# Quadratic Equations : Previous Year's Questions

**Â One Mark QuestionsÂ **

**Q.1Â Â Â Â Â Â Isa solution of the equationÂ **

**[AI 2008]**

**Sol.**

We have Â

When

Therefore, is not a solution of the given equation.

**Q.2 Â Â Â Â Is Â a solution of the equationÂ **

**[AI 2008]**

**Sol.**

We have Â

When

Therefore, is a solution of the given equation.

Â

**Q.3 Show that is a solution of equation**

**[Foreign 2008]**

* Sol. *We have Â

WhenÂ

Therefore ,Â is a solution of the given equation.

Â

**Q.4 For what value of k are the roots of the quadratic equation real and equal.**

**[Delhi 2008]**

* Sol.* We have Â

HereÂ

SinceÂ

For real and equal roots, D = 0

So, at the equation has real and equal roots.

Â

**Q.5 For what value of k are roots of the quadratic equationÂ equals and reals.**

**[AI 2008C]**

* Sol. *We have

Here

Since

For real and equal roots, D = 0

So, at equation has real and equal roots.

**Q.6 For what value of k does have equal roots.
[AI 2008C]
**

* Sol. *Here Â

SinceÂ

Now for equal and real roots D = 0

Then

Because Â for the given equation Â becomes a linear equation.Â So, is rejected.

Therefore, onlyis solution for the equation having real and equal roots.

Â

**Q.7 Find the discriminant of the quadratic equation **

**[AI 2009]**

* Sol. *We have Â

Here Â

Since

So, the discriminant of quadratic equation is

**Q.8. Write the nature of roots of quadratic equationÂ **

**[Foreign 2009]**

* Sol. *We have

Here , b =,

Since

As , the equation has real and equal roots.

**Two Marks QuestionsÂ **

**Q.1 Solve for **

**[Foreign 2005; Delhi 2006 C]**

* Sol.* We have Â

Here, ,

Since

[Because ]

Now

Then values of x are or

Â

**Q.2 Solve for **

**[Delhi 2006]**

* Sol.Â *We have Â

**Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â **

**Q.3Â **A two digit number is such that the product of its digits is 35. When 18 is added to the number, the digits interchange their places. Find the number.

**[Delhi 2006]**

*Sol.*Let digit at tenâ€™s placeÂ

Digit at unitâ€™s placeÂ

Therefore, Number

Now Â ,â€¦â€¦â€¦..(1)

A.T.Q.

[using (1)]

Because, so it can be rejected

Putting in eq(1) , we get

Therefore, number

Â

**Q.4 Using the quadratic formula; solve the equation **

**[AI 2006]**

* Sol.*Â We have Â

Here

SinceÂ

Therefore,

**Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â **

**Q.5Â **The sum of two natural numbers is 8. Determine the numbers if sum of their reciprocals is 8/15

**[AI 2006]**

* Sol.*Let one number

Therefore, other natural number

A.T.Q.

â€¦â€¦â€¦â€¦..(1)

So Â Â (By (1))

A.T.Q.

When , other number

When, other number

Therefore, Numbers are 3 and 5

**Q.6 Solve for **

**[Foreign 2006]**

**Sol.**Â

A = , B = , C =

Since Â

Therefore,

**Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â **

**Q.7Â **Two numbers differ by 3 and their product is 504. Find the numbers.

**[Foreign 2006]**

* Sol.*Â Let one number = x

Therefore, other number = x + 3

A.T.Q. Â

When, other number

When, other number

Therefore, Numbers are 21, 24 or â€“ 24 , â€“ 21

Â

**Q.8 Rewrite the following as a quadratic equation in x and then solve for x. **

**[AI 2006 C]**

* Sol. *We have Â

**Q.9 Find the value of p so that the quadratic equation has two equal roots.**

**[Delhi 2011]**

* Sol. *We have Â

Here

For equal roots

Since

But [Because for p=0 quadratic equation becomes Â linear equation]

So, is the root of the equation.

Â

**Q.10 Find the value of m so that the quadratic equation has two equal roots.**

** [AI 2011]**

* Sol. *We have Â

HereÂ

For equal roots D = 0

Since

But [Because for m = 0 Â quadratic equation becomes Â linear equation]

So, is the root of the equation.

Â

**Q.11 For what value of k does the quadratic equation has two equal roots.**

**[AI 2011]**

* Sol. *We have Â

Here

For equal rootsÂ

But [Because for Â k = 5 Â quadratic equation Â becomes linear equation]

So is the root of the equation.

Â

**Q.12 Find the value of k for which the roots of the quadratic equation are equal.**

**[Delhi 2013]**

* Sol*. We have Â

Here Â

Â for real and equal roots

But [Because for Â k = 4 Â quadratic equation becomes linear equation]

So, Â is the rootÂ

*o*f equation.

**Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â **

**Q.13Â **Solve for

**[Delhi 2013]**

* Sol*.Â We have Â

,

Â orÂ

**Q.14 Solve for **

**[AI 2013]**

* Sol. *We have Â

orÂ

Â

**Q.15 Solve for **

**[Foreign 2013]**

* Sol. *We have Â

Â orÂ

**Â Three Marks QuestionsÂ **

**Q.1 A passenger train takes 2 hours less for a journey of 300 km. It its speed is increased by 5 km/hr from its usual speed. Find the usual speed of the train.**

**[Delhi 2005 C, 2006]**

* Sol. *Let usual speed of the train = x km / hr

Distance = 300 km

Time taken = Â [Because ]

It speed = (x + 5) km/hr

Then time taken =

A.T.Q.

[reject x=-30 because speed cannot negative)

Therefore, usual speed = 25 km / hr

Â

**Q.2 A speed of a boat in still water is 11 km / hr. It can go 12 km upstream and return downstream to the original point in 2 hr 45 minutes. Find the speed of the stream.**

**[AI 2006]**

* Sol. *Let speed of the stream = x km / hr

Speed of the boat in still water = 11 km / hr

Therefore, up stream speed = (1 â€“ x) km / hr

and downstream speed =(11 + x) km / hr

Distance = 12 km

Time taken for downstream direction =

Time taken for upstream direction =

A.T.Q.

2 hr 45 min

[Reject x=-5 because speed cannot be negative]

So x = 5 km/hr is the speed of the stream.

Â

**Q.3 A fast train takes 3 hours less than a slow train for a journey of 600 km. If the speed of the slow train is 10 km / hr less than that of the fast train. Find the speeds of the two trains..**

**[Foreign 2006]**

* Sol.* Let speed of fast train = x km / hr

Speed of slow train =

A.T.Q.

Â [rejected]

So, speed so of the that train = 50 km / hr

And speed of the slow train = 40 km / hr.

Â

**Q.4 Seven years ago Varunâ€™s age was five times the square of Swatiâ€™s age. three years hence Swatiâ€™s age will be two-fifth of Varunâ€™s age. Find their present ages.**

**[Delhi 2006 C]**

* Sol.* Let Varunâ€™s present age = x years.

And Swatiâ€™s age = y years

**Case 1 :**Seven years ago

Varunâ€™s age was (x â€“ 7) years and

A.T.Q.

Â â€¦â€¦â€¦..(1)

**Case II :**Â Three years hence

Varunâ€™s age will be (x + 3) years and

Swatiâ€™s age will be (y + 3) years

A.T.Q.

,

y = 9 Â [Rejecting Â y=11/2 because age cannot be in fraction]

Putting y=9 in (1),We get

Therefore, Present age of Swati = 9 years and Varun = 27 years.

Â

**Q****.5 A 2-digit number is such that product of its digits is 18. When 63 is subtracted from the number, the digits interchange their places. Find the number.**

**[AI 2006 C]**

* Sol*. Let digit at unitâ€™s place = x

And digit at tenâ€™s place = y

Therefore, number = 10y + x.

A.T.Q.

And

Â [at using (1)]

[Reject x=-9]

When x = 2,

Therefore, number =92

**Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â **

**Q****.6Â **A train covers a distance of 90 km at a uniform speed. Had the speed been 15 km / hr more, it would have taken 30 minutes less for the journey. Find the original speed of the train.

**[AI 2006 C]**

* Sol*.Let uniform speed of the train = x km/hr

Distance = 90 km

Therefore, Time taken = Â hr

If speed was (x + 15) km / hr

Then time taken =

A.T.Q.

[Rejecting x=-60 because speed cannot be negative]

Therefore, x = 45 km / hr

Hence uniform speed of the train = 45 km/hr

Â

**Q.7 The difference of two numbers is 5 and the difference of their reciprocals is Find the numbers.**

**[Delhi 2007]**

* Sol.Â *Let one number is x

Therefore, other number = x + 5

A.T.Q.

Hence, numbers are -10, -5 or 5, 10

**Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â **

**Q.8Â **By increasing the list price of a book by Rs. 10. A person can buy 10 less books for Rs. 1200. Find the original list price of the book.

**[Delhi 2007]**

* Sol.Â *Let list price of the book = Rs. X

Total cost = Rs. 1200

Therefore, number of books =

If list price of the book = Rs. (x+10)

Then number of books =

A.T.Q.

Â [Because price cannot be negative so rejected]

Therefore, list price of the book = Rs. 30Â Â Â Â Â Â Â Â Â Â Â Â

Â

**Q.9 The numerator of a fraction is one less than its denominator. If three is added to each of the numerator and denominator, the fraction is increased by .Â Find the fraction.**

**[AI 2007]**

* Sol.Â *Let Denominator = x

Therefore, numerator = x â€“ 1

Fraction =

A.T.Q.

orÂ Â [Rejecting x = -7]

Fraction is

**Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â **

**Q.10Â **The difference of squares of two natural number is 45. The square of the smaller number is four times the larger number. Find the numbers.

**[AI 2007]**

* Sol.*Let larger number be = x

Smaller number be = y

A.T.Q.

â€¦â€¦â€¦â€¦.(1)

Also

Â [using (1)]

So larger numbers is 9.

Putting x=9 in (1),We get

[Rejecting y=-6]

So Numbers are: 9 and 6

Â

**Q.11 Find the roots of the following equation**

**, **

*[Delhi 2008]*

** Sol.** We have

Therefore, required roots are 2 , 1

Â

**Q.12 Find the roots of the following equation **

**[Delhi 2011]**

** Sol.** We have Â

Here a =, b = -5 , c =

SinceÂ

Or

So required roots are Â or

**Q.13 Solve the following quadratic equation for **

**[AI 2012]**

* Sol.*Â We have

Here A = 1, B = - 4a , C =

Since D =

Therefore,

Therefore, required roots are

**Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â
**

**Q.14Â **If the sum of two natural number is 8 and their product is 15. Find the numbers

**[AI 2012]**Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â

* Sol. *Let the first natural number = x

A.T.Q.

x + y = 8 â€¦â€¦â€¦â€¦â€¦..(1)

Where, y is another natural number

So,First natural number = x

Second natural number =8 â€“ x [By equation (1)]

A.T.Q.

Â Â or Â

Now if x = 3 then another number is 5

If x = 5 then another number is 3

Â

**Q.15 for what value of k, are the roots of the quadratic equation equal ?**

**[Foreign 2013]**

* Sol.* We have

Here Â

For equal roots D = 0

Since Â

or

[ If k=4 then quadratic equation becomes linear Equation ]

is the required root.

**Five Marks QuestionsÂ **

Â

**Q.1 In a class test, the sum of the marks obtained by p in Mathematics and science is 28. Had he got 3 more marks in maths and 4 marks less in science, the product of marks obtained in the two subjects would have been 180. Find the marks obtained in the two subjects separately.**

**[Delhi 2008]**

** Sol. **Let marks obtained in mathematics be x and marks obtained in science be y.

A.T.Q.

â€¦â€¦â€¦â€¦â€¦.(1)

Also

Â [using (1)]

or

Therefore, marks obtained in mathematics = 12 or 9

If marks obtained in mathematics = 12

marks obtained in science =

If marks obtained in mathematics = 9

marks obtained in science =

Â

**Q.2 The sum of the areas of two squares is 640m ^{2.}**

**If the difference in their perimeters be 64m. Find the sides of the two squares.**

**[Delhi 2008C, Delhi 2008]**

* Sol.* Let side of bigger squareÂ = x m

And side of smaller square = y m

A.T.Q.

Â â€¦â€¦â€¦â€¦â€¦.(1)

And

Substituting in equation (1), we get

or

If y = 8, then

Therefore, sides of squares are 8 m and 24 m

**Q.3 A motor boat whose speed is 18 km/hr in still water takes 1 hr more to go 24 km upstream and than to return downstream to the same spot. Find the speed of stream .**

**[AI 2008]**

* Sol.Â *Speed of boat = 18 km / hr

Let speed of stream = x km/hr

Upstream speed Â km/hr

Downstream speed Â km/hr

Time taken to cover 24 km upstream = hrs.

Time taken to cover 24 km downstream = hrs.

A.T.Q.

x = 6, x =- 54

[ Because speed cannot be Â negative]

Therefore, speed of the stream = 6 kmÂ / hr

**Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â **

**Q.4Â **Two water taps together can fill the tank in hrs. The tap of larger diameter takes 10 hrs. less than smaller one to fill the tank. Find the time in which each tap can separately fill the tank.

**[AI 2008]**

* Sol.Â *Let the two water taps fill the tank separately in (x) and (x-10) hrs

Since, both the taps together fill the tank in hr

Therefore,

Therefore, , 25

Rejecting , we get xÂ = 25

Therefore, two taps separately fill the tank in (x) and (x â€“ 10) i.e. 25 hrs and 15 hrs.

Â

**Q.5 A peacock is sitting on the top of a pillar which is 9 m high. From a point 27 m away from the bottom of the pillar a snake is coming to its hole at the base of the pillar. Seeing the snake the peacock pounces on it. If their speeds are equal at what distance from the hole is the snake caught?**

Â

Let the distance covered by peacockÂ

Therefore, Distance covered by snake

In Â right angle

AndÂ

Therefore, the snake is caught at 12 m from the hole.

Â

**Q.6 A person on tour has Rs. 4200 for his expenses. If he extends his tour for 3 days Â he has to cut down his daily expenses by Rs. 70. Find the duration of the tour.**

**[AI 2008]**

* Sol. *Let number of days tour = x

Total expenses = Rs. 4200

Therefore, daily expenses = Rs.

If number of days of tour = x + 3

Then daily expensesÂ = Rs.

A.T.Q.

Â or Â [Rejecting x=-15 because days cannot be negative]

So, number of days of tour is 12 days.

Â

**Q.7 A trader bought a number of articles for Rs. 900,five articles were found damaged. He sold each of the remaining articles at Rs. 2 more than what he paid for it. He got a profit of Rs. 80 on the whole transaction. Find the number of articles he bought .**

**[Foreign 2009]**

* Sol.Â *Number of articles = x

Cost of each articles = y

A.T.Q.

Â â€¦â€¦â€¦â€¦â€¦.(1)

Â (rejected), x = 75

Therefore, No of articles = 75

**Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â **

**Q.8Â **Two years ago a manâ€™s age was three times the square of his sonâ€™s age. Three years hence his age will be four times his sonâ€™s age. Find their present ages.

**[Foreign 2009]**

* Sol.Â *Age of man = x years

Age of son = y years

A.T.Q.

â€¦â€¦â€¦â€¦â€¦â€¦.(1)

Also

â€¦â€¦â€¦â€¦â€¦â€¦..(2)

Â [ Using (1)]

(rejected, age can never be in fraction form)

Putting y=5 in (1),We get

Hence, Present age of Son is 5 years and Present age of Father is 29 years.Â

Â

**Multiple choice questions **

Â

**Q.1 The roots of the equationwhere p is constant, are**

**[Delhi 2011]**

**(a)
(b)
(c)
(d) **

* Sol. *(b) We have

Â

Â

**Q.2 The roots of the equation , where m is a constant, are -**

**[AI 2011]**

**(a)
(b)
(c)
(d)
**

** Sol.** We have

Â

**Q.3 The roots of the quadratic equation , WhereÂ is a constant, are**

**[Foreign 2011]**

**(a)
(b)
(c)
(d) **

* Sol.* (b)

Â

**Q.4 If the quadratic equation has two equal roots, then the values of m are**

**[Foreign 2012]**

**(a)
(b) 0, 2
(c) 0, 1
(d) -1, 0**

* Sol. *We have

Here Â

For equal rootsÂ Â D = 0

Â