The chapter “Continuity and Differentiability” is central in Class 12 Mathematics because it links limits, continuity, and tangent-slope behavior of functions. Board questions typically test continuity/differentiability at a point using limits and derivatives, while competitive exams often ask conceptual understanding via inverse functions, mean value theorem ideas, and carefully constructed piecewise functions.
20
Minutes
15
Questions
1 / -0
Marking
Q1. Let
. Then ?
does not exist
Q2. Let
. For which real numbers is differentiable at ?
Q3. Let be continuous on and differentiable on with and for all . The best possible upper bound for is:
Q4. Consider
. Which statement about differentiability at is correct?
Both and are differentiable at
Neither nor is differentiable at
is differentiable at but is not
is not differentiable at but is
Q5. Does there exist a differentiable function such that
? Choose the correct statement.
Yes, such functions exist
No, no such function exists
Yes, such a function exists but must be discontinuous at
Such a function exists only if
Q6. Let for and . Then
does not exist
Q7. Let for and for . For which value of is differentiable at ?
Q8. Let and let be the inverse function of . Then
Q9. For real , define for and . For which values of is differentiable at ?
Q10. Let be continuous on and differentiable on . Suppose exists (finite). Which conclusion is valid?
may fail to be differentiable at even if exists
is differentiable at and
is differentiable at but need not equal
the limit must be
Q11. Let for and . Which of the following is true about at ?
is not continuous at .
is continuous at but not differentiable at .
is differentiable at and .
is differentiable at and .
Q12. Find real numbers such that the function defined by for and for is differentiable at .
Q13. Let for all real . Which of the following statements is correct?
is decreasing on .
is increasing on and attains its global maximum at .
has no stationary points.
is even and attains its minimum at .
Q14. Assertion (A): If is differentiable on an interval , one-to-one on , and for all , then the inverse function is differentiable on .
Reason (R): For , .
Both A and R are true and R is the correct explanation of A.
Both A and R are true but R is not the correct explanation of A.
A is true but R is false.
A is false but R is true.
Q15. Let and for , and . Which of the following statements about differentiability and continuity of derivatives at is correct?
Both and are differentiable at and both and are continuous at .
Neither nor is differentiable at .
Both and are differentiable at ; is not continuous at whereas is continuous at .
is not differentiable at but is differentiable at and is not continuous at .