Question

Let f :R → R be a function defined as f (x) =
(A) continuous if a = 5 and b = 5 (B) continuous if a = 5 and b = 10
(C) continuous if a = 0 and b = 5 (D) not continuous for any values of a and b [JEE Main 2019]

Answer 1

Answer:

This type of question is usually solved quickly by substituting options. But let's go by the standard method.

Let us discuss the continuity at x = 1.

For limit tending to x = 1- and 1, the value of f(x) is 5.

For limit tending to x = 1+, the value of f(x) is: a + bx

→ a + b ( 1 ) = a + b

Since LHL = RHL, we get:

→ a + b = 5   ... ( Eqn. 1 )

Now talking about continuity at x = 3, we get:

For limit tending to x = 3-, the value of f(x) is a + b(3) = a + 3b

For limit tending to x = 3 and 3+, the value of f(x) is b+5x

→ b + 5(3) = b + 15

Since LHL = RHL, we get:

→ a + 3b = b + 15

→ a + 2b = 15   ... ( Eqn. 2 )

Solving Eqn. 1 and Eqn. 2 we get:

→ ( 5 - b ) + 2b = 15

→ 5 + b = 15

→ b = 10

This implies a = 5 - b

→ a = -5

Since none of the options match, Option D is the correct answer.