Q: Prove that √3 − √2 is an irrational number.
A: √3 = 1.732; √2 = 1.414; √3 − √2 = 0.318 = 1/π, which is irrational.

Q: Show that the function f : Z → Z given by f(x) = 2x − 1 is one-to-one.
A: f(1) = 1, so f is one-to-one.

Q: Show that c > ℵ⁰, the cardinal of the natural numbers.
A: As c is the speed of light, and nothing can travel faster than light, therefore c > ℵ⁰.

Q: Decide whether or not the groups H and K are isomorphic.
A: H is isomorphic but K is not.

Q: Solve the differential equation

A:

Q: A committee of three men and four women is to be chosen from seven men and eight women. In how many different ways can this be done?

A:

This committee can be formed in 1½ ways or 3 ways if the people are doubled.

"The commutative law of addition is when you add a column of figures from bottom to top and from top to bottom and get two different answers."

"This series is not an arithmetic progression therefore it is a geometric progression."

"The minimum value of cosh x occurs when h = 1."

"An average is a thing that hens lay their eggs on — for example, 'My hens lay four eggs a week on average.'"

"The basic postulates of geometry are a pencil, a ruler, and a rubber."

"The image of a right angle under an axial symmetry of the plane is a left angle."

"90° is the boiling-point of a right-angled triangle."

"An angle is a triangle with only two sides."

"A circle is a round straight line with a hole in the middle."

"A curve is the longest distance between two points."

Examiner: Why is a thicker conductor necessary to carry a current in AC as compared to DC?
Student: An AC current goes up and down (drawing a sinusoid) and requires more space inside the wire, so the wire has to be thicker.

Examiner: How will you tell if that wall outlet carries AC or DC?
Student: I will put my finger in. If it is pushed away, it is AC. If it gets stuck, it was DC.

Examiner: How will you reverse direction of an induction motor?
Student: I will remove the four bolts at the base, turn the motor around, and put back the bolts.

Examiner: How do you start a synchronous motor?
Student: Vrrrrrrrmmmmmmmmmmmmmmmmmmmm (in rising pitch)
Examiner: Stop! Stop!
Student: Vrrrrrrrmmmmmmmmmmmmmmmmmmmm (in falling pitch)

Examiner: How do you limit surge current within an integrated circuit?
Student: By using a miniature circuit breaker.

Examiner: Why does a capacitor block DC but allow AC to pass through?
Student: See, a capacitor is like this ---| |--- , OK. DC Comes straight, like this ---------->, and the capacitor stops it. But AC, goes UP, DOWN, Up DOWN and jumps right over the capacitor !

Examiner: What is a step-up transformer?
Student: A transformer that is put on top of electric poles.
Examiner (smiling): And then what is a step-down transformer?
Student (hesitantly): Uh — A transformer that is put in the basement or in a pit ?
Examiner (pouncing): Then what do you call a transformer that is Installed on the ground ?
(student knows he is caught — can't answer)
Examiner (impatiently): Well ?
Student (triumphantly): A stepless transformer, sir!