JEE Mathematics Continuity and Differentiability MCQs

Question. f(x) = a sin| x | +be^|x| is differentiable at x = 0, when :
(a) a + b = 0
(b) a – b = 0
(c) a = 0
(d) b = 0

Answer: A

Question. Statement 1 : f (x) = sin x + [x] is discontinuous at x = 0.
Statement 2 : If g (x) is continuous and h (x) is discontinuous at x = a, then g (x) + h (x) will necessarily be discontinuous at x = a.
(a) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.
(b) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.
(c) Statement -1 is False, Statement-2 is True.
(d) Statement -1 is True, Statement-2 is False.

Answer: A

Question. Statement 1 : | x³ | is differentiable at x = 0.
Statement 2 : If f (x) is differentiable at x = a then | f (x) | is also differentiable at x = a.
(a) Statement -1 is True, Statement-2 is False.
(b) Statement -1 is False, Statement-2 is True.
(c) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.
(d) None of these

Answer: A

Question. If f (x) = x^1/x – 1 for all positive x ≠1 and f is continuous at 1, then f(1) equals:
(a) 0
(b) e
(c) e²
(d) 1/e

Answer: A

Question. Let [x] denotes the greatest integer function and f(x) = [tan² x], then :
(a) f(x) is continuous at x = 0
(b) f(x) is discontinuous at x = 0
(c) f(0) = 1
(d) None of these

Answer: A

Question. The function f (x)= log(1 ax) log(1 bx)/x is not defined at x = 0, the value of which should be assigned to f at x = 0, so that it is continuous at x = 0, is :
(a) a + b
(b) log a – log b
(c) a – b
(d) log a + log b

Answer: A