UPSC CDS 2 2021 Math Paper

Given quadratic equation,
ax2−4ax+15=0... (i)
One root of the equation is ‌

3

2

.
Put x=‌

3

2

into Eq. (i), we get
‌a(‌

3

2

)2−4a(‌

3

2

)+15=0
⇒a⋅‌

9

4

−2⋅a⋅3+15=0
⇒‌‌‌

9a

4

−6a+15=0
⇒‌‌‌

9a−24a+60

4

=0
⇒‌‌−15a+60=0
⇒‌‌a=‌

60

15

=4
The quadratic equation is
4x2−16x+15=0‌
⇒4x2−10x−6x+15‌‌=0
⇒2x(2x−5)−3(2x−5)‌‌=0
⇒‌‌(2x−5)(2x−3)‌‌=0
x‌‌=‌

5

2

‌ or ‌‌

3

2

Sum of the squares of the roots
‌‌=(‌

5

2

)2+(‌

3

2

)2
‌‌=‌

25

4

+‌

9

4

‌‌=‌

34

4

=‌

17

2

Hence, sum of the squares of the roots is ‌

17

2