Quadratic Equations — Class 10 Maths PYQs

Which of the following equations is a quadratic equation?

The equation $x + \frac{1}{x} = 3$ ($x \neq 0$) is expressed as a quadratic equation in the form of $ax^2 + bx + c = 0$. The value of $a - b + c$ is :

If $x = \sqrt{x}$, ($x \neq 0$) is expressed as a quadratic equation in the form $ax^2 + bx + c = 0$, then the value of $a + b + c$ is :

If $(\sqrt{x} + 1)^2 = x^2 + 2\sqrt{x}$ is expressed as a quadratic equation in the form of $ax^2 + bx + c = 0$, then the value of $a - b + c$ is :

If $(x + 1)^3 = x^3 + 1$ is expressed as a quadratic equation in the form $px^2 + qx + r = 0$, then the value of $p - q + r$ is :

Which of the equations among the following is/are quadratic equation(s) ?
$q_1 : x^2 + x = (x+1)^2$, $q_2 : x-1= x^2 -1$, $q_3 : x^4 = x^2$, $q_4 : \sqrt{x} = x^2\sqrt{x} +1$

Which of the following equations is/are quadratic equation(s) ?
$q_1: (y+1)^2 = 2y$,
$q_2: (y-1)^2 = y^2$
$q_3 : (y+1)^3 = (y-1)^3$,
$q_4:1+\sqrt{y} = (\sqrt{y} +1)^2$

Which of the following equations is/are not quadratic ?
$q_1: (x-1)^2 = x^2$,
$q_2: (x-1)^3 = x^3$
$q_3: (x+1)^3 = 3x^2$,
$q_4: (\sqrt{x}+1)^2 = 2\sqrt{x}$

If $x = 0.3$, is a root of the equation $x^2 -0.9k = 0$, then $k$ is equal to :

A quadratic equation whose roots are $(2 + \sqrt{3})$ and $(2 - \sqrt{3})$ is:

A quadratic equation whose roots are $(3 - \sqrt{2})$ and $(3 + \sqrt{2})$ is :

If the zeroes of the quadratic polynomial $x^2 + (a + 1) x + b$ are $2$ and $-3$, then

If 'p' is a root of the quadratic equation $x^2 - (p + q) x + k = 0$, then the value of 'k' is

If $x = 5$ is a solution of the quadratic equation $2x^2 + (k - 1)x + 10 = 0$, then the value of $k$ is:

Value of k for which $x = 2$ is a solution of the equation $5x^2 - 4x + (2 + k) = 0$, is

If $y = 1$ is one of the solutions of the quadratic equation $py^2 + py + 3 = 0$, then the value of $p$ is :

The quadratic equation whose roots are $7$ and $\frac{1}{7}$ is :

If $x = - 1$ is a root of the equation $ax^2 – bx + 3 = 0$, then :

If $x = \sqrt{3}$ is a solution of the equation $ax^2 + \sqrt{3}x - 12 = 0$, then

Two water taps together can fill a tank in $3\frac{1}{3}$ hours. The tap of larger diameter takes $5$ hours less than the smaller one to fill the tank separately. Find the time in which each tap can fill the tank separately.

Sum of the areas of two squares is $468$ m$^2$. If the difference of their perimeters is $24$ m, find the lengths of the sides of the two squares.

The sum of two numbers is $15$. If the sum of their reciprocals is $\frac{3}{10}$, find the two numbers.

In a $2$-digit number, the digit at the unit's place is $5$ less than the digit at the ten's place. The product of the digits is $36$. Find the number.

Three consecutive integers are such that sum of the square of second and product of other two is $161$. Find the three integers.

A dealer sells an article for ₹75 and gains as much percent as the cost price of the article. Find the cost price of the article.

Find two consecutive negative integers, sum of whose squares is $481$.

The sum of the squares of two consecutive even numbers is $452$. Find the numbers.

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

The sum of the squares of two consecutive odd numbers is 514. Find the numbers.

While designing the school year book, a teacher asked the student that the length and width of a particular photo is increased by $x$ units each to double the area of the photo. The original photo is 18 cm long and 12 cm wide. Based on the above information, answer the following questions:
(I) Write an algebraic equation depicting the above information.
(II) Write the corresponding quadratic equation in standard form.
(III) What should be the new dimensions of the enlarged photo?
OR
Can any rational value of $x$ make the new area equal to $220\text{cm}^2$

A garden designer is planning a rectangular lawn that is to be surrounded by a uniform walkway.
The total area of the lawn and the walkway is $360$ square metres. The width of the walkway is same on all sides. The dimensions of the lawn itself are $12$ metres by $10$ metres.
Based on the information given above, answer the following questions:
(i) Formulate the quadratic equation representing the total area of the lawn and the walkway, taking width of walkway $= x \operatorname{m}$.
(ii) (a) Solve the quadratic equation to find the width of the walkway 'x'.
OR
(b) If the cost of paving the walkway at the rate of ₹50 per square metre is ₹12,000, calculate the area of the walkway.
(iii) Find the perimeter of the lawn.

A train travels at a certain average speed for a distance of $54$ km and then travels a distance of $63$ km at an average speed of $6$ km/h more than the first speed. If it takes $3$ hours to complete the journey, what was its first average speed ?

Two pipes together can fill a tank in $\frac{15}{8}$ hours. The pipe with larger diameter takes $2$ hours less than the pipe with smaller diameter to fill the tank separately. Find the time in which each pipe can fill the tank separately.

The denominator of a fraction is one more than twice the numerator. If the sum of the fraction and its reciprocal is $\frac{16}{21}$, find the fraction.

A train travels a distance of $90$ km at a constant speed. Had the speed been $15$ km/h more, it would have taken $30$ minutes less for the journey. Find the original speed of the train.

At present, Sourav's age is $3$ years more than the square of his son Ravi's age. When Ravi grows to his father's present age, Sourav's age would be $6$ years less than $13$ times the present age of Ravi. Find present ages of Ravi and Sourav.

The sum of the areas of two squares is $52$ cm$^2$ and difference of their perimeters is $8$ cm. Find the lengths of the sides of the two squares.

The numerator of a fraction is $3$ less than its denominator. If $2$ is added to both numerator and denominator, then the sum of the new fraction and the original fraction is $\frac{29}{20}$. Find the original fraction.

A train travelling at a uniform speed for $360$ km would have taken $48$ minutes less to travel the same distance if its speed were $5$ km/h more. Find the original speed of the train.

Find two consecutive odd numbers, sum of whose squares is $650$.

A faster train takes one hour less than a slower train for a journey of $200$ km. If the speed of the slower train is $10$ km/hr less than that of the faster train, find the speeds of the two trains.

The sum of the areas of two squares is $640$ m$^2$. If the difference in their perimeters is $64$ m, find the sides of the two squares.

Venkat can row a boat in still water at the speed of 12 km/h. He ferries tourists 15 km upstream and 18 km downstream in 3 hours. Find the speed of the stream.

By selling an article for ₹$48$, a trader loses as much percent as half of the cost price of the article. Calculate the cost price and loss amount of the article.

Two water taps together can fill a tank in $8\frac{8}{3}$ hours. The tap of larger diameter takes $4$ hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

The roots of the quadratic equation $x^2 + x - p (p + 1) = 0$ are :

The roots of the quadratic equation $(x – 1)^2 = 16$ are :

A rectangular floor area can be completely tiled with $200$ square tiles. If the side length of each tile is increased by $1$ unit, it would take only $128$ tiles to cover the floor.
(i) Assuming the original length of each side of a tile be $x$ units, make a quadratic equation from the above information.
(ii) Write the corresponding quadratic equation in standard form.
(iii) (a) Find the value of $x$, the length of side of a tile by factorisation.
OR
(b) Solve the quadratic equation for $x$, using quadratic formula.

If the sum and the product of the roots of the quadratic equation $ax^2 + 6x + 4a = 0$ are equal, then 'a' is equal to :

If the quadratic equation $ax^2 + 2bx - c = 0$ ($a \neq 0$) has real and equal roots, then which of the following is true?

Verify that roots of the quadratic equation $(p-q)x^2 + (q - r)x + (r - p) = 0$ are equal when $q + r = 2p$.

Find the nature of roots of the equation $3x^2 - 4\sqrt{3}x + 4 = 0$.

Find the value of $p$ for which the quadratic equation $(2p + 1)x^2 - (7p + 2)x + (7p - 3) = 0$ has equal roots. Also, find these roots.

Find the value of 'k' for which the quadratic equation $(k + 1) x^2 - 2 (3k + 1) x + (8k + 1) = 0$ has real and equal roots.

Find the value of 'k' for which the quadratic equation $(k + 1)x^2 - 6(k + 1)x + 3(k + 9) = 0, k \neq - 1$ has real and equal roots.

Find the smallest value of $p$ for which the quadratic equation $x^2 - 2(p+1)x + p^2 = 0$ has real roots. Hence, find the roots of the equation so obtained.

Express $x - \frac{1}{x} = 3$ as a quadratic equation in standard form and hence find its roots. Also, find the value of 'a' for which the equation $x + \frac{1}{x} = a$, when expressed as a quadratic equation, has real and equal roots.