Quadratic Equations

CBSE Class 10 — Maths — Previous Year Questions

Q1.MCQ1 mark(2020)Easy

The roots of the quadratic equation $x^2-0.04=0$ are

Q2.MCQ1 mark(2020)Easy

The value of $\lambda$ for which $(x^2+4x+\lambda )$ is a perfect square, is

Q3.MCQ1 mark(2020)Easy

The quadratic equation $x^2-4x+k=0$ has distinct real roots if

Q4.MCQ1 mark(2020)Easy

The value(s) of $k$ for which the quadratic equation $2x^2+kx+2=0$ has equal roots, is

Q5.MCQ1 mark(2023)Medium

The least positive value of $k$, for which the quadratic equation $2x^2+kx-4=0$ has rational roots, is
(A) $\pm 2\sqrt{2}$
(B) $2$
(C) $\pm 2$
(D) $\sqrt{2}$

Q6.MCQ1 mark(2023)Easy

Which of the following quadratic equations has sum of its roots as $4$?
(a) $2x^2-4x+8=0$
(b) $-x^2+4x+4=0$
(c) $\sqrt{2}{x}^2-\frac{\sqrt{2}}{2}x+1=0$
(d) $4x^2-4x+4=0$

Q7.MCQ1 mark(2023)Easy

The roots of the equation $x^2+3x-10=0$ are:

Q8.MCQ1 mark(2023)Easy

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

Q9.MCQ1 mark(2023)Easy

If the quadratic equation $a{x}^2+bx+c=0$ has two real and equal roots, then $c$ is equal to

Q10.Assertion-Reason1 mark(2023)Medium

**Statement A (Assertion):** If $5+\sqrt{7}$ is a root of a quadratic equation with rational co-efficients, then its other root is $5-\sqrt{7}$.

**Statement R (Reason):** Surd roots of a quadratic equation with rational coefficients occur in conjugate pairs.

Q11.MCQ1 mark(2024)Easy

If the roots of equation $a{x}^2+bx+c=0$, $a\ne 0$ are real and equal, then which of the following relation is true?

Q12.MCQ1 mark(2024)Easy

The quadratic equation $x^2+x+1=0$ has ______ roots.

Q13.MCQ1 mark(2024)Easy

If the discriminant of the quadratic equation $3x^2-2x+c=0$ is $16$, then the value of $c$ is:

Q14.MCQ1 mark(2024)Easy

The ratio of the sum and product of the roots of the quadratic equation $5x^2-6x+21=0$ is:

Q15.MCQ1 mark(2025)Easy

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

Q16.MCQ1 mark(2025)Easy

If $\frac{x}{12}-\frac{3}{x}=0$, then the values of $x$ are:

Q17.MCQ1 mark(2025)Medium

The value of '$a$' for which $a{x}^2+x+a=0$ has equal and positive roots is:

Q18.VSA2 marks(2020)Easy

Solve for $x$:
$$6x^2+11x+3=0$$

Q19.VSA2 marks(2022)Medium

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

Q20.VSA2 marks(2022)Medium

Solve the following quadratic equation for $x$: $\sqrt{3}{x}^2+10x+7\sqrt{3}=0$

Q21.VSA2 marks(2022)Medium

The product of Rehan's age (in years) $5$ years ago and his age $7$ years from now, is one more than twice his present age. Find his present age.

Q22.VSA2 marks(2022)Easy

Solve the quadratic equation: $x^2+2\sqrt{2}x-6=0$ for $x$.

Q23.VSA2 marks(2022)Medium

If the sum of the roots of the quadratic equation $k{y}^2-11y+(k-23)=0$ is $\frac{13}{21}$ more than the product of the roots, then find the value of $k$.

Q24.VSA2 marks(2022)Medium

If $x=-2$ is the common solution of quadratic equations $a{x}^2+x-3a=0$ and $x^2+bx+b=0$, then find the value of $a^{2}b$.

Q25.Case-Study2 marks(2023)Medium

OR part of Q36: Can any rational value of $x$ make the new area equal to $220$ cm${}^2$?

Q26.VSA2 marks(2023)Easy

Find the sum and product of the roots of the quadratic equation $2x^2-9x+4=0$.

Q27.VSA2 marks(2023)Easy

Find the discriminant of the quadratic equation $4x^2-5=0$ and hence comment on the nature of roots of the equation.

Q28.VSA2 marks(2025)Easy

Find the value(s) of '$k$' so that the quadratic equation $4x^2+kx+1=0$ has real and equal roots.

Q29.SA3 marks(2020)Medium

In a flight of 600 km, an aircraft was slowed due to bad weather. Its average speed for the trip was reduced to 200 km/hr and time of flight increased by 30 minutes. Find the original duration of flight.

Q30.SA3 marks(2020)Medium

Solve for $x$: $\frac{1}{x+4}-\frac{1}{x+7}=\frac{11}{30}$, $x\ne -4,7$.

Q31.SA3 marks(2020)Medium

In a flight of 600 km, an aircraft was slowed down due to bad weather. The average speed of the trip was reduced by 200 km/hr and the time of flight increased by 30 minutes. Find the duration of flight.

Q32.SA3 marks(2020)Medium

Find the values of $k$, for which the quadratic equation $(k+4)x^2+(k+1)x+1=0$ has equal roots.

Q33.SA3 marks(2022)Hard

Find the value of $p$ for which the quadratic equation $p(x-4)(x-2)+(x-1{)}^2=0$ has real and equal roots.

Q34.SA3 marks(2022)Medium

Had Aarush scored 8 more marks in a Mathematics test, out of 35 marks, 7 times these marks would have been 4 less than square of his actual marks. How many marks did he get in the test?

Q35.SA3 marks(2023)Medium

Find the value of '$p$' for which the quadratic equation $px(x-2)+6=0$ has two equal real roots.

Q36.SA3 marks(2023)Medium

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

Q37.SA3 marks(2023)Medium

If $\alpha$ and $\beta$ are roots of the quadratic equation $x^2-7x+10=0$, find the quadratic equation whose roots are ${\alpha }^2$ and ${\beta }^2$.

Q38.SA3 marks(2024)Medium

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.

Q39.LA4 marks(2020)Medium

In a flight of 600 km, the speed of the aircraft was slowed down due to bad weather. The average speed of the trip was decreased by 200 km/hr and thus the time of flight increased by 30 minutes. Find the average speed of the aircraft originally.

Q40.LA4 marks(2020)Medium

₹9,000 were divided equally among a certain number of persons. Had there been 20 more persons, each would have got ₹160 less. Find the original number of persons.

Q41.LA4 marks(2020)Medium

Sum of the areas of $2$ squares is $544$ m${}^2$. If the difference of their perimeter is $32$ m, find the sides of two squares.

Q42.LA4 marks(2020)Medium

A motorboat whose speed is $18$ km/h in still water takes $1$ hour more to go $24$ km upstream than to return downstream to the same spot. Find the speed of the stream.

Q43.LA4 marks(2022)Medium

A $2$-digit number is such that the product of its digits is $24$. If $18$ is subtracted from the number, the digits interchange their places. Find the number.

Q44.LA4 marks(2022)Medium

The difference of the squares of two numbers is $180$. The square of the smaller number is $8$ times the greater number. Find the two numbers.

Q45.Case-Study4 marks(2023)Medium

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 $220$ cm${}^2$?

Q46.Case-Study4 marks(2024)Medium

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.



Assuming the original length of each side of a tile be $x$ units, answer the following questions:
(i) Write 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.

Q47.Case-Study4 marks(2025)Medium

Case Study 1:

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 $=x$ 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.

Q48.LA5 marks(2023)Medium

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?

Q49.LA5 marks(2023)Medium

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.

Q50.LA5 marks(2024)Medium

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.

Q51.LA5 marks(2024)Medium

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

Q52.LA5 marks(2024)Medium

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

Q53.LA5 marks(2024)Medium

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

Q54.LA5 marks(2025)Medium

The perimeter of a right triangle is 60 cm and its hypotenuse is 25 cm. Find the lengths of other two sides of the triangle.

Q55.LA5 marks(2025)Medium

A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken 3 hours more to cover the same distance. Find the speed of the train.

Q56.LA5 marks(2025)Medium

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.

Q57.LA5 marks(2025)Medium

The time taken by a person to travel an upward distance of 150 km was $2\frac{1}{2}$ hours more than the time taken in the downward return journey. If he returned at a speed of 10 km/h more than the speed while going up, find the speeds in each direction.

Q58.LA5 marks(2025)Medium

The perimeter of an isosceles triangle is 32 cm. If each equal side is $\frac{5}{6}$ of the base, find the area of the triangle.

Q59.LA5 marks(2025)Medium

The sides of a right triangle are such that the longest side is 4 m more than the shortest side and the third side is 2 m less than the longest side. Find the length of each side of the triangle. Also, find the difference between the numerical values of the area and the perimeter of the given triangle.

Q60.LA5 marks(2025)Medium

Express the equation $\frac{x-2}{x-3}+\frac{x-4}{x-5}=\frac{10}{3}$; $(x\ne 3,5)$ as a quadratic equation in standard form. Hence, find the roots of the equation so formed.