CBSE Class 12 Mathematics Application of Derivatives VBQs Set 02

Fill in the blanks.

Question. The slope of the tangent to the curve \( y = x^3 - x \) at the point \( (2, 6) \) is _____________ .
Answer: 11

Question. The maximum value of \( f(x) = x + \frac{1}{x}, x < 0 \) is _____________ .
Answer: -2

Question. The rate of change of the area of a circle with respect to its radius r, when r = 3 cm, is _____________ .
Answer: \( 6\pi \text{ cm}^2\text{/cm} \)

Question. If \( f(x) = \frac{1}{4x^2 + 2x + 1} \), then its maximum value is _____________ .
Answer: \( \frac{4}{3} \)

Very Short Answer Questions:

Question. If the rate of change of volume of a sphere is equal to the rate of change of its radius, find the radius of the sphere.
Answer: \( \frac{1}{2\sqrt{\pi}} \text{ units} \)

Question. Find the interval in which the function f given by \( f(x) = 7 - 4x - x^2 \) is strictly increasing.
Answer: \( (-\infty, -2) \)

Question. At what points on the curve \( x^2 + y^2 - 2x - 4y + 1 = 0 \), the tangents are parallel to y-axis?
Answer: \( (-1, 2) \text{ and } (3, 2) \)

Question. It is given that at \( x = 1 \) the function \( x^4 - 62x^2 + ax + 9 \) attains the maximum value on the interval \( [0, 2] \). Find the value of a.
Answer: \( a = 120 \)

Question. Find the least value of \( \lambda \) such that the function \( (x^2 + \lambda x + 1) \) is increasing on \( [1, 2] \).
Answer: \( \lambda = -2 \)

Short Answer Questions–I:

Question. The contentment obtained after eating x-units of a new dish at a trial function is given by the function \( C(x) = x^3 + 6x^2 + 5x + 3 \). If the marginal contentment is defined as rate of change of \( C(x) \) with respect to the number of units consumed at an instant, then find the marginal contentment when three units of dish are consumed.
Answer: 68 units

Question. Prove that the function \( f(x) = \tan x - 4x \) is strictly decreasing on \( \left( -\frac{\pi}{3}, \frac{\pi}{3} \right) \).
Answer: \( f'(x) = \sec^2 x - 4 \). For \( x \in \left( -\frac{\pi}{3}, \frac{\pi}{3} \right) \), \( 1 \le \sec x < 2 \), so \( 1 \le \sec^2 x < 4 \), hence \( f'(x) < 0 \). Function is strictly decreasing.

Question. Find the value of a for which the function \( f(x) = \sin x - ax + b \) increasing on R.
Answer: \( (-\infty, -1) \)

Question. Show that the function \( f(x) = 4x^3 - 18x^2 + 27x - 7 \) is always increasing on \( \mathbb{R} \).
Answer: \( f'(x) = 12x^2 - 36x + 27 = 3(4x^2 - 12x + 9) = 3(2x - 3)^2 \ge 0 \). Hence always increasing.

Question. Prove that \( f(x) = \sin x + \sqrt{3} \cos x \) has maximum value at \( x = \frac{\pi}{6} \).
Answer: \( f'(x) = \cos x - \sqrt{3} \sin x \). Setting \( f'(x) = 0 \Rightarrow \tan x = \frac{1}{\sqrt{3}} \Rightarrow x = \frac{\pi}{6} \). \( f''(x) = -\sin x - \sqrt{3} \cos x < 0 \) at \( x = \frac{\pi}{6} \).

Question. Show that the function f defined by \( f(x) = (x - 1) e^x + 1 \) is an increasing function for all \( x > 0 \).
Answer: \( f'(x) = (x-1)e^x + e^x(1) = xe^x \). For \( x > 0 \), \( xe^x > 0 \). Hence increasing.

Short Answer Questions–II:

Question. Find the equations of the tangent and the normal to the curve \( y = \frac{x-7}{(x-2)(x-3)} \) at the point where it cuts the x-axis.
Answer: \( x - 20y - 7 = 0 \) and \( 20x + y - 140 = 0 \) respectively

Question. A ladder 13 m long is leaning against a vertical wall. The bottom of the ladder is dragged away from the wall along the ground at the rate of 2 cm/sec. How fast is the height on the wall decreasing when the foot of the ladder is 5 m away from the wall?
Answer: \( \frac{5}{6} \text{ cm/sec} \)

Question. Find the intervals in which the function \( f(x) = 3x^4 - 4x^3 - 12x^2 + 5 \) is (a) strictly increasing (b) strictly decreasing.
Answer: (a) \( (-1, 0) \cup (2, \infty) \) (b) \( (-\infty, -1) \cup (0, 2) \)

Question. Find the point on the curve \( 9y^2 = x^3 \), where the normal to the curve makes equal intercepts on the axes.
Answer: \( \left( 4, \frac{8}{3} \right) \) and \( \left( 4, -\frac{8}{3} \right) \)

Question. Find the equations of the normals to the curve \( y = x^3 + 2x + 6 \) which are parallel to the line \( x + 14y + 4 = 0 \).
Answer: \( x + 14y - 254 = 0 \) and \( x + 14y + 86 = 0 \)

Question. Prove that the semi-vertical angle of the right circular cone of given volume and least curved surface area is \( \cot^{-1} \sqrt{2} \).
Answer: [Result proved following optimization of surface area with respect to volume]

Question. Find all the points of local maxima and local minima of the function \( f(x) = -\frac{3}{4}x^4 - 8x^3 - \frac{45}{2}x^2 + 105 \).
Answer: Local maxima at 0, -5; and local minima at -3

Question. Using differentials, find the approximate value of \( \sqrt{0.082} \).
Answer: 0.2867

Long Answer Questions:

Question. Find the dimensions of the rectangle of perimeter 36 cm which will sweep out a volume as large as possible, when revolved about one of its side. Also, find the maximum volume.
Answer: Length = 12 cm, breadth = 6 cm and maximum volume = \( \frac{216}{\pi} \text{ cm}^3 \)

Question. Find the angle of intersection of the curve \( y^2 = 4ax \) and \( x^2 = 4by \).
Answer: 90°

Question. The volume of a sphere is increasing at the rate of 3 cubic centimeter per second. Find the rate of increase of its surface area, when the radius is 2 cm.
Answer: \( 3 \text{ cm}^2\text{/sec.} \)

Question. Find the local maxima and local minima, of the function \( f(x) = \sin x - \cos x, 0 < x < 2\pi \), Also find the local maximum and local minimum values.
Answer: Local maximum value = \( \sqrt{2} \), local minimum value = \( -\sqrt{2} \)

Question. Find the value of p for which the curves \( x^2 = 9p(9 - y) \) and \( x^2 = p(y + 1) \) cut each other at right angles.
Answer: \( p = 0, 4 \)

Question. Find the point on the curve \( y = \frac{x}{1 + x^2} \), where the tangent to the curve has the greatest slope.
Answer: (0, 0)

Question. Find the absolute maximum and absolute minimum values of the function f given by \( f(x) = \cos^2 x + \sin x, x \in [0, \pi] \).
Answer: Absolute maximum value = \( \frac{5}{4} \) at \( x = \frac{\pi}{6} \) and \( \frac{5\pi}{6} \), absolute minimum value = 1 at \( x = 0, \frac{\pi}{2} \) and \( \pi \)

Question. Find the equation of tangents to the curve \( y = \cos (x + y), -2\pi \le x \le 2\pi \), that are parallel to the line \( x + 2y = 0 \).
Answer: \( x + 2y = 0 \)

Question. Determine the intervals in which the function \( f(x) = x^4 - 8x^3 + 22x^2 - 24x + 21 \) is strictly increasing or strictly decreasing.
Answer: \( (1, 2) \cup (3, \infty) \); \( (-\infty, 1) \cup (2, 3) \)

Question. Find the equation of the normal at a point on the curve \( x^2 = 4y \) which passes through the point (1, 2). Also find the equation of the corresponding tangent.
Answer: \( x + y - 3 = 0 \); \( x - y - 1 = 0 \)

Question. A manufacturer can sell x items at a price of Rs left( 5 - \frac{x}{100} \right) \) each. The cost price of x items is Rs left( \frac{x}{5} + 500 \right) \). Find the number of items he should sell to earn maximum profit.
Answer: 240 items

Question. A wire of length 34 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a rectangle whose length is twice its breadth. What should be the lengths of the two pieces, so that the combined area of the square and the rectangle is minimum?
Answer: 16 m, 18 m

Question. If \( y = x^4 - 10 \) and x changes from 2 to 1.99, then what is the change in y?
(a) 0.32
(b) 0.032
(c) 5.68
(d) 5.698
Answer: (a)

Question. The maximum slope of curve \( y = -x^3 + 3x^2 + 9x - 27 \) is
(a) 0
(b) 12
(c) 16
(d) 32
Answer: (b)

Question. The maximum value of \( \frac{\log x}{x} \) in \( [2, \infty) \) is
(a) 0
(b) 1
(c) \( \frac{1}{e} \)
(d) e
Answer: (c)