Riddle 11 [With Answers]

Puzzle 1: What kind of room has no doors or windows?

Answer: Mushroom

Puzzle 2: What kind of tree can you carry in your hand?

Answer: A Palm

Puzzle 3: Which word in the dictionary is spelled incorrectly?

Answer: Incorrectly

Puzzle 4: If you have me, you want to share me. If you share me, you haven't got me. What am I?

Answer: Secret

Puzzle 5: What gets broken without being held?

Answer: A promise

Voltage Source across Resistor

 

V(t) = V_m sin ωt

i(t) = I_m sin ωt

And P(t) = v(t) i(t)

P(t) = V_m sin ωt I_m sin ωt

Hence, average power is

It is clear that if current or voltage waveform has a frequency of 50 Hz then power waveform has a frequency of 100Hz.

Riddles 11

Riddle 1: What kind of room has no doors or windows?

Riddle 2: What kind of tree can you carry in your hand?

Riddle 3: Which word in the dictionary is spelled incorrectly?

Riddle 4: If you have me, you want to share me. If you share me, you haven't got me. What am I?

Riddle 5: What gets broken without being held?

First order and first degree differential equations and their geometrical interpretations

A first order and first degree differential equation involves the independent variable x (say), dependent variable y (say) so, it can be put in any one of the following forms:

dy/ dx = f(x, y) or f (x, y) = 0, or f(x, y) dx + g(x, y)dy = 0

Where f(x, y) and g(x, y) are functions of x and y.

Geometrical interpretation

The general from of a first order and first degree differential equation is f(x, y, dy/dx) = 0 … (i)

We know that the tangent of the direction of a curve in Cartesian rectangular coordinates at any point is given by dy/dx, so the equation in (i) can be known as an equation which establishes the relationship between the coordinates of a point and the slope of the tangent i.e., dy/dx to the integral curve at that point. Solving the differential equation given by (i) means finding those curves for which the direction of tangent at each point coincides with the direction of the field. All the curves represented by the general solution when taken together will give the locus of the differential equation. Since there is one arbitrary constant in the general solution of the equation of first order, the locus of the equation can be said to be made up of single infinity of curves.

Riddles 10 [With Answers]

Puzzle - 1:

A car has to carry an important person across the desert.

There is no petrol station in the desert and the car has space only for enough petrol to get it half way across the desert.

There are also other identical cars that can transfer their petrol into one another.

How can we get this important person across the desert?

Answer: We need 4 cars (including the car with the important person).

All 4 cars start full.

At 1/6th of the way, all cars are 2/3rds full. One car sacrifices itself and fills up two of the other cars. Two cars are now full, one is 2/3rds full. (An empty car is left behind.)

At 2/6th of the way, two cars are 2/3rds full. One car is one third full, and sacrifices itself to fill up one of the other cars. One car is now full, one is 2/3rds full. (An empty car is left behind.)

At half way, one car is 2/3rds full. One car is one third full, and sacrifices itself to fill the other car, which is now full and can make the other half of the journey.

Puzzle - 2:

Joey leaves his house in the morning to go to day camp.

Just as he is leaving his house he looks at an analog clock reflected in the mirror.

There are no numbers on the clock, so Joey makes an error in reading the time since it is a mirror image. Joey assumes there is something wrong with the clock and rides his bike to day camp.

He gets there in 20 minutes and finds that just as he gets there the day camp clock has a time that is 2 1/2 hours (2 hours and 30 minutes) later than the time that he saw in the mirror image of his clock at home.

What time was it when he got to day camp?

(The clock at camp and the clock at home were both set to the correct time.)

Answer: First subtract 20 minutes from 2 1/2 hours to compensate for his 20 minute bike ride to give a difference of 2 hours and 10 minutes.

To be a "Mirror Effect" it must be mirrored around 12 o'clock (when the hands are straight up), or around 6 o'clock (when the hands are pointing up and down), as we know he left in the morning, it must be 6 o'clock.

So, divide that 2 hours and 10 minutes by 2 and this will give you the center-point (65 minutes) for compensating for the mirror.

By adding that 65 minutes to 6 o'clock you get the time he left home (7:05), and the time he saw in the mirror (4:55).

Furthermore, by re-adding the 20 minutes from when he left (7:05), you get what time he got to camp (7:25).

Puzzle - 3:

In front of you are several long fuses.

You know they burn for exactly one hour after you light them at one end.

But the entire fuse does not burn at a constant speed. For example, it might take five minutes to burn through half the fuse and fifty-five minutes to burn the other half.

With your lighter and using these fuses, how can you measure exactly three-quarters of an hour of time?

Answer: Light both ends of one fuse.

At the same time light one end of a second fuse.

The first fuse will finish in half an hour.

At that point the second fuse will be half done (in time, not necessarily in distance) and you immediately light its other end. The second half hour will now take only quarter of an hour.

Total time: half an hour plus quarter of an hour equals three-quarters of an hour.

Puzzle - 4:

You have two straight lengths of wood.

How can you cut one of them so that one of the three pieces is the average length of the other two?

Answer: Put the two pieces end to end in a straight line

Then the average length of the three cut pieces has to be one third of this total length.

So we simply cut one third of the way along the longer piece.

Puzzle - 5:

A girl, a boy, and a dog start walking down a road.

They start at the same time, from the same point, in the same direction.

The boy walks at 5 km/h, the girl at 6 km/h.

The dog runs from boy to girl and back again with a constant speed of 10 km/h. The dog does not slow down on the turn.

How far does the dog travel in 1 hour?

Answer: 10km. Because the dog's speed is 10 km/h.

Diazonium Salts - Methods of Preparation

The diazonium salts have the general formula RN⁺₂X⁻ where R stands for an aryl group and X^- ion may be Cl⁻, Br⁻, HSO⁻₄, BF⁻₄ etc.

Primary aliphatic amines form highly unstable alkyldiazonium salts.

Primary aromatic amines form arenediazonium salts which are stable for a short time in solution at low temperatures (273 - 278 K).

Preparation: Benzenediazonium chloride is prepared by the reaction of aniline with nitrous acid at 273-278K.

Nitrous acid is produced in the reaction mixture by the reaction of sodium nitrite with hydrochloric acid. The conversion of primary aromatic amines into diazonium salts is known as diazotisation.

Due to its instability, the diazonium salt is not generally stored and is used immediately after its preparation.

Voltage Source across Inductor

V(t) = ωL I_m cos ωt

V(t) = X_LI_m cos ωt

Where; X_L = ωL; X_L is known as inductance of circuit and has a unit of “ohm”.

V(t) = XLI_m sin (ωt + 90⁰)

And instantaneous power

p(t) = v (t) * i(t)

= V_m cos ωt I_m sin ωt

So average power is 

Hence in positive half cycle of the power, inductor takes energy from the source and in the negative cycle inductor delivers energy to the source. Hence not power dissipation is zero.

From above mathematical expressions we can draw phasor diagram of rms value of voltage and current.

From vector diagram it is clear that in an inductor current lags to voltage by  (or voltage leads to currents by 90⁰).

Riddle 10

Riddle - 1:

A car has to carry an important person across the desert.

There is no petrol station in the desert and the car has space only for enough petrol to get it half way across the desert.

There are also other identical cars that can transfer their petrol into one another.

How can we get this important person across the desert?

Riddle - 2:

Joey leaves his house in the morning to go to day camp.

Just as he is leaving his house he looks at an analog clock reflected in the mirror.

There are no numbers on the clock, so Joey makes an error in reading the time since it is a mirror image. Joey assumes there is something wrong with the clock and rides his bike to day camp.

He gets there in 20 minutes and finds that just as he gets there the day camp clock has a time that is 2 1/2 hours (2 hours and 30 minutes) later than the time that he saw in the mirror image of his clock at home.

What time was it when he got to day camp?

(The clock at camp and the clock at home were both set to the correct time.)

Riddle - 3:

In front of you are several long fuses.

You know they burn for exactly one hour after you light them at one end.

But the entire fuse does not burn at a constant speed. For example, it might take five minutes to burn through half the fuse and fifty-five minutes to burn the other half.

With your lighter and using these fuses, how can you measure exactly three-quarters of an hour of time?

Riddle - 4:

You have two straight lengths of wood.

How can you cut one of them so that one of the three pieces is the average length of the other two?

Riddle - 5:

A girl, a boy, and a dog start walking down a road.

They start at the same time, from the same point, in the same direction.

The boy walks at 5 km/h, the girl at 6 km/h.

The dog runs from boy to girl and back again with a constant speed of 10 km/h. The dog does not slow down on the turn.

How far does the dog travel in 1 hour?

Number Puzzle 4 [With Answers]

1. Which number replaces the question mark?

Answer: 10.

Explanation: Working in columns, the sum of the numbers in each column equals 23.

2. Which number replaces the question mark?

Answer: 7.

Explanation: In each square, multiply the top and bottom numbers together to give a 2 digit result, and put the letters with the numerical values of each digit in the left and right hand spaces.

3. Which number replaces the question mark?

Answer: 13

Explanation: Working in rows, from left to right, double the left hand number to get the middle number, and add 3 to this to get the right hand number.

4. Which number replaces the question mark?

Answer: 38

Explanation: Starting at the top, add 4 to the first number to get the second, then add 5, 6, 7 etc.

5. Which number replaces the question mark?

Answer: 3.

Explanation: Using the lower two circles as a source, the values in corresponding segments of the upper left circle equal the sums of the numbers in the lower two circles. The values in the upper central circle equal the products of the values in the lower two circles, and the upper right circle equals the difference between values in the lower two circles.

Formation of differential equations

Consider a family of exponential curves (y = Ae^x), where A is an arbitrary constant for different values of A, we get different members of the family. Differentiating the relation (y = Ae^x) w.r.t.x, we get dy/dx = Ae^x

Eliminating the arbitrary constant between y = Ae^x and dy/dx = Ae^x, we get dy/dx = y. This is the differential equation of the family of curves represented by y = Ae^x

Thus, by eliminating one arbitrary constant, a differential equation of first order is obtained.

Now consider the family of curves given by y = A cos 2x + B sin 2x ... (1)

Where A and B are arbitrary consists.

Differentiating (1) w.r.t, x we get dy/ dx = - 2Asin 2x + 2Bcos 2x ... (2)

Differentiating (2) w.r.t. x we get d²y/ dx² = - 4Acos ax - 4Bsin 2x ... (3)

Eliminating A and B from equations (1) and (2) (3), we get

d²y/ dx² = - 4y ⇒ d²y/ dx² + 4y = 0

Here we note that by eliminating two arbitrary consists, a differential equation of second order is obtained.

Step I: write the given equation involving independent variable x (say), dependent variable y (say) and the arbitrary constants.

Step II: obtained the numbers of a arbitrary constants in step in step I. let there be n arbitrary consists.

Step III: differentiate the relation in step in times with respect to x.

Step IV: eliminate arbitrary constants with the help of n equations involving differential coefficient obtained in step III and an equation in step I.

The equation so obtained is the desired differential equation.

Example: Show that the differential equation that represented all parabolas having their axis symmetry coincident with the axis of x is yy₂ + y₁² = 0.

Solution: The equation that represents a family of parabolas having their axis of symmetry coincident with the axis of x is y² = 4a(x - h) ... (1)

This equation contains two arbitrary constants, so we shall differentiate twice to obtain second order differential equation.

Differentiating (1) w.r.t x we get

2y dy/dx = 4a ⇒ y dy/ dx = 2a ... (2)

Differentiating (2) w.r.t, x we get

y d²y/ dx² + (dy/ dx)² = 0 ⇒ yy₂ + y₁² = 0

Which is the required differential equation.

Example: Find the differential equation of all non-horizontal lines in a plane.

Solution: The equation of the family of all non-horizontal line in a plane is given by

Ax + by = 1 ... (1)

Where a, b are arbitrary constants such that (a ≠ 0)

 Differentiating (1) w.r.t, x we get

a dx/dy + b = 0

Differentiating this w.r.t y we get 

ad²x/ dy² = 0

⇒ d²x/ dy² = 0

Hence, the differential equation of all non-horizontal lines in a plane is d²x/ dy² = 0.

Example: Find the differential equation of all non- vertical lines in a plane.

Solution: The general equation of all non-vertical lines in a plane is (ax + by = 1) where (b ≠ 0).

Now,

ax + by = 1

a + b dy/dx = 0            [differentiating w.r.t.x]

b d²y/ dx = 0                [differentiating w.r.t.x]

d²y/ dx² = 0                  [∵ b ≠ 0]

Hence, the differential equation is d²y/ dx² = 0

Solution of a differential equation: The solution of a differential equation is a relation between the variable involved which satisfies the differential equation. Such a relation and the derivates obtained therefore when substituted in the differential equation, makes left hand right hand sides identically equal.

General solution: The solution which contains as many as arbitrary constants as the order of the differential equations is called the general solution of the differential equation.

For example, y = Acos x + Bsin x is the general solutions one arbitrary constant.

Particular solution: Solution obtained by giving particular values to the arbitrary constant in the general solution of a differential equation is called a particular solution.

Example: Show that xy = ae^x + be^{- x} + x² is a solution of the differential equation  x d²y/ dx² + 2 dy/dx - xy + x² - 2 = 0

Solution: We are given that xy = ae^x + be^{- x} + x² ... (1)

Differentiating w.r.t.x, we get x dy/ dx + y = ae^x - be^{- x} + 2x

Differentiating again w.r.t.x, we get xd²y/ dx² + dy/dx + dy/dx = ae^x + be^{- x} + 2

xd²y/ dx² + 2dy/dx = ae^x + be^{- x} + 2 ... (2)

Now x d²y/ dx² + 2dy/dx - xy + x2 - 2

= [ae^x + be^{- x} + 2] - [ae^x + be^{- x} + x²] + x² - 2

= 0 [using (1) and (2)]

Thus, (xy = ae^x + be^{- x} + x²) is a solution of the given differential equation.