UPSC CDS 2 2022 Math Paper

1. For quadratic equation ax2+bx+c=0
x=‌

−b±√(b2−4ac)

2a

2. (a+b)(a−b)=a2−b2
3. (a+b)2=a2+b2+2ab
Using above formulas to solve
qx2−2px+q=0
By above formula,
x=‌

−b±√(b2−4ac)

2a

Here, a=q,b=−2p and c=q
x=‌

2p±√(4p2−4q2)

2q

⇒x=‌

p±√p2−q2

q

‌‌‌⋅⋅⋅⋅⋅⋅⋅(i)
Statement: 1
x=‌

√p+q+√p−q

√p+q−√p−q

, where p>q
Rationalizing the denominator
⇒‌

√p+q+√p−q

√p+q−√p−q

×‌

√p+q+√p−q

√p+q+√p−q

Using the identities (2) \& (3)
⇒x=‌

p+q+p−q+2√(p+q)(p−q)

p+q+p−q

⇒x=‌

p+q+p−q+2√(p2−q2)

p+q+p−q

⇒x=‌

p±√p2−q2

q

Hence, statement 1 is correct
Statement: 2
From equation (2), we can say that,
x=‌

p+√p2−q2

q

Hence, statement 2 is also correct.
Statement 3:
We have, x=‌

p±√p2−q2

q

For any value of p&q
x=‌

√p+q

√p−q

is not possible
Hence, statement 3 is wrong