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.
The probability of hitting the target by a trained sniper is three times the probability of not hitting the target on a stormy day due to high wind speed. The sniper fired two shots on the target on a stormy day when wind speed was very high. Find the probability that
(i) target is hit.
(ii) at least one shot misses the target.
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2.
A racing track is built around an elliptical ground whose equation is given by \[ 9x^2 + 16y^2 = 144 \] The width of the track is \(3\) m as shown. Based on the given information answer the following:

(i) Express \(y\) as a function of \(x\) from the given equation of ellipse.
(ii) Integrate the function obtained in (i) with respect to \(x\).
(iii)(a) Find the area of the region enclosed within the elliptical ground excluding the track using integration.
OR
(iii)(b) Write the coordinates of the points \(P\) and \(Q\) where the outer edge of the track cuts \(x\)-axis and \(y\)-axis in first quadrant and find the area of triangle formed by points \(P,O,Q\).
View Solution
3.
A rectangle of perimeter \(24\) cm is revolved along one of its sides to sweep out a cylinder of maximum volume. Find the dimensions of the rectangle.
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4.
Sports car racing is a form of motorsport which uses sports car prototypes. The competition is held on special tracks designed in various shapes. The equation of one such track is given as

(i) Find \(f'(x)\) for \(0<x>3\).
(ii) Find \(f'(4)\).
(iii)(a) Test for continuity of \(f(x)\) at \(x=3\).
OR
(iii)(b) Test for differentiability of \(f(x)\) at \(x=3\).
View Solution
5.
Find the domain of \(p(x)=\sin^{-1}(1-2x^2)\). Hence, find the value of \(x\) for which \(p(x)=\frac{\pi}{6}\). Also, write the range of \(2p(x)+\frac{\pi}{2}\).
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6.
If \[ P = \begin{bmatrix} 1 & -1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2 \end{bmatrix} \quad \text{and} \quad Q = \begin{bmatrix} 2 & 2 & -4 \\ -4 & 2 & -4 \\ 1 & -1 & 5 \end{bmatrix} \] find \( QP \) and hence solve the following system of equations using matrix method:
\[ x - y = 3,\quad 2x + 3y + 4z = 13,\quad y + 2z = 7 \]
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