Class 10 Maths · Chapter 4

Quadratic Equations

61 Class 10 Maths questions on Quadratic Equations from the NCERT textbook, NCERT Exemplar and CBSE board papers 2026, each with its answer and a step-by-step solution.

CBSE Class 10 Maths previous year papers, with answers →

Questions · page 3 of 4

1. Q41 CBSE 2026
   The ​value ​of $p$ ​for which roots of ​the quadratic equation $x^2-px+6=0$ are rational, ​is

   1. (a)$1$
   2. (b)$-5$
   3. (c)$25$
   4. (d)$\sqrt{5}$
   Show answer

   (b) $-5$

   Step-by-step solution →
2. Q42 CBSE 2026
   If the ​roots ​of ​the ​quadratic equation $\sqrt{3}{x}^2-kx+2\sqrt{3}=0$ ​are ​real ​and ​equal, then the value(s) of $k$ is/are :

   1. (a)$\pm \sqrt{24}$
   2. (b)$0$
   3. (c)$4$
   4. (d)$-5$
   Show answer

   (a) $\pm \sqrt{24}$

   Step-by-step solution →
3. Q43 CBSE 2026
   Verify ​that ​roots of ​the ​quadratic ​equation $(p-q)x^2+(q-r)x+(r-p)=0$ are equal when $q+r=2p$.
   Show answer

   The discriminant is $b^2-4ac=(q+r-2p{)}^2$, which becomes $0$ when $q+r=2p$. A zero discriminant means the roots are equal, so the statement is verified.

   Step-by-step solution →
4. Q44 CBSE 2026
   Each root of a quadratic equation $a{x}^2+bx+c=0$ is $2$ ​more ​than each ​of the roots of the ​equation $3x^2-2x=1$. ​Find the roots ​of the ​quadratic equation $a{x}^2+bx+c=0$.
   Show answer

   The roots of $a{x}^2+bx+c=0$ are $3$ and $\frac{5}{3}$.

   Step-by-step solution →
5. Q45 CBSE 2026
   Find two ​consecutive ​negative integers, ​sum ​of whose ​squares is $481$.
   Show answer

   The required integers are $-16$ and $-15$.

   Step-by-step solution →
6. Q46 CBSE 2026
   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.
   Show answer

   The speed of the stream is $4$ km/h.

   Step-by-step solution →
7. Q47 CBSE 2026
   Solve ​for $x$: $\frac{x-2}{x-3}+\frac{x-4}{x-5}=\frac{10}{3}$; $x\ne 3,5$.
   Show answer

   $x=6$ or $x=\frac{7}{2}$.

   Step-by-step solution →
8. Q48 CBSE 2026
   A person on tour ​has $\text{₹}5400$ ​for his expenses. If ​he extends ​his tour by $5$ days, ​he has to cut down his daily expenses by $\text{₹}180$. ​Find the original duration of the tour ​and daily ​expense.
   Show answer

   The original duration of the tour is $10$ days and the original daily expense is $\text{₹}540$.

   Step-by-step solution →
9. Q49 CBSE 2026
   Express $\frac{24}{18-x}-\frac{24}{18+x}=1$ as ​a quadratic equation in standard ​form ​and ​find ​the ​discriminant of the quadratic ​equation, so obtained. Also, find the roots of the ​equation.
   Show answer

   Standard form : $x^2+48x-324=0$; discriminant $D=3600$; roots $x=6$ and $x=-54$.

   Step-by-step solution →
10. Q50 CBSE 2026
    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.
    Show answer

    Standard form: $x^2-3x-1=0$; its roots are $x=\frac{3+\sqrt{13}}{2}$ and $x=\frac{3-\sqrt{13}}{2}$; the equation $x+\frac{1}{x}=a$, i.e. $x^2-ax+1=0$, has real and equal roots for $a=2$ or $a=-2$.

    Step-by-step solution →
11. Q51 CBSE 2026
    A motor ​boat, whose speed in still ​water is $20$ km/h, takes $1$ ​hour more to go $48$ km upstream than to return ​downstream to the same point. ​Find ​the speed of ​the stream.
    Show answer

    The speed of the stream is $4$ km/h.

    Step-by-step solution →
12. Q52 CBSE 2026
    The total cost of a ​certain piece of cloth was ₹$2100$. During ​special sale time, the shopkeeper offered $2$ ​m extra cloth for free, thus reducing the ​price of ​cloth per metre by ₹$120$. What was the original per metre price of the cloth and ​its length?
    Show answer

    The original price of the cloth was ₹$420$ per metre and its length was $5$ m.

    Step-by-step solution →
13. Q53 CBSE 2026
    The ​sum of ​squares of two positive numbers is $100$. If ​one number exceeds ​the ​other ​by $2$, ​find the numbers.
    Show answer

    The two positive numbers are $6$ and $8$.

    Step-by-step solution →
14. Q54 CBSE 2026
    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.
    Show answer

    The speed of the faster train is $50$ km/hr and the speed of the slower train is $40$ km/hr.

    Step-by-step solution →
15. Q55 CBSE 2026
    A ​person on a ​tour ​has ₹ $4,200$ ​for expenses. If he extends his tour for $3$ days, he has ​to cut down his ​daily expenses by ₹ $70$. Find the original duration ​of ​the tour.
    Show answer

    The original duration of the tour is $12$ days.

    Step-by-step solution →
16. Q56 CBSE 2026
    The ​sum of ​the ​areas ​of ​two squares is $640{\text{m}}^2$. If the difference in their perimeters ​is $64$ m, find the sides of the two squares.
    Show answer

    The sides of the two squares are $24$ m and $8$ m.

    Step-by-step solution →
17. Q57 CBSE 2026
    The ​area of a right-angled triangle is $600{\text{cm}}^2$. If ​the ​base of the triangle exceeds the altitude by $10$ cm, find ​all the three ​dimensions ​of the triangle.
    Show answer

    The altitude of the triangle is $30$ cm, the base is $40$ cm and the hypotenuse is $50$ cm.

    Step-by-step solution →
18. Q58 CBSE 2026
    Three ​consecutive positive integers ​are ​such ​that sum of square of the first and the ​product of the other ​two is $67$, find the integers.
    Show answer

    The three consecutive positive integers are $5$, $6$ and $7$.

    Step-by-step solution →
19. Q59 CBSE 2026
    $ABCD$ is a rectangle of dimensions $80\text{ cm}\times 60\text{ cm}$. Another rectangle $PQRS$ is drawn ​inside $ABCD$ leaving space of equal width $x$ ​cm along the edges of $ABCD$. If area $PQRS$ is ​half ​of the ​area $ABCD$, then ​find the value ​of $x$.
    Show answer

    $x=10$ cm.

    Step-by-step solution →
20. Q60 CBSE 2026
    A ​train covers a distance of $90$ km ​at a uniform speed. ​Had ​the speed ​been $15$ km/h more, it ​would have ​taken $30$ minutes less ​for the same journey. Find the original speed of the train.
    Show answer

    The original speed of the train is $45$ km/h.

    Step-by-step solution →