Test: Continuity and Differentiability- Assertion & Reason Type Questions - Commerce MCQ

= 2θ

= 2 sin^-1 3x

A is true. R is false.

Consider the functions f(x) = sin x and g(x) = |x| both of which are continuous in R.

gof(x) = g(f(x)) = g(sin x) = |sin x |.

Since f(x) and g(x) are continuous in R, gof(x) is also continuous in R.

Hence A is true.

R is the correct explanation of A.

A differentiable function is always continuous. Hence R is true.

A continuous function need not be always differentiable.

For example, |x| is continuous at x = 0, but not differentiable at x = 0.

Hence A is false.

Hence R is true.

f(x) = tan² x is not defined when x = π/2.

Therefore f(π/2) does not exist and hence f(x) is not continuous at x = π/2.

A is false.

So f(x) is not continuous at x = 2. Hence R is true.

A differentiable function is always continuous.

Since f(x) = [x] is not continuous at x = 2, it is also not differentiable at x = 2.

Hence A is true.

R is the correct explanation of A.

This is the definition for modulus function and hence true.

Hence R is true.

Since f is continuous at x = 0,

Here f(0) = 3,

∴ -k = 3 or k = -3

Hence A is true.

R is the correct explanation of A.

Hence A is true.

At x = 0, LHD ≠ RHD.

So f(x) is not differentiable at x = 0.

Hence R is false.

∴ R is true.

∴ k = 7

Hence A is false.