NCERT Class 10 Chapter 5 Quadratic Equations Official CBSE Board Sample Problems Short Answer (For CBSE, ICSE, IAS, NET, NRA 2022)

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Question

If- 5 is a root of the quadratic equation and the quadratic equation has equal roots, find the value of k

Solution

is the root of the quadratic equation

Now, given that equation has equal roots

i.e.. has equal roots

i.e.. has equal roots

( For equal roots, D = 0, i.e.. ]

Question

If and are roots of the quadratic equations, find the values ofaand b.

Solution

Given quadratic equation is . . (i)

( x = is the root of equation (i) )

… (ii)

[ is the root of equation (i) ]

. . (iii)

Putting the value of b from (iii) in (ii) , we get

Putting a = 3 in (iii) , we have

Question

A two digit number is four times the sum of the digits. It is also equal to 3 times the product of digits. Find the number.

Solution

Let ones digit of number = x

Let tens digit of number = y

Number will be

According to question,

… (i)

and … (ii)

Putting from (i) in (ii) , we get,

and )

The required number .

Question

Solve for x:

Solution

Question

If the roots of the quadratic equation are equal, prove that.

Solution

For equal roots, discriminant, D = 0

Question

The following picture shows the shape of a certain grass patch. If the area of the patch is , find k.

The Shape of a Certain Grass Patch

Solution

The total area

Since the area is

Since the length cannot be negative, .

Question

Using the quadratic formula, solve the quadratic equation:

Solution

and

Question

Find 2 numbers that sum to 21 and the sum of the squares is 261.

Solution

The numbers are 15 and 6.

Hence, and

Question

Find the roots of the following equations:

Solution

On simplifying we get

Question

If p and q are the roots of the equation , then find the value of p and q.

Solution

Since p and q are the roots of this equation, Hence: . Hence,

Hence,

Question

Find the value of k for which the quadratic equation has two real and equal roots.

Solution

For equal roots

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