Match the column:
Column-I | Column-II |
(i) If y = 2[x] + 9 = 3[x + 2], where[.] denotes greatest integer function, then [x + y] is equal to | [A] –1 |
(ii) If = ek/2 then k is equal to | [B] 0 |
(iii) If three successive terms of a G.P. with common ratio r, (r > 1) forms the sides of a triangle then [r] + [–r] is equal to (where [.] denotes greatest integer function) | [C] 2 |
(iv) Let f(x) = (x2 – 3x + 2)(x2+3x+2) and α, β, γ are the roots of f '(x) = 0, then [α]+[β] + [γ] is equal to (where [.] denotes greatest integer function) | [D] 3 |
Text Solution
Verified by Experts(i) [D]; (ii) [C]; (iii) [A]; (iv) [A]
Ans.
(i) [D]
(ii) [C]
(iii) [A]
(iv) [A]
Sol. (i) y = 2[x] + 9 = 3[x + 2]
2[x] + 9 = 3[x] + 6 ⇒ [x] = 3
⇒ y = 2 × 3 + 9 = 15
Now 
[x + y] = 
([x] + y) = 
(3+15) = 3
(ii) 
(1 ∞ form)
= 
= 
(put x = 1/t)
= 
= e 1 = e k/2 ⇒ k = 2
(iii) Let sides of a triangle are a, ar, ar
2 Since r > 1 ∴ ar
2 is the greatest side
⇒ a + ar > ar 2 ⇒ r 2 – r – 1 < 0
⇒ 
also r > 1
⇒ 1 < r < 
∴ [r] = 1 and [–r] = –2
⇒ [r] + [–r] = –1
(iv) f(x) = (x – 1) (x – 2) (x + 1) (x + 2)
⇒ f(–2) = f(–1) = f(1) = f(2) = 0
∴ by rolles theorem the equation f '(x) = 0
has roots in (–2, –1), (–1, 1) and (1, 2)
∴ [ α ] = –2, [ β ] = –1 or 0 , [ γ ] = 1
also f '(0) = –4 – 2 + 4 + 2 = 0
so [ β ] = 0
∴ [ α ] + [ β ] + [ γ ] = –1
Prepare Smarter with CGP Edu
Get practice questions, solutions, and test series in one place.
Write a Review
Share your experience with this question and solution.
Commentary
Send your comment, doubt, correction, or feedback to admin.
Similar Questions
Explore conceptually related problems