Homework Statement
If the divisor P is a prime of the form 4q+1 then the number -1 or P-1 is certainly a residue.
Homework Equations
The Attempt at a Solution
First the book told me to prove that 1 and -1 are the only two remainders that are their own reciprocals modP...
Homework Statement
Calculate the output impedance of the emitter-follower circuit called a pass transistor. Assume that beta=200
**See attached diagram**
Homework Equations
The Attempt at a Solution
Not really sure how this works, I thought it would just be 1k cause that is the...
Homework Statement
The absolute max voltage rating for the base-emitter voltage is the lowest of all values. This low value means the B-E junction must be protected from a high reverse voltage. A diode is used for an npn transistor. Explain its function and conditions under which...
Homework Statement
Make a modification so that the battery is charged by the DC supply at a current of 10mA
**See attached diagram**
Homework Equations
The Attempt at a Solution
Well I know that I would put in a resistor in parallel with the diode. However, I don't know how...
Homework Statement
sketch the output waveforms when a 5Vp sine wave with a frequency of 100 Hz is applied to each of the following circuits **see attached diagrams**
Homework Equations
The Attempt at a Solution
Homework Statement
What would happen if the load resistor was short-circuited. What would happen if the load resister was removed?
-see attached diagram-
Homework Equations
P=Iin^2*R
Pz=IzVz
=(Iin-Iout)Vz=[((Vin-Vz)/R)-Iout]Vz
The Attempt at a Solution
I'm not sure about...
Homework Statement
Given a sine wave signal (20V PP) connected to a 10 kilo-Ohm resistor. Calculate the current through the resistor in terms of average, rms, and peak currents. What wattage rating should the resistor have?
Homework Equations
Vrms=(1/Sqrt(2))(Vp)
V=IR
The Attempt...
Homework Statement
Design a RC circuit so it has the same V vs time characteristics
*See attached diagram*
Homework Equations
The Attempt at a Solution
I really have no idea how to approach this, but I do know,
RC circuit:
Vc(t)=Vb(1-e^-(t/RC))
RL circuit...
Homework Statement
Design an RC circuit that has the same Voltage vs. Time characteristics as the RL circuit shown
*See attached Diagram*
Homework Equations
The Attempt at a Solution
Not really sure how to approach this, but all i know is the equations:
RC circuit...
Homework Statement
Sketch a graph of Vout vs time after the switch is closed
*See attached diagram*
Homework Equations
The Attempt at a Solution
I am not really sure how to do this, because I don't know if it makes a difference whether or not the resistor is before the...
Homework Statement
Find the current through R2 (see attached figure)
Homework Equations
The Attempt at a Solution
Not really sure if I did this right, would like if someone would double check my work:
First I tried Using Kirchhoff equations for a node and I got:
I1-I2+I3=0
Next I broke...
Homework Statement
A voltmeter whose resistance is 1000 ohms measures the voltage of a worn out 1.5V battery as .9V. What is the internal Resistance
Homework Equations
The Attempt at a Solution
In order to do this, I Solved for the Equiv Resistance of the circuit (leaving in...
Homework Statement
Looking at the open circuit demonstrated in figure 2 (see attachment), what is the current through the ammeter? What is the voltage at point p?
Homework Equations
The Attempt at a Solution
I am assuming if it an open circuit, I would think that the current...
Homework Statement
How do you measure the output of a voltage divider with real voltmeters that have effective resistances
Homework Equations
The Attempt at a Solution
I know that for an ideal voltmeter (one with infinite resistance), the voltage divider is R2/(R1+R2), but I...
Homework Statement
What is the diffraction-limited object size (at 25 cm) imposed by the numerical aperture of the eye (if the eye is a diffraction-limited optic)? Use 4mm for the iris diameter and 550 nm for the wavelength.
Homework Equations
Rayleigh criterion states: for angular...
Homework Statement
(2) For the combination of one prism and 2 lenses shown (Fig. 1), find the location and size of the final image when the object, length 1 cm, is located as shown in the figure. [Hint: Treat the prism as a mirror, but you have to take into account the image shift caused by...
Homework Statement
a) A beam of light in the air strikes the surface of a smooth piece of plastic having an index of refraction of 1.55 at an angle with the normal of 60 degrees. The incident light has equal component E-field amplitudes parallel and perpendicular to the plane of incidence...
Homework Statement
How to show that there exists a triangle whose defect is greater than 14 degrees
Homework Equations
The Attempt at a Solution
No idea what to do here.... something about the angle of parallelism
Homework Statement
Prove that a scalene triangle exists
Homework Equations
The Attempt at a Solution
I really don't know how to approach this proof.... should I do a direct proof or maybe by contradiction? Any help would be great...
Homework Statement
Given a line l, a point A on l, and a point B not on l. Then every point of the ray AB (except A) lies on the same side of l as B.
Homework Equations
The Attempt at a Solution
I understand why this is true, however I'm having some trouble wording my proof...
Homework Statement
Consider a function g : (a, b)-->R. Assume that g is differentiable at some point c in
(a,b) and that g'(c) is not = 0. Show that there is a delta > 0 so that g(x) is unequal to g(c) for all x in V_delta(c)\{c}intersect(a,b)
Homework Equations
The Attempt at a...
Homework Statement
Let f : (a, b)--> R be differentiable on (a,b), and assume that f'(x) unequal 1 for all x in (a,b).
Show that there is at most one point c in (a,b) satsifying f(c) = c.
Homework Equations
The Attempt at a Solution
I think that We need to use the mean value...
Homework Statement
Let f : [0; 1] --> R be continuous on [0, 1], and assume that the range of f is contained
in [0; 1]. Prove that there exists an x in [0, 1] satisfying f(x) = x.
Homework Equations
The Attempt at a Solution
Well i am almost positive I need to use the...
Homework Statement
Let G be a planar graph with n vertices, q edges, and k connected components.
If there are r regions in a planar representation of G, prove that:
n − q + r = 1 + k
Hint: Use induction on k. The base case is Euler’s formula.
Homework Equations
The Attempt at a...
Homework Statement
Let G be a non-empty graph of order n whose vertices have degrees d1, . . . , dn.
The line graph of G is defined as follows: the vertices of L(G) are the edges of G, and two vertices of L(G) are adjacent if they share an endpoint in G. Prove that the size of L(G) is...
Homework Statement
Consider a function f : R--> R, and assume that there is a c is in (0, 1) so that
|f(x) - f(y)|<= c|x -y|
for all x, y in R.
(a) Show that f is continuous on R.
(b) Given a point y1 in R define a sequence by yn+1 = f(yn). Prove that yn is a Cauchy sequence
(and...
Homework Statement
Let f : A --> R be a function, and let c in A be an isolated point of A. Prove that f
is continuous at c
Homework Equations
The Attempt at a Solution
I'm kind of confused by this problem.... if c is an isolated point, then the limit doesn't exist. So I can't...
Homework Statement
Let f, g : A \rightarrow R be functions and let c \in R be a limit point of A. Assume that lim(x-> c): f(x) = L
and lim(x->c): g(x) = M. Use the epsilon-delta definition to prove:
lim(x->c) f(x)g(x)=LM
Homework Equations
The Attempt at a Solution
alright in...
Homework Statement
Show if K contained in R is compact, then supK and inf K both exist and are elements of K.
Homework Equations
The Attempt at a Solution
Ok we proved a theorem stating that if K is compact that means it is bounded and closed.
So if K is bounded that means...
Homework Statement
Let E be a nonempty subset of R, and assume that E is both open
and closed. Since E is nonempty there is an element a \in E. Denote the set
Na(E) = {x > 0|(a-x, a+x) \subsetE}
(a) Explain why Na(E) is nonempty.
(b) Prove that if x \in Na(E) then [a-x, a+x] \subset...