Question

Consider f(x) = sin^–1[2x] + cos^–1([x] – 1) (where [.] denotes greatest integer function.) If domain of f(x) is [a, b) and the range of f(x) is {c, d} then `a + b + (2d)/c` is equal to ______. (where c < d) 

Options

• 2.00

• 3.00

• 4.00

• 5.00

Answer 1

Consider f(x) = sin^–1[2x] + cos^–1([x] – 1) (where [.] denotes greatest integer function.) If domain of f(x) is [a, b) and the range of f(x) is {c, d} then `a + b + (2d)/c` is equal to 4.00. (where c < d) 

Explanation:

f(x) = sin^–1[2x] + cos^–1([x] – 1)

–1 ≤ [2x] ≤ 1 and –1 ≤ [x] – 1 ≤ 1

⇒ –1 ≤ 2x < 2 and 0 ≤ [x] ≤ 2

⇒ `-1/2 ≤ x < 1` and 0 ≤ x < 3

⇒ 0 ≤ x < 1 Domain [0, 1)

⇒ [x] = 0

⇒ 0 ≤ 2x < 2

⇒ [2x] = 0 or 1

Now f(x) = sin^–1[2x] + cos^–1(–1)

= `(0 or π/2) + π = π or (3π)/2`

⇒ `a + b + (2d)/c` = 1 + 3 = 4