Assume that when adults with smartphones are

Deriving the confidence interval
${\displaystyle P(-{\mathrm{\Phi }}_{\alpha }^{-1}<X<{\mathrm{\Phi }}_{\alpha }^{-1})=1-2\alpha }$
Let ${\displaystyle P\sim \mathcal{N}(0,1)}$
$\begin{array}{rlrlr} & P(-{\mathrm{\Phi }}_{\alpha }^{-1}<X<{\mathrm{\Phi }}_{\alpha }^{-1})& & & \left(1\right)\\ & P(X<{\mathrm{\Phi }}_{\alpha }^{-1})-P(X<-{\mathrm{\Phi }}_{\alpha }^{-1})& & P(A<X<B)=P(X<B)-P(X<A)& \left(2\right)\\ & \mathrm{\Phi }\left({\mathrm{\Phi }}_{\alpha }^{-1}\right))-\mathrm{\Phi }(-{\mathrm{\Phi }}_{\alpha }^{-1}\left)\right)& & P(X<A)=\mathrm{\Phi }\left(A\right)& \left(3\right)\\ & \mathrm{\Phi }\left({\mathrm{\Phi }}_{\alpha }^{-1}\right))-(1-\mathrm{\Phi }\left({\mathrm{\Phi }}_{\alpha }^{-1}\right)\left)\right)& & \mathrm{\Phi }(-x)=1-\mathrm{\Phi }\left(x\right)& \left(4\right)\\ & \alpha -1+\alpha & & f\left(f^{-1}\right)=id& \left(5\right)\\ & -1\times (1-2\alpha )& & & \left(6\right)\end{array}$

1a. 

Show from first principles, i.e., by using the definition of linear independence,
that if μ = x + iy, y ̸= 0 is an eigenvalue of a real matrix
A with associated eigenvector v = u + iw, then the two real solutions
Y(t) = eat(u cos bt − wsin bt)
and
Z(t) = eat(u sin bt + wcos bt)
are linearly independent solutions of ˙X = AX.

1b. 

Use (a) to solve the system (see image)
 

$f ( x ) = 7 x^{2 }+ 2$ and $g ( x ) = 5 x^{3 }+ 7 x^{2 }+ 6 x$, find the derivative of the quotient $\frac{f ( x ) }{g ( x ) }$.

Consider the incomplete character table for a group given below:
$$\begin{array}{|cccccc|}\hline & (1)& (1)& (2)& (2)& (2)\\ & 1& a& b& c& d\\ x_1& 1& 1& 1& 1& 1\\ x_2& 1& 1& -1& -1& 1\\ x_3& 1& 1& -1& -1& -1\\ x_4& 2& -2& 0& 0& 0\\ \hline\end{array}$$
All the conjugacy classes are there.
1) What is the order of the group?
2) How many characters are missing?
3) Find the missing character and complete the table.
4) Find the order of the Kemel of the missing character.

An MP3 manufacturer claimsthat 65% of teenagers 13 to 16 years old have their own MP3 players.
A researcher wishes to test the claim and selects a random sample of 80 teenagers. She finds that
57 have their own MP3 players. At � = 0.05, should the claim be rejected?

Find k$k$ such that the following matrix M$M$ is singular.

M=⎡⎣⎢−225+k−5933−4114⎤⎦⎥

set-builder form {y I x is a natural number and y >2}