# NCERT Class 10 Chapter 5 Quadratic Equations Official CBSE Board Sample Problems Short Answer

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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.**

### 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