NCERT Solutions For Class 12 Mathematics Chapter 6 Applications of Derivatives

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NCERT Solutions for Class 12 Mathematics Chapter 6 Application of Derivatives covers important concepts of determinants, rate of change of quantities, tangents and normals, increasing and decreasing functions, Approximations, Maxima and minima and many more. The word “Derivative” comes from “derive” meaning to get or obtain something from something else. A derivative is an expression that provides us with the rate of change of a function related to an independent variable.

The chapter Calculus with chapters Continuity and Differentiability and Application of Derivatives Class 12 has a weightage of 10 marks in the CBSE Class 12 examination. Questions related to increasing or decreasing functions, tangents and normals, maxima and minima are generally asked in the examination. Simple problems demonstrating basic principles and understanding of derivatives are also included.

Download PDF: NCERT Solutions for Class 12 Mathematics Chapter 6

NCERT Solutions for Class 12 Mathematics Chapter 6 Application of Derivatives

Important Topics in Class 12 Mathematics Chapter 6 Application of Derivatives

Rate of change of quantity – If we have a function y = f(x), then the rate of the change of function is defined as dy/dx = f'(x).

Further, if the two variables x and y are varying to some other variable, say if x = f(t), and y = g(t), then using the Chain Rule, we have: dy/dx = (dy/dt)/(dx/dt) where dx/dt isn’t equal to 0.

Increasing and Decreasing Functions – Consider a function f that is continuous in [a,b] and differentiable on the open interval (a,b), then the function can be determined to be increasing or decreasing in the following way.

f is increasing in [a,b] if f'(x) > 0 for each x in (a,b) f is decreasing in [a,b] if f'(x) < 0 for each x in (a,b) f is a constant function in [a,b], if f'(x) = 0 for each x in (a,b)

Finding tangents and normals for a given curve is necessary to find the maxima and minima of the function, in turn.

A tangent at a point on a curve is a straight line that touches the curve at that specific. Its slope is equal to the gradient or derivative of the curve at that point. A normal is a straight line at a point on the curve that intersects the curve at that particular point and is perpendicular to the tangent at that point.

NCERT Solutions For Class 12 Maths Chapter 6 Exercises

The detailed solutions for all the NCERT Solutions for Chapter 6 Application of Derivatives under different exercises are as follows:

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CBSE CLASS XII Related Questions

1.
An amount of ₹ 10,000 is put into three investments at the rate of 10%, 12% and 15% per annum. The combined annual income of all three investments is ₹ 1,310, however, the combined annual income of the first and second investments is ₹ 190 short of the income from the third. Use matrix method and find the investment amount in each at the beginning of the year.
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2.
Let \( \vec{a} \) and \( \vec{b} \) be two co-initial vectors forming adjacent sides of a parallelogram such that:
\[ |\vec{a}| = 10, \quad |\vec{b}| = 2, \quad \vec{a} \cdot \vec{b} = 12 \] Find the area of the parallelogram.
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3.
Find the general solution of the differential equation \[ x^2 \frac{dy}{dx} = x^2 + xy + y^2 \] OR
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4.
Prove that:
\( \tan^{-1}(\sqrt{x}) = \frac{1}{2} \cos^{-1}\left( \frac{1 - x}{1 + x} \right), \quad x \in [0, 1] \)
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5.
Find: \[ I = \int (\sqrt{\tan x} + \sqrt{\cot x}) dx. \]
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6.
Three students run on a racing track such that their speeds add up to 6 km/h. However, double the speed of the third runner added to the speed of the first results in 7 km/h. If thrice the speed of the first runner is added to the original speeds of the other two, the result is 12 km/h. Using the matrix method, find the original speed of each runner.
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