Answer: Option D. -> $20m+50n\ge 800$
:
D

Total value of m $\$20$ notes and n $\$50$ notes = 20m + 50 n
The least amount of money the woman can have in her possession is $\$800$.
Therefore, $20m+50n\ge 800$
The correct choice is D.

Answer: Option B. -> All values greater than 3
:
B
$2-4x<-6$
$\Rightarrow 2(1-2x)<-6$
$\Rightarrow (1-2x)<-3$ (on division by 2)
$\Rightarrow (2x-1)>3$ (on multiplication with -1), multiplying with a negative number inverts the inequality.
Hence, B is the correct choice

Answer: Option B. -> $(-2,4)$
:
B

$LetP=(4-x)(x+2)$
Find the points on the number line where P=0
$P=0\Rightarrow xϵ\{-2,4\}$
Split the number line into 3 using these points.
$(-\infty ,-2),(-2,4)\&(4,\infty )(-\infty ,-2):(4-x)>0\&(x+2)<0\Rightarrow P<0(-2,4):(4-x)>0\&(x+2)>0\Rightarrow P>0(4,\infty ):(4-x)<0\&(x+2)>0\Rightarrow P<0$
$\therefore P>0,whenxϵ(-2,4)$

Answer: Option B. -> $2x-3y>4$
:
B
We know that an inequality remains unchanged if the entire inequality is multiplied by a positive constant. If an inequality is multiplied by a positive constant, the ratios of the coefficients of x, coefficients of y and the constant terms will be equal.
It can be observed that in option B,
Ratio of coefficients of $x=\frac{6}{2}=3$
Ratio of coefficients of $y=\frac{-9}{-3}=3$
Ratio of constant terms =$\frac{12}{4}=3$
Hence, B is the correct choice.

Answer: Option D. -> All of these
:
D

${\log }_e$ 144 = ${\log }_e{12}^2$ = $2{\log }_{e}12$ (option A)

Factors of 144 = $2^4$ $\times$ $3^2$

            ${\log }_e$ 144 = ${\log }_e$ ($2^4$ $\times$ $3^2$)

                                  = ${\log }_e{2}^4$ + ${\log }_e{3}^2$

                                  = $4{\log }_{e}2$ + $2{\log }_{e}3$ (option B)

            ${\log }_e$ 144 = ${\log }_e$ ($2^3$ $\times$ 18) = ${\log }_e\left(2^3\right)$ + $\log 18$

                           = $3{\log }_{e}2$ + $\log 18$ (option C)

Answer: Option D. -> All of these
:

From the graph, it can be observed that the solution region for the inequality lies beneath each line. Hence, the maximum value of the y co-ordinate of the solution occurs at the point of intersection of the two lines.
To find the point of intersection, solve the equations of the lines y= -15x+3000 and y=5x
Equating the right sides of both equations directly, we get
 5x= -15x+3000
$\Rightarrow 20x=3000$
$\Rightarrow x=150$
So, y=5x=750
Hence, the maximum possible value of b is 750.
 

Answer: Option A. -> -1
:
A
$3x-5\ge 4x-3$  
$\Rightarrow -5+3\ge 4x-3x$  
$\Rightarrow -2\ge x$
$\Rightarrow x\le -2$
Hence, -1 is not a solution

Answer: Option C. -> 4
:
C and D

Given that Chris spends $\$3$ on one adult ticket and $\$2$ per ticket on x children's ticket. Hence, total amount spent = 3 + 2x
Further, it is given that Chris spends at least $\$11$ and at most $\$14$. This can be represented by the inequality
$11\le 3+2x\le 14$
$\Rightarrow 8\le 2x\le 11$
$\Rightarrow 4\le x\le 5.5$
Since x is an integer, the possible values for x are 4 or 5.

Answer: Option D. -> $k\le 5$
:
D
k cannot exceed 5 means that k can be less than or equal to 5.
Hence, D is the correct choice.

Answer: Option C. -> 5
:
C
$A=lo{g}_{2}lo{g}_{2}lo{g}_{4}256+2lo{g}_{2_{\frac{1}{2}}}2$
$=lo{g}_{2}lo{g}_{2}lo{g}_4{4}^4+2\times \frac{1}{\left(\frac{1}{2}\right)}lo{g}_{2}2$
$=lo{g}_{2}lo{g}_{2}4+4=lo{g}_{2}lo{g}_2{2}^2+4$
$=lo{g}_{2}2+4=1+4=5$