Question

Let a, b ∈ R, b ≠ 0. Define a function

F(x) = `{{:(asin  π/2(x - 1)",", "for"  x ≤ 0),((tan2x - sin2x)/(bx^3)",", "for" x > 0):}`

If f is continuous at x = 0, then 10 – ab is equal to ______.

Options

• 13

• 14

• 15

• 16

Answer 1

Let a, b ∈ R, b ≠ 0. Define a function

F(x) = `{{:(asin  π/2(x - 1)",", "for"  x ≤ 0),((tan2x - sin2x)/(bx^3)",", "for"  x > 0):}`

If f is continuous at x = 0, then 10 – ab is equal to 14.

Explanation:

If function is continuous at x = 0

Then, LHL = RHL = f(0)

LHL = –a

RHL = `lim_(x→0) (tan2x - sin2x)/(bx^3)`

Since, sinx = `x - x^3/(3!) + x^5/(5!) + ....`

tanx = `x + x^3/3 + (2x^5)/15 + ...`

∴ RHL = `lim_(x→0^+) ((2x)^3/3 + (2x)^3/6)/(bx^3) = (8/3 + 8/6)/b = 4/b`

∴ f(0) = `asin[π/2(0 - 1)] = asin((-π)/2)` = –a

⇒ LHL = RHL = f(0)

⇒ `4/b` = –a

⇒ –ab = 4

⇒ 10 – ab = 14