**(I) Choose the correct option from the given alternatives:**

**Question 1.**

respectively……..

(a) 2, 1

(b) 1, 2

(c) 3, 2

(d) 2, 3**Answer:**

(d) 2, 3

**Question 2.**

**Question 3.**

**Question 4.The differential equation of all circles having their centres on the line y = 5 and touching the X-axis is**

**Question 5.**

(a) circles

(b) parabolas

(c) ellipses

(d) hyperbolas

Answer:

(a) circles

Hint:

**Question 6.**

**Question 7.**

**Question 8.**

**Question 9.**

**Question 10.**

**Question 11.The solution of the differential equation dy/dx= sec x – y tan x is…….**

(a) y sec x + tan x = c

(b) y sec x = tan x + c

(c) sec x + y tan x = c

(d) sec x = y tan x + c

**Answer:**

(b) y sec x = tan x + c

Hint:

**Question 13.**

**Question 14.The decay rate of certain substances is directly proportional to the amount present at that instant. Initially, there are 27 grams of substance and 3 hours later it is found that 8 grams left. The amount left after one more hour is……**

**Question 15.If the surrounding air is kept at 20°C and the body cools from 80°C to 70°C in 5 minutes, the temperature of the body after 15 minutes will be…..**

(a) 51.7°C

(b) 54.7°C

(c) 52.7°C

(d) 50.7°C

**Answer:**

(b) 54.7°C

**(II) Solve the following:**

**Question 1.**

Solution:

The given D.E. is

Differentiating both sides w.r.t. x, we get

Solution:

Solution:

y = 3 cos(log x) + 4 sin (log x) …… (1)

Differentiating both sides w.r.t. x, we get

Solution:

Differentiating both sides w.r.t. y, we get

**Question 3.Obtain the differential equation by eliminating the arbitrary constants from the following equations:**

Differentiating again w.r.t. x, we get

This is the required D.E.

**(ii) y = a sin(x + b)Solution:**

y = a sin(x + b)

This is the required D.E.

Differentiating both sides w.r.t. x, we get

Differentiating twice w.r.t. x, we get

This is the required D.E.

**Question 4.Form the differential equation of:(i) all circles which pass through the origin and whose centres lie on X-axis.Solution:**

Let C (h, 0) be the centre of the circle which pass through the origin. Then radius of the circle is h.

This is the required D.E.

**(ii) all parabolas which have 4b as latus rectum and whose axis is parallel to Y-axis.Solution:**

Let A(h, k) be the vertex of the parabola which has 4b as latus rectum and whose axis is parallel to the Y-axis.

Then equation of the parabola is

(x – h = 4b(y – k) ……. (1)

where h and k are arbitrary constants.

Differentiating both sides of (1) w.r.t. x, we get

This is the required D.E.

**(iii) an ellipse whose major axis is twice its minor axis.Solution:**

Let 2a and 2b be lengths of the major axis and minor axis of the ellipse.

Then 2a = 2(2b)

∴ a = 2b

∴ equation of the ellipse is

This is the required D.E.

∴ a = 4√k, b = 6√k

∴ l(transverse axis) = 2a = 8√k

and l(conjugate axis) = 2b = 12√k

Let 2A and 2B be the lengths of the transverse and conjugate axes of the required hyperbola.

Then according to the given condition

2A = a = 4√k and 2B = b = 6√k

∴ A = 2√k and B = 3√k

∴ equation of the required hyperbola is

This is the required D.E.

**Question 5.Solve the following differential equations:Solution:**

Solution:

Solution:

**(iv) x dy = (x + y + 1) dxSolution:**

This is the general solution.

Solution:

Solution:

**Question 6.Find the particular solution of the following differential equations:Solution:**

Solution:

This is the general solution.

When x = 2, y = 1, we have

This is the linear differential equation of the form

**(iv) (x + y) dy + (x – y) dx = 0; when x = 1 = ySolution:**

Solution:

**Question 7.Solution:**

**Question 8.The normal lines to a given curve at each point (x, y) on the curve pass through (2, 0). The curve passes through (2, 3). Find the equation of the curve.Solution:**

Let P(x, y) be a point on the curve y = f(x).

∴ equation of the normal is

This is the general equation of the curve.

Since, the required curve passed through the point (2, 3), we get

**Question 9.The volume of a spherical balloon being inflated changes at a constant rate. If initially its radius is 3 units and after 3 seconds it is 6 units. Find the radius of the balloon after t seconds.**

Solution:

Let r be the radius and V be the volume of the spherical balloon at any time t.

Then the rate of change in volume of the spherical balloon is which is a constant.

Hence, the radius of the spherical balloon after t seconds is (63t+27 units.

**Question 10.A person’s assets start reducing in such a way that the rate of reduction of assets is proportional to the square root of the assets existing at that moment. If the assets at the beginning are ₹ 10 lakhs and they dwindle down to ₹ 10,000 after 2 years, show that the person will be bankrupt in years from the start.Solution:**

Integrating both sides, we get