Match the following columns :
Column-I | Column-II |
(i) If ax2 + bx + 6 = 0 does not have two distinct real roots where a∈R, b∈R. then least value of 3a + b is | [A] 4 |
(ii) The number of solutions of | [x] – 2x | = 4 where [⋅⋅] is greatest integer value ≤≤ x | [B] 42 |
(iii) The arithmetic mean of two positive numbers is 6 and their geometric mean G & harmonic mean H satisfy relation G2 + 3H = 48, then product of two numbers is | [C] –2 |
(iv) If a1, a2, a3 …a21 are in A.P. and a3 + a5 + a11 + a17 + a19 = 10 then the value of ai is | [D] 32 |
Text Solution
Verified by Experts(i) [C]; (ii) [A]; (iii) [D]; (iv) [B]; (i) [x]
Ans.
(i) [C]
(ii) [A]
(iii) [D]
(iv) [B]
Sol. (i) D ≤ ≤ 0
Now f(0) = 6 > 0
∴ f(x) ≥ ≥ 0 ∀ x ∈ R
∴ f(3) ≥ 0
9a + 3b + 6 ≥ 0
3a + b ≥ – 2
(ii) case(i) [x] –2x = 4
⇒ – [x] –2 {x} = 4
⇒ 
= {x}
∴ 0 ≤ – [x] – 4 < 2
4 ≤ – [x] < 6
⇒ –6 < [x] ≤ – 4
∴ [x] = – 5, or – 4
∴ {x} = 
if [x] = – 5
{x} = 0 if [x] = – 4
∴ x = 
or – 4
case (ii)
If [x] –2x = – 4
– [x] –2{x}= – 4
[x] + 2 {x} = 4
0 ≤ 4 – [x] < 2
⇒ 2 < [x] ≤ 4
[x] = 3 or 4
∴ {x} = 
when [x] = 3
= 0 when [x] = 4
∴ x = 
or 4
(iii) ∴ 
⇒ a + b = 12 …..(i)
Also ab + 
= 48 ..… (ii)
from (i) & (ii)
ab = 32
(iv) (a 1 + 2d) + (a 1 + 4d) + (a 1 + 10d) …+ (a 1 + 18d) = 10
⇒ 5(a 1 + 10d) = 10
⇒ a 1 + 10d = 2 ……(i)
Now 
= a 1 + a 2 + a 3 …….. + a 21
= 
[a 1 + a 21 ]
= 21 (a 1 + 10d)
= 42
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