UPSC NDA I 2021 Mathematics Solved Paper

Concept
General Quadratic Equation
Ax2+Bx+C=0
Product of roots (αβ)=‌

C

A

Sum of roots (α+β)=‌

−B

A

Root find by using this formula, x=‌

−B±√B2−4AC

2A

Calculation:
Given: 4x2+2x−1=0
By comparing with Ax2+Bx+C=0
A=4,B=2,C=−1
Product of roots =‌

C

A

αβ=‌

−1

4

β=‌

−1

4α

Sum of roots =‌

−B

A

α+β=‌

−2

4

=‌

−1

2

α+β=‌

−1

2

α+‌

−1

4α

=‌

−1

2

4α2+2α−1=0
α=‌

−b±√b2−4ac

2a

α=‌

−2±√22−4(4)(−1)

2(4)

α=‌

−1±√5

4

Consider positive value Let α=‌

−1+√5

4

β=‌

−1

4α

=‌

−1

4

(1+√5)
putting the value of α, weget
β=−2α2−2α
‌‌=−2(‌

−1+√5

4

)2−2(‌

−1+√5

4

)
‌‌=‌

−3

4

+‌

√5

4

+‌

1

2

−‌

√5

2

‌‌=‌

−1

4

(1+√5)
So, option 1 is correct.