Consider the differential equation $${{\text{x}}^2}\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{x}}^2}}} + {\text{x}}\frac{{{\text{dy}}}}{{{\text{dx}}}} - {\text{y}} = 0.$$ Which of the following is a solution to this differential equation for x

Consider the line integral $$I = \int_{\text{c}} {\left( {{{\text{x}}^2} + {\text{i}}{{\text{y}}^2}} \right){\text{dz,}}} $$ where z = x + iy. The line c is shown in the figure below

The value of $$I$$ is

A

$$\frac{1}{2}{\text{i}}$$

B

$$\frac{2}{3}{\text{i}}$$

C

$$\frac{3}{4}{\text{i}}$$

D

$$\frac{4}{5}{\text{i}}$$

The figure shows the plot of y as a function of x

The function shown is the solution of the differential equation (assuming all initial conditions to be zero) is

A

$$\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{x}}^2}}} = 1$$

B

$$\frac{{{\text{dy}}}}{{{\text{dx}}}} = {\text{x}}$$

C

$$\frac{{{\text{dy}}}}{{{\text{dx}}}} = - {\text{x}}$$

D

$$\frac{{{\text{dy}}}}{{{\text{dx}}}} = \left| {\text{x}} \right|$$

It is desired to concentrate a 20% salt solution (20 kg of salt in 100 kg of solution) to a 30% salt solution in an evaporator. Consider a feed of 300 kg/min at 30°C. The boiling point of the solution is 110°C, the latent heat of vaporisation is 2100 kJ/kg and the specific heat of the solution is 4 kJ/(kgK). The rate at which the heat has to be supplied in (kJ/min) to the evaporator is

A

3.06 × 105

B

6.12 × 105

C

7.24 × 105

D

9.08 × 105

Singular solution of a differential equation is one that cannot be obtained from the general solution gotten by the usual method of solving the differential equation.

A

True

B

False

Consider the differential equation $$\frac{{{\text{dy}}}}{{{\text{dx}}}} = 1 + {{\text{y}}^2}.$$
Which one of the following can be a particular solution of this differential equation?

A

y = tan(x + 3)

B

y = tan x + 3

C

x = tan(y + 3)

D

x = tan y + 3

Consider the following second-order differential equation: y" - 4y' + 3y = 2t - 3t²
The particular solution of the differential equation is

A

-2 - 2t - t2

B

-2t - t2

C

2t - t2

D

-2 - 2t - 3t2

Consider the differential equation $$\left( {{{\text{t}}^2} - 81} \right)\frac{{{\text{dy}}}}{{{\text{dt}}}} + 5{\text{ty}} = \sin \left( {\text{t}} \right)$$ with y(1) = 2π. Thereexists a unique solution for this differential equation when t belongs to the interval

A

(-2, 2)

B

(-10, 10)

C

(-10, 2)

D

(0, 10)

Consider the following statements about emulsion SBR and solution SBR. I. Solution SBR has higher content of non-rubber materials than emulsion SBR II. Solution SBR has higher molecular weight but broader molecular weight distribution than emulsion SBR III. Solution SBR has higher cis-1,4-polybutadiene content than emulsion SBR Which of the following statements are true?

A

I, II, III

B

I, II

C

III only

D

I only

Suppose you have four test tubes labelled as 'A', 'B', 'C' and 'D'. 'A' contains plain water, 'B' contains solution of an alkali, 'C' contains solution of an acid, and 'D' contains solution of sodium chloride. Which one of these solution will turn phenophthalein solution pink?

A

Solution 'A'

B

Solution 'B'

C

Solution 'C'

D

Solution 'D'

Solution X is percent alcohol by volume, and solution Y is 30 percent alcohol by volume. How many milliliters of solution Y must be added to 200 milliliters of solution X to create a solution that is 25 percent alcohol by volume?

A

600

B

800

C

300

D

250