopkeeper gains as much percent as the cost price. Find the cost price of the article.

A shopkeeper buys a certain number of books for ₹ 960. If the cost per book was ₹ 10 less, the number of books that could be bought for ₹ 960 would be 8 more. Taking the original cost of each book to be ₹ x, find the value of x.

A piece of cloth costs Rs. 35. If the piece were 4 m longer and each meter costs Rs. 1 less, the cost would remain unchanged. How long is the piece?

Rs. 480 is divided equally among ‘x’ children. If the number of children were 20 more, then each would have got Rs. 12 less. Find ‘x’.

Some students planned a picnic. The budget for the food was Rs. 480. As eight of them failed to join the party, the cost of the food for each member increased by Rs. 10. Find how many students went for the picnic.

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.

The ages of two brothers are 10 years and 14 years. In how many years time will the product of their ages be 285?

Three years ago, the father’s age was the square of his son’s age. 21 years hence, the father’s age will be twice the age of his son’s age. Find their present ages.

The sum of first ‘n’ even natural numbers is given by S_n = n (n + 1). Find n if S_n = 930.

Five years ago, a woman’s age was the square of her son’s age. Ten years hence, her age will be twice that of her son’s age. Find:

1. the age of the son five years ago.
2. the present age of the woman.

A can do a piece of work in x days and B can do it in (x – 6) days. If working together, they can do it in 4 days, find the value of x.

An area is paved with square tiles of certain size and the number of tiles required is 512. If the tiles had been 2 cm smaller each way, 800 tiles would have been needed to pave the same area. Find the size of larger tile.

A swimming pool is filled with three pipes with uniform flow. The first two pipes operating simultaneously, fill the pool in the same time during which the pool is filled by the third pipe alone. The second pipe fills the pool five hours faster than the first pipe and four hours slower than the third pipe. Find the time required by each pipe to fill the pool separately.

Vikas wishes to fit three rods together in a shape of the right triangle. The hypotenuse is to be 2 cm longer than the base and 4 cm longer than the altitude. What should be the lengths of the rods?

A boat can go 4 km upstream and 10 km downstream in 6 hours. If the speed of stream is 2 km/h, find the speed of boat in still water.

If 3 is a root of the quadratic equation x² – px + 3 = 0 then p is equal to ______.

If x = 3 is a root of the quadratic equation x² + kx − 8 = 0, then the value of k is ______.

The value of k for which the equation kx² − 6x + 1 = 0 has equal roots is ______.

The value of k for which the equation 8x² − kx + k = 0 has equal roots, is ______.

The quadratic equation x² + x + 1 = 0 has ______.

The solution of the equation x² − 6x − 7 = 0, x ∈ N is/are ______.

The roots of the equation x² + x − (a + 1) (a +2) = 0 are ______.

The discriminant of the equation 3x² − 5 = 0 is ______.

If `x^2-(a + 1/a) x + 1 = 0` then the values of x are ______.

If the equation (1 + m²) x² + 2cm x + c² − a² = 0 has repeated roots then the correct relation is ______.

The roots of the quadratic equation px² – qx + r = 0 are real and equal if ______.

Assertion: The roots of the equation x² − 4x + 3 = 0 are 1 and 3.

Reason: If ax² + bx + c = 0, a ≠ 0 then roots are given by `x =(-b+-sqrt(b^2-4ac))/(2a)`.

Assertion: If a and B are the roots of the equation 13x² − 7x + 1 = 0 then `alpha + beta = 7/13`

Reason: Every quadratic equation has two real roots.

Assertion: Equation 3x³ + x² − 1 = (x + 1)³ is quadratic.

Reason: The equation ax² + bx + c = 0, a ≠ 0 is quadratic.

Assertion: The equation 9x² − 6x + 1 = 0 has equal roots.

Reason: The equation ax² + bx + c = 0, a ≠ 0 has equal roots if b² − 4ac > 0.

Assertion: The equation x² + 4x − 19 = 0 has both real roots.

Reason: The equation ax² + bx + c = 0, a ≠ 0 has both real roots if b² − 4ac = 0.

In the following question, two statements (i) and (ii) are given. Choose the valid statement.

1. The equation x² + x + 1 has two real roots.
2. Every quadratic equation can have at most two real roots.

In the following question, two statements (i) and (ii) are given. Choose the valid statement.

1. If α and β are the roots of the equation ax² + bx + c = 0, a ≠ 0 then α + β = `-b/a.`
2. The equation ax² + bx + c = 0 is quadratic for all real values of a, b, c.

In the following question, two statements (i) and (ii) are given. Choose the valid statement.

1. The equation x² – 6x – 7 = 0 has no real root.
2. The equation 4x² – 6x + 1 = 0 has equal roots.

In the following question, two statements (i) and (ii) are given. Choose the valid statement.

1. The roots of equation 6x² − 5x + 1= 0 are `1/2 and 1/3.`
2. The sum of two numbers is 7 and the sum of their reciprocals is `7/12`. Then smaller number is 3.

Calculate the discriminant:

`sqrt3 x^2 - 2sqrt3x - sqrt3 = 0`

Find the value of k for which the quadratic equation 9x² − 3kx + k = 0 has equal roots. 

Find the value of k for which the equation x² − 2(1 + 3k) x + 7(3 + 2k) = 0 have equal roots.

Find the roots of the equation by factorization:

`10x -1/x = 3`

Find the roots: ax² + (4a² − 3b) x − 12ab = 0

Find the value of k for which the given quadratic equation has real and distinct roots: 

x² − kx + 9 = 0

If the roots of the equation (b − c) x² + (c − a) x + (a − b) = 0 are equal, then prove that 2b = a + c.

Separate 18 into 2 parts such that twice the sum of their squares is five times their product.

The perimeter of a right-angled triangle is five times the length of the shortest side. The numerical value of the area of the triangle is 15 times the numerical value of the length of the shortest side. Find the lengths of the three sides of the triangle.

The hypotenuse of a right-angled triangle is 5 m. If the smaller side is doubled and the longer side is tripled, the new hypotenuse is `6sqrt5` m. Find all the sides of the triangle.

Find the roots of the quadratic equation 5x² − 24x − 5 = 0.

If one root of the quadratic equation 3x² − kx − 2 = 0 is 2, find the value of ‘K’.

Find the value of ‘k’ for which the equation 5x² − 4x + 2 + k (4x² − 2x − 1) = 0 has real and equal roots.

Find the value of ‘k’ for which the equation kx² + kx + 1 = −4x² – x has real and equal roots.

If the roots of the equation (c² – ab)x² – 2(a² – bc)x + b² – ac = 0 are real and equal prove that either a = 0 (or) a³ + b³ + c³ = 3abc

In a flight of 2800 km, an aircraft was slowed down due to bad weather. Its average speed is reduced by 100 km/h and time increased by 30 minutes. Find the original duration of the flight.