UPSC NDA 2 2021 Mathematics Solved Paper

(d) Since, we know that if a quadratic equation ax2+bx+c=0 has real roots of equal magnitude and opposite sign.
Then,
b=0
and product of roots <0
In the given quadratic equation,
3x2−(k2−5k)x+3k2−5k=0
a=3,b=−(k2+5k),c=3k2−5k
By Eq. (i), ‌‌b=0
⇒‌‌‌−(k2+5k)‌‌=0
⇒‌k(k+5)‌‌=0
‌∴‌k‌‌=0,−5
By Eq. (ii), Product of roots <0
‌

c

a

<0
⇒‌‌‌

3k2−5k

3

<0
⇒‌‌k(3k−5)<0
∴‌‌0<k<‌

5

3

From (i) and (ii) no such values of k exists.