Plus Two Maths Chapter Wise Previous Questions Chapter 6 Application of Derivatives are part of Plus Two Maths Chapter Wise Previous Year Questions and Answers. Here we have given Plus Two Maths Chapter Wise Previous Chapter 6 Application of Derivatives.

## Kerala Plus Two Maths Chapter Wise Previous Questions and Answers Chapter 6 Application of Derivatives

Question 1.

a. f (x) is a strictly increasing function, if f ’ (x) is

i. positive

ii. negative

iii. O

iv. None of these

b. Show that the function f given by f(x)=x^{3}-3x^{2}+4x, x∈ R is strictly increasing.** [March-2018] **Answer:

a. positive

b. f(x) = x

^{3}-3x

^{2}+4x

f(x) is strictly increasing for all values of x at which

f(x)=3x

^{2}-6x+4

= 3(x-1)2+1>0 for all x∈R

∴ f(x) is strictly increasing on R.

Question 2.

a. Find the slope of the tangent to the curve y = (x-2)^{2} at x=l.

b. Find a point at which the tangent to the curve y = (x-2)^{2} is parallel to the chord joining the points A(2,0) and B(4, 4).

c. Find the equation of the tangent to the above curve and parallel to the line AB. **[March-2018] **Answer:

a. 2(x – 2) = 2(1 – 2) = -2

b. y = (x-2)^{2 }Diff. w.r.to x =2(x-2) dx

slope of the tangent = 2(x-2)

slope of the chord joining (2,0) and

Since the tangent is parallel to the chord. Slope of the tangent = Slope of the chord

i.e; 2(x-2) = 2⇒x-2=1 ⇒x=3

When x=3, y=(3-2)^{2} = 1

Hence the required point is (3,1)

c.

Slope = 2 dx

y= (x-2)^{2 } =2(x-2)

2(x-2) = 2 = 2x-4=2

2x=6 x=3

When x=3, y=(3-2)^{2} =1

At (3,1) the tangent is parallel to AB.

The equation of the tangent to the given curve at (3,1) is

y-1=2(x-3) .

y-1 = 2x-6

y-2x-1+6 = 0

y-2x+5 = 0

Question 3.

Slope of the normal to the curve y^{2} = 4x at (1, 2) is

a. 1

b.

c. 2

d. -1

b. Find the interval in which 2x^{3} + 9x^{2}+12x-1 is strictly increasing.

**OR **a. The rate of change of volume of a sphere with respect to its radius when radius is 1 unit .

a. 4π

b. 2π

c. π

d. π/2

b. Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.

**[March-2017]**

Answer:

a. y

^{2}= 4x

b. Let d(x) = 2x

^{3}+ 9x

^{2}+ 12x – 1

f(x) is strictly increasing for all values of x at which f (x) >0.

⇒ 6x

^{2}+ 18x+12 > 0

⇒ 6(x

^{2}+ 3x + 2) > 0

⇒ x

^{2}+ 3x + 2 < 0

⇒ (x + 2)(x + l)< 0

⇒ -2 < x < -1

∴ d(x) strictly increasing for in the intervals (-∞, -2) and (-1, ∞)

**OR**

b. Let one number be x. Then the other number is 16 – x.

Let S = x

^{3}+ (16-x)

^{3 }0<x< 16.

S is minimum where

^{ }0 and

= 6 x 8 + 6(16-8) = 48 + 48=96>0 dx

∴ S is minimum when x = 8

∴ The two numbers are 8 and 8.

Question 4.

a. The slope of the tangent to the curve given by

x =1-cosθ,y=θ-sine at θ= is

(i) 0

(ii) -1

(iii) 1

(iv) Not defined

b. Find the intervals in which the function f(x) = x^{2} – 4x + 6 is strictly decreasing.

c. Find the minimum and maximum value, if any, of the function f(x)=(2x-1)^{2} +3.

**OR **a. Which of the following functions has neither local maxima nor local minima?

(i) f(x)=x

^{2}+x

(ii) f(x) =log x

(iii) f(x)=x

^{3}-3x+3

(iv) f(x)=3+|x|

b. Find the equation of the tangent to the curve y = 3x

^{2}at (1,1).

c. Use differential to approximate

**[March-2016]**

ANS:

ANS:

a (iii).

Question 5.

a. Which of the following functions is always increasing?

(i) x + sin 2x

(ii) x – sin 2x

(iii) 2x + sin 3x

(iv) 2x – sin x

b. The radius of a cylinder increases at a rate of 1 cm/s and its height decreases at a rate of 1 cm/s. Find the rate of change of its volume when the radius is 5 cm and the height is 15 cm.

If the volume should not change even when the radius and height are changed, what is the relation between the radius and height?

c. Write the equation of tangent at (1,1) on the curve 2x^{2} + 3y^{2} = 5. **[March-2015] **Answer:

Question 6.

a. Which of the following function is increasing for all values of x in its domain?

A. Sin x

B. log x

C. x^{2}

D. | x |

b. Find a point on the curve y = (x-2)^{2 }at which the tangent is parallel to the chord joining the points (2,0) and (4,4).

c. Find the maximum profit that a company can make, if the profit function is given by p(x) = 41 – 24x – 6x^{2}

**[March-2014] **Answer:

Question 7.

a. Find the slope of the normal to the curve y = sinθ at θ=

b. Show that the function x^{3} – 6x^{2} + 15x + 4 is strictly increasing in R.

c. Show that all rectangles with a given perimeter, the square has the maximum area.** [March-2013] **Answer:

c.

Question 8.

i. Show that the function f(x) = x^{3}– 3x^{2}+ 6x-5 is strictly increasing on R.

ii. Find the intervals in which f(x) = sinx + cosx; 0 ≤ x ≤ 2π is strictly increasing or strictly decreasing.** [June -2012] **Answer:

Question 9.

a. The slope of the tangent to the curve y = x^{3} – 1 at x = 2 is ……..

b. Use differential to approximate

c. Find two numbers whose sum is 24 and whose product is as large as possible. ** [March – 2012] **Answer:

Question 10.

Find the approximate value of (82)^{1/4} upto 3 places of decimal.

b. Find two positive numbers such that their sum is 8 and the sum of their squares is minimum.**[June-2011] **Answer:

Question 11.** **a. The radius of a circle is increasing at the rate of 2 cm/sec. Find the rate at which the area of the circle is increasing when the radius is 6 cm.

b. Prove that/(x) = log sin x is strictly increasing in and strictly decreasing in .

c. Find the maximum and minimum value of f (x)=x

^{3}-6x

^{2}+9x+15.

**[March-2011]**

Answer:

Question 12.

Consider the curve y=x^{3}-3x + 2

i. Find a point on the curve whose x- cordinate is 3.

ii. Find the slope of the tangent and normal at this point

iii. Write the equation o the tangent and normal at this point. **[March 2010] **Answer:

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