This Class 12 Mathematics worksheet on Application of Derivatives, along with its detailed solutions, is prepared mainly for CBSE board exam preparation and objective-based revision. The worksheet contains 75 multiple-choice questions, all strictly aligned with NCERT. It covers increasing and decreasing functions, monotonicity, maxima and minima, first and second derivative tests, points of inflexion, tangents and normals, angle of intersection of curves, related rates, optimisation problems, and behaviour of trigonometric, exponential, and logarithmic functions. Overall, the worksheet reflects the exact style and difficulty level of one-mark calculus questions asked in board examinations Application of Derivatives WS 1….
I am writing about this worksheet because Application of Derivatives is one chapter where students often know the rules of differentiation but struggle to apply them correctly. Small mistakes in sign analysis, interval selection, or interpreting derivative conditions can easily change the final answer. A solution-based worksheet like this helps students understand not just the correct option, but also the reasoning behind it. Analysing these questions carefully makes exam preparation more systematic and helps avoid common traps in MCQs.
Structure of the Application of Derivatives Worksheet
The worksheet consists of 75 MCQs, each carrying one mark, followed by a full solution section. The solutions clearly show derivative steps, sign analysis, and logical conclusions. Questions range from direct concept checks to application-based problems that require careful interpretation.
The overall difficulty level ranges from moderate to high, making this worksheet suitable for revision after completing the chapter.
Increasing and Decreasing Functions
A large portion of the worksheet focuses on monotonicity. Students are tested on:
- Using the first derivative test
- Finding intervals where f′(x) is positive or negative
- Analysing monotonic behaviour of polynomial, trigonometric, exponential, and logarithmic functions
- Interpreting results over given intervals
These questions are among the most frequently asked MCQs in board exams Application of Derivatives WS 1….
Maxima and Minima Using Derivatives
The worksheet strongly emphasises maximum and minimum value problems. Questions include:
- Finding critical points by solving f′(x) = 0
- Applying the first and second derivative tests
- Identifying local and absolute maxima or minima
- Evaluating maximum or minimum values of functions
The solution explanations help students clearly see why a point is a maximum, minimum, or neither.
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Second Derivative Test and Point of Inflexion
Several MCQs test understanding of the second derivative. Topics covered include:
- Using f″(x) to confirm maxima or minima
- Identifying points of inflexion
- Understanding when f″(x) = 0 does not indicate an extremum
These questions are conceptual and require clarity rather than lengthy calculations.
Tangents, Normals and Angle of Intersection
The worksheet includes multiple questions on tangents and normals, such as:
- Finding slope of tangent at a given point
- Conditions for parallel or perpendicular tangents
- Angle of intersection of curves
- Equation-based reasoning using derivatives
These questions test both algebraic skill and conceptual understanding.
Related Rates and Rate of Change Problems
Several MCQs are based on related rates. Students encounter problems involving:
- Rate of change of area, volume, and surface area
- Motion-related problems using derivatives
- Geometrical situations involving cones, circles, and spheres
These questions help students connect calculus with real-life applications.
Optimisation and Word Problems
Optimisation problems form an important part of the worksheet. Questions include:
- Maximum and minimum values in practical situations
- Shortest distance and least value problems
- Maximisation of area, volume, or expressions
These MCQs require correct interpretation of the situation before differentiation.
Behaviour of Trigonometric and Exponential Functions
The worksheet also tests monotonicity and extremum of functions involving:
- Trigonometric expressions like sin x, cos x, tan x
- Exponential and logarithmic functions
- Mixed algebraic–trigonometric forms
These questions are common scoring areas if derivative rules are applied correctly.
