The chapter “Application of Derivatives” is crucial for CBSE Class 12 as it builds on the idea of derivative as an instantaneous rate of change and uses it to solve real problems: maxima-minima, tangents/normals, monotonicity, convexity, and optimization. These concepts also appear frequently in competitive exams because they test both computation and the ability to reason using derivative properties (like Darboux’s theorem and convexity).
15
Minutes
10
Questions
1 / -0
Marking
Q1. Use linear approximation (tangent at ) to estimate the value of .
Q2. Find the point(s) on the curve that are nearest to the point .
Q3. A rectangle has one vertex at the origin and the opposite vertex on the curve in the first quadrant. Find the width and height of the rectangle with maximum area.
width , height
width , height
width , height
width , height
Q4. Consider the statements
(A) If is differentiable on and for all , then is one-to-one on .
(R) The derivative has the intermediate value (Darboux) property; hence if never vanishes on it cannot change sign there, so is monotonic on .
Both (A) and (R) are true but (R) does not correctly explain (A).
(A) is true but (R) is false.
Both (A) and (R) are true and (R) is a correct explanation of (A).
(A) is false and (R) is true.
Q5. For define . For which values of is differentiable at and convex on ?
all
Q6. Find the equation of the tangent to the curve at the point where .
Q7. Find all real values of the parameter for which the tangent to the curve at passes through the origin.
only
Q8. Let . For which real value(s) of does the normal to the curve at the point pass through the origin?
No real value of
Q9. Assertion (A): If is twice differentiable on and for all , then the equation has at most two distinct real roots.
Reason (R): If for all , then must be a quadratic polynomial with positive leading coefficient.
Both A and R are true and R is a correct explanation for A.
Both A and R are true but R is not a correct explanation for A.
A is true but R is false.
A is false but R is true.
Q10. Let
Determine which of the following is true about at .
is not differentiable at .
is differentiable at but is not continuous at .
is differentiable at and is continuous at .
is differentiable everywhere and is unbounded in every neighbourhood of .