Vaibhav,   Sorry, I forgot about this question over the weekend, as I got pretty busy. Perhaps you already have decoded the solution, but based on what you wrote, I will give you my interpretation of what is going on.     From the solution you present, it seems that...

Vaibhav and Richard,   The solution given by Richard above shows a correct approach to the problem, but makes one small but crucial misstep.  In the line where the modulus of the expression (z + 3/z) is computed, the given formula is not quite correct.  Letting |z|=R,...

Sammy,   The matrix of L w.r.t. the standard basis is gotten by simply taking the coefficients of x as one column, and the coefficients of y as the next column.   M = (3  -1; 1  1; -1  0)   To get the matrix w.r.t. the new bases of R2 and...

Sammy,    Recall that the kernel of a transformation is the subset of the domain which maps into the zero vector.  If the kernel is itself the zero vector, then the transformation is injective (one-to-one).   So, determine the vectors (x y)T in R2 whose image...

Hi Rola,   To show that B={v1,v2,v3} is an orthogonal set, simply compute the inner products (dot products) of each pair of vectors and show it is 0.   To show that B is orthonormal, additionally, compute the magnitude of each vector and show it is 1.   To...

Hi Sarah,   Indeed, this problem looks quite tricky at first.  However, if we carefully inspect what we know, maybe we can find a solution.     Let's first name the points P1, P2, ... , P41, in such a way that the x-coordinates are in ascending order, x1 ≤...

Shayoni,   Let's write down a couple of observations first.  If we let P denote the perimeter of the triangle, note that   b+c-a = P-2a a+c-b = P-2b a+b-c = P-2c   Let's furthermore assume, for convenience, that a≤b≤c, so that   -2a...

Hello Polo,   You will want to use the formula   q = m⋅C ⋅ ΔT   where m is mass in grams, C is the specific heat of liquid water, and ΔT is the change in temperature. q is heat and is measured in Joules.   Regards,  Has...

Hi Lucy,   This is a residue calculus question, so you will want to find a complex-valued extension of the integrand or else a complex-valued integrand whose real part coincides with the integrand, and then integrate it over an appropriately chosen contour and do some estimates on the...

Lucy,   Some things to think about...   a)  What is the residue of f(z) = 1/z4 at z=0?   b)  I can't make out what limit you are trying to represent.   c)  If you choose w=0 for the pole for simplicity, and you represent f and g...

Hello Savitri,   I assume you meant to write "if an unknown polynomial..." in which case, let us denote the unknown polynomial by f(x).   We can encode the two conditions on the polynomial as follows:   f(x) = q1(x)⋅(x-1) + 2 f(x) = q2(x)⋅(x-2)...

To Someone in San Antonio, You haven't stated what it is you want to integrate, only the vector field and the surface. I'll have to assume you mean to integrate curl F ⋅ dS over the surface S stated, so that Stokes' Theorem can be invoked.   In that case, the...

Justin,   I would suggest computing the probability that the player loses.   So, compute   P(losing) = P(No H in 12 turns)⋅P(losing | No H) + P(1 H)⋅P(losing | 1 H) + ⋅⋅⋅ + P(12 H)⋅P(losing | 12 H)   and then take the complementary...

Bilal,   I believe you are seeking a solution to the differential equation   θ'(t) = 2sin θ.   This is a separable equation, so separate variables and integrate to obtain    ∫ csc θ dθ = ∫ 2dt            ...

Hi Sarah,   Since f is a one-to-one function, (x,y) is on the graph of f if and only if (y,x) is on the graph of f-¹.  See if you can take it from there.   Regards, Hassan

Hi Paislei,   Assuming that each dog is four-legged, and each bird two-legged, then the total number of legs is four times the number of dogs plus two times the number birds.   Let d = number of dogs, and b = number of birds she owns.  We'd get a two equations:   4d...

Ishadri,   These types of problems (where you repeat some portion of a number) involve factoring out the "repeated portion" and then examining the other factor.   So, let's say the number you repeat is represented in digits as ABCD.  We'd have:   ABCDABCD...

Hello Kyle,   Just to make sure I understand the problem, by N5 do you mean the null graph on five vertices?     If so, the set V would contain all the possible pairs of vertices from a set of five vertices.  The standard description of a graph has the vertex...

Hi Bob,   I can see why this problem was confusing for you.  The information about the repetition of 2^X is incorrect as stated, at least in my interpretation of what you have written.     What is incorrect is the starting point of the pattern, and what is missing...

Hello Sun,   Just for your reference (and since I don't have too much time right now), first find the eigenvalues of the matrix.  Do this by solving   (2 - r)(-2 - r) + 5α = 0,   the solutions will be your eigenvalues (r = ±√(4-5α) for your reference)...