Question

IIT mains

Answer 1

Find all the integral values of a for which the quadratic equation
(x-a)(x-10) +1 =0 has integral roots

Answer 2

options?

Answer 3

There are no options for this this came in very early in IIT . With options a new born baby would do that

Answer 4

lol...then y do u ask?

Answer 5

i didn't get what u mean?

Answer 6

I tried to think of a approach to start this question

x^2 -(a+10)x + 10a +1 =0
Since integral roots will always be rational it means that D should be a perfect square

D= a^2 -20a +96

D=(a-10)^2 - 4

Answer 7

Or 4 = (a-10)^2 -D

So here we would need the difference between two perfect squares to be 4

Answer 8

How to proceed further or is my analogy incorrect

Answer 9

what are your A, B, C values ?

Answer 10

i get A = 1
B = -(10+a)
C = 1+10 a

Answer 11

Why we need them

Answer 12

m trying to approach the question in tow parts...
first...i find the range of 'a' for which the equation is satisfies...that is there are real roots...
and in the second part...i try to find range when the roots would be integral...
after finding these range...we can take the intersection of these ranges and write the integers in the range as answer

Answer 13

because we want
\(B^2-4AC\)
to be a perfect square

Answer 14

for (x-a)(x-10) +1 =0
to have integer roots

Answer 15

so expnd
(x-a)(x-10) +1 =0

Answer 16

See above i calculated it ^^

Answer 17

a^2 -20a +96

is correct!

Answer 18

(a-10^2 + 4 = D is also correct.

Answer 19

answer is 12

Answer 20

* (a-10)^2 + 4 = D

Answer 21

so we need
a^2 -20a +96 to have integer values

Answer 22

thats just
(a-8)(a-12)

so integer values of a are just 8,12

Answer 23

tanmay is half correct 12 ,8 are the answers i saw them

Answer 24

you'll get both, 12,8

Answer 25

\[a^2-20a+96=0\]
completing the square...we get \[(a-10)^2=4\]
a=12

Answer 26

consider negative root too

Answer 27

Yes consider negative

Answer 28

a-10 = -2

Answer 29

Was it so easy? :)

a-10 = \(\pm \sqrt{4}\)

a-10 = + 2 or a-10 =-2

Answer 30

yep

Answer 31

Yeah compared to iit level it was , they put some questions of 11th std

Answer 32

In mains not in advanced