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NCERT Questions for Class 12 Maths Chapter 6 – Application of Derivatives
Important questions on Class 12 Maths Applications of Derivatives will assist students in preparing for and excelling in the Class 12 Maths examination. These questions were developed by experts utilising the latest curriculum. Students are urged to diligently prepare for the board examinations by practicing the essential questions for each chapter in the 12th standard mathematics curriculum offered by StudyMaterialsOnline.
Important Questions with Solutions of Class 12 Maths Chapter 6 – Application of Derivatives
1) Determine the maximum and minimum values, if applicable, of the function f(x) = -|x-1| + 7∀x ∈ R
Ans – Given f(x) = -|x+1| + 3
|x+1| > 0
⇒ -|x+1| < 0
Maximum value of g(x) = maximum value of -|x+1| + 7
⇒ 0 + 7 = 7
Maximum value of f(x) = 3
There is no minimum value of f(x).
2) Find the interval over which the function f(x)=cos x, 0 ≤ x ≤ 2π is decreasing.
Ans – The given the function is f(x)=cos x, 0 ≤ x ≤ 2π.
If it is a strictly decreasing function then f'(x) < 0.
Differentiating w.r.t. x, we get
f'(x) = -sin x
Now
f'(x) < 0
⇒ -sin x < 0
⇒ sin x > 0 i.e.,(0,π)
Hence, the given function is decreasing in (0,π).
3) Find the interval on which the function f(x) = 1/x is strictly decreasing.
Ans – Given function is
If it is a strictly decreasing function then f'(x) < 0.
The intervals are (-∞,-1), (-1,1), (1,∞)
f'(0) = 0
∴ Strictly decreasing in (-1,1)
4) Find the sub-interval of the interval (0,π/2) such that the function f(x) = sin 3x is increasing.
Ans – Given f(x) = sin 3x
Differentiate the above function w.r.t x,
f'(x) = 3cos 3x
f(x) will be increasing when f'(x) > 0
In the first quadrant, the cosine function is positive and in the second it is negative.
Case 1:
Case 2:
5) A particle follows the curve with equation 6y = x3 + 2. Find the points on the curve at which y co-ordinate changes 8 times faster than the x coordinate.
Ans – Equation of the curve is 6y=x^3+2.
Differentiating both sides w.r.t t,
When x = 4
When x = -4
∴ Points on the curve are (4, 11) and (-4, -31/3).
6) A man 2 meters tall moves away from a lamp post 6 meters high at a constant pace of 5km/hr. Determine the speed at which his shadow length increases.
Ans – Consider AB be the lamp post.
Let at any time t, the man CD is x meters away from the lamp post and that the length of his shadow is 9 meters.
Clearly ∆ABE and ∆CDE are similar,
⇒ 3y = x + y
⇒ 2y = x
Differentiate both sides w.r.t x
7) Determine the equation of all lines that have a slope of zero and are tangents to the curve
Ans – Given the equation of the curve,
At every point (x,y), the slope of the tangent to the given curve is given by,
If the slope of the tangent is 0, i.e. m = 0, then we have:
The equation of the tangent through (1,1/2) is given by,
8) Determine the equation of the tangent at 
to the curve √x + √y = a.
Ans – Given equation of the curve is √x + √y = a
Differentiate with respect to x in order to determine the slope of the tangent of the given curve.
At slope m is-1
Equation of the tangent is given by y – y1 = m (x – x1)
9) Prove that at the point where the curve crosses the y-axis, 
touches the curve y = be(-x/a).
Ans – The abscissa of the point at where the curve crosses the y axis i.e., x=0
∴ y = b.e-x/a = b [∵ e0=1]
∴ On the curve with the y-axis, the slope point of intersection is (0,b).
At (0,b), the slope of the given line is given by,
Also at (0,b) the slope of the curve is,
Since m1 = m2 = (-b)/a
Drawing conclusions from above, we see that the line touches the curves at the point where they intersect the y-axis.
10) Determine the equation of the tangent to the curve at a position where t = π/2, given by x = a sin3t, y = b cos3t.
Ans – The provided equation of the curves are x = a sin3t, y = b cos3t
Differentiate w.r.t t,
So,
When t = π/2,
So, when t = π/2 then x = a and y = 0.
⇒ at t = π/2 or at (a,0), equation of tangent of given curve
y – 0 = 0(x – a) or y = 0