CAT Exam Model Paper 3 with solutions for free online practice

Given, 4x2−16x+

λ

4

=0
⇒ 16x2−64x+λ=0...(i)
It is given that α and β are the roots of Eq. (i)
∴α=

−(−64)+√(−64)2−4×16×λ

2×16

and β=

−(−64)−√(−64)2−4×16×λ

2×16

⇒α=

64+8√64−λ

32

and β=

64−8√64−λ

32

⇒α=

8(8+√64−λ)

32

and
β=

8(8−√64−λ)

32

⇒α=

8+√64−λ

4

and β=

8−√64−λ

4

Since, 1< α < 2 and 2 < β < 3
∴ 1<

8+√64−λ

4

<2 and 2<

8+√64−λ

4

<3
⇒4<8+√64−λ<8 and 8<8−√64−λ<12
⇒4−8<√64−λ<8−8 and 8−8<√64−λ<12−8
⇒−4<√64−λ<0 and 0<√64−λ<4
On squaring each term, we get
16 < 64 − λ < 0 and 0 < 64 − λ < 16
⇒ 16 − 64 < − λ < 0 − 64 and 0 − 64 < − λ < 16 − 64
⇒ − 48 < − λ < − 64 and − 64 < − λ < − 48
⇒ 48 > λ > 64 and 64 > λ > 48 [∵48 > λ > 64 not possible]
Hence, λ can take 15 values.