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### Math Assignment Class XII Ch -06 | Application of Derivatives

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## Math Assignment Class XII Ch - 09

## Differential Equations

**Extra questions of chapter 09 Applications of Derivatives, class XII with answers and hints to the difficult questions, strictly according to the CBSE Board . Important and useful math. assignment for the students of class XII****MATHEMATICS ASSIGNMENT OF CHAPTER 06**

**STRICTLY ACCORDING TO THE PREVIOUS CBSE SAMPLE QUESTION PAPERS AND CBSE BOARD PAPERS**

## RATE OF CHANGE OF QUANTITIES

**Question 1**

**A particle moves along the curve x2 = 2y. At what point,
ordinate increases at the same rate as abscissa increases?**

**Answer (1, 1/2)**

**Solution Hint:**

**Differentiating the given equation
w. r. t. t**

**Now putting dy/dt = dx/dt and
find the value of x.**

**Putting the value of x in the
given equation and find the value of y**

**Question 2**

**The side of an equilateral triangle is increasing at the
rate of 2 cm/s. At what rate is its area increasing when the side of triangle
is 20 cm?**

**Answer: 20√3 cm ^{2}/s**

**Question 3**

**Volume of sphere is increasing at the rate of 3cm ^{3}/s.
Find the rate of increase of surface area, when radius is 2 cm**

**Answer: 3cm ^{2}/s**

**Question 4**

**For the curve y = 5x – 2x ^{3}. If x increases at the
rate of 2 unit/sec., find the rate of change of the slope of the curve when x =
3.**

**Answer: -72 unit /sec**

**Question 5**

**A man 2m high walks at a uniform speed of 6km/h away from
the lamp post 6m high. Find the rate at which the length of his shadow increases.**

**Answer : 3km/h**

**Solution Hint**

**AB = 6m, CD = 2m dy / dt = 6 km/h, **

**Let AC = y, CE = x, AE = x + y, **

**To find : dx/dt = ?**

△ABE ~ △ CDE

∴

⇒ y = 2x

Differentiating on both side w.r.t. t we get

△ABE ~ △ CDE

∴

⇒ y = 2x

Differentiating on both side w.r.t. t we get

## INCREASING & DECREASING OF FUNCTIONS

**Question 6**

**Without using derivative show that the function f(x) = 4x ^{3} – 18x^{2}
+ 27x – 7 is always increasing in R**

**Question 7**

**Find the interval in which the function f(x) is (i) Strictly Increasing (ii) Strictly Decreasing**

**Answer**

**Strictly Decreasing in (-2, 6)**

**Question 8**

**Find the intervals in which the function f(x) = x ^{4 }/ 4 – x^{3} – 5x^{2} + 24x + 12
is**

**i) Strictly increasing ii) Strictly decreasing**

**Answer: **

**Strictly Decreasing in (-∞, 3)⋃(2, 4)**

**Question 9**

**Find the interval in which the function:**

**f(x) = sin ^{4}x + cos^{4}x, 0 ≤ x < Ï€/2**

**is strictly increasing or strictly decreasing.**

**Answer:**

**Strictly increasing in (Ï€/4, Ï€/2)**

**Strictly decreasing in (0, Ï€/4)**

**Question 10**

**Find the intervals in which f(x) = sin3x - cos3x, 0<x<Ï€, is strictly increasing or strictly decreasing.**

**Answer:**

**Strictly increasing in (0, Ï€/4) ⋃ (7Ï€/12, 11Ï€/12)**

**Strictly decreasing in (Ï€/4, 7Ï€/12) ⋃ (11Ï€/12, Ï€)**

**Question 11**

**Prove that f(x) = x**

^{2}-x + 1 is neither increasing nor decreasing strictly on (-1, 1)

**Question 12**

**Find the interval in which the following functions are strictly increasing and strictly decreasing : f(x) = 4x**

^{3}– 6x^{2}- 72x + 30**Answer:**

**Strictly increasing in (-∞, -2)⋃ (3, ∞).**

**Strictly Decreasing in (-2, 3)**

**Question 13**

**Find the interval in which the following functions are strictly increasing and strictly decreasing : f(x) = 2x ^{3} – 12x^{2} + 18x + 5**

**Answer:**

**Strictly increasing in (-∞, 1)⋃ (3, ∞).**

**Strictly Decreasing in (1, 3)**

**Question 14:**

**Find the interval in which the following functions are strictly increasing and strictly decreasing :**

**Answer:**

**Strictly increasing in **

**Strictly Decreasing in **

**Question 15**

**Find the intervals in which the function given by f(x) = sin3x, x∈ [0, Ï€/2] is increasing and decreasing.**

**Answer**

**Increasing on [0, Ï€/6]**

**Decreasing on [Ï€/6, Ï€/2]**

## MAXIMA AND MINIMA

**Question 16**

**If , then find the maximum value of f(x)**

**Ans: 3/4**

**Solution Hint:**

**f(x) is maximum if 4x ^{2}
+ 2x + 1 is minimum**

**g(x) = 4x ^{2} + 2x + 1 **

**g'(x) = 8x + 2**

**For critical point g'(x) =0 ⇒ x = -1/4**

**g''(x) = 8 > 0**

**⇒ g(x) is minimum at x = -1/4**

**Minimum value of g(x) = 3/4**

**⇒ f(x) is maximum at x = 3/4**

**Question 17**

**Find the maximum value of **

**Ans: 1/e**

**Solution Hint**

**Find the derivative of f(x) we get **

**f ' (x) = 0 ⇒ x = 0 and e (critical points)**

**But at x = 0 logx is not defined so we have x = e only**

**Now find f '' (x) we get **

**At x = e, f ''(x) < 0 ⇒ f(x) is maximum at x = e**

**Max. value of f(x) = f(e) = 1/e (∵ Loge = 1)**

**Question 18**

**Find the least value of f(x) = e**

^{x}– e^{-x }**Answer: 2**

**Question 19**

**Without using the derivatives, find the maximum and minimum values if any for the function f(x) = sin2x + 5.**

**Answer: Max. value = 6, Min Value = 4**

**Question 20**

**Of all the rectangles each of which has perimeter 40 meters, find one which has maximum area. Also find the maximum area?**

**Solution Hint:**

**Let sides of rectangle = x and y**

**Perimeter = 2(x + y)**

**ATQ : 2(x + y) = 40 ⇒ x + y = 20 ⇒ y = 20 - x**

**A = Area = xy = x(20 - x) ⇒ 20x - x**

^{2}**A' = 20 - 2x**

**A' = 0 ⇒ 20 - 2x = 0 ⇒ x = 10**

**A'' = -2 < 0 ⇒ A is maximum at x = 10**

**If x = 10 then y = 10 ⇒ Rectangle is a square**

**Maximum area = xy = 10 x 10 = 100 sq m**

**Question 21**

**At what point, the slope of the curve y = - x**

^{3}+ 3x^{2}+ 9x - 27 is maximum? Also find the maximum slope.**Answer: Maximum slope = m = 12 at x = 1**

**Solution Hint**

**y = - x**

^{3}+ 3x^{2}+ 9x - 27**Slope = m = dy/dx = -3x**

^{2}+ 6x + 9**Now using second derivative test to find the maxima and minima.**

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